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Constrained stochastic differential equations Storm, Andrew 1992

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Constrained Stochastic Differential Equations by A N D R E W S T O R M B.Eng.(Hons), The University of Adelaide, 1980 B.Sc.(Hons), The University of Adelaide, 1985 M.Eng.Sc, The University of Adelaide, 1985 A T H E S I S S U B M I T T E D I N P A R T I A L F U L L F I L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F D O C T O R O F P H I L O S O P H Y in T H E F A C U L T Y O F G R A D U A T E S T U D I E S I N S T I T U T E O F A P P L I E D M A T H E M A T I C S We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A June 1992 © A n d r e w Storm, 1992 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department The University of British Columbia Vancouver, Canada DE-6 (2/88) Abstract. This work uses techniques from convex analysis to study constrained solutions {u, ?/) to stochastic differential equations in Hilbert spaces. The process u must stay in the domain of a given convex function (p, and ?/ is a bounded variation process. The constraint is expressed by a variational inequality involving u and TJ, and is equivalent to 7/ G d^{u), where ^{u) = (p{ut) dt. Both ordinary and partial stochastic differential equations are considered. For ordinary equations there are minimal restrictions on the constraint function ip. By choosing <^  to be the indicator of a closed convex set, previous results on reflected diffusion processes in finite dimensions are reproduced. For stochastic partial differential equations there are severe restrictions on the constraint functions. Results are obtained i f is the indicator of a sphere or a halfspace. Other constraint functions may be possible, subject to a technical condition. Contents. Abstract i i Acknowledgement. iv 1. Introduction. 1 2. Technical Background. 9 3. Constrained Stochastic Differential Equations. 23 3.1 An Itô type inequality for convex functions. 23 3.2 Lipschitz Drift and Diffusion Coefficients. 28 3.3 Monotone Drift and Diffusion Coefficients. 46 3.4 Examples and Interpretation. 60 4. Constrained Stochastic Partial Differential Equations. 65 4.1 Abstract Case. 69 4.2 Linear Operators on Sobolev Spaces. 93 References. 105 Acknowledgement. I would like to thank my supervisor, Dr. Ulrich Haussmann, for his assistance during the preparation of this thesis. Chapter 1. Introduction. In this thesis I consider some problems concerning stochastic differential equations in infinite dimensions which originate in two areas of mathematics: variational inequalities involving parabolic partial differential operators, and reflected solutions of stochastic differential equations. A suitable abstract framework may be found which includes both types of problem, and a penalty method common in both settings may be used to find a solution. Variational inequalities were introduced to deal with problems such as obstacle problems in elasticity, flow or diffusion through semi-permeable membranes, or minimal surface area problems with inequality constraints. Some free boundary problems, such as the Stefan problem, may also be cast as variational inequalties. The books of Duvaut & Lions [1], Glowinski, Lions & Trémolières [1], Kinderlehrer & Stampacchia [1], and Baiocchi & Capelo [1] contain many applications and references. To motivate the formulation of a stochastic variational inequality which will be used throughout the thesis, I will begin with an example from Duvaut & Lions [1], Chapter 1. Consider a bounded open region O C R^, with boimdary dO sufficiently regular to justify any necessary computations. For t > 0 and x e O let u(t, x) represent the pressure of a fluid which percolates through O. Assume that at the boundary of O there is a semi-permeable membrane which allows fluid to enter O (with some resistance to the flow ), but does not allow it to leave. Assume that a function h{x)is given for X G dO which represents the fluid pressure immediately outside O. Let T be the natural surface measure on dO, and for brevity write diu for Within O, u satisfies the diffusion equation ~ - A u = fit,x) (1.0.1) for some forcing function / . Two cases must be considered to determine the boundary conditions for u: (1) u(t,x) > h(x). The pressure inside is higher tiian the pressure outside, but the fluid flow is blocked by the membrane. Since there is no flow across the boundary, = 0, where ^ = Z)i=i(^i^)^t is the normal derivative of u with respect to the outward normal n = (rei ,n2,«3). (2) u{t, x) < h{x). The membrane allows the fluid to flow in: assume that for some constant k, Note that the regions where u{t, x) > h{x) and u{t, x) < h{x) are not known a priori, and must be determined as part of the problem. To obtain a variational formulation of this problem, let vhea sufficiently nice function on O ( e.g., V e C^iO) ), multiply (1.0.1) by v - u and integrate over O: f ^{v - u)dx - f {Au){v - u) dx = f f{v-u)dx Jo Jo Jo ^ ( 1 ^ ' ^ - ^ ) + ^ - - - ^ ) | ^ ^(da;) = {f,v-u). (1.0.2) hi this equation (•, •) is the inner product in lJ{0), and a{u, w) = Y^%i JQ diudiW dx. The choice of V — as test function is obscure, but will become clearer shortly. The boundary conditions may be written concisely by introducing a function on R x dO: J O -.ryhix), ^ ( ^ ' ^ ) = = \ l A ; ( ^ - / i ( x ) ) ^ -.vKKx). The derivative V'l of with respect to its first variable is I ( ^ _ J 0 •  ^ > K^)^ ^ 1 ^ ^ ' ^ ^ - \ A; ( r - / i (x ) ) : 7 - < % ) , so the boundary conditions may be summarized as = 'ip\{u{x), x) for x G dO. Note that t}) is convex in r for each x, so V r , 5 e R V a ; e dO, i^i{r,x){s- r) + ip{r,x) < ip(s,x). This may be used to rewrite the boundary integral term in (1.0.2): - / ^{<^)-<^))ndx)= f ViKa;) ,x) ( t ) (x) -u(a ; ) ) r (drr ) JdO JdO < I il){v{x),x)-i/;{u{x),x)T{dx) JdO = 'Pi'") -where by definition (fi{u) := Jg^ •^(w(a;), x)) T{dx). The original problem may now be formulated as: find u such that (^~,v-uit)^+a(u{t),v~u{t)) + <p{v)~<p{u{t)) > (f(t),v-u{t)) M v. This must be completed by specifying the spaces in which u and v live. For this problem it is natural to look for u[i) = u{t, •) 6 H^(C>) and to require that (/?(u) < oo, which means that Jo ^ c?r < 00. The test functions v should also be in H^{0), but only v for which ip{v) < oo need be considered. It is only a small step from this to obtain the abstract problem considered by Brézis [1], p66, or Duvaut & Lions [1], p47. Let H be a Hilbert space and V a Banach space which is continuously and densely imbedded in tt Identifying H with its dual produces the triplet of spaces V H V*. In the example H = lu^{0) and V = 11^(0). Let (•, •) denote the pairing between V and V*; i.e. for veYmd^eY*, {^,v) = ^{v). Fix T > 0 and assume given (1) A map a : V X V R which is continuous and linear in its second variable. Identify a with A : V 1-^  V* by a{u, v) = {Au, v). (2) A lower semicontinuous convex function ip : V HH. ] R U { O O } with non-empty domain D^:={ueY: </?(n) < oo}. (3) A f i m c t i o n / : [0,T]H^ V*. The problem is then to find a function -u : [0,T] V with derivative u' : [0,T] i -^ V* such that u{t) e t-a.e. and a.e. {u'{t), V - u{t)) + a {u{t), v - u{t)) + (p{v) - < />«0 ) > ( / (O? - ^(*)) V u G Z»<^ . (1.0.3) Thismustbe completed by adding integrability requirements on / , u and u', and further assumptions on a ( continuity, coercivity, monotonicity, etc. ) are needed. Equation (1.0.3) may be written as a.e. {f{t)-Au{t)-u'(t),v~u{t)) + (p{u{t))<(p{v) Mv^D^ (1.0.4) which is equivalent to the inclusion f{t)-Au{t)-u\t)ed'.p{u{t)) a.e., (1.0.5) where by definition the subdifferential d(p(u) is {C € V* : (C, v - u) + ip{u) < ip{v) ^ve D^}. If stochastic forcing terms are allowed in addition to / then u' will in general have no meaning. Instead the equation ( or inclusion ) must be integrated over time. At this point I will change notation slightly and write Ut for n(t), ft for f{t), etc. Define Then with a constraint Ct ••= ft - Aut - u[. ut+ Aus ds+ j Cs ds = uq + I fs ds Jo Jo Jo G 6 d(p{u,). From the definition of d<p, (Cs.Us - v) +(p{v) - <p{us) > 0 \/veD^, so / {us - VsXs) ds+ / <f{vs)ds- / (p{us)ds (1.0.6) Jo Jo Jo is increasing in t for all bounded measurable v : [0,T] t-^ D^. To understand the nature of C a little better, return to the motivating example. The constraint function is (p{u) = Jg^ij){u(x),x)T{dx). Since i / ' is differentiable in its first variable, (p is Gâteaux differentiable with derivative Vc^ given by {V(p(u),v) = jQQil:i{u{x),x)v{x)T{dx), and d(p(u) is single valued and equal to V(p(u). But d(p{u) = {(} and •^i{u{x),x) = so {(, v) = - / g ^ dT. Thus C is a fimctional on li^{0) which measures the flow of fluid at dO. To obtain a stochastic problem, allow / and uq to be random, and introduce martingale forcing terms m and / Q B{us) dwg, where m is a martingale in H , is a Brownian motion in a suitable Hilbert space G, and 5 : V i -^ > C ( G ; H ). From a mathematical point of view, these are the most general types of forcing terms which the theory will allow. For physical motivation, imagine that u is the temperature in some region. The Brownian motion w could model a random heat source for which the energy outputs produced over disjoint time intervals are independent and have a Gaussian distribution. The B{u) term allows one to model situations in which the heat production depends on u or on the temperature gradient Vu. Similarly, m may be another Brownian motion or an integral with respect to Brownian motion. Due to the martingale forcing terms, J j Cs ds cannot be expected to be absolutely continuous in general, but must be replaced by a boimded variation process r/t. As will be seen shortly, r] plays the part of the local time of u, "at the boimdary of O " in the motivating example, and even in the simplest case of reflected one dimensional Brownian morion, the local rime is of bounded variarion but is not absolutely conUnuous. The constraint (1.0.6) must be changed to: / {us- Vs,dr)s)+ I (f{vs)ds- f ip{us) ds (1.0.7) Jo Jo Jo is increasing in t for all bounded measurable v : [0,T] i-> D^. The abstract form of a stochastic variational inequality which I have adopted can now be stated. Let V , H and V * be a triplet of spaces as before, and let (O, .F, {Tut e[0, T]), P) be a stochastic basis. Assume given (1) A progressively measurable V*-valued process / . (2) A n H-valued martingale m. (3) A Brovmian motion win a Hilbert space G. (4) Amap A : V h ^ V*. (5) A m a p 5 : V ^ £ ( G ; H ) . (6) A lower semicontinuous proper convex function (p on V . (7) An .T^o-measurable V-valued random variable uq with uo G a.s. Problem: find a progressively measurable V-valued process u and a cadlag adapted V*-valued process rj such that almost surely ut G D^p V t and f{ut) dt < oo, rj is of boimded variation in V*, almost surely u is integrable with respect to 77, and u and 7/ satisfy a.s. ut+ I A{us)ds+ I B{us)dws + rit = uo+ f fgds + mt V i € [0,T] (1.0.8) Jo Jo Jo and a.s. V t e [ 0 , T ] f {u,-Vs,dT}s)+ f (p{vs)ds-f ip{us)ds>0 Vw G C([0,T]; V ) . ^0 Jo Jo (1.0.9) The constraint has been weakened slightly: (1.0.9) is easier to work with than (1.0.7), and is equivalent to (1.0.7) i f (p is the indicator of a convex set. Further conditions on A and B are necessary. I have used monotonicity and coercivity conditions derived from Pardoux [1] and Krylov & Rozovskii [2]. These are stated in §4.1. In this abstract setting I have only obtained solutions for very special constraint functions: (p must be the indicator of a sphere in H or of a half-space in V*. Other constraint sets or fimctions may be allowed i f A and B are linear operators on the Sobolev space Hj(C?). If the triplet is collapsed by setting V=H , then A : U ^ M, B : M >^ £ ( G ; H ) and : H 1-^  R . I need to have non-empty interior, and A and B must be defined at least on D^, so A and B caimot be just densely defined, and (1.0.8) becomes an ordinary stochastic differential equation in Hilbert space, with a constraint. If A and B are Lipschitz maps on ÏÏ ( or just on ), then (1.0.8) and (1.0.9) can be solved for any lower semicontinuous proper convex (p, provided has non-empty interior. The Lipschitz assimiption can be weakened to a monotonicity condition on A and B by using the technique of Yosida approximations. The method of solution that I have used is to first solve, for e > 0, an approximate version of (1.0.8) in which rjt is replaced by Pdul) ds, and is an approximation to d(p. This method and the formulation I have used are derived from the paper by Haussmann & Pardoux [1]. They took M = L'^{0) for O = (0,1), UHO) C V C B\0), and <f the indicator of the set K = {u e Y : u > 0 a.e.}. Haussmann [1] extended this result to allow O C R", using techniques derived from Krylov & Rozovskii [1]. hi both these papers TJ is regarded as a random measure on O x [0,T]. Nualart & Pardoux [1] and Donati-Martin & Pardoux [1] solved similar problems with O = (0,1) and the same constraint set, but with a white noise forcing term. Nualart & Pardoux first solve a deterministic problem by penalization, and then make a change of variable to solve their stochastic problem. Donati-Martin & Pardoux apply the penalization technique directly to their stochastic problem, hi passing to the limit as e ^  0 they use a comparison theorem and results from Nualart & Pardoux. Rascanu [1], [2] solved a stochastic variational inequality in a very abstract setting, but with a much weaker formulation derived from the weak formulation for deterministic problems found in Brézis [1]. Return for a moment to the deterministic problem (1.0.4) and integrate over t to obtain / {u[ + Aut,vt - Ut) dt+ / ip(vt)dt- / <f{ut)dt> / {ft,Vt - Ut) dt Jo Jo Jo Jo for suitable v : [0,T] i - ^ V . Use the formal integration by parts Jq{u[ - v[,ut - vt) dt = \\uj — VJI"^ — ^\uo — vo\^ to write this as / {v[ + Aut,Vt-Ut)dt+ I ip{vt)dt- I (p(ut)dt Jo Jo Jo > J {ft,vt-ut)dt+-\uT-vj\'^--\uo-Thus vol' I {v^ + Aut,Vt-ut)dt+ f (p{vt)dt- I (p{ut)dt (1.0.10) J O Jo Jo > / {ft,'Vt-Ut)dt- -\uq-Vq\^ Jo ^ The weak formulation in Brézis is: find u : [0,T] V such that jj^ «/'(^t) dt < oo and u satisfies (1.0.10) for all V with ^{vt) dt < oo and v' £ V*. Beginning from this, Rascanu considers the "stochastic differential inclusion" dut + Aut dt + dip{ut) dt S ftdt + dmt (1.0.11) (c f . 1.0.5 ) and defines a weak solution of (1.0.11) as an adapted process u in (fi x ( 0 , T ) ; V ) such that u € D^f, P x dt a.e. and /•T E 1 2 {v't + Aut - ft,vt + mt + nt- Ut) dt^ + {\nj\'^} + ^E{\VO-UO\'^}+ES^J (p{vt + mt + nt)dt^ >ES^J^ (p{ut) dt^ for all n G M^'^W) such that m + n € L2(fi x (0,T); V ) a n d all v G L2(fi x (0,T); V ) such that vt = vo + / Q WS ds foTVo € L^(ft,.Fo,P; E I ) and an adapted process w G L ^ ( 0 x (0,T); V * ) . Rascanu [1] also defines a strong solution to (1.0.11) as a pair {U,TJ) satisfying (1.0.8) with 5 = 0, but with T] absolutely continuous: rjt = Cs ds, and ( G dip{u) F x dt a.e.. Thus a strong solution in Rascanu's sense is a solution in my sense. By a simple application of Itô's formula he shows that i f {u, TJ) is a strong solution in his sense then w is a weak solution in his sense. The same proof, using the Itô formula from Gyongy & Krylov [2], shows that i f (u, rj) is a solution to (1.0.8) and (1.0.9) ( with 77 bounded variation ), then u is a weak solution in Rascanu's sense. If (p is the indicator of a closed convex set K, tiien (1.0.9) reads, for t = T, a.s. r{us-v,,dvs)>0 y V e C{[0,T]; K) (1.0.12) ^0 This form of a stochastic variational inequality has been used to study reflected diffusion processes in finite dimensions by Bensoussan & Lions [1], Menaldi [1] and Menaldi & Robin [1]. It can be shown that (1.0.12) implies a.s. Tjt = I \{usedK)dr)s and \r)\t = [ l{us e dK) d\T]\s. Jo Jo ( See Proposition 3.1 in §3.4. ) This type of constraint on u and TJ appears in the Skorokhod problem for reflected diffusions: seeTanaka [1], Lions & Sznitman [1] and Saisho [1]. The organization of this thesis is as follows. Chapter 2 contains some technical material needed to understand the thesis. Chapter 3 discusses constrained "ordinary" stochastic differential equations in Hilbert space i.e., the operators A and B are defined on sets with non-empty interior. In this setting the penalization technique works smoothly, with very few restrictions on the constraint function. The key tool is the "Itô inequality" proved in §3.1. Chapter 4 discusses constrained stochastic partial differential equations, using the framework of a Hilbert triplet V ^ H ^ V*. Severe restrictions are imposed on the constraint sets, as I do not have an Itô inequality for processes in a Hilbert triplet. Chapter 2. Technical Background. This chapter contains a brief summary of material which is needed to understand the thesis. The material divides naturally into three areas: vector measure theory and integration in Banach spaces, stochastic integration in Hilbert spaces, and monotone operators on Banach spaces. All Banach spaces (including all Hilbert spaces) in this thesis are real. 2.1 Integration in Banach spaces. Most of this material may be found in Diestel & Uhl [1]. Let (X , n) be a finite measure space and B a Banach space. Denote the pairing between B and its dual B* by angle brackets (•,•). A function 5 : X H-^ B is an T-measurable simple function i f s{x) = X) iL i for a, e B and e (Ip denotes the characteristic function of F). A function / : X B is strongly T-measurable i f there exists a sequence (5„ )„ of .F-measurable simple fimctions with lim„^oo Sn(a;) = f{x)^x e X, f is J"-measurable ii f-'^{B) € for all Borel sets 5 C B, and / is weakly T-measurable i f (/(•)) C) is .F-measurable for all C 6 B*. It is a simple matter to check that strong measurability measurability =^  weak measurability. The ftmction / is separably valued if / ( X ) is separable i.e., there exists a countable subset (a„)„ of B with / ( X ) c {(an)n}-Theorem (Pettis). / : X B is strongly measurable iff f is weakly measurable and separably valued. (Diestel & Uhl [1], p42. Theorem 2.) Thus all three notions of measurability coincide if B is separable. It is more common to define / as strongly measurable i f it is the /i-almost everywhere limit of a sequence of simple functions. Then / is only measurable, and weakly measurable, with respect to J"^, the /x-completion of T. Pettis' theorem becomes: / is strongly /"-measurable <^ f is weakly ./'^-measurable and almost separably valued ( i.e., / ( X \ N) is separable for some null set If s is a simple function as above define s djj, := X ^ ^ ^ am{Fi). A function / : X i->^  B is Bochner integrable i f it is strongly measurable and there exists a sequence ( s„ )„ of simple functions such that l im„^oo | | / - 5„ | | dp, = 0. The Bochner integral of f is J-^f dp := lim„_,.oo Jx dp. Theorem. / : X B is Bochner integrable iff it is strongly measurable and ||/|| dp < oo. (Diestel & Uhl [1], p45. Theorem 2) The Bochner integral enjoys most of the properties of the usual Lebesgue integral, such as the dominated convergence theorem and Fubini's theorem, with the obvious modifications ( replace absolute value with the norm in B, etc. ). The notable exception is the Radon-Nikodym theorem. A readable discussion of Bochner integration may be found in Hille & Phillips [1]. See also Yosida[l]. For 1 < p < oo, L ' ' ( X ; B) denotes the set of ( equivalence classes of ) strongly measurable / : X 1-^  B with J-^ dfi < oo. L°°(X; B) is the set of ( equivalence classes of ) strongly measurable bounded / : X i -^ B. If / : X B* then / is weak* measurable i f {/{•), a) is measurable for all a G B. If {/{•), a) is also integrable for all a then a closed graph argument shows that there exists a unique element Jjj / dfi of B*, called the Gel'fand or weak* integral of / , such that l^j fdn,a^ = J{f,a)dfi V a e B . For 1 < p < oo, weak*-LP(X; B * ) is the set of equivalence classes of weak* measurable / : X H ^ B * such that ||/||p : = i n f | ^ y h^df^j : / i : X ^ R i s measurable and < V a ; G x | < oo. If | |/(-)|| is measurable then ||/||p = (/^ \\f{x)\\P fiidx)^^^. This will always be true if B is separable. Weak*-LP(X; B* ) is a Banach space with the norm || • \\p. Weak*-L°°(X; B* ) is the Banach space of equivalence classes of weak* measurable boimded / : X B*. A set function i / : i-> B is a countably additive vector measure if, for all pairwise disjoint sequences {En)n C 5", v {li'^^En) = ^{^n) in the norm topology of B. If is a positive measure on T then v is called fi-continuous if lim^(£;)^o ^{E) = 0: this is written as < / i . According to a theorem of Pettis, a coimtably additive vector measure v on n sigma field is /i-continuous iff /x(^) = 0 v{E) = 0. (Diestel & Uhl [1], plO, Theorem 1). The variation of v is the set function \ v\onJ^ defined by W\iE) := sup | g IIKA.OII : Ai G J', Ul,Ai = E md Ai nAj = iè if z ^ j j . The measure z/ is of bounded variation if | i / | (X) < oo, in which case is a finite countably additive positive measure on J^. A Banach space B has the Radon-Nikodym property if, for every finite measure space (X, J", /x) and every /x-continuous bounded variation vector measure v : B, there exists a function geV (X , p.; B) such tiiat i^iE) = J^gdpM E e T. Theorem ( Dunford & Pettis ). Separable dual spaces have the Radon-Nikodym property. (Diestel & Uhl [1], p79, Theorem 1). Theorem ( Phillips ). Reflexive spaces have the Radon-Nikodym property. (Diestel & Uhl [1], p82, Corollary 4). If / : X B is strongly measurable and g : X B* is weak* measurable then x i-^ {f{x),g{x)) is measurable. If, for 1 < p < oo, / e L P ( X ; B ) and 5 € w e a k * - L P ' ( X ; B * ) then X {f{x),g{x)) is integrable and {f,g) Jx{f(x),g{x)) p{dx) is linear and continuous with respect to both / and g. This shows that weak*-LP'(X; B * ) may be imbedded in 17(X; B ) * . In fact: Theorem. For I < p < 00, weak*-L' ' ' (X; B * ) is isometrically isomorphic to L P ( X ; B ) * . ( A & C lonescu-Tulcea [1], Theorem 7, p94 and Theorem 9, p97) Theorem. For 1 < p < 00, L P ( X ; B ) * ^ L P ' ( X ; B * ) iff B* has the Radon-Nikodym property. (Diestel & Uhl [1], Theorem 1 p98) If X is a compact Hausdorff space let C ( X ; B ) be the space of continuous functions f :X \-^B. Let M ( X ; B * ) be the space of regular, countably additive B*-valued Borel measures of bounded variation on X . ( A vector measure is regular i f its variation is regular.) Proposition. If B is reflexive or if B* is separable dien C ( X ; B ) * ^ M ( X ; B * ). If X = [0,T] for T > 0, then for û e M([0,T]; B* ) define P : [0,T] ^ B * by u{t) := i>([0, t]). This allows M([0 ,T] ;B*) to be identified with BV([0 ,T] ;B*) := {z/ : [0,T] ^ B * : 1/ is right continuous on [0,T), has left limits on (0,T], and is of bounded variation}. One particular case which will arise later is: i f {^,F,P) is a probability space then L P ( f i ; C([0,T]; B ) ) * = weak*-LP'(Û; BV([0,T]; B * ) ) . 2.2 Stochastic Integration in Hilbert space. The general references for this section are Métivier [1] and Métivier & Pellaumail [1]. Henceforth, "measurable" will mean "strongly measurable" unless otherwise noted. Fix T > 0 and let (fi, F, {Ft)te[o,T]-,P) be a stochastic basis satisfying the usual conditions: F is P-complete, the filtration (/t)ie[o,T] is right continuous, and Fq contains all the P-nuU sets of F. Let B denote the Borel sets in [0,T], and Bt the Borel sets in [0,t]. Let B be a Banach space and a; : X [0,T] 1-^  B a stochastic process. Then x is measurable i f it is measurable with respect to the product field F x B, xis adapted i f x(-, t) is /"t-measurable, and x is progressively measurable ( or just progressive ) i f a; |Qx[o,t] J^t x measurable for all t. A set A is progressive if \A is a progressive process. The progressive sets form a cr-field U contained 'm.T xB. A process x is cadlag if for all u, x{u!, •) is right continuous on [0,T) and lim^ts x{u, r) exists for all 5 G (0,T]. A process x is continuous i f a;(a;, •) is continuous for all u. Theorem, Any adapted cadlag process is progressively measurable. (This may be proved directly, and is a consequence of the stronger result given in the next theorem.) A function r : fi i-> [0,T] is a stopping time i f { r < G V< G [0,T]. A stopping time T is predictable i f there exists a sequence of stopping times (rn)„ such that (1) lim„^oo = r , (2) Tn < Tn+i V n and (3) r„ < r on {r > 0} V n . Given a stopping time r, define the cr-field J'r ••= {A e T : An {T < t} e J='t yt}. If a < r then J^^ C J^t- Given a process x and a stopping time r , x^ denotes the random variable x{tLi, If ^ is progressive then Xt is J>-measurable. For stopping times a and r, the stochastic interval {a, r ] is the set {(w,/) : (r{uj) < t < r(w)} C 0 x [0,T]. The optional a-field C? on O x [0,T] is the a-field generated by the stochastic intervals {a, r J for arbitrary stopping times a and r. The predictable (T-field •p on J7 x [0,T] is the cr-field generated by the stochastic intervals {a, r ] for predictable times (T and arbitrary r . (See Dellacherie [1], IV.T3, p67.) A process is optional if it is measurable with respect to O, and predictable i f it is measurable with respect to V. Theorem. Any adapted cadlag process is optional, and O is equal to the a-Geld generated by the real adapted cadlag processes. (Dellacherie [1], IV.T26, p81) Theorem. Any adapted left continuous process with right limits is predictable, and V is equal to the a-field generated by such processes, and to the cr-Reld generated by the real adapted continuous processes. (Dellacherie [1], IV.T22 and IV.T23, pp78,79) Note ÛiSLtV C O CU C T X B. Theorem. T = a{n), whereU := {Fx(s,t] : 0 < s < t <T and F e . F j u { F x { 0 } : F G f'o}. TZ is called the set of "predictable rectangles". If / G (fi, JT, P ; B) let E {/} denote the Bochner integral f{io) P{du). If ^ is a a-field contained in J" and / G L ^ ( f i , T,P; B) then the conditional expectation E {f\G} may be defined as the unique element of L ^ ( f i , a , P ; B ) such that E { I Q E {/l^}} = E { 1 G / } V G 6 This holds for arbitrary Banach spaces by virtue of the strong measurability assumption on / and does not require that B have the Radon-Nikodym property ( see Métivier [1] §8 ). A process a; is a martingale i f it is cadlag, x{-,t) e L^(f i , / " t ,P ; B) V i and E{xt\Fs} = Xs^ s < t. For 1 < p < oo, M^{E) is the space of B-valued martingales with E {||a;||P} < oo. M'P{B) is a Banach space with the norm \\x\\p = E{\\xj\\'''}^^'^. M^''^{M) is the closed subspace of A^^(B) consisting of continuous martingales. If H is a Hilbert space then it is easy to check that a cadlag process in H is a martingale iff (x, /i)H is a real martingale for all h e K For any process x and any stopping time r, let x'^ denote the stopped process xj := xtAr-Theorem (Doob). If x is a martingale and a and T are stopping times then (a) E{xr\T„} = x^ifcT<T. (b) is a martingale ( with respect to {Tt)t )• The finite time interval causes some of the usual technicalities connected with this theorem to disappear. In particular, every martingale is uniformly integrable, all stopping times are bounded, and the hypothesis "a; is of class D " is redundant. A process a; is a locally p-integrable martingale {x e Alfoc(B) ) i f there exists an increasing sequence of stopping times (r„)„ with lim„_,oo Tn = T such that x^" G A^^(B) V n . M^^^iB) is the set of continuous processes which are locally in ^^^''^(B ). A digression into functional analysis is necessary before proceeding with martingale theory. Let A and B be Banach spaces, and B ( A , B ) the space of continuous bilinear functionals on A X B. For a G A and 6 G B define a linear functional a ® 6 on B(A, B) by (a g) 6, A) = A(a, b). The tensor product A (gi B is defined to be the subset of B(A ,B)* consisting of all finite linear combinations X ^ ^ ^ Aj-a,- (gi bi for A, G K and (a,, bi) G A x B. Define a norm || • | | i on A ® B by The topology on A ® B generated by || • | | i is called the projective topology. The completion of A ig B with respect to || • | | i will be denoted by A (âiB. Theorem. Every element U G A ÔiB is the sum of an absolutely convergent series oo where < oo and limi^oo\Wi\\ = Hnii^oo \\bi\\ = 0. This is Theorem 6.4 in Chapter 3 of Schaefer [1]. If G is a Banach space and B(A, 1; G) the space of continuous bilinear maps from A x B into G, then every element A in B(A, B; G) may be factored as A(a , b) = Â(a ® b) for A € £ ( A ê i B ; G), and this establishes an isomorphism between B(A, B; G) and £ ( A (g)iB; G). In particular, B ( A , B) may be identified with (A (giiB)* by choosing G = R. Since B(A, B) may also be identified with £(A;B*), one may write {A'a,b) = A{a,b) = (  , a ® 6) for A ' G >C(A;B*), A G B ( A , B ) and  G (A ê iB)* . These identifications allow some convenient notational shuffling, particularly in writing Itô's formula. If H is a Hilbert space then trace on H (81H is the continuous linear extension of the map h ® k {h,k). If G is another Hilbert space and A and B are in £(G; H ) , the map A ® 5 : G ® G i - ^ H ê i H defined by (A ® B){f ® g) = A{f) ® B{g) is continuous for the projective topology and so extends to G Ôi G. The action of A^ B onU e G will be denoted by {A ® B, U). A n element 7^ G G ÔiG is positive i f (C (8 C, t^) > 0 V C G G*, and f7 is symmetric if {C®^,U) = f/) V C e e G*.If î7 isposi t ivethenrracef /= yc^lli. Define an inner product (•, •)2 and norm \\U\\2 = y/iU, U)^ on G ® H by i 3 where U = X),- / , ® hi and V = 9j ^ kj. This expression for (U, V)2 is independent of the representations used for U and V. The completion of G ® H with respect to || • II2, which is a Hilbert space, will be denoted by G (82ÏÏ. It may be shown that \\U\\2 < {\U\\u so G ®iH C G ®2H. Given Û G G ® ÏÏ, define a finite rank operator U from G into Why Ug= Y^i{fi, g)hi, where U = fi ® hi. Conversely any continuous finite rank operator U may be written in the above form, and so defines an element f/ G G 181 H. This equivalence extends to G (§>iH and G ^2^.: G (âiH may be identified with the trace class or nuclear operators £^(G; H ) , and G ®2lK may be identified with the Hilbert-Schmidt operators >C (^G; H ). For more details see Schaefer [1] and Reed & Simon [1]. Let m be a martingale in X ^ ( ï ï ) . There exists a unique real predictable increasing process (m) with (m)o = 0 such that - {m)t is a martingale, (m) is called the compensator or Meyer process of m . There also exists a unique predictable bounded variation H ®i H-valued process ((m)) with((m))o = 0suchthatmi(gimt-((m))jisamartingaleinH(8)iHand/race((m)) = (m), and there exists a predictable H ®i H-valued process Q™, with values in the positive symmetric elements of H ê i H , such that traceQ'^ = 1 and ((m))t = / J d{m)s. Given two martingales m and n in A1^(H), define {m,n) := \{{m+Ti) - {m- n)) and ((m,n)) := \{{{m+n)) - ((m - n))). These are the unique predictable bounded variation processes starting at zero such that (m, n) - (m, n) and m®n- ((m, «)) are martingales. Let (n„)„ be an increasing sequence of partitions of [0,T] by stopping times: n„ = (r^)^^o with 0 = < r f < . . . < r^^,^ < r^^ = T, such that n „ C n„+i and a.s. lim„_,,oo maxA;(r^ - Tfc_i) = 0. Given m and n in M'^{W), there exists aimique real cadlag adapted bounded variation process [m, n] such that [m, n]i = - Jfirn^ ^ y^r^M - "^T;t"_iAt, nr^^ - nrj^_.,At) and such that (mt^rii) - [m, n]t is a martingale. If « = m then [m] := [m,m] is increasing and is called the quadratic variation of m. If either m or n is continuous then so is [m, n] and [m,n] = (m,n). Also, there exists a unique adapted cadlag bounded variation H®iH-valued process [m, n] such that kn A;=l converges in probability in M iS>2^ to [m, n] and ® - | m , w|t is a martingale in H (âiM. [m] := [m, m]isthefen5orçMa<ira»'cvana?w«ofm. Ifmorwiscontinuousthenlm, n | = {{m,n)). To reduce the ramiber of angle brackets in use, [ T O , n] will be preferred to ((TO, n)), and [m, n] to ( T O , n), whenever these coincide. Let G and ÏÏ be separable Hilbert spaces and let H , V), or just 5 for short, be the set of simple P-measurable C{G; E )-valued processes. Given m e J M ^ ( G ), define a semi-norm on E by Take equivalence classes in E modulo || • ||m-null processes to get a normed space and complete this to get a space A ^ ( G , ÏÏ, V, TO). This is a Hilbert space with irmer product i^^y) = K l l ^ y\\m ~ 11^  ~ y\\m)- If a; is an £ ( G ; H)-valued process with x{g) predictable for all 5- G G then {x ® x,Q"^) may be shown to be a predictable H êiH-valued process. If E \^J^ trace{xt ® xt,Q^) d{Tn)t^ < oo then x G A\G,W,V,m). Let x := Er=i rilF.x(r., . .] for Ti G C(G; H), 0 < r,- < Si < T and Fi G Fr, and define J^^^^ x dm := J27=i ^F,{TiTntAsi -TiTOtAr,)- Then J x dm G M^{E.), and a standard computation shows that EI I xdm \ = E\ f trace {xt (g) xt, d{{Tn))t)\ = J(o,r] „ U o J \\m-Since V = cr(Tt), processes of the form given for x are dense in £, and the map x J x dm extends to an isometry between A'^{G, H , V, m) and A^^(II). If n = J x dm e A^^(H) then For local martingales m,n e Mi^^{G), processes (m), ((m)), [m], [mj, {m, n),... etc. may be defined by localizafion, and the local martingale / x dm may be defined for locally integrable predictable processes x exactly as in the real case. Theorem (Burkholder-Davis-Gundy). For any number p > 1 there exist numbers and Ap (depending only onp) such that for any m G Mlg^{E. ) and any stopping time r, A n important special case of a martingale in Hilbert space is Brownian motion. Fix the Hilbert space G and choose S e G ®iG. A continuous G-valued process w is a Brownian motion in G with covariance S i f (w, g) is a real Brownian motion for all 5 G <G and yf,geG\ls<t, E {{wt - w,, f){wt - w,,g)} = (t - s){S, f®g). Then w G M^'''(G) and [wjt = St. Define a bilinear maponZ:(G; H ) by ( X , y ) ^ := trace{X®Y, S) and let \X\^ := ^/(X,X)^. Define A{'w) to be the completion of C{G; H ) with respect to | • |„ modulo | • |w-null elements. Then A{w) is a Hilbert space, and it is easy to check tiiat A'^{G ,W ,V,w) = 'L'^{il x [0,T], 7?,P x dt; A{w)). As shown by an example in Métivier [1], A(to) may contain unbounded operators. The restriction to trace class covariances is required to guarantee that w is a ' 'genuine' ' process. A more natural object is a "cylindrical Brownian motion" or a "p-cylindrical martingale" as defined in Yor [1] or Métivier & Pellaimiail [1]. I have preferred to use the more concrete object with trace class covariance, although the results in Qiapters 3 and 4 could be applied to cylindrical Brownian motions with only notational changes. A G-valued process ^ is a semimartingale i f it may be written as 2 = v+m, where m G Mf^^{G) and u is an adapted cadlag process with paths of bounded variation in G. The integral J x dz may and {n)t = trace{{m))t = / trace {xg ® Xg, Q™} d{m)s a, :pE {[m]?/^} < E { ^^^Jm^f^} < A ^ E {[m]?/^} . be defined for locally bounded predictable £(G; H )-valued processes as J xdv + J x dm, just as in the real case. Given a semimartingale z, the quadratic variation processes [z] and {zj may be defined as := kt l^ - koP - 2 / iz,_,dz,) J{o,t] and {zjt := zt® zt - zo® zo- / Zs- ®dzs- I dzs ® Zg-. J{o,t] J(o,t] in the definition of {zj, an element ^ e <G is regarded as an element of £{G;G ®2G) by z : g i-^ z ® g or z : g g ® z. [z]is a cadlag adapted increasing process with [z]o = 0, and {zj is a positive symmetric G (giiG-valued cadlag bounded variation process with [z]t = tracelzjf If (II„)„ is an increasing sequence of partitions of [0,T] by stopping times as before, then for all t E^=i ktAT^" - ^tATj^_J' converges in probability to [z]t, and Y,k=i i^t/^r^ - ^iAT^"_J ® (•^ tAT^ - ^tAr^_J converges in probability in G ® 2 G to {zjt. If z is continuous then there is a imique decomposition z = v + m with m a continuous local martingale and v continuous, and [z] = [m]. A n essential tool in stochastic analysis is: Theorem (Itô's Formula). Let z be a œntmuous semimartingale in E. and suppose that ip -.Ut-^R has Frechét derivatives (p' : M i-^ E and (p" : E t-^ £ ( H ) = (H ®iE)*, such that (p, ip' and tp" aie bounded on bounded sets inE, tp and ip' are continuous (for the norm topologies ), and (p" is continuous from E into (ÏÏ lâiH)* with the weak* topology. Then a.s. (p{zt) = (p{zo) +J^i(p'{zs),dzs) + ^ {(p"iz,),dlz}s) V i . This is a special case of Theorem 4.1, p l9 of Pardoux [1]. If (p"{h) is regarded as an element of JC{E ) then the second integral should be written as ^ / J trace {(p"{zs) dlzjs)- The assumptions are stronger than needed. Métivier [ 1 ] has a proof (without assuming that z is continuous) under the strong assumption that (f"{h) £ E i§i2ÏÏ. If (p(h) = \h\'^ then Itô's formula reduces to the definition of [z]. Other versions of Itô's formula are presented when they are needed at the start of Chapters 3 and 4. 2.3 Monotone Operators and Convex Analysis. Most of this material may be fomid in Deimling [1], Barbu & Precupanu [1] or Brézis [2]. Let B be a Banach space, B* its dual and 2^^ the collection of subsets of B*. A map F : D cB>-^2^ is monotone i f \/x,yeD, {C-V,x-y)>0 ^CeFx, rj^Fy. This will be abbreviated to {Fx - Fy, x - y) >OonD x D. Example: Duality Maps. Define J : B 2** by Jx := { C G B * : | |C | |* = ||x||and(C,a;) = | | x | n . Then {Jx - Jy,x - y) > {\\x\\ - \\y\\y > 0, so / is monotone. Note that J{rx) = rJ{x) Vx e B, r G R. If B is a Hilbert space then J is the canonical isomorphism of B onto B*. By identifying a monotone F with its graph a corresponding notion of a monotone subset of B X B* may be defined. A monotone map F is an extension of F i f the graph of F contains the graph of F. A monotone map F is maximal monotone if it has no proper monotone extension i.e., there is no monotone F witii graph(F) c graph(F) and F F . If {7 c B is open md F : U 2^ then F is locally bounded i f for every xq e U there exist numbers r > 0 and c < oo such that IICII* < c whenever \\x — xo\\ < r and C € Fx. o The notation D will mean the interior of the set D. Theorem. Let B be a Banach space and F : D C B 2^ a monotone map. Then F is locally 0 bounded on D, and F has a maximal monotone extension. A Banach space B is sfrjc//y convex i f ||a;|| = \\y\\ = lmdx^y ^ ||é'a; + ( l - e)y\\ < 1 V0 G (0,1), i.e. the unit ball is "nowhere flat". B is locally uniformly convex i f V x with ||a;|| = 1 and V e > 0 3 ^ > 0 , depending onx and e, such that \\y\\ = 1 and \\y - a;|| > e ^ \\y - x\\/2 < 1 - S. Local uniform convexity implies strict convexity. The duality map J is single valued iff B* is strictly convex, in which case || • || is Gâteaux differentiable on B \ {0} ( which, by definition, means that B is smooth ), and Jx = ||a;|| V | | x | | = ^V(||a;|p). J is single valued and strictly monotone, i.e. {Jx - Jy, x - y) = 0 =^ X = y,iffB and B* are strictiy convex. Theorem ( Troyanski [1] ). / / B is reflexive then there exists a norm on B, equivalent to the original norm, under v/hich both B and B* are locally uruformly convex. Theorem. Let B be reflexive and F : Dp c B 2"* maximal monotone. For every x e E and e > 0 there exists a solution y to the inclusion OeJ{y-x) + €Fy. (2.3.1) This solution is unique if both B and B* are strictly convex. Let B and F be as in the theorem, and choose J corresponding to a norm under which B and B* are strictly convex. This is always possible, by the theorem of Troyanski. For e > 0 and a; G B let RcX denote the unique solution y to (2.3.1). Define F^x := -\j{ReX - x) = \J{x - R^x). Then R^ and F^ are related by J{R,x -x) + eF,x = 0 and F,x e FR,x (2.3.2) is a ( single valued ) monotone map : B B*, called the Yosida approximation to F. R^ iB I-* Dpis called the resolvent of F. Let Rp := {( £ B* : 3x e Dp s.t. ( £ Fx} hethe range of F, and coDp the closure of the convex hull of Dp. It can be shown that there is a unique element F°x of Fx which minimizes IICII* for ( £ Fx. A map F : B i-> B* is demicontinuous i f it is continuous for the norm topology on B and the weak* topology on B*, and F is bounded i f it maps bounded sets into bounded sets. Theorem. Let B be reflexive, F : Dp c B i-^ 2^ maximal monotone, and F^ and R^ as above. Then (1) R,x—>xase—^ 0"" yx£côDp. (2) Dp and Rp are convex (3) \Fcx\ — ^ 00 as e — ^ 0" ifx^ Dp (4) F.x^F^x as€ —> 0^ Va; G Dp. (5) Fc is boxmded, demicontinuous, and maximal monotone. Let D C B be convex and (p : D R a lower semicontinuous convex ftmction. By allowing <fi to take values in R U {+00} =: R, (p may be defined on all of B, with domain D^ := {x £ B : (p{x) < 00}. cp is proper i f D^ ^ 0. Generalizing the idea of tangent line, for xq £ D^ any C G B* satisfying 'Pix)><p{xo) + {C,x-xo) V X G F > < ^ is called a subgradient of (p at xq, and the setof all subgradients of at xq is called the subdifferential of (p at xo, denoted by d(p(xo). The set of all a; G D^ for which d(p{x) ^Ç)is denoted by Dg^. It is immediate from the definition that dip is monotone. If B = R then dip{x) = [ip'_{x), ip+ix)], where (p'_{x) and (p'+{x) are the left and right hand derivatives of (p at x. The subdifferential dip is single valued at x iff ip is Gâteaux differentiable at x. Theorem. Let B be a reflexive Banach space and ip a lower semicontinuous proper convex fiaiction on B. Then (1) (p is continuous on D^. (2) dip is maximal monotone. (3) = Dd^ andD^ = Dd^. (4) 3 c G R and^ G B* such thatip{x) >c+{^,x) Va; G D^. Since dip is maximal monotone the Yosida approximations (dip)^ exist. In this special case there is a close connection between {dip)c and a smoothed version of ip. For e > 0 define Let be the resolvent of dip and let (3^ be the Yosida approximation to dip. Then by definition o := -J{x - TT^x) and fi^x G dip{wex). (2.3.3) Theorem. (a) ipc is Gâteaux differentiable on B and = Vipc. (b) The infimum defining ipc{x) is attained at ir^x, and (2.3.4) ip,{x) = -\\^,x\\ï + ip{Tr,x) V a ; G B (2.3.5) (c) liin ip^{x) = ip{x) V X G B (d) ip{n,x) < ip^{x) < ip{x) V x G B V e > 0. See Barbu & Precupanu [1] Theorem 2.3, Chapter 2, pl21, for a proof. (2.3.6) (2.3.7) A result which wil l be needed within the guts of the thesis is: Proposition. For any e,6 > 0 and any u,v eB, {P,u - psv, u-v)> 4(i,u\\î + 8\\fisv\\î - (€ + è)\\l3,u\\*\\fi8v\\* (2.3.8) Proof: Use (2.3.3), the definition of the duality map J and the monotonicity of dip: = — (is'V, TTeU — U-¥ U — V + V — TTsv) = -{l3cU, u - TT^u) - {j3sv, V - Ksv) + {P^u - Psv, u- v) whence {f3cU - fisv, u- v)> e\\f}^u\\* + è\\Psv\\ï - {(3cU, v - ttsv) - {/Ssv, u - t:^u) > e\\(3,u\\ï + 6\\f3sv\\ï - (6 + 6)\\l3M\*\\Psv\\*. If B is a Hilbert space then /?e is the Frechét derivative of (pc, and (3^ is Lipschitz continuous with constant 1/e. In fact, since x = e/3^x + tt^x, \\x - yf = \\e/3,x - e^.y + n^x - Tr^ ylP = e'^WP.x - (3,y\f + Ww.x - 7r,yf + 2e{(3,x - P,y, n.x - •K.y) >^\\l3,x-li,yf + p,x-T^M?. The inequality follows from fi^x 6 d(f{-K^x) and the monotonicity of d^p. An important and instructive special case occurs when (p is the indicator of a closed convex set K, i.e. hoo :x ^ K Then ip^x) = j-^dKixY where dxix) := inf - y\\;y G K} is the distance-to-A' function, Ve > 0 TT^x = TTx is the metric projection of x onto K, (i^x = ^J{/3x) where /3x = x — irx, and dxix) = \\x - zx\\ = \\f3x\\ = e\\l3,x\\*. If x e K then d(p{x) = {0}. If a; G dK and C € d(p{x) then {Ç,y-x)<0'iyeK. The existence of such Ç is guaranteed by the Hahn-Banach theorem 0 if K The set := {y e B : (C, y) = {C, x)} is called a supporting hyperplane for i^ f at a;, C is called a support functional of iiT with support point x, and 5</3(a;) is also written as Nk{x) and is called the normal cone to A at x. A key ingredient in the proofs in Chapters 3 and 4 is a lower boimd on (C, x - a) for ( G d(p{x). The result actually holds for monotone F. Proposition. Let B be a Banach space and F : Dp c B 2^* monotone with Dpi^. For any 0 a £ Dp there exist numbers r > 0 and R < oo such that ( C , X - a) > rllCII* -R{\ + ||x||) V x G i ^ F , C e Fx. o Proof: By local boundedness of F on Dp there exist r > 0 and c < cx3 such that \\r]\\* < c whenever rj e Fy and \\y - a|| < r. Fix x £ Dp and C G F x . For any € > 0 3y G B with \\y\\ < 1 and (C, y) > IICII* - f- Then for any 6 F(a + ry), 0 < {C - V,^ - (a + ry)) {C,x-a)>r{C,y) + {'n,x)-{7j,a+ry) >r\\C\\*-re-c\\x\\-c{\\a\\ + r) >r\\C\\*-R{l + \\x\\)-re if R = c{l + r+ \\a\\). Since this holds for all € > 0, the result follows. A slightly finer result is possible for subdifferentials. Proposition. Let B be a Banach space and ip a lower semicontmuous proper convex function on o o B with D^^il). For any a G there exists a number r > 0 such that (C, x-a)> rllCII* + <pix) - cp{a) - 1 V x G Dg^, C € dcp{x). (2.3.9) If (p is the indicator of a convex set K then {C,x-a)>r\\C\\* yx edK,Ced(p{x) = NKix). (2.3.10) o o Proof: Fix a G D^. There exists an r > 0 such that y G D^^ if \\a — y\\ < r. Since (p is continuous o on D^, r may be chosen such that >p{y) < (p{a) + 1 i f \\a - y\\ < r. Fix x G -Dat^  and ( € d(p{x), and for e > 0 choose y G B with \\y\\ < 1 such that {(, y) > ||C||* - e. Then (C, X- (a + ry)) > (p(x) - (p{a + ry) > ip{x) - ip{a) - 1 => {C,x-a)> r\\C\\* -r€ + <p{x) - ip{a) - 1, which results in (2.3.9) after letting e ^ 0.. If = IK then (p{x) = (p{y) = 0 for x , y G K, so that o (C, X - {a + ry)) > 0 and (2.3.10) is immediate. The result is trivial i f a; G A', since then Ç must equal 0. Chapter 3. Constrained Stochastic Differential Equations. Throughout this chapter H is a Hilbert space with inner product (•, •) and norm | • |, 9? is a o lower semicontinuous proper convex fimction on H with =f 0, and (fi , {Tt, t e [0,T]),P) is a stochastic basis satisfying the usual conditions. Relative to this basis, assume given (1) A Brownian motion w in a Hilbert space G, with covariance S G G A{w) is the space of integrands for w which map G into H , as described in Chapter 2. The norm on A{w) is denoted by | • | „ . If e £ ( G ; H ) then \B\l = trace{B ® B, S). (2) A progressively measurable ïï-valued process / . (3) A continuous H-valued martingale m with [m]< = hg ds for some progressively measurable process h. (4) An /"o-measurable H-valued random variable uo, with uo € -D<^  almost surely. Under either Lipschitz or monotonicity assumptions on A and B I prove the existence and uniqueness of continuous adapted H-valued processes u and r] satisfying a.s. ut+ f A(us)ds+ I B{us)dws + T]t = uo+ f fgds + mt V i G [ 0 , T ] (3.0.1) ^0 Jo Jo with a.s. ut € D^'it, (p{ut) dt < 00, t] is of bounded variation, and a.s. V i e [ 0 , T ] , f iu,-Vs,dT],)> f (fi(us) ds - f <p{vs) ds V G C([0,T]; H ) . Jo Jo Jo (3.0.2) 3.1 An Itô type inequality for convex functions. This section comprises a proof of an inequality which is analagous to Itô's formula, and which will be a crucial tool in the next section and in Chapter 4. The proof uses the following well known result. Lemma 3.1 (Integration by Parts). If g and h are continuous and of bounded variation on [a, b], then for any r and s in [a, b] with r < s, g{s)h{s)-g{r)hir)= f g{t)dh{t)+j h(t)dg(t). (3.1.1) J{r,s] J{r,s] Theorem 3.1 (Itô's Inequality). Assume that H is a separable Hilbert space, u is a continuous semimartingale in H relative to some hxed stochastic basis, f is in C^ (R) and ip is a Gâteaux differentiable convex function on H with Lipschitz continuous derivative (3. Then almost surely mut)) < f{ip{uo))+ [\f'{<p{usm{u,),dus) (3.1.2) where k is the Lipschitz constant for fi. The assumptions on ip imply that it is actually continuous and Fréchet differentiable. As described in Chapter 2, any lower semicontinuous proper convex function (p may be approximated, for e > 0, by functions ip^, satisfying the assumptions of Theorem 3.1. The theorem is actually applied to such approximations in sections 3.2 and 4.2, and the assumptions of the theorem were chosen with this application in mind. Note that J^ f dfi will always denote /^ ^ j,j / dfx. Proof. The proof proceeds along the same lines as the usual proof of Itô's formula. The first step is to establish a Taylor's formula for f oip.To Ms end, fix a; and y in M and define V'x.i/ on [0,1] by i^a:,yit) = fiHx + t{y-x))). Then €,yit) = f'M^ + Kv - ^))W^ + t{y - x)), y-x) = f'{^{xt)mxt),y-x) where xt:= x + t{y - x). Note that 11-> {/3{xt), y - x)is increasing since f3 is monotone, and that t f'{ip{xt)) is of bounded variation (it is actually continuously differentiable). Thus V'x.jy is of bounded variation and one may define a measure px,y on [0,1] by /x,,,((r, s]) := i^'^^yis) - V ; , ( r ) , ^({0}) = 0. The Taylor's formula for tf>x,y is obtained by applying the fundamental theorem of calculus and an integration by parts: V ' . , , ( l ) -V 'x , , (0 )= [\'.,yit)dt=i,',^yiO)+ f\l-t)fi,,yidt). (3.1.3) Jo Jo The inequality needed to prove (3.1.2) is obtained from this by getting an upper bound for the integral involving fix,y The key observation is the following: define another measure Xx,y on [0,1] by Ax,y((r, s]) = (/J(x,) - fi{xr), y - x), A({0}) = 0. Since (5 is monotone, A((r-, s]){s - r) = {(^{xs) - /3{xr), Xg - Xr) > 0. The Lipschitz assxmiption on 13 implies that Aa;,v(( ,^ s]) < K\y — x\ \Xs — Xr\ = K\y — x\^{s — f) . Integrating with respect to A^ ;,,^  then produces / 9{t)X^,y{dt)<K\y-x\^ f g{t)dt (3.1.4) Jo Jo for any positive Borel function p on [0,1]. This inequality yields one for fix,y by expanding the definition of iJ,x,y, integrating by parts, and using the continuous differenfiability of f'{(p{xt)): M.,v((^ s]) = f'{ip{xs)){(3{xg), y-x)- f{^{xr)mxr), y - x) (3.1.5) = / f'{ip{x^))K,y{dt) + [ (/3(xt), y - x) df'{ip{x,)) J(T,S] J{r,s] = [ n<p{xt))Xx,y{dt)+ f iP(xt),y-xfnvix,))dt. J{r,s] J(r,s] From this and (3.1.4), - t)px,y{dt) Jo = f\l-t)f'{<p{xt))K,y{dt)+ f {l-t)i(3{xt),y-xff"{cp{xt))dt Jo Jo <^\y-x\-' [\l-t)\f{^{xt))\ dt+\y-x\'' f\l-tmxt)f\f"{ip{xt))\ dt. Jo Jo Substitute this into (3.1.3) and note that V'^ ,y(0) = f{<p{x)), i'xM = fifiv)) and i^'x^yiO) = f'{ipix)){P{x),y-x)toget my))-mx)) < f'i^ix)mx),y-x) (3.1.6) + l\l-t)\ficp{x,))\dt Jo + \y-x\' [\l-t)\P{x,)\'\f"{^{x,))\ dt. Jo To obtain (3.1.2) from this, fix i > 0 and let TT = {0 = io < < • • • < tn-i = be any partition of [0, t]. Then by telescoping and using (3.1.6), fi^iu,)) - f{<p{uo)) = 2 / ( V ( t . w ) ) - fivM) (3.1.7) «•=0 n - l n-1 + T^c Jo n-l + i=0 where = + s(uti^ i -Notice that [ \ l - s)\fXip{u\))\ds < [ \ l - s)\f'{^{ut,))\ ds+ [ \ l - s)\f'iipiui)) - ds Jo Jo Jo = llfifM)] + J \ l - s)\f{^{ui)) - f'{^{ut,))\ds and / \ i - 5 ) | / 3 K ) | 2 | / x < ) ) | d . Jo = \muu)\v"(^M)\+^'(1 - . ) m<)\V"iv{<))\ - m^u)\V'\v(ut,m ds. Substitute these into (3.1.7) to get n-l fiifiut)) - muo)) < J^f{ipMmut,),nt,^, - Ut,) (A) + E ( f + ^ l /^KJlV'Cv 'C^*.) ) ! ) - uuf (B) i=0 ^ ^ + Junk-where n-l Junk = E ^ I ^ W - Utf f\l - S) \rW{u\)) - as i=o + E l ^ w - ^ d ' / V - ^ ) ( I / ? K ) I V > ( < ) ) I - I / 3 ( « 0 I V > K ) ) I ) Now choose a sequence of partitions {•Kk)k with mesh(7r;t) —> 0 as A; ^  CXD and let A^, B^, and J^„^. correspond to TT^. By well known results ( e.g. Métivier [1] Theorem 26.3 and Corollary 26.7), > [\f'i^{Us))0{Us),dUs) Jo and [ f i/'iv'i^.))!+\w{us)?\nv{u,))\d[u],, in probability. It remains to show that J^„^ converges to 0 in probability. For this, fix e > 0, choose c > 0 such thatP([ti]< > c) < €/3,choose > 0 such that i^(c-i-z/)/2 < eand define J2 K.i-^tf-H Since Y^ue^^k ~ ""i^ l^  ~^ f^ '^ in probability as A; oo, 3K such that P(fiA:) < c/3 V/s > K. Now define G{u;) := {us{u)); s G [0,t]} and note that G{u) is compact in EL The points ul = (1 - s)ut. + suti^j^ are not in G{oj), but are in := {(1 - s)u + sv: s e [0,1], u,v e G(a;)}, which is also compact since the map (5, u,v) (1- s)u+sv is continuous. By uniform continuity on compacts, 3 7(w) > 0 such that u,v e G{u)) and \u- v\ < 7(0;) implies that f'{<f{u)) - nv{v))\ + \\P{uWiviu))\ - \f3(vWiviv))\ < V. (3.1.8) As K - lit, I = sht;^i - tiij < l^ ti^ i - wjj, 3 (^u;) > 0 such that | < - UiJ < 7(0;) i f U+i - U < K'^)-Define Ak := {uj : meshTTjt < S(u))} . Then Aj, C Ak+i, P (UkAk) = 1, and for u e Ak, j \ l - s) ( K | / ' ( ^ ( < ) ) - nvM)\ + |1/9(<)| V ' ( ^ (< ) ) l - l/5(^t.)l V ' (¥ ' («t . ) ) l | ) JO < 1/ / l-sc?5 = î / /2 . Jo Let F := {cj : [ti]t < c} so P(J5) > 1 - e/3. Choose iîT such that V{BK) < e/3 and V{AK) > 1 - e/3, and let F := E n B'j^ n AR.For u) e F md^ k > K, jLk{^)\ < ^ E - ^ ^ ( M t + ^ ) < ^(c+^) < e. Since P(F) > 1 - €, P (|/^„fc| > f) < e V A; > as required. Corollary 3.1. If p is the distance function to a closed convex set K, and (3{u) := u - 7r(w) where TT is the projection onto K, then for any a > 2 and any continuous semimartingale u, a.s. pimr < piuoT + a j\p{ugY-^(3{ug),dug) + \a{a-l) f piug)^-U[ul > 0. JO ^ Jo (3.1.9) Proof: p does not have a Lipschitz continuous derivative, but its square does, hi fact i f ip{u) := |/3(ti)^ then (p'(u) = wliich is Lipschitz with constant 1. Note also that \(3{u)\ = p{u) and that p'{u) = (3{u)/p{u) for p{u) > 0. For a = 2 or a > 4, (3.L9) may be obtained from (3.L2) by using this (f and f(x) = {2x)°'l'^. For 2 < a < 4, (3.L9) may still be obtained formally in this way, but some qualifications to the proof must be made since / will no longer be twice differentiable. First, t p{xt), with Xt as in the previous proof, is convex, non-negative, and possibly zero on an interval. Also, since p is differentiable away from dK, p{x.) fails to be differentiable at at most two points ( where xt enters or leaves K ). If p'{xt) is defined to be zero at these points then and it is easy to check that p{s) - p(r) = ftPi^t) Thus f'((p{xt)) = ^p{xt)°'~'^ is absolutely continuous, and so is still of bounded variation. For the proof of Theorem 3.1 to go through, one just needs to check that (3.1.8) still holds. Since / '(v(-)) 's continuous, this just amounts to checking continuity of |/?(-)P/"(¥'("))- But | /3(M)P/"(<^(U)) = a{a - 2)p{u)°'~'^, which is continuous for a>2. This result corrects an error which appeared in Menaldi [1]. Menaldi assumes that p°' is twice differentiable i f a > 2. This is not true in general. While choosing a > 2 provides sufficient smoothing at dK ( where p{u) = (i),{or u e E. \ K, p" will fail to be twice differentiable i f /? is not differentiable at u. For a simple example take E = and let iîf be a square. For any point u such that iru is one of the comers of the square, /3 will not be differentiable at u, and p" will not be twice differentiable. The error was corrected in Menaldi & Robin [1], where an inequality similar to (3.1.9) appeared, but they do not provide a clear justification of their inequality. 3.2 Lipschitz Drift and Diffusion Coefficients. Within this section the drift coefficient A : t-^ E and the diffusion coefficient B : Ij^ A{w) are assumed to be Lipschitz. That is, 3c> 0 such tiiat \A{u) - A{v)\ +\B{u) - B{v)\^ < c\u - v\ y u,v eD^. (3.2.1) The main result in this section is Theorem 3.2 below. The integrability assumptions on uo, f and / i are rather strong, but are relaxed later in Theorem 3.3 using continuity of the solutions with respect to the data. Theorem 3.2. If, for some a > 4, UQ £ L '" ( f i , /"o,?; H), (p{uo) e TQ,F; R), f e L " ( f i X [0,T];H), and h e L"/2(f2;L2([0,T]; R)), then there exist unique predictable processes u and T] satisfying a.s. ut+ f A(us)ds+ I B{us)dws + rit = uo+ f f^ds + nit V < G [0,T] (3.2.2) Jo Jo Jo with u e L«(f i ;C([0 ,T] ;H)) and r] e L«(fi; C([0,T]; H)) n weak*-L«/2(Û;BV([0,T]; H)). Moreover, a.s. Ut € V i € [0,T], (p{ui) dt < oo and a.s. V i e [ 0 , T ] , f {us-Vs,dr]g)> [ cf>{us)ds-f <p{vg)ds V v € C([0,T]; H) . Jo Jo Jo (3.2.3) Proof: Uniqueness. Let {u, rj) and (w, Q be two solution pairs as in the theorem. Then a.s. ut-vt+ I A{ug) - A(vs) ds + f B{us) - B(vg)dwg + T]t - Ct = 0 V f e [ 0 , T ] . Jo Jo Apply Itô's formula to get Wt-vt\'' + 2 / {A(us) - A{vs),Us - Vs)ds+ / {us - Vg) o {B{us) - B{vs))dws Jo Jo + 2 f {u,-Vs,drts-dQ= I \B{us)-B{v,)\lds. Jo Jo From the constraint (3.2.3), / {us - Vs,dr}s- dC.s)> / (f{us)ds- I (p{vs)ds+ / ip{vs)ds- / (f{us)ds = 0, Jo Jo Jo Jo Jo and applying this and the Lipschitz condition on A and B produces Wt - %P < c I Us - Vs\'^ ds -2 (us - Vs) o {B(us) - B{vs)) dw^. Jo Jo The stochastic integral is a martingale, so taking expectations produces E{\ut-vt\^} <c I E{\u,-Vs\'] ds Jo and applying Gronwall's lemma yields rE{\ut-vt\'} dt = 0. Jo Since u and v have paths continuous in H it follows that almost surely ut = vt for all t, and so rit = Ct for all t, almost surely. Proof: Existence. Throughout the proof c, ci,C2, etc. denote numbers which may depend on Uo, f, m, w, A and B, but not on e. Also, c is used as a generic constant, i.e. the value of c may change from line to line. Young's inequality: ab < 8aF+ , V a , 6 , ^ > 0 and l<p,q<oo with - + - = 1 (3.2.4) and the inequalities E k r < ( E k | ) ' < n ^ - i E | a . r (3.2.5) i=l i=l i=l will be used repeatedly. For semimartingales X and Y, a simple consequence of Young's inequality which will be useful is {X + Y\<{l + 6)[X\ + {l + \)[Y\ V<5>0. (3.2.6) Step 1: Construction of the penalized solution. From Chapter 2, for every e > 0 and u € H there exists a unique solution TTJM G DQ^P to the inclusion ir^u — tt + ed(p{7r^u) 3 0. The penalty operator /3e is defined by P,iu) := ^{u - TT.ti), (3.2.7) and the smoothed constraint function is defined by ip,{u) := i n f { ^ | t i - + if{v) : v e D^}. (3.2.8) The inf is attained aiv = TV^U. Recall the following relations from §2.3: 0, {u) e d<p(K,u) Vw € H (3.2.9) (p,{u) = ^\f3c{u)f + ^ ( T T . U ) V U 6 H (3.2.10) 'Pi'^cu)<if,{u)<<f{u) V u e H (3.2.11) limejo Mf^) = Vu € H (3.2.12) Vu e H , /3,(u) is the Frechét derivative of (p, at u (3.2.13) €^\P,(u)-fi,{v)f + \Tr,u-7r,v\^<\u-vf V u , u € H (3.2.14) 1. e. is Lipschitz with constant 1, and Pc is Lipschitz with constant 1/e. o Given (3.2.9), for any a e I^^ apply (2.3.8) with a; = TT^U and ( = /3t(u): (/3,(u),7reU - a) > r\Pc{u)\ + «^(Tr^u) - ip{a) - 1 u - a) = ((3c{ii), Il - TT.u) + (/3((u), TTjU - a) > r|y9,(u)| + e |A(u)p + ^ (T T ^ U ) - < (^a) - 1 A lower boimd on (/'(TT^U) is needed at this point. Recall from §2.3 that (p is bounded below by an affine function ( this follows by applying the Hahn-Banach Theorem to the epigraph of ^ ). Thus for some cj > 0, (p{u) > - c ( l + \u\). Since TT^ is Lipschitz with constant 1, (p{7r^u) > - c i ( l + ITTEUI) > - C 2 ( l + ITTEOI + |-u|). To show that (iTTeODe is bounded, fix V e Dg^. Then |7re0| < |t7| + {n^vl, which is boimded since ir^v v as e ^  0. Thus (p{Tr,u)>-c{l + \u\) V u e H . (3.2.15) and (/3,(ti), u - a) > r|/3e(îi)| + e|/3£(îi)|^ - c|w| - c Vu 6 H . (3.2.16) The approximate, or penalized solution is constructed as the solution to the equation u'^.+ f A{ul)ds+ I B{ul)dws+ I P^{ul)ds = UQ+ I f^ds + mt (3.2.17) Jo Jo Jo Jo Before doing this, however, A and B must be extended to all of EL This is easily done by defining A{u) := A(7ru) and B{u) := B{Tru) for u ^ D^, where is the metric projection of u onto D^. Since TT is Lipschitz with constant 1 and 1^= u for u £ D^p, A and B are now defined on all of H and are still Lipschitz. Assuming that A and B have been extended in this way, (3.2.17) has a unique predictable solution in L"(f i ;C([0 ,T] ; H)) . This follows from the usual argument using Banach's fixed point theorem. Step 2: Bounds on and Pdu'^)-E | g P | u , ^ r + {^f^mu\)\d^ ' + (^J\\(3,{n^,)f dt^ ^ I < c (3.2.18) with c independent of e. o Proof: Fix a £ and apply Itô's formulate \ul - ap : \ul-a\'^ + 2 f {A{ul),ul- a)ds + 2 f (u^ - a) o B{ul) dw^ + 2 I {(i,{u\),u\- a) ds Jo Jo Jo = | u o - a p + 2 / {fs,ul - a) ds + 2 / {ul-a,dms) Jo Jo I- J B{u')d-Now use (3.2.16), (3.2.6) with ^ = 1 and the estimates (1) K - a\^ > \\u'f - \a\^ and (2) \{Aiu),u-a)\ < \A{u)\\u-a\ < 0^(1 + \u\)\u - a\ < c^il + Hf to get ^ + 2r \f3ciul)\ds + 2 6 1 ^ \l3,{ul)\' ds <c+\uo-a\'^ + 2 f \f,\\ul-a\ds + 2[m]t Jo + c I \u^^\ + \ul\^ ds + 2 I {u\- a,dms)-2 f {u^ - a) o B{u\) dw,. Jo Jo Jo Raise this to the power a/2, use (3.2.5), take the sup over / and take expectations: E { « ^ P K r } + E | ( ^ ' i / 3 , « ) i d . + E ^moi'ds < c E { | u o - a n + E { ( \fs\\ul-a\ds \ a/2" + E {[m]r] + cE (3.2.19) a/2") \u'f ds Note that a/2>i r / {u^ - a, drrir) Jo ^ ^ ^ E sup / {ul - a) 0 B{u^^) dwr Jo a/2' E \ul\ + \ul\^ds] ' \ < E T + Jo \\Us < Ci + C2E lull'^ds J <c{t,a)E\^^\u\Yd^ <<^\[^{Tl\uT]às. From Burkholder's inequality, / {u\ — a, drrir) Jo a/2^ < C2E \u\ — d[m]. c«/4" < C 2 E { ^ ^ P | < - a r / V ] , - / ^ } < C 3 E { g P i < r / ^ [ m ] r / ' ' } + C 4 E { M / ^ } and c ,E sup < C2E / {ul - a) 0 B{ul) dw, Jo a/2 I .a/4' ul-a\\l + \ul\yds 0 / < C4E <C5 + C 4 E ^ ^ P | sup I Ia/2 ( / ' a/4' Finally, E I ( / \ fs\\< - a\ dsY'^ < E I gP |^^ _ a|«/2 i/^i ^ E { f ^ P | < r } - H c E | ( j r ' | / , | r f , y | . T \ « + cE<{ I \ft\dt 0 / < c + Substitute these estimates into (3.2.19) to obtain E / { ^ u P | ^ r } + E | ( £ | / 3 e « ) | r f . l + E { | u o r } + E \ < c ^ '^mdt] | + E { [ m ] ? / ^ ^ ^ (3.2.20) Ignore the second two terms on the left hand side of (3.2.20) for the moment, and apply Gronwall's lemma to get E { g ? K r } < c i + E { K n + E ( / \Mdt] U E J r f ^ } v with c independent of e. Now return to (3.2.20) to get E mnl)\ dt a/2' < c and E \Jo J j Note that the last inequality implies that E a/2' \ul - TTttt^ P dt a/2-< C. (3.2.21) Step 3: For e > 0 and (5 > 0, 2" E / sup < - ut uf - -KsUt l / 2 \ Proof: For e > 0 and ^ > 0, u\ - ul + f A{ul) - A{ul) ds + /* B{u\) - B{ul) dw^ + f - (is{<) ds = 0 . Jo Jo Jo Apply Itô's formula to this to get 14 - uf\' + 2 A A « ) - Aiul), ul - 4) ds + 2 f\ul - 4) o {B{ul) - B{ui)) dw, Jo Jo + 2 f\pc{ul) - Ps(u% nl - 4) ds = f \B{u\) - B{nl)\l ds. (3.2.22) Jo Jo Since the subdifferential dip of ip is monotone, 0 < {h^ — hy,x — y) for silx,y € Dg^, G dip(x) and hy e d(p{y). Thus, from (3.2.9), 0 < (/3eK)-/35(ti*),7r,tt^-7r5U*) = (/3,(u^) - (is{u% - ef3,{u') -u' + ôfisin')) = (/3,(u^) - (3e(u'), - u') + (€ + S){/3c{u^), /3s(n')) - e|/3,(u^)p - ê\Ps(u')\^. Use this and the Lipschitz condition on A and B (equation 3.2.1) to get ft f e | / ? , « ) p ds + 2 f m{<)? ds (3.2.23) Jo Jo <c f\4-ul\^ds + 2{e+è) fmu% (is{ul)) ds - 2 / * « - uf ) o ( 5 « ) - 5(uf )) dw^. Jo Jo Jo Take the supremum over t, expectations and use Burkholder's inequality to get ]<c^ j\[\u\-u'f] ds + 2{e + ë)Ei^j\lic{ul)\\(i5{ui)\ds + C2E t \ l / 2 " But C2E I | < - ni\Us^"'^ < \E { SUP |< - + C3E | £ | < - ufp} so E { - I ' } ^ (\ul - uf |2} + 2(e+ ^)E | £ | /3 ,« ) | | / Î5 (uf ) | cf . j . Gronwall's lemma applied to this yields E { sup|4 _ ut\'} < cs{e + 6)E \Pc{ul)m{4)\dt^ . 34 Use (3.2.7) e/3f(u) = u - •K^U and the second tenn in (3.2.18): /•T E • " ' < c, (e {supi^J _ ; r ,u^ |^ îâ , (n ,^ ) |d i |+E | g P | 4 - i:e4\jWu\)\dt < C6 ( e { \u\ - TT.u^f } '^ ' + E { |uf - ^e<?] 1/2 as required. Step 4: = 0 and lim E t->0 Proof: This is the crux of the proof, and the point where the Itô's inequality is needed. Apply (3.1.2) to (p^{u\f' to obtain: ^,{u\f + 2 [\,{ul){A{ul),Mul))ds + 2 fMulM{ul)\'ds Jo Jo < V e K f + 2 / v,(ul){fg,/3,(ul))ds Jo + 2 f {ip,{ul)P,iul),dms)-2 f <f,{ul)P,{ul)oBiul)dWg Jo Jo + - f\^,{nl)mul)\ids+- f\^,{ul)\d[m]s ^ Jo ^ Jo + f\(3,{ul)\'\Biul)\lds+ fmul)\Ulm]s. Jo Jo To simplify the notation a little define d^ := - •K^U^ ~ e/3f^{ul). Then from (3.2.10) (3.2.24) 1 Recall that [m]t = hg ds and multiply (3.2.24) by to get e V ( 4 ) ' + - f\d't\'ds<e'cp,{uof-2 f <p{7r,ul)\dl\'ds f Jo Jo + f\dl\'\Aiul)\ds+2e f\^(^,ul)\\dl\\A{ul)\ds Jo Jo + f\dl\'\fg\ds + 2e f\^{n,ul)\\dl\\fs\ds Jo Jo A [ ^'^'[ | < / . ( x , < ) | | f i « ) | ^ ds ^2JQ \dl?^^ds + e \ip{T:,ul)\hs ds + 2€ f (MOdl, drug) - 26 Ç M<)dl ° dwg. Jo Jo (3.2.25) At this point some bounds on (p{ncu) are needed. A lower bound has already been obtained: <f(7r,u)>-c{l + \u\) V u G H . (3.2.15) A n upper bound may be obtained from (3.2.9): a - TTeu) < ip{a) - v?(7reu), so <p(n^u) < (p(a) + {f3,(u), ir^u - a) = <p{a)-e\(3,iu)\'' + {/3,{u),u-a) < >p{a) + {Pc{u),u- a) (3.2.26) and thus \<p{-Kcu)\ < max{c2(l + \u\), C3 + {f3c{u), u - a)} < c + c\u\ + \{pciu),u - a)\. (3.2.27) Using these estimates, take the supremiun over t, take expectations, and use Burkholder's inequality in (3.2.25) to get E (3.2.28) \ + E + E 1 ^ (1 + K | ) | d | | 2 d i | + E | ^ (1 + K | ) | d , f d/} y\i+\ui\f\d^,\' dtj+'^{Jo'-^ '^'] + E { ^ ^ \dlf\ft\dt^ + \dl\il + |<|)|/,| rfij + E + dt^ + \ul\f dt'j + E \dl\il + \ul\f dt^ \dl\^ht dt^ + eE + \ul\)ht rfij +^{J^ 1^ *1(1 + l^'tDht dt^ + eE + E + eE + eE Note that ^ V , ( u 0 ^ l r f , f ( l + k ? l ) 2 d / j I I <^e(tin'l^tfd?j I < E I g P , | ^ , ( 4 ) | f^ ^"" dt^ (3.2.29) and eE<{ 1^ ^,iu^,Y\d'i\\l+\ul\fdt^ ' I < 2 _ E { s u P e V . K ) ' } (3.2.30) ^cE\^Ç^\d\\\\^\n\\fdt\^. Now use Young's inequality (equation (3.2.4) ) in the fonn ah < ^aP + c(p,q)e'^/Pb\ and Holder's inequality to get (1) E + \u^,\M\' dt^ < -^^{J^ K\' dt^ + ceE + \u^,\r dt^ (2) E + \ul\)\dlf rfij < ^ E \dl\' di} + ce'E | j j V + \ul\)' dt^ (3) E {J^ \dim\ dt) < \di\^ dt]. C.3E i / . r dt (4) E + |u,^|)^|d,^P d/} < ^ E \dl\' dt] + ceE {(1 + |u,^|)^ d/} (5) E | d , f d . } <J^E{J^ |d , t d.} . ceE dt} (6) e E | £ ( l + |u,^|)2|d,V*} < j ^ E y j d l \ U t ] + ce'/'Ey\l + \ul\f/'dt (7) eE I |d,^|(l + luimi dt] < ^ E | |d , f di} + ce" 1/2 E + \ u i \ ) m M dt] < ^ E y ^ i d , f df I + c e E | j j V + l^^l)'^*} ^ Ey^MUt] 1/2 (9) e E y \ l + \ul\)htdt^ < e E y \ l + \ul\fdtj ' E^Ç^h\dt^ (10) Ey\d\\{\^\u\\)htdt\^ < - L _ E | ^ V t l ' r f ^ } + c 6 i / 3 E | £ ( i + K i ) ^ r f i } ^ E y \ u t ^ Substitute (3.2.29) and (3.2.30) and these estimates into (3.2.28) and use (3.2.18) to obtain ^ E { ^lPe\ciuir} + 1 e \dl\Ut'^ < 2^E {^c{u,f] 4- c{e^e^ + . ^ / ^ . ^ 3 ) . Then for e sufficiently small E { s u P e V ( u D ^ } < « V 3 + { v ^ e K f } and Ey\d\\Ut^ < c6^/3 + 4 e 3 E { v P e K ) 2 } . To deal withthe(/5£(uo)tenn, use (3.2.15) and (3.2.11): -0(1+1^0!) < (/5(;re-uo) < </'e('«o) < V'('"o). Then Iv^el^o)! < c ( l + |uo| + ^ {UQ)) and VPe(tto)^  < c ( l + \uo\^ + ( / ' (UQ)^). Given (/?(uo) S L 2 ( Û ) and (3.2.12), dominated convergence yields \imE{ip,{uof]=E{<p{uof]. Now it follows that E for e sufficiently small. This proves one of the conclusions of step 4, and expands to ( ^ K t f + ^ V ' M ) ) ' } < (3-2.31) To show that lim^-^o E | ^^P |dt |^ j = 0, note first that the same argument as used above for UQ shows that v ( T X ) ' < < l + k t l ' + ¥ ' . ( ^ t ) ' ) . Then s ^ P e V ( ^ X ) ' } < « ' + c e 2 E { s u P | u , f } + c E { f u P e V ( t ^ D ' } < « ^ ^ (3.2.32) 38 for e sufficiently small. Use the estimate (a + 6)^ > ^a^ - 6^  in (3.2.31): , , l / 3 > E | s u p Q | ^ e | 2 ^ ^ ^ ( ^ ^ ^ . ) y | > E { g p Q K ^ r - e V ( ^ , < ) 2 ) } and apply (3.2.32) to get as required. Step 5: Completion of the proof. It is immediate from steps 3 and 4 that {u'^)^ converges to an element u in L^( f l ; C([0,T]; H)) as € 0. Since the adapted processes are a closed subset of L^( f i ; C([0,T]; H)) , u is also adapted, and hence predictable by path continuity. The integrability of u may be improved by using (3.2.18). Note that L'^(f i ; C([0,T]; H)) ^ L " ( 0 ; L°«([0,T]; E)) ^ weak*-L"(Û;L°°([0,T];E)) = L" ' ( f i ;L i ( [0 ,T] ;E) )* so (3.2.18) implies that (u^)^ is weak* compact in this space. If w'^  û weak* along a subnet in weak*-L"(fi;L°°([0,T];E)) then ^ Û weak* in weak*-L2(fi; L°°([0,T]; E)); but u strongly in this space, so û = -u and u e L° ' ( f i ; C([0,T]; E ) ) . Define rj" by Jo Since the total variation of TJ^ on [0,T] is |/3,(u^)| dt, (3.2.18) shows that (r/') is bounded in L«/2(fi; BV([0,T];H)).But L"/2(f i ; BV([0,T]; E)) = L"/2(f i ; C([0,T]; H)*) ^ weak*-L"/2(fi; C([0, T]; E)* ) = L'*/(«-2)(fi;C([0,T];E))*, so that (7/^ )f is boimded, and hence weak* compact, in the latter space. Then, along a subnet, ri' ^ fi weak* as e ^ 0, with fj e weak*-L"/2(û; C([0,T]; H)*). Identify the random measure fj with a process TJ by TJ{U}, t) := ^(o;, [0, t]). Then TJ has paths which are cadlag and of bounded variation, but rj is not yet known to be adapted. Since A and 5 are Lipschitz, the maps u /g A ( M S ) rfs and u H-> J^B{US) dws are continuous from C ( [ 0 , T ] ; H)) into itself, so A{ul) ds A{u,) ds and B{ul) dw, B{u,) dw^ in L ^ ( û ; C ( [ 0 , T ] ; H)), and as before one may conclude that A(us) ds and Jo B(u,) dws are actually in L « ( f i ; C ( [ 0 , T ] ; H)). From the penalized equation ( 3 . 2 . 1 7 ) , rj^ also converges to an element ^ in L2( f i ;C( [0 ,T ] ;H) ) , and ut+ I A{us)ds+ I B{us)dws + it = V'0+ / fsds + mt. Jo Jo Jo It follows from this that ^ G L " ( f i ; C ( [ 0 , T ] ; H)). To identify 77 with ^, choose g in C ( [ 0 , T ] ; H) n B V ( [ 0 , T ] ; H) and X in L 2 ( 0 ; E ) . By the weak* convergence of 7/^ , ^ E | X ^ (ffi,dr?t)| = l i m E | x ^ (fft,cir/o}. ( 3 . 2 . 3 3 ) But for any / and 5f in C ( [ 0 , T]; H) n B V ( [ 0 , T]; H) die integration by parts formula / ifu dgt) + I {gt, dft) = ih,gr) - ifo,go) Jo Jo holds so l i m E | x J\gt, drjt)^ = l i m E | x ((gfr, ^ ) - (50, v'o) - j \ v t , dgt)) } • ( 3 . 2 . 3 4 ) Since 77^  = 0 and 7/^  ^ ^ in V{n; C ( [ 0 , T ] ; H ) ) , 77^  ^  in L 2 ( 0 ; H ) and, combining ( 3 . 2 . 3 3 ) and ( 3 . 2 . 3 4 ) and passing to the limit as e ^ 0 produces E | x J\gt, d77t)| = E | x ( ( f f T , e x ) - ^ ^ ( 6 , dgt)) | . This holds for all X in R) so a.s. I {gt,dT}t) = {gT,ij)- f {Ct,dgt) Jo Jo with the null set depending on g. Fix s in [ 0 , T) and / i in H and let r 1 ; 0 < i < 5 = i 1 - 7i(i - 5) •,s<t<s+ 1/n [0 [s+l/n <t <T. Then hfj^ converges pointwise to /il[o,5](<) asn ^ 00, hf"' is continuous and of bounded variation, and as 71 ^ 00 / ihfl^,drjt)^{h,rj,), Jo and I i^uhdfn = -{h,n£''ldt) ~{h,Q. Substitute / i / " for g in (2.33) and let n ^ oo to see that {h, TJS) = (h, G ) almost surely. The null set here depends on h as well as 5, but H is separable so rj^ = almost surely with the null set depending on s. Then almost surely rjs = for all rational s in [0,T), but ^ is continuous and ri is right continuous so a.s. = 6 V 5 G [ 0 , T ) . To extend this to s = T let rn^jO ; 0 < i < T - l / n \n{t-T)+l ; T - l / n < i < T . Then as n ^ oo / V / r , d 7 ? t ) (h,dvi{T}))={h,rrT- m-), Jo {hf?,^T) = {h,^T) V n , and ' (6 , / id/r) — ( / ^ , ^ T ) , /o so almost surely (h, TJY — f/r-) = {h, ^ T ) — {h, ^j) = 0, and since H is separable, TJT = rjr- almost surely. Now a.s. TfT = ijT- = lim m = lim = as required. To check that a.s. ut G D^Mt, note that since u ' converges strongly to u and E I \u\ — TT^Ujp 1^0, {•Kt'^^)i also converges strongly to u, and -Kc^t € Dg^p, so a.s. ute Dd^ = D^ V i e [ 0 , T ] . It remains only to show that a.s. ip{ut) < oo and to establish the constraint (3.2.3). To this end, choose X e L°°(fi) with X > 0 a.s., and recall that (f is lower semicontinuous with fiu) > - c ( l + \u\) for some c > 0. Then | x c ( l + \ut\) + ip{ui) dt] < E | x ^ lim^inf (c(l + |7r,-«^|) + ^(Tr^u^)) dt] < l i m ^ i n f E J x c(l + \TC,UI\) + (fi(Tr,ul)dt] = E | x c( l + |w(|) dt] + lim inf E S^X j\{7r,ul) dt 41 E so /•T E • < l i m i n f E J x ip{7r,ul)dt^. But from the definition of TJ", since /3f (uj) € 9vj(7reuj), a.s. / (TTE-U '^ - vt, dr)l) = I {•K,U\ - Vt, fic{u\)) dt Jo Jo > l (p{Tr,ul)dt- I ip(vt)dt V u G C ( [ 0 , T ] ; H ) . Jo Jo Use the strong convergence of XTT^U'^ to Xu and the weak* convergence of T/^ to rj to see that E | X ^ {Tr,u't-vt,dr]',)^ j\ut-vudTit)^ Vt; G L ^ C ^ ; C([0,T]; H ) ) . Thus E | x ^ («i - Vt, drjt)^ = lim^inf E | x (TT.U^ - u*, drjt)^ > H m ^ i n f E | x J <P(T,UI) - (p(vt) dt^ > E | X J ^ cp{ut)-<p{vt)dtY whence, since E j j f Jo{ut - vt, rfï/t)| is finite, E X is also finite, and thus yT / <p{ut) dt < oo a.s. Jo (in fact (p(ut) dt G L i ( f i ) ), and a.s. / {ut-vt,dr]t)> f (p{ut) - f (p{vt)dt. Jo Jo Jo This argument may be repeated for / J rather than j j , so that i f X> is a countable dense set in C([0,T];M)then a.s. yte[0,T]nq, [ {n,-Vs,dT],)> [ <f{us)ds- [ ip{vs)ds V u G P . Jo Jo Jo This extends by path continuity to (3.2.3). Theorem 3.3 . The conclusions of Theorem 3.2 remain true if, for some a > 2, UQ e L°'{a,Fo,'P; ÏÏ) withuo e 'D^ a.s., f e L " ( 0 ; L H [ 0 , T ] ; U)) and me M'^iU) withd[m] « dt. If (u, TJ) is the solution to (3.2.2) corresponding to UQ, f and m, then the maps UQ I-> ( M , TJ), f ^ (u,r]),andm ^ {U,T]) are continuous from L°'{Çl,Fo;M),h°'{^;V{[0,T];M)) andA^^(H) , respectively, into ( L " ( f i ; C( [0 ,T] ; H ) ) , L ' * / 2 ( 0 ; C( [0 ,T] ; H ) ) * ) , with the weak* topology on L « / 2 ( f i ; C ( [ 0 , T ] ; H ) ) * . Proof: If iu^,T]^) and {u^.rf) are the solutions to (3.2.2) corresponding to (uo,/^,m^) and {ul,p,m^), respectively, then u]-ul+ f A{u\)-A{ul)ds+ f B{u\)-B{ul)dwg + rj]-ri', Jo Jo = ul-ul + I fj - fg ds + m] - ml. Jo Apply Itô's formula to get \u\ - + 2 / * {A{ul) - A{u% ul - u"^ ds + 2 f [u] - u^, drj] - drj]) Jo Jo = \ul-ul\' + 2 j\u\-ulf]-fg) ds Jo + 2 f {ul - ul dm] - dm',) - 2 f\ul - u',) o {Biu]) - B{ul)) dw, Jo Jo + m^ -m'^- J B{u^) - B{u^) dw . From (3.2.3), {ul - u^drjl - dr/^) > 0, so ignore this tenn, apply (3.2.1) and (3.2.6) to get \ul - < \ul - ul\^ + c I \ul- ul\^ ds Jo + 2 f\fl-f^\\ul-ul\ds + 2[m'-m% Jo l\ul-uld{m^-m%) +2 f {ul ~ u]) o {B{ul) - B{ul))dw. Jo Jo + 2 Raise this to the power a/2, use (3.2.5), and take the supremmn over t to obtain '^^\ui-uir<c. \ul-ulr+y\ul-ul\Us [m' - m^r,l' + \^j\u\ - uld{m' - m%) r{ul-ul)o{B{ul)-B{ul))dw. Jo [j\fl-fl\\<-ul\ds^ c/2 a/2 On this use E u] - 4\'dsY'^ < cE If Jul - ulrds'^ <cEl^^p\ui-uirf' y\fl-fAds^ <C2Ei^y\u\-ul\U[m^-m'\)l'''^ <^2E[f^y,-ulrl\m'~mX"} <\E[^^y,-ul\^)^cE[[m^-mX''] c.Eii ljul~ul\\f}~f]\dsy^' j\ul-uld{m'-m\) Jo 4^ a/2) and / ' ( . ; - . ? ) o ( f l ( „ ; ) - B ( „ ; ) ) < i » , Jo a/2' < C2E !^y^ \ui - uiiusY'^ < { supi^i _ ^ 2 | . | ^  | £ sup _ ^ 2 | . to obtain { - ulr} < c (E {\ul - u ^ r } + E - f^\c^«)^|+E {[m^ -^ / E { - P | , I _ , 2 | . J + c Gronwall's lemma then produces E { , ^ ^ p K - « ? r } < c E { i 4 - u ^ r } + E v ds. \ft-ft\dt^ \Jo / + E[[m'-mX^^' (3.2.35) This will establish continuity of u with respect to the data. To get continuity of T?, let {u, rj) be the solution corresponding to UQ, f and m, apply Itô's formula to \ut - ap, use the estimate / ius-a,dT]s)>r\î]\t+ I ip{us)ds-T<fi{a)-i: Jo Jo > r|7/|T - ci / \us\ dt - C 2 , Jo obtained in Chapter 2 and proceed as in step 2 ofTheorem 3.2.1 to get E { gP \n t r + \V\T^'} <c(^l+ E{\uon + ^^[J^ \ft\dt^ | + E {[mp'}^ . (3.2.36) Now take sequences {ul^)n C L " ( f i , Fo\ H) with tig e a.s. for each n, C L«(n ;L i ( [0 ,T] ;H) ) and (m")„ c Af?(H) , and suppose < ^ UQ, / " / and m " ^ m in the relevant spaces as n ^ oo. Let (u", 7/") be the solution to (3.2.2) corresponding t o « , r , m " ) . By (3.2.35), (u")„ is Cauchy in L"(J7; C([0,T]; H)) , and so converges to a predictable process u in this space. By (3.2.36), (?/")„ is bounded in L(«/2)'(fi; C([0,T]; H))*, and so converges weak* along a subsequence in this space to some element rj. The same reasoning as used in step 5 of the proof to Theorem 3.2 will show that 7/ is in L°'(n; C([0,T]; H)) n weak*-L"/2(j|. BV([0,T]; H)) and is predictable, and that u and rj satisfy (3.2.2) and (3.2.3). To this point it is only known that some subsequence of (?/")« converges to TJ. Suppose that the entire sequence does not converge weak* to 77. Then there is a weak* neighbourhood Uofi] and a subsequence (7/')^ of {'r)"')n such that rf for any i. But {'rf)i is also boimded, and so a further subsequence (77-')j will converge weak* to some process (. As above, u and ( must solve (3.2.2) and (3.2.3), and the uniqueness part ofTheorem 3.2 then implies that C = i]- This contradicts if for any i, so the entire sequence (r/")n must converge weak* to 77. This establishes the continuity claims ofTheorem 3.3. To establish the existence claims, fix UQ, f and m as in the statement of Theorem 3.2. It is clear that there exists a sequence ( / " )„ of progressively measurable simple functions converging t o / i n L « (f i ;Li([0,T]; H)). To construct an approximating sequence for UQ, let {xi)i be a countable dense set in D^. Define = {w; \UQ - xi\ < 2-"}, let F ; = F ^ \ U^^El and note thatP (UiEi) = 1 since e 'D^ a.s.. Now choose such that P (\uo - J^iti Xi^E'J > 2~") < 2"" and let ug = J^Zi XilE'„- Then < e L°°(H), < (^ug) e L°°(K) and as 71 ^ 00, ug Uo almost surely and in L'^(H). To approximate m, let ht = d[m]/dt and define m'^ := l{hs < n) dm-s. Then [77î"]t = Jo hsl{hg < n) ds, m " ^ m in A t ^ as 71 ^ 00, and /i? = htlih. < n) so e L°°(Û x [0,T]). Now UQ, f and 772" satisfy the assumptions of Theorem 3.2, so there exist solutions u" and 77„ which, as shown above, converge to a solution u and 77 corresponding to uo, f and m, with u e L ^ ( f i ; C([0,T]; H)) and 77 e L^/^j^ft; C([0,T]; H))*. From the Lipschitz continuity of A and B and (3.2.2) it follows that 77 e L " ( f i ; C([0,T]; H)) . 3.3 Monotone Drift and Diffusion Coefficients. It is possible to relax the assumption of Lipschitz continuity made in §3.2 to a monotonicity condition similar to one often encountered for unbounded operators. Specifically, suppose that as o before v' : H R is a lower semicontinuous proper convex function with non-empty, but now A: DACE.^B.WVAB : DB CB.^ A(w) satisfy, for some A > 0 and c > 0, Restriction on Domains: D^CDACDB. (3.3.1) Hemicontinuity: r iA(u + rv), u;) : R i-^- R is continuous yu,v,wÇ. EL (3.3.2) Monotonicity: 2{A{u)-Aiv),u-v) + X\u-v\^>\B{u)-Biv)\l y u,v e DA- (3.3.3) Maximal Monotonicity: I + eAis onto H for all e sufficiently small. (3.3.4) Growth of B: \B{u)\ < c(l + \u\) V u e DA. (3.3.5) Growth of A: For some p>l,\A{u)\ < c( l + \U\P) V u e D^. (3.3.6) Jacod [1] and Gyongy & Krylov [1] have studied imconstrained equations in R " under similar assumptions, but with much more general forcing terms tiian I use here. One situation in which my assumptions are satisfied arises from Galerkin approximations to some stochastic PDE's. A n example is given in §3.4. The proof of Theorem 3.4 below replaces A and B by their Yosida approximations A^ and Be. These are Lipschitz, so Theorem 3.3 may be used to get a "penalized solution" u^ The analysis is similar in many ways to that in §3.2, since the penalty term there, (3^, is itself a Yosida approximation. o There is no continuity assumption on B, but (3.3.3) actually forces B to be continuous on DA-In fact, the left hand side of (3.3.3) is {Fu - Fv, u - v) for the monotone map F =2A+ XL Any 0 monotone map is demicontinuous on the interior of its domain, so if u e DA and u„ u, then Fun Fu weakly and ( F u „ - F u , u „ - u) 0. Now (3.3.3) forces | 5 ( u „ ) - B(u)\^ -> 0. Assumption (3.3.4) is required to guarantee the existence of the resolvent := (/+ and thus of the Yosida approximations Ac := ARc and Be := BRc. Assumption (3.3.1) is needed for the definition of to make sense, since -.E. DA- A monotone operator satisfying (3.3.4) is sometimes called hypermaximal monotone ( see Deimling [1] ), but this is actually equivalent to maximal monotonicity for operators on a Hilbert space. Under assumption (3.3.3), however, it is A+^I which is monotone rather than A itself so (3.3.4) requires A+^Itohe maximal monotone. Any monotone operator has a maximal monotone extension, and it would be sufficient to use this extension to construct the Yosida approximations, but B must also be extended simultaneously so that (3.3.3) continues to hold. While A and B may be extended to a "maximal domain" on which (3.3.3) is valid, it is not clear that this extension for A is maximal monotone. Thus assumption (3.3.4) must be made explicitly. If B is Lipschitz then the monotonicity condition (3.3.3) may be weakened to 2(A(«) - A{v),u- v) + Xi\u- v\'^ > 0 \fu,veDA-Given this, let F be a (possibly multivalued) maximal monotone extension of A + and let A := F - ^I. As explained in Step 1 of the proof of Theorem 3.2, B may be assumed to be defined on all of ÏÏ. If | f i | is the Lipschitz constant for B, then 2(l(w) - À{v),u-v) + {Xr + \B\^)\u - > \B{u) - fi(a)|^ ^ u,v e D^. Now  is an extension of A and (3.3.3) holds with A replaced by À and A = Ai + | f ip . Condition (3.3.5) is automatic. Two lemmas are needed before proceeding to Theorem 3.4. The first concerns the Yosida approximations and is well known for monotone operators (see Deimling [1], Proposition 11.3, pi04, for example), but the hypotheses being used here produce slightly different results. The second lemma shows that (3.3.3) implies a quadratic lower bound on A. Definition 3.1. R, := {I + tA)-^, A, := AR„ and B, := BR^. Lemma 3.1. (a) R,{u) = u - €A,{u) V U G H (3.3.7) (b) R„ A, and B, are all Lipschitz, with\R,{u) - R,{v)\ < 2\u - v\ (3.3.8) for all ( sufficiently small. (c) For all u e DA, \A,{u)\ < -^^\A{n)\, (3.3.9) \u-R,{u)\<-^=L==\A{u)l \B{u) - fi,(u)U < V(\Aiu)\ 47 and Ac{u) —> A{u) strongly as e 0. (d) For sufficiently small and positive e and 6, 2{Ac{u) - A5{v), Rc{u) - Rs{v)) + X\Re{u) - Rsiv)]^ > \Be{u) - Bs{v)\l. (3.3.10) and 2(A,(u) - Asiv),u- v) - \Be{u) - Bsiv)\l (3.3.11) > e\A,iu)\'^ + S\As{v)\'^ - 2X\u - v\'^ - 2(e + S){A,(u), Asiv)) V u , t ; e H. Proof: (a) (I + €A)Re{u) = u =4> Rc{u) + eAe(u) = u. (b) For u and t; in EE, using (a), R,{u) - R,{v) + eA,{u) - eA^iv) = u-v. (3.3.12) Take the inner product of this with Re{u) - Rc{v) : \R,{u) - R,(v)\'^ + €{AR,(u) - AR,(v), R,{u) - R,(v)) = (R,(u) - R^Çv), u - v). To deal with the second term on the left hand side, substitute Rt{u) and Rc(v) for u and v respectively in (3.3.3): e{ARc{u) - ARe{v),Rc{u) ~ R,{v)) > l | 5 , ( u ) - Be{v)\l - y | i ? e (^ ) - Rc{v)\\ so that ( l - Y)\M^)-Re{v)\'+'^\Be{n)-Be{v)\l < {Reiu)-Rc{v),u-v) < \Re{u)-Rc{v)\\u-v\. Ignore the second term on the left hand side to see that \R,{u) - R,{v)\ < — - v \ < 2\u - v\ for e sufficiently small. Now use this and ignore the first term to get \Bc{u)-Be{v)\l<-^~^\u-v\\ ^ U 2 / To show that A^ is Lipschitz, take the inner product of (3.3.12) with Ae{u) - Ac{v): {Ae{u) - Ac{v), Re{u) - Rc{v)) + e\Aciu) - Ac{v)\^ = (A,(u) - Ac{v), u - v). As before, (3.3.3) yields Then {AR,{u) - AR,{v), R,{u) - R,{v)) > ^\B,{u) - B,{v)\l - ^\R,{u) - R,{v)f >-^mu)-R,{v)\\ e\A,iu) - A,iv)\^ < {A,(u) - A,iv),u- v) + -{R^iu) - R,iv) <i\A,(u)-A,{v)\' + ^ Ju-v\' + A so \A,iu) - A,iv)\' < 2€' / i A ^ 2 ( 1 -\u — v\ \u — v\ . (c) Assume u is in DA, set v = R^u) in (3.3.3) and use (a): \B{u) - B,iu)\l < Mu - R,(u)\^ + 2(Aiu) - A,(u) , u - R,(u)) = e^X\A,iu)\^ + 2e(Aiu),A,iu)) - 2e\A,(u)f < e^X\A,iu)\^ + e\Aiu)\' + e\A,(u)\' - 2e\A,iu)\^ = e\A{u)\'- e(l - eX)\A,(u)\\ Then for e sufficiently small, \B(u) - B,{u)\l < e\A{u)\l and \A,{u)\'< ^ \ A { u ) \ \ Applying (a) again produces \u-R,{u)\' = e'\A,{u)f<^^\A(u)\\ To show that A^{u) —> A{u) as e ^  0, first substitute Rc{u) for M in (3.3.3): 2iA,iu) - Aiv), R,{u) -v) + X\R,{u) - v\^ > \B,{u) - Biv)\l. (3.3.13) Now \B(u) - Be{u)\^ < y/è\A{u)\ implies B,{u) B{u) in A{w), and \u - R,{u)\ < --^^=^\A{u)\ implies Rc{u) ^  u in EL Also, \A^{u)\ < -^^=\A{u)\ implies (A^{u))^ is weakly compact. Assume then that Ac{u) —> h weakly along a subnet, and pass to the limit along the subnet in (3.3.13) to get 2{h - A(v), u-v) + X\u- > \B(u) - B(v)\l > 0. Setv = u - tw fort > 0 and any to in H to get 2t(h - A{u - tw), w) + \t'^\w\^ > 0, whence {h - A{u - tw),w) > - — \wf. Let i ^ 0 and use the hemicontinuity of A (equation (3.3.2)) to get (h - A{u),w) > 0. Since this holds for all tf; in H, h = A{u). This shows that A{u) is the only possible weak limit for a subnet of (Ae(u))e, so Ae{u) A(u) without a subnet. Also, from the bound on {Ac(u))c, limsupj_,o |yle(u)| < |A(u) | , and by weak lower semicontinuity of the norm, lim infe^o IAt( 'M) | > | A ( ' M ) | . Thus | A ( ; ( « ) | |A(u) | , but weak convergence plus convergence of norms in a Hilbert space implies norm convergence, so Ac{u) A{u) strongly, (d) Substitute Rt{u) and Rs{v) for u and v respectively in (3.3.3) to get (3.3.10),and use the relations Rc{u) = u - €Ac{u) and Rs{v) = v - 6As{v) from part (a) (equation (3.3.7)) to get 2iAe{u)-A6{v),Rc{u)-Rs{v)) = 2{Ac{u) - As{v), u-v- eAc{u) + 6As{v)) = 2{Ac{u) - Asiv), u-v)- 2€\Ae{u)\^ - 26\Asiv)\^ + 2(e + 6)(Ae{u), As{v)), and \Re{u) - Rs{v)f =\u-v- {eA,{u) - 6As{v))f = \u- up + \eA,{u) - SA5{v)f - 2{eA,{u) - êAs{v), u-v) . <\u- up + \eA,{u) - êAs{v)f + \eA,iu) - SAs{v)f + \u- up = 2\u - up + 2\eA,{u) - SAs{v)f < 2\u - up + 4e^\A,{u)\^ + 4S^\As(v)\\ Substituting these into (3.3.10) produces 2(A,{u) - A8(v),u-v) - \B,(u) - Bs(v)\l > 26|A,(u)p + 2<5|A5(u)p - 2(6 + ê){A,iu), Asiv)) - 2X\u - up - 4e^X\A,{u)\^ - 4S^X\As{v)\^ > e\A,iu)\' + ê\Asiv)\' - 2X\u - up - 2{e + S){A,{u), As{v)). The last step follows from 2e - ie'^X = 2^(1 - 2eA) > e for all e sufficiently small. Lemma 3.2. Under assumption (3.3.3), for any a e DA there exist numbers Ai > 0 and c > 0 such that, for allu e DA, (a) {A{u),u-a) > - A i j u p - c. (3.3.14) (b) (A.iu), u-a)> ^\A,(u)\'^ - A i | up - c for all e sufficiently small. (3.3.15) Proof: (a) Substitute a for v in (3.3.3): {A(u),u — a) > —-^lu — ap + (A{a),u — a) > -A|-u|2 - A|a|2 -\A{a)\\u\ -\A{a)\\a\ > - A i | u p - c (b) Substitute i?£(u) for win part (a), use (3.3.7) and note 2(yle(u),u) > -e\A,{u)\'^ - \\u\'^: {A,{u), R,{u) -a)> -Ai|i<:,(u)p - c => (>le(u),u - a) + (Ae(u),i?e('^) - > - A l i u - eA^(u)\'^ - c ^ {A,{u),u-a) > €\A,{u)\^ - Ai|wp + 2Aie(u,yle(u)) - Xi€'^\A,(u)f - c >{e- 2Aie2)|A,(u)p - 2\i\u\^ - c >i\A,{u)\'-2XM'-c for e sufficiently small. Theorem 3.4. Suppose that UQ is an To-measurable H valued random variable with uo G D^ a.s., / is a progressively measurable process in H and m is a continuous martingale in H with d[m] <C dt. If for some a > 2p, UQ G L"(n , j ro ; H), / G L«(f i ;Li([0,T]; H)), and m G X" ' '=(H) , then there exist unique predictable processes u and T] such that a.s. Ut + / A(ug)ds+ B{ug)dws + T]t = uo+ fsds + mt V i G [ 0 , T ] , (3.3.16) Jo Jo Jo with u G L'^(fi; C([0,T]; H)) andrje L2(fi; C([0,T]; H)) n weak*-L"/2(0; BV([0,T]; H)). Moreover, a.s. Ut € D^\/t £ [0,T], <fi{ut) dt < oo and, a.s. V t G [ 0 , T ] , [ {us~Vs,dT]s)> f ip{us)ds- f <f{v,)ds V t; G C([0,T]; H) . Jo Jo Jo (3.3.17) Proof: Uniqueness. The uniqueness proof is the same as for Theorem 3.2 except for one minor detail. If {u, rj) and {v, ^) are two solution pairs, then applying Itô's formula and the constraint as before produces \ut-Vt\^ + 2 I {A{us)-A{v,),Us-Vs)ds + 2 I (u, - v^) o {B(us) - B(vs)) > Jo Jo < f\B{us)-B{vs)\lds. Jo Use the monotonicity condition (3.3.3) on tliis to get \ut - Vt\'^ ^ \'"'s - Vs\^ ds-2 / {Us- Vs) 0 {B{us) - B{vs)) dws. Jo Jo Now proceed as in Theorem 3.2. Proof: Existence. This follows the same pattern as in Theorem 3.2: establish the existence of approximate solutions u^ and T]\ get some bounds on {u^)c and {T]^)^ which will establish weak* compactness of (7?')e, and show that {u^)c converges strongly. Part (c) of Lemma 3.1 (equation (3.3.9) ) and the growth condition (3.3.6) are the key ingredients in this last step. Step 1; Construction of the penalized solutions. For any e > 0 and sufficiently small, and B^ exist and are Lipschitz. Since a > 2p > 2, by Theorem 3.3 there exist unique predictable processes and T/^ satisfying a.s. / A,(ul)ds+ I B,{ul)dws + r}t = uo+ I f^ds + rut V ^ e [ 0 , T ] (3.3.18) Jo Jo Jo with G L«(f i ;C([0 ,T] ;H)) and 77 e L'^(fi ; C([0,T]; H)) n weak*-L«/2(fi; BV([0,T]; H)). Also, and TJ" satisfy a.s. u\ e D^Mt, v'(^t) dt < 00 and a.s. V i e [ 0 , T ] , f\ul-Vs,dTjl)> [\iul)ds- f <p(v,)ds V u € C([0,T]; H) . Jo Jo Jo (3.3.19) Step 2; Bounds o« (u^)^ and (77^ )^ . < c v + E \Jo Ol^ds a/2' (3.3.20) \Jo 't\dt + E { M r } l + E{\uon + E Proof: Apply Itô's formula to - ap to get \u't - ap + 2 / ( A e « ) , ul-a)ds + 2 I « - a) 0 B,(ul) dw, Jo Jo + 2 / « - a, dr]l) = \uo - ap + 2 / ( / „ < - a) ds Jo Jo + 2 / {ul — a,dms)+ m - Be{u\)dws . Jo I Jo It (3.3.21) The constraint (3.3.19) may be written as Î/^|[O,J] € d^ti^'^) where $t('u) := ip{us) ds. Since o o a e D^, the constant funcrion o is in so applying (2.3.8) and the lower bound (3.2.15) yields r « - a, dvl) > r\v% + W^') - $(a) - 1 Jo > r\r)% - ci - ci / \ul\ds Jo > r\Tj'\t - C2 - C2 I K\''ds. Jo Use this and (3.3.15), 1^^^  - > i - \a\^ and [X + Y]t < 2[X]t + 2[Y]t in (3.3.21) to get ^ K f + j\\Mnl)\'ds + r\r,^t < |uo - ap + ci + C2 / \ul\^ds + 2 f {fs,ul-a)ds Jo Jo + 2[m]i + 2 / \B,inl)\ids Jo + 2 / « - a, dmg) - 2 / {u^ - a) o B,{ul) dw^. Jo Jo Now proceed as in step 2 ofTheorem 3.2, but note that E J sup I s<t I « - a) o B^{ul)dw, Jo < c i E \ul-a\''mul)\lds <c,Ehj^^\ui-a\f y^mui)\ids \ T + C 3 E \MO\ids) J to obtain E a / 2 ' (3.3.22) l + E{ |uor} + E V a / 2 ' Now from (3.3.5) and (3.3.8), \B,{uX)\ = \BR,{u\)\ < c i ( l + \R,{u\)\) < 02(1 + |u^|), so C 2 E | £ sup |^.,a ^ I (^ Ij l f i , « ) | ^ ds C 3 E | ( ^ ^ | f i , « ) | 2 d 5 ) I < C 4 E | £ ( l + | < | ) " d 5 | < C5 + C 5 E Substitute this in (3.3.22) and use Gronwall's lemma to get E { Wr} < c h + E {\uon + \ft\dt " | + E { H ^ / ^ } ' whence, from (3.3.22) E{\vrr} < c and E T \ « / 2 ^ 0 e\Mul)\Us Step 3: is Cauchy in C([0,T]; H)) Proof: For non-negative and sufficiently smdl e and è, Jo Jo Itô's formula applied to this yields \ul - uf p + 2 [\A,{UI) ~ Asiul),ul - 4) ds + 2 [\ul - uldvl - rfr/f) Jo Jo = f \B,{4) - Bs{ui)\lds - 2 A< - 4) o {B,i4) - Bs{4))dws. Jo Jo From the constraint (3.3.19), /^(u^ - , drjl - drjl) > 0, so this term may be ignored. Do this and apply (3.3.11) to get ul~4\'+ fe\A,iul)\Us+ f è\A5{4)? (3.3.23) Jo Jo <2X I \4~4fds + 2{e + 6) f iAiul),Asi4))ds-2 I {4 - 4) o {B,{ul) - Bs{4)) dw^. Jo Jo Jo Use Burkholder's inequality in the usual way to get E { g P - 4) 0 - Bs{4)) dws } (3.3.24) { - ^'1'} + [ [ m o - BsiOll ds^ . < ^ E I - u. The last term is finite given the boimds from step 2 and the Lipschitz character of B^, so the stochastic integral in (3.3.23) is a martingale and thus E{\UI~ 4\'] <2\[E{\UI-4\'} ds + 2{e-H ^)E [[ \A,{4)\\As{4)\ds} . Gronwall's lemma now yields E 1^ "^  \4 - 4f dt^ < c{e + S)E [[\A,{4)\\M4)\dt^ • (3.3.25) 54 From (3.3.19), (p{ul) dt < oo a.s., so u' e c DA^ y. dt-a.e.. Thus (3.3.9) and (3.3.6) may be combined to get, for e sufficiently small, \A,{u't)\ < 2\A{ul)\ < c( l + \u't\P) P X dt-a.e. (3.3.26) From (3.3.25) E y^\u't - ul\^ dt] <ci{e-i-6)Ey\l + \u't\P)^dt] x E | J^^( l + |uf | P ) ' d i | < C2ie + 6). The last step follows from a > 2p and the bounds from Step 2. This shows that (•u^ )^  is Cauchy in L^( f l x [0,T]; H), and so converges strongly to some element u in this space. Now use (3.3.7) to see from Step 2 that E y^lul-R^iuDlUt] <ce, so that {R^{u^))^ also converges strongly to '«inL^(r i x [0,T]; H). Return to (3.3.23), take the supremum over t, expectations and use (3.3.24) to get E { - ^ ' 1 ' } ^ { ^ UP\ul - ds + C2{e + ^ )E \A.{nmAs{ut)\dt] + cEy^\B,{u^t)-Bs{ut)\ldt]. Apply Gronwall's lemma to this to get E ' ~ ' ' g P K - 4 p } <c(e+<5)E|j^ \A,iumAs(ul)\dt] + cEy \B,{ul) - Be{uf)\ldt] . (3.3.27) From (3.3.10), fT E y \B,iul)-Bs{nt)\ldt] < 2E l ^ " " |A, (uD - Asiut)\\R,iu^t) - Rs{<)\dt] + >^^{J^ " ^^(4)1' dt < 2 E | j f \A,{u^,)-Asiu',)\'dt] x E J j f \R,iul)-Rsiu',)\'dt] + XE y^\R,(ul) - Rsi4)\''dt] . But E I JQ \Rc{u^t) - -R5(uf )p df I ^ 0 since {R^{u^))^ converges strongly to u, and E y ^ \A,{ul) - As{ut)\' dt] < 2E 1^ "^  dt] \As{ut)\' dt which is bounded as shown by (3.3.26) and Step 2. Thus E { jJ\B,{u't) - Bs{uf)\l dt} 0, and from (3.3.27), {u'), is Cauchy inlJ{n; C([0,T]; H)). Step 4; Completion of the Proof. From Step 3, ^ u in C([0,T]; H)), with u predictable. Using the bound from Step 2 as in Theorem 3.2 shows diat u is in L '*(fi; C([0,T]; H)) . Also, for each e, a.s. u\ e D^^t, so a.s. Ut G D^\/t. Using Step 2 again, and arguing as in Theorem 3.2, (r]'^)c is bounded and hence weak* compact in L (« /2) ' ( f i ;C([0 ,T] ;H))* = weak*-L«/2(fi; BV([0,T]; H)), and so converges weak* along a subnet to an element rj. From (3.3.26), a > 2p and (3.3.20), iA,{u'))^ is bounded in L 2 ( f i x [0,T]; H), so A,{u^) x weakly along a subnet. Since the map v H^- VS ds is linear and continuous from (fi x [0, T]; H) to L 2 ( f i ; C([0,T]; H)), A,{nl) ds ^  x . ds weakly along a subnet in L 2 ( f i ; C([0,T]; M)). As shown at the end of Step 3, E { Jo\B,(u't) - Bs{ul)\l dt} ^ 0 as e, (5 ^ 0, so B,(u'.) ^ ^ strongly in A(w), and J^B,{ul)dws Jo^dws strongly in L 2 ( f i ; C([0,T]; H)). From the penalized equation (3.3.18) it now follows diat ??' ^ C weakly in L^(n; C([0,T]; H)) , and a.s. ut+ / Xsds+ / ^dws + Ct = uo+ / fsds + mt V / G [0,T]. Jo Jo Jo One may show that Ç, = rjhy arguing exactly as in Theorem 3.2. A well known argument (the "method of monotonicity"; see Pardoux [1], ppll6-119) will identify x as A{u) and ^  as B{u). Considerable simplifications are possible here due to the strong convergence of {u^)t and (^^(u^))^. Define, for any v in Ui^l x [0,T]; H), X, := 2E y \ A , i 4 ) - M^t), R.{u\) - Vt) dt^ + ^ E j^^ \R,{u\) - | -E[[\B,{u\)-B{vt)\ldtY Note first that > 0 in virtue of (3.3.3). Then from the strong convergence of {Bf_{u^))^ and (i?e(îi^))£ and the weak convergence of (Ae(u'))^, along a subnet, 0 < l i m i n f X , = 2 E | y ( X i - A ( u t ) , ' U t - - y t ) d * | + A E | j ^ \ut - Vt\'^ dt^ (3.3.28) -E[[\it-B{vt)\ldtY Set u = u to get E {/^ ^ \it - B{vt)\l dt} = 0; i.e. ^  = B{u) in h{w). Now (3.3.28) becomes ^[J\x - A{vt),ut - vt)dt] > - ^ E \ut - vt\Ut] . (3.3.29) Set t; = u + r^ r for r > 0 and any z e l^i^ x [0,T]; ET): E { J\xt - A{ut + rzt),zt) dt] <Y^{J^ l-^tl' dt] . Now let r ^ 0 and use the hemicontiuity of A to get E y\xt-A{ut),zt)dt]<0. This holds for arbitrary z,sox = A{u) P x dt-a.e., and a.s. Xs ds = A{us) ds V t A l l that remains to be done is to establish the constraint (3.3.17). Recall that (p{u) > - c ( l + \u\) for some c > 0 and use (3.3.19), Fatou's Lemma and the lower semicontinuity of ip to get, for non-negative X e L°°(fi) and t; G C([0,T]; H) , EyJ^{ut-vt,dr]t)]+Eyc{l + \ut\)dt] (3.3.30) = lim (E | X J\u^t - Vu dril)] ^^{^ [ <^ + > l i m i n f E | x J ^ fp{u\) + c{\ + \u\\)dt]-Ey ^{vt)dt] > E | x y ^ \immf{<p{u\) + c{l + \u\\))dt]-Ey ip{vt)dt] > E | x ^ ^{ut) + c ( l + \ut\) dt] - E | x 2^ ip{vt) dt] . Thus E [x Q{ut - Vt, drjt)] >E[X ^{ut) - ip{vt) dt^ , which shows that a.s. T T T T /p ip{ut) dt < oo and that a.s. JQ {ut - Vt, drjt) > JQ <f{ut) dt - ip{vt) dt. This extends, as in Theorem 3.2, to a.s. V t e [ 0 , T ] I {u,-v„drj,)> I <p(u,)ds-f <fi{v,) ds V-y G C([0,T]; H) , Jo Jo Jo which is (3.3.17). As in §3.2 it is possible to prove continuity of the solutions with respect to the data, but the integrability conditions cannot be relaxed. Theorem 3.5. Let a = max(2p,4). With notation as in Theorem 3.4, the maps UQ {uyV)' f {u,T]) and m {U,T}) are continuous from l/'{9,,To,W), L'*(fi; L^([0,T]; H)) and A1^(H), respectively, into {y{n; C([0,T]; C([0,T]; H))*), with the weak* topology O7iL2(fi;C([0,T];H))*. Proof: Use (3.3.5) and (3.3.14) and proceed as in Step 2 of Theorems 3.4 and 3.2 to get E ( f u p K r } + E { M ? / ^ } < c 1^1 + E { K H + E \Mdt + E [m] a/2 T (3.3.31) If (u^,7?^) and {u^,7]^) correspond to {ul,f^,m}) and (•«§, Z^, m^) then \ul-4\' + 2 f\A{ul)-A{ul),ul-ul)ds + 2 f\ul-uldvl-dr,j) Jo Jo = \ul-4\' + 2 f\ul-nlJl-f^)ds+U-m'- [ B{u\) - B{ul) dw, Jo I Jo J t + 2 f\u\-ul,dm\-dml)-2 l\u\ - u]) ^ {B{u\) - B{ul))dws. Jo Jo From the constraint (3.3.17), jl{u\ — u^^drjl - drjl) > 0, so this term may be ignored. Note that [X - Y]t = [X]t + \Y]t - 2[X, Y]t, and that by die Kunita-Watanabe inequality ( Dellacherie & Meyer [1], vol.2, VII.53 ), \[X,Y]t\ < [ X j ^ ^ f y j ^ l Apply this with X = / B{u^) - B(u^) dw, Y = m^ - m^, and (3.3.3) to get \ul-4\'<c f\u]-ul\Us (3.3.32) Jo + 1 4 - ul\' + 2 f \u\ - ulWfl - I ds + c[m' - m% Jo at \ 1/2 \Biul)-Biul)\ldsj + 2 [\ul - ul dm] - dml) - 2 f\u] - 4) o (B{ul) - B(ul)) dw,. Jo Jo It is easy to check that the stochastic integrals are martingales, so E{\uj - 4 n < c fE{\u] - ul\'} ds + CIE{\UI- ul\'} + E {[m^ - m']j} Jo \ + E + E {[m'-m']j}'^'Ey\ + \ul\U\u^t\'dt^ l /2\ From Gronwall E y ^ \ul - u^tf dt] < c(E{\ul-ul\^}+E{[m^-m^]T} (3.3.33) + E { suP(|,i |2 + | .2 |2)}^/^E I \fl - m dt^^ ' Given sequences {u^, f'^,Tn^) converging to {UQ-, / , m) it will follow from (3.3.31) that E { f^P(|uJ|2 + |u2|2)} and E { / J 1 + I^JI^ + l^^p dt} are bounded. Thus from (3.3.33) it may be concluded that (u")„ converges to u in x [0,T]; H). Return to (3.3.32), take the sup over t, take expectations, use Burkholder's inequality and Gronwall's lemma to get , T X 2-1 1/2 Use (3.3.5): E { g P | 4 - ^ ? P } < c ( E { i 4 - . ^ ^ P } + E | Q | / , ^ - / ,^ |d t ) I (3.3.34) + c,Ei^yju]-u'A'\B{uj)-B{u't)\idt^ ' I c^EUj\u]-u'A''\B{u])-B{ul)\ld^ ' I ^1/2' •'^Ut < C 2 E | ^ ^ P ( 1 + K | + |U?|)(J^ | U Î - U , ^ | < C2E { g P ( l + luJi + \u^\fy'\y^\u] - n'A' dt]'^' . From (3.3.33) and (3.3.34) it now follows that u" u in L2(fi; C([0,T]; H)). From (3.3.31) there is a subsequence (T/'), converging weak* to TJ. Since u^ € C -D^ P x dt a.e., it follows from (3.3.31), (3.3.6) and (3.3.5) that (A(u"))„ is bounded in x [0,T]; B) and (fi(u"))„ is bounded in A{w). Thus A(-u") ^ x weakly in L 2 ( 0 X [0,T]; H) and fi('M") ^ ^ weakly in A('«;). The weak convergence of 5(u") forces a slight change in the identification of x and ^. From (3.3.3) for any v 6 L'^(Q, x [0,T]; H), E I jj"^ | f i « ) - B{vt)\ldt] < 2E - A{vt),u2 - vt)dt] + A E | ^ ^ K - u « p d t | . 59 Use the weak lower semicontinuity of the norm in the Hilbert space A{w) to get, along a subsequence, < 2 E | ^ (Xt-A{vt),ut-vt)dt] + XEy^ \ut-vt\Ut] Now proceed as in Theorem 3.4 to identify x as A{u) and ^ as B{u), and u and ?/ as the solutions to (3.3.16) and (3.3.17) corresponding to wo> / and m. The same argument as used in Theorem 3.3 will show that the entire sequence (77")n converges weak* to TJ. 3.4 Examples and Interpretation. The constraint which appeared in Theorems 3.2 to 3.5 can be given a further interpretation if ip o is an indicator function. Specifically, assume that iîT C H is closed and convex with K ^ 0, and let (p{u) = IK{U). For convenience define € := {v G C([0,T]; E.) : vt e K Vt}. Then the constraint becomes: a.s. u e £ and Vt JQ{US - Vg, drjs) > 0 V v G Proposition 3.1. IfueS andrje BV([0,T]; M), then (1) / {ut-vudrjt)>0 yve€ Jo is equivalent to (2) ^ V e £, j {us - Vs,dTjs) is increasing int. Jo and either of these imply 0)\rj\t= f l{u,edK)d\rj\s Vt , Jo where \Tj\tisthe variation of 7/ on [0, t]. Proof: That (2) implies (1) is obvious. To show that (1) implies (2), assume that (2) is false. Then there exist a function v e £ and numbers r,s G [0,T] with r < s such that /o ( M * - Vt,drjt) > J^int - Vt,drjt). Define U , : = ( ^ * ; t G ( r , . ] l Ut ;otherwise Then û( G A V t and / {ut-vt,drjt)= {ut-Vt,dTjt) <0. (3.4.1) ^0 J{r,s] By Lusin's theorem there exists a sequence {g"-)n C C([0,T]; H) with ||5f"|| < f^^\vt\ V n and 5f" ^ û d\Tj\-a.e.. If := TTK O g"- then (t?")„ C 5 and z;" ^ w d\rj\-a.€.. From (3.4.1) it now follows by dominated convergence that J^iut - v^,drjt) < 0 for all n sufficiently large. This contradicts (1). To prove (3), define the open set U := {t e [0,T] : ut e K}. Let / be one of the component intervals of U and choose r < s e I. Note that there is a measurable function : [0, T] H with \hi\ = 1 Vf such that rjt = JQ d\r]\s. (This is just the Radon-Nikodym theorem, which is valid in a Hilbert space, plus an additional argument to show that |/it | = 1 Vf. Rudin [1] Theorem 6.12 does this for H = C, using an argument which generalizes immediately to arbitrary separable Hilbert spaces.) Let p{x, dK) be the distance of x to dK; then x i-^ p{x, dK) is continuous, as is f p{ut, dK). Fromthedefmitionof M and the choice of rand s it follows that ^  := M{p{ut,dK) : t e [r,s]} > 0. Let _ ( Ut ;t < rort > s ''^•=\ut + ^ht teir,s] Then / {ut-vt,dr)t)= -7:{ht,dTjt)=--{\T]l-\r]\r). Jo J(r,s] 2 2 Approximate v as before by functions (u")„ C S with ^ u d\T)\-a.e.. From (1), / T f'^ _ S 0 < lim / {ut - v'^,dr]t) = {ut - vt, drjt) = --{\v\s - \r}\r) " 7o Jo 2 so \TJ\S < \rj\r. But \rf\. is increasing, so \rj\s = \ri\r. This shows that |?7| is constant on each of the component intervals of U, and thus | 7 , | , = f\{u,eK)d\r)U= fl{u,edK)d\r,\s Vf . Jo Jo Note also that Vt = fl{us e dK)drjs. Jo By choosing </? to be differentiable. Theorem 3.3 may be used to produce a result for monotone drift terms without a boimdedness assumption such as (3.3.6). The following simple proposition is needed. Proposition 3,2, Assume that : H H^- E is convex, contmuous and Gâteaux differentiable with derivative f3, and define $ : C([0,T];H) ^ K by $(u) = Jg(p{ut)dt. Then $ is Gâteaux differentiable, and (V$(u) , v) = /o^(/9(u<), vt) dt Proof: It is well knovm that r i - ^ {ip{u + rv) - (p{u))/r is increasing on E \ {0}. Thus for 0 < r < 1, v(u+rv) — (p(u) ip{u) - cp{u -v)< -f-^ < (p{u+v) - (p{u). r If u,v e C([0,T]; H) then t (p{ut + vt) - <p{ut) and t H-^ (p(ut) - (fi{ut - vt) are continuous and therefore integrable. Dominated convergence and the differentiability of v yields r^0+ r Jo r^o+ r Jo Now in Theorem 3.3 assume that (p = <f) + ii, where (j) is convex, continuous and differentiable o on H with derivative /3, and i> is convex and lower semicontinuous with ^ 0. From Barbu & Precupanu [1] Corollary 2.5, pl28, dip = (3 + dtj). Thus in (3.2.2), r/t = (i{us) ds + where a.s. \f t e [0,T] [\us-Vs,dQ> f-tP(u,)ds- f-ip{vs)ds V« € C([0,T]; H) . Jo Jo Jo The result is that an extra drift term /3 has been included, which is monotone but need not be Lipschitz, and need not satisfy a growth condition such as (3.3.6). Of course, the same trick works in Theorem 3.4. Slightly more generally, assume that is open and that (p is continuous and differentiable on D^, and let u and 7/ be the solutions to (3.2.1) & (3.2.3) or to (3.3.16) & (3.3.17). If ut e D^\ft then one may conclude as in Proposition 3.2 that rjt = (3(us) ds, where (3 is the derivative of (f. However, i f ut touches dD^ then d^{u) is no longer single valued and r/ need not be absolutely continuous. For example, suppose that ut^ e dD^ and choose g e dl-^^{ut„). (The Hahn-Banach theorem guarantees that dl^^{ut^) is non-empty.) Let vt '•= fol^{us)ds and let •.= Pt + gl[t^^T^{t). Then for V e C([0,T];H), / {ut - vt,dvt)= / (ut-vt,f3{ut))dt> / (p{ut)dt- I (f{vt)dt Jo Jo Jo Jo I {ut-vt,dnt)= j {ut-vt,f}{ut))dt+{uto-vto,g) Jo Jo > / (p(ut)dt~ / (p{vt)dt+{ut^-vt,,g). Jo Jo The right hand side is -oo if vt^ ^ D^, while {ut^ -vta,g)>'A i f vt^ e D^, so either way / (ut - Vt,diJ.t) > / 'fi(ut)dt- / (fi(vt)dt V v . Jo Jo Jo Thus both v and p are in d^{u). Of course /x is not continuous and so is not a candidate for the solution to (3.2.1), but the possibility that there are singular measures without point masses in 9$(u) has not been ruled out. and To give a simple example, consider two particles constrained to move on a straight line under a mutually repulsive force. Let ip{x,y) := - ln{x - y), which is convex and differendable on = {{x, y) e : X > y}. If and are independent standard Brownian motions then by Theorem 3.3 there exist imique solutions yt) and rjt to (xt, yt) + r]t = {xo, yo) + {wj,w^) with a.s. Xt > yt^t, - ln{xt - yt) dt < oo and a.s. yt f {xs-vlys-vl)-dr)s> f Hv]-vl)-\n{xs-ys)ds V ( u \ t ; 2 ) 6 C([0,Tj; E ^ ) . JQ JO Up until the time when the particles first hit (if ever), rj will be absolutely continuous: T]t= , ds Jo -ys Xs- ysj fort < T where r = inf{s : Xg = ys}. Many other choices for (p are possible. If the particles are not constrained to move in a straight line then the appropriate "potential" (p is no longer convex. Finally, I need to show die existence of operators satisfying the assumptions in §3.3. Let V ^ H ^ V * be a triplet of spaces as described in the introduction or at the start of Chapter 4 , and let A : V V * and B : Y £ ( G ; H ) be operators satisfying assumptions (4 .1.1) to (4.1.5) in §4.1 . Let (e, ) , £ N be an orthonormal basis for H consisting of elements of V . For each n G N , define i2„ : E " ^  V and P„ : V * ^ E " by RnX := XiCi and (P„C)i := (C^ i ) ( so P„ = < ). Define A „ : E " E " and B „ : R ' ' £ ( G ; E ' ' ) by An := PnARn and B „ := PnBR,,, so diat (Anx) • y = {A{Rnx), Rny) and Bn{x){g) • y = {B{Rnx){g), R^y). It is immediate from (4.1.2) that An is hemicontinuous. From (4.1.1), \Anx\ < Cn\\ARnx\\* < c{l + \\Rnx\\P-'^) < c( 1 + |a;l""^), so (3.3.6) is satisfied, and (3.3.5) follows from (4.1.5) in the same way. For the monotonicity of An and Bn, (4.1.4) implies that 2{ARnX - ARny, RnX - Rny) + MRnX - Rny\^ > \B{RnX) ~ P(i2„2/)| 2{AnX - Any,x -y) + X\x - y\^ > \B{Rnx) - B{Rny)\l. Let 5" 6 G (âi G be the covariance of the Brownian motion w and let 5 = a/gf; (gi gi for positive nimibers {ai)i with X); cr; < oo and {gi)i an orthononnal basis for G. Then I oo = E E iB{Rnx){9i) - B{Rny){9i), eif \B{Rnx) - B{Rny)\l=J2''i\B{Rnx){9i) - B{Rny){9i)\' i=l n ^ E ^ ' E ( ^ ( ^ " ^ ) ( ^ ' ) - B{Rny){9l),e^f = J2^i\Bn{x)(gi)-Bniy)igi)\' I = trace {{B^x - B^y) ® {B^x - B^y), S) = \BnX - Bny\l^ where | • |„,^ is the norni in A(G, R " , V, w), so (3.3.3) holds. To check (3.3.4), let F = / + eA for e sufficiently small, and note F„ := P„FRn = In + eA„ where / „ is the identity on R". From (4.1.4), F " is monotone for e sufficiently small. Note that (4.1.3) may be assumed to hold for a = 0, so 2{Au, u) + X\u\'^ > 0\\U\\P and {Fu, u) > C | | U | | P for c=9e/2mde < 2/A. Then iFnX,x) {FRnX,RnX) ^ | | ^ ^ ^ , , , -^^—r~i— = ^  r~i > c , , > c a;r ^ oo as oo. \x\ \x\ \x\ Thus Fn is coercive, monotone and hemicontinuous, and so is onto R " ( Deimling [1] Theorem 11.2 ). Note that (3.3.1) is automatic as = -Ds„ = R" . so all that remains to do is to find some suitable constraint functions. If : H R is Isc proper convex then v'n := o -Rn : R " R is O 0 clearly Isc proper convex on R", and 0 i f I)<^  7^  0. If : V R then ^ „ := o R^ is still Isc o and convex, and D^^ may be non-empty. For example, assume that V i s a Hilbert space and that the imbedding i : V ^  H is compact ( e.g., take V = n^{0) and H = V?{0) for C c E'^ ). Let ((•, •)) be the inner product in V and let L be the canonical isomorphism of V onto V*. Then is linear and continuous, and T := iL~^i* is a compact self adjoint operator on EL By the Hilbert-Schmidt theorem there exists an orthonormal basis {fk)k for H and positive numbers 0 as A; -> 00, such diat Tfk = Okfk- If Ck := L~'^i*fk tiien Cfc S V , ick = Okfk, and {•s/\kek)k is a basis for V , o where \k = 11 Ok. Suppose that <^  : V R has non-empty interior and choose a e D^. Then O 0 Efc=i \/Âfc((a, Cfc))efc e for n sufficiently large, so = (\Afc((a, ^A:) ) )^! e D^^. Chapter 4. Constrained Stochastic Partial Differential Equations. In this chapter I discuss constrained stochastic partial differential equations, in the framework of a "Hilbert triplet". Let V be a real separable reflexive Banach space which is continuously and densely imbedded in a Hilbert space H. If i is the imbedding of V into ÏÏ then the adjoint i* : H* i-^- V*. Identifying H with its dual in the usual fashion produces the triplet The canonical examples are H = ïJ{0) for a bounded open region O in and V = W ^ ' P ( C ) or W j ' ^ ( O ) . The space H is called the "pivot space" for the triplet. The iimer product on H will be denoted by (•, •) and the norm by | • |. The norms on V and V * will be denoted by || • || and || • ||* respectively. Angle brackets (•, •) wil l be used to denote the pairing between a Banach space and its dual. Subscripts will be used occasionally to indicate the spaces involved, although these will usually be V and V * . Some restrictions on the constraint functions or the spaces involved are necessary. Two cases which may be handled abstractly (i.e., without reference to specific spaces V and H ) are discussed in §4.1. These are when the constraint function is the indicator either of a sphere in H or of a half-space in V * . Results for other sets or for functions other than indicators may be obtained i f the drift and diffusion coefficients are linear partial differential operators acting on Sobolev spaces. This is done in §4.2. The proofs use the penalty method as in Chapter 3. The principal difficulty is that the Itô inequality of §3.1 is only valid for semimartingales in H , but i f the drift term contains an unboimded operator then the penalized processes wil l have a bounded variation part which is only of bounded variation in V * , not H . Two ways of overcoming this problem are to restrict attention to constraint sets whose distance function is twice differentiable, or to use the technique of "pivoting" from P.D.E. theory to reduce the penalized processes to semimartingales in H . The first option allows the use of the Itô formula in Pardoux [ 1 ], and is followed in §4.1. The second option allows the Itô inequality to be used, but forces a restriction on the spaces. Two versions of Itô's formula are needed: Pardoux's version for twice differentiable functions, and Krylov & Rozovskii's version for | • p. Both are quoted below, along with a result of Gyongy & Krylov [2]. Theorem 4 . 1 . Let zbea continuous semimartingale in H and ( a weak* progressively measurable process in V * . Suppose that Ut := / (sds + Zt Jo is equal P x dt a.e. to a progressively measurable V-valuedprocess û, and that for some p > I, a.s. \\û\\ e LP([0,T]) and \\(\\* € LP ' ( [0 ,T] ) . Then u is mdistmguishable from an adapted continuous process in H and a.s. \ut\^ = \uo\'^ + 2 [ {û„Qds + 2 f {u„dzs) + [z]t V t € [0,T]. (4.0.1) Jo Jo This is Theorem 1.3.1 of Krylov & Rozovskii [2]. Since C is only weak* measurable, (s ds should be interpreted as a Gel'fand integral. A result which allows / Cds to be replaced by a non-absolutely continuous term was obtained by Gyongy & Krylov [2]. I have quoted their result even though it is not needed in this thesis, as a corollary to it may be useful in future work. Theorem 4.2. Let m be a locally square integrable martingale inU, Aa real continuous increasing adapted process with A(0) = 0 and ( a weak* progressively measurable process in V * . Suppose that Ut := / Cs dAs + mt Jo is equalP x dA a.e. to a progressively measurable V-valued process û, and that almost surely \\û\\, IICII* and ll^ll IICII* are integrable with respect to dA. Then u is indistinguishable from an adapted cadlag process in H and a.s. \utf=\uof + 2[{ûs,Cs)dAs + 2f{us-,dm,) + [m]t V t G [0,T]. (4.0.2) ^0 Jo This is a special case of Theorem 1 in Gyongy & Krylov [2]. As before, ( dA should be interpreted as a Gel'fand integral. Corollary 4 . 1 . Let m be a locally square integrable martingale in H and 77 a continuous adapted Y*-valued process with paths of bounded variation in V * . Suppose that Ut := Tjt + mt is equal P x d\T]\ a.e. to a progressively measurable process û inV and that a.s. \\û\\ is integrable with respect to d\r]\. Then u is indistinguishable from an adapted cadlag process in H and a.s. \utf = \uof + 2f{û{s),r]{ds)) + 2f{us-,dms) + [m]t V t 6 [0,T]. Jo Jo 66 Proof: Let B be the Borel sets in [0,T] and let \r]\{uj, t) be die variation of rj{u, •) on [0, t]. The corollary will follow from Theorem 4.2 i f it can be shown that 7/t = Cs d\ri\s, where C and |r/| have die required measurability properties. It may be assimied that E {\ri\j} < oo. The general case will follow from this by localization. By the Radon-Nikodym theorem, for every oj e Q. there exists a strongly B measurable V*-valued function ^(w, •) such that riB{t)r,{uj,dt)= C lB{mu,,tM{u,dt) SBeB. (4.0.3) Jo Jo Any continuous / : [0, i] i->^  V may be approximated by the step functions n - l r: := fohi[0,t/n]) + J2fiJ/nU(tJMJ + l)M). Since i] is adapted, / {rs, dT]s) = (/o, Vt/n) + y \ {ftj/n,Vt{j+l)/n " Vtj/n) Jo is .Frmeasurable, and thus so is /(,*(/«, df/s) = lim„^oo Jlif"^ AVs) ( die existence of the limit follows from (4.0.3) and dominated convergence). Let D be a countable dense set in die unit ball of C([0, T]; V ). Then |7/| (a>, f ) = supy ç^, , dr}s), so is adapted. Note also that continuity of 77 implies continuity of | T ; | . Define a collection of sets G:=[G xB:uj^ J {V1G{UJ, dt), r/(w, dt)) is /"-measurable V T; e V | . It is easy to check, using the representation (4.0.3) and dominated convergence, that (7 is a a-field. Let £ :={F X I : F e J'mdlisminterval in [0,T]} . For any interval / there exists a sequence of continuous real valued functions ( /„ )„ on [0,T] with fnit) 1 / (0 for all / as n -> 00 and \fn(,t)\ < 1^ n, t. By dominated convergence, r {vlF{ij)fn{t), 77(0), dt)) = IF(OJ) C {vfn{t), e(u;, t)) \rj\{u, dt) Jo Jo ^1F{OJ) j {v\j{t),au,t))\jj\{u,dt) Jo = / {vlFxl{i^,t),7]{u,dt)). Jo Since oj i->- {vlF(u!)fn{t), r](ùj, dt)) is measurable, so also is a; i - ^ JQ {V1FXI{U}, t),r]{u>, dt)) for all V e V and F X / € Thus é : c ^ a n d s o J ^ X ; B = a{e) C G i.e., G = F x B. Given the integrability condition E {|T/|T} < oo, for any G e T x B one may define p{G) -=^[1^ 1 G d7?| = 10(0^, * )^ (^ , dt)V{du) 6 V * as a Gelfand integral. Define also X(G) :=Ey\Gd\v\Y Then A is a finite positive measure on f x B and < X{G), so yti is a coimtably additive V*-valued measure of finite total variation, with /x < A. Consider the restrictions of p and A to the optional cr-field O. By the Radon-Nikodym theorem there exists a V*-valued strongly O-measurable process C with n{G) = J^(dX for all G £ O. For any stopping time r the stochastic interval [0, r ] is optional, so considering the definitions of fi, X and C, for any v eY, ^{{v,T]r)} = {v,p{lO,T})) = ES^(v, Cd\r]\^] V stopping times r. But both (t;, 7/) and {v, J ( d\T]\) are optional, so by the optional section theorem ( see IV.86 and IV.87 of Dellacherie & Meyer [1] ), (7;, 77) and (v, J C< |^^ |) are indistinguishable. Since V is separable, 77 and / C d|77| are indistinguishable. By extending an argument of Rudin [1] Theorem 6.12 to separable Banach spaces with the Radon-Nikodym property, it may be assimaed that = 1 V (w, t). Thus the integrability conditions in Theorem 4.2 reduce to a.s. û G L^(d|77|). Theorem 4.3. Let m be a continuous locally square integrable martingale in H , and h and C progressively measurable processes in H and V * respectively. Assume that a.s. \h\ e L ^ ( [ 0 , T ] ) anda.5. ||C||* € U'{[0,T]), and that Ut := hs + Cs ds + mt Jo is equal P x dt a.e. to a progressively measurable Y-valuedprocess û with a.s. \\û\\ 6 LP([0, T ] ) . Let (f : M 1-^ R be a twice Fréchet differentiable function such that (1) ip, ip' and if" are bounded on bounded subsets of H. (2) (p and (p' are continuous ( for the norm topologies ) (3) ip" is continuous from H into ( E lâiE)* with the weak* topology (4) The restriction of ip' to V maps V info V , and is continuous from V with the norm topology to Y with the weak topology. (5) There exists a number c > 0 such Oiat \\<p'(u)\\ < c( l + \\u\\) V u e V . Then a.s. ip{ut) = ip{uQ) + j {ip'(us),hs + Cs)ds + J ((/^'(u,), dm,) + - ((/^"(u^),4ml,) . V i . This is Theorem 4.2 p65 of Pardoux [1]. I have strengdiened die hypodieses very slighUy and stated the hypotheses on u in the same maimer as in Theorems 4.1 and 4.2. A further restiction on V and V * is also necessary: ' 'Hypothèse II", p63, of Pardoux [1] must be satisfied. A sufficient condition for this ( Theorem 4.1 in Pardoux ) is that both V and V * be uniformly convex. 4.1 Abstract Case. Let G be a Hilbert space and w a Brownian motion in G with covariance 5 G G Let A{w) be the space of integrands for w which map G into E, as described in Chapter 2. Let K denote either a sphere in E with radius p> 0 and centre a G V , or a half space in V * ; K = {heE;\fi-a\<p} or, for some fixed A; G V , i(' = { C G V * ; ( C , A ; ) < l } o r I{ = {CeY*; {C,k)<-1}. o If i i ' is a half space let a be any point in V n /T . Assume given a number p>2 and operators A : V i-> V * and B : Y A(w) satisfying, for numbers 0 > 0, C > 0, A > 0 and c > 0, Growth of A: ||A(u)||* < C ( l + | | u | | P - ^ ) V U G V . (4.1.1) Hemicontinuity of A: r (A(u + rt;),u;) is continuous Vu,u,uj G V . (4.1.2) Coercivity: 2{A{u),u-a) + \\u\'^ + c>0\\u\\P + \B{u)\l V U G V . (4.1.3) Monotonicity: 2{A(u)-A{v),u-v) + X\u-v\'^ >\B{u)~ B{v)\l V U , U G V . (4.1.4) Growth of B\ \B{u)U<C{l + \\u\\) yueV. (4.1.5) hi addition, i f p = 2 then B will be assumed to be linear and to satisfy \u o B{u)\w < c|Mp and \u o B{v)\,u, < c\v\\\u\\\/u,v G V . (4.1.6) As in §3.3, the monotonicity condition forces B to be continuous from V into A{w). The linear operators defined in §4.2 satisfy these assumptions. A non-linear example is given at the end of the proof to Theorem 4.4. Given (4.1.1), the point a in (4.1.3) may be replaced by any point b in V , since {A{u), u- b) = {A{u), u- a) + {A{u), b - a) > {A(u),u - a) - ci - C2\\u\\P-'^ >{A{u),u-a)-cs-^\\ur. o Thus, without loss of generality, a may be assumed to be in V fl K. If iiT is a sphere then assume a is the centre of K. Let (f2, F, (Ft, t e [0, T]), P) be a stochastic basis satisfying the usual conditions, and let V be the predictable cr-field on ÎÎ x [0,T] for this filtration. Assume that w is adapted to this filtration and suppose given (1) A n .Fo-measurable H-valued random variable UQ. (2) A progressively measurable V*-valued process / . (3) A continuous H-valued martingale m with [m]t = / Q hg ds for some progressively measurable process h. Let Q be the H ê i H-valued process with traceQ = 1 and {mjt = /g Qs d[m]s. Theorem 4 .4 . Assume that a > 4, uo e L " ( f i , J'o;H), uo e K a.s., f e L " ( f i ; L°°([0,T]; V*)) and, for some t > 1, h e L " / ^ ( f i ; L'[0,T]). If K is a sphere then there exist unique continuous adapted M-valued processes u and TJ such that a.s. ut e K ^ t, u e V P x dt a.e., j] has paths of bounded variation, E | ( | j K r d t y ^ ' | - f E { s u P | ^ , | « / 2 } <<^, (4.1.7) E { ^ ^ P k r / ' } H - E { M ? / ^ } < o o , (4.1.8) a.s. ut+ I A(u,)ds+ f B(us)dws + i]t = uo+ f fsds + mt V t G [0,T]. (4.1.9) Jo Jo Jo and a.s. Vt G [0,T] j (u, - v,,dr]s)>0 Vt; G C([0,T]; H) such that tJj G A V t. Jo (4.1.10) If K is a half-space in V* then there exist unique processes u and rj such that u is adapted, continuous in E, a.s. Ut 6 Kyt,u G Y P x dt a.e. and u satisfies (4.1.7); TJ is adapted, continuous and of bounded variation in Y, E { g P | k r / ' } + E { H ^ / ^ } < o o , (4.1.11) u and T] satisfy (4.1.9) and a.s. V«G [ 0 , T ] f {us-Vs,dTjs)>Q V u G C([0,T]; V* ) such that u^ G iiT (4.1.12) Jo Proof: Uniqueness. The only change from Theorem 3.4 is that the Itô formula given in Theorem 4.1 must be used. Proof: Existence. Step 1: Construction of the penalized solution. If i i ' is a sphere, define ip^u) := inf {ji\u - h\'^ : h e K} = j-^diuy, where d is die distance to K in H . If K is a half space, define <^E(U) := inf {^| |u - Cll* C € K] = Yed{uf, where d is the distance to K in V*. The penalty operator is defined as in Chapter 3, but since </? is an indicator, (3^ = ^/3 where /3 := €V<p,. Then /3 : H H or /3 : V* V , d{u) = \p{u)\ or diC) = ||/3(C)||, and f3 is monotone and continuous. By using the imbeddings i and i*, fi may be considered as a continuous monotone map from V into V*. For the simple sets under consideration it is also possible to compute fi explicitiy: /?(u) = [(1 - PI\U - a\)y Q](U - a)ii K = {U eB.:\u - a\ < p}. (4.1.13) /5(0 = p i p [ ( ( f c , C ) - l ) V O ] f c i f i i ' = {C€ V*:(A; ,C) < l } . (4.1.14a) /^(C) = 0 + 1) V 0]A; i f iir = {C G V* : (fc,C) <-l}. (4.1.14b) One may check that (4.1.1), (4.1.2), (4.1.3) and (4.1.4) continue to hold with A replaced by A + for any e > 0. Thus, by Theorem II.2.1, Corollary II.2.1 and Theorem II.2.2 of Krylov & Rozovskii [2], pi253, the penalized equation u\+ I A{u\)ds+ j B{u\)dws+ j -fi{ul)ds = uo+ [ fsds + mt (4.1.15) Jo Jo Jo ^ Jo has a unique solution u" : il x [0,T] H , with u" adapted, continuous in E, u" e Y P x dt a.e. and E | / J dt} < oo. One may also use Theorem 3.1, pl05, of Pardoux [1] to get a solution to (4.1.15), at the expense of a local Lipschitz condition on B. Step 2: Bounds on {u% and E { g ? K r } + E | ( j J ^ K i r a / 2 ' dt + E -d{u\)dt a / 2 ' < C (4.1.16) with c independent of e (and recall that (/(w) = |/?(w)| or ||/?('w)||). Proof: Use Theorem 4.1 to get \u\-a\^ + 2 j {A{ul),ul- a)ds + 2 f (u^ - a) o B{ul) dws + 2 j {-I3{u\),u\ - a) ds Jo Jo Jo ^  = 1 ^ 0 + 2 / {fs,ul - a) ds + 2 / (ul - a,dms)+ m - B{u)dw . Jo Jo I J it Use the estimates (1) From (2.3.10) for some r > 0, | ( / ? « ) , < - a) > f d « ) (2) [m-f B{u') dw\ = [m]t + / J |fi(ti^)|2. d^ - 2 [m, / B{u') dw]^ (3) 2 { / „ u ^ - a ) < 2 | | / , | | * | | a | | + 2 | | / , | H | < <2 | | / , |H | a | | + - | | < | | P + ci < ^ i K i r + < i + i i / . i i ^ ' ) «II* {A)\u\-a\''>\\u\\''-\a\' and (4.1.3) to obtain \\u\\'' + \E J\Krds + 2r F^\d{ul)ds < 1^0 - ap + ci + C2 / \\fs\\tds + \ j \ul\^ds+[m]t Jo Jo - 2 m , / B{u^)d'W +2 I (ul - a, dnis) - 2 f (ul - a) o B{ul) dws-J it Jo Jo Use Yomig's inequality, (4.1.5) and x^ < 1 + for x > 0 and p > 2 to get - 2 m,J B{u')dw ^<ct[m]t + ^  J^\B{ul)\ids <c^[m]t + LJ\l + \\ul\\fds < Ci[m]t + C2 + - \\<\\^ds, \\u\\'' + \E J\\u\rds + 2r F^\d{u\)ds <c(^ + \uo-a\'^ + J \\ft\V'dt+ J K p d s + [rn]i + 2 j (ul - a,dms) -2 f (u^ - a) o B(u',) dw^. Jo Jo from which A localization argument is needed before taking expectations, as there is very little information on integrability of the penalized solution. For any stopping time r , take the supremum over t < T to obtain r\\M\l'dt+[m]'r]+c n^l'dt Jo J Jo J\ul-a,dms) +csup J\ul-a)oB{ul)dw < c 1+ + ,sup Raise this to the power a/2, and take expectations to get E i + E { K r } + E j (^ ^^  + c , E ^ s u p < c T . \ « / 2 1 , _ \ I / rr ïdt + E 7 2 + cE / (ul - a, drris) Jo V 2 ' . c , E - p / {ul-a)o B{ul)dw, Jo luil'^ds .12' (4.1.17) « / 2 " l c ,E sup ft « / 2 ^ f / j {u\ - c, drus) U c E ^ Jo V The stochastic integrals are dealt with using Burkholder's inequality in a by now familiar fashion. For die " d m " integral: t j (4.1.18) > ^ i E { s u P K r } + C 2 E { M / ^ } + C3 The cases p = 2 and p > 2 must be handled separately for the " dw" integral. Forp = 2use(4.1.6) a/2") ( / rr \ < cE / {ul-a)oBiul)dw, Jo <cE<[ 1^ ( |<P + | | a | | K | ) 2 d i < C 4 + C 4 E | S U P K P / 2 ( / j ^ P r f t ) ' ^ ^ ' ! < C 4 + ^ E { s u P | , e | . } ^ , E | ( ^ y ^ | , . | 2 , Forp > 2, from (4.1.5), / {ul-a)oB{ul)dw, Jo < C5 E ^ \ul\\l + \\ul\\)Ut a/2' a/4' \ + E{[ I WulW'dt^ a/4^ \ To deal with the first term, let x be in (0,1). Then The penultimate step follows from Young's inequality with triple conjugate indices p\ = 2/x, P2 = P and p3 = 2p/{p{2 - x) - 2). The last step follows for suitable choices of CQ and C7. One may always ensure that 2p(l - x)/{p - 2) < a by choosing x sufficiently close to 1. Since p(l - x)/[p(2 - x) - 2] < 1 for any p > 2 regardless of x, and < 1 + a for any a > 0 and 0 < 6 < 1, For the second term. Thus Finally note that Substitute these estimates into (4.1.17) to obtain ^ { S ? K r } ^ E { ( £ i K i r d <c (da ta ) + c E | £ s u p | ^ ^ r d t } U E H y -d{u\)dt (4.1.19) where da ta=l + E { K r } + E { ( C\\fs\\tds\ ' | + E { H ? / ' } . Define stopping times r„ := inf | t € [0,T]; g P | < | + £ d . = n | and replace r by i A r„ . A l l expectations in the previous computations are finite, and E { } < c( data ) + cE ^up^^^ \uT ds] <c(da ta ) + c £ E { s u p ^ ^ J < r } d . . Apply Gronwall's lemma : E { t < ? J ^ t T } < c(data) where c is independent ofn as well as e. Since r„ -+ T almost surely as n ^ oo, Fatou's lemma yields E { g P K r } < ^ ( d a t a ) and it follows immediately from this and (4.1.19) that E f / rT dt] > < c ( data ) and E « / 2 -1 \ -d{ul)dt } < c(data) 0 e / as required. Step 3; Convergence ofd^u"). limE{^^Pd{uir^'} = 0. Proof; This will be done first for a half space. The modifications necessary for a sphere will be given later. For some 7 > 2 define <^  : H R by (^(u) := d{uy. From (4.1.14), f3{u) = d{u)k/\\k\\ and Vd(u) = /3{u)/d{u) = k/\\k\\ foTu e M.\K. Since 7 > 2, 0 is twice differentiable with derivatives W<f){u) = 7 d ( w ) ^ - i ^ = jd{uy-^p{u) and vV(w) = 7(7 - i)d{uy-^— ^ — ii^ii i i^ i r Apply Theorem 4.3 formally to (f>{u'^) and note that d{uo) = 0 a.s. to obtain M2 rt d«y + îîlîi j [ d « ) ^ - ^ f c ) ds+j^J^ d{uiy-'k o B{ul) dWs + J ^ [ ]d{uiy ds j\{4r-\fs,k) ds+^^ J\d(uir-'k,dms) (4.1.20) 7 + i 7 ( 7 - i ) f « r - ( l ^ « ^ , o . ) « . 17(7 - 1 ) / rf«r-^ o « ^ o S K ) , This must be justified since ç!) does not satisfy Pardoux's condition ||Vç!i(u)|| < c( l+ ||u||). To this end define, for any positive integer n, . s _ J a;''' ;0 < a; < n 5n(xj : - I ^ 7 + ^ n^-i(a; _ n) + ^7(7 - l)n^-2(a; - \n < x and ç!>n(«) := 6'n(d('ti)). Then Çn and ^„ are with V<A„(u) = 5r;(d(u)) ^ and y'Uu) = g''{d{u))^^® ^ Since 5^(x) < c(n)(l + a;) and d(u) < c||u|| (infact d('ti) < c||u||*), \\VMu)\\<c{n){l+\\u\\) so Pardoux's Itô formula applies and MO + 9M4)){A{ul), k) ds + j\n{dK))k o B{ul)dws ^ ^ J ^ 9Mul))d{ul) ds = P I [ g'nidKWs, k)ds+-^^ j\g'M<))K dm,) ^l[9'M<)){~u®j^,,Qs)d[m] \ [ 5 ; V « ) ) ( p 0 B{u\) ® ^ o B{ul), S) ds. As n —> 0 0 , Qn (x) Î x^, g'^(x) Î 7 x ^ - 1 and g'Ji{x) Î 7(7 - l)x'^-^ for all x (in fact unifonnly on compacts). Since almost surely u e C([0,T]; H) and \d{u)\ < c|u|, fif„(d(w|)), g!^{d{ul)) and g"{d{ul)) are almost surely boimded uniformly in n and s. Simple convergence arguments then yield (4.1.20). Apply some obvious estimates in (4.1.20) and take the supremum over t to get ^^Pd{uir+^J\{uirdt • rT rT / d{uir-'\\A{um*dt+ / diuir-'\\M\*dt Jo Jo C d{u\y-^\k o B{u\)\^ dt+ C d{u\y-^ d[m]t Jo Jo (4.1.21) + sup j\{uiy-^koB{ul)dWs + g P J\diuiy-'k,dm,) From here the analysis is a little different for p = 2 than for p > 2. For p > 2 use (4.1.1) and d[m]t = ht dt, take expectations and use Burkholder's inequality to get E I ^'^Pdiu'.V \+E^- C d(ulV dt \ (4.1.22) {^l d{utr]+ES^-j\{u^tr t] E diuir-\i+wu^twr' dt]+Ediuiy-'m* dt] y ^ d{uiy-^ht dt j + E I y ^ diulf-^-^ht dt yy{u\y-\i+\\u\\\fdt]+EI y\{u\f'^-\i + E X 1 /2 -t 1 / 2 ^ \ + E For some a; e (0,7 - 1) +H\\rdt E y\iuiy-'\\fth < E { gPd(u,^)^ j\{u\y-^-'\\ft\V dt] (4.1.23) <E\^^^d{u\ry\{u\ydt^ ' y^^mï dt f + ce^-^-^E -t\\r' dt 0 / x+l The last step follows from Young's inequality with conjugate indices 7/a;, 7/(7 - a; - 1) and 7. Choose X such that 7 / ( 3 ; + 1) = p'. Then 7 - a; - 1 = 7/p, so if 7 < ap'/2, f will be in L^(fi;LP'([0,T];V*))and By the same analysis E + ce ilv [ j \ { u \ y - \ i + \\u\\\Y~' dt^ < - l - E ( g P d K r } + ^ ^ E | | J d K r d i } + ce^-^-^E -y(p-i) 37+1 "\ Choose 7(p - l)/{x + 1) = p, i.e. ^/{x + 1) = p' again. If 7 < ap'/2 then 7(p - 1) < ap/2, so {u'), will be bounded in L ^ ( P - i ) ( f i ; LP([0 ,T] ; V)) and E [[ diuiy-^ui\r' dt^ < ^ E { f ^ P d K r } + Y ^ T T ^ '^^^^^^ Next, for 7/2 - 1 < a; < 7 - 2 ( so diat 7/2(7 - x - 1) > 1), E diu'tf-^-^ht dt 1 /2" < E I g P d « ) " diulf^-^-^-^K dt y /2" T 7 / i f d t 0 2+2J;—r 27 < 12ci E / i f ' ^ - ^ di 2-»2i—i 2 Choose a; such that 7/(2 + 2a; - 7) = i . Then 7 - a; - 1 = ^(1 - \) and E / i f ' ^ - ^ dt 4 = E /ij(// so for 7 < a, /•T E r d(uD'^-^/^t rf*) ' ^ ' 1 < y ^ E { ^ ^Pd(ttD^} + | y ' ^ d « r dt^ + ce?(i-^). (4.1.24) By the same analysis E [di^f^-^i + Wuiwydt^ ' \<A_^[^^Vd{u\y] + ^ E + ce^-^-^E ax. ( l + | K | | ) - - - 7 df 2*2x—1 ' Choose X such that 27/(2 + 2a; - 7) = p and note that now 7 - a; - 1 = 7(1 - 2lp)l2. Then E ax. 2+2x—I \ 2 (1+ dt = E T \ "T /P" {i + \\u\\\Ydt which is bounded independently of e for 7 < ap/2, so E M 7^  i(u\f~'-^{\*\\u;\\fit^' | < ^ E (4.1.25) Once more; for 0 < a; < 7 - 2, fT E 1^ d{u\r-^htdt]<E\^^^d{u\)y\{u'tV'''~^htdt] f^d{u\ry d{u\yd^ ' y hf^dt x+2 2 + ce^-^-^E ^ ( ^ hr^ dt The last step uses conjugate indices 7/x, 7/(7 - a; - 2) and 7/2. Choose x to get 7/(x + 2) = i . Then for 7 < a, E | | ^ ^ d « r 2 / . , d t } < J _ E { s u p , ? ( „ J ) 7 | + J _ E | ^ ^ l d ( « ^ ^ ^ (4.1.26) Similarly E y \ { u t y - \ i + \\ut\\rdt] < Y ^ E { s ^ P d ( t i n ^ } + Y ^ E | ^ ' " l d « ) - d i A i + IKII) .^0 ) ^ di 1+2 1 \ "2-Now choose x such that 2 7 / ( 1 + 2) = p so that 7 - a; - 2 = 7(1 - | ) . Then for 7 < ap/2, E (4.1.27) Inserting all of these estimates into (4.1.22) yields rT E { ^ ^^d{uiy] + E -^d{uiy dt] < c (fl^ + e i (^ - i ) + e^^^'p^ + €^(^"7) + 2.^ \ J which imphes the resuU. The restrictions on 7 are 7 < aminjp ' /^ , l ,p /2} . Asp> 2, then p' < 2 and the relevant condition is 7 < ap'/2. hi particular, 7 = a/2 is a valid choice. For p = 2 the estimates obtained above for 1/2 j rd(u',V-^(l+M\\?dt^ and EU^diulf-H^+KWfdt are useless. Instead the assiunption (4.1.6) must be used to get \k 0 B(u)\ < c\u\. From (4.1.21) one obtains /•T E {f^Pd(uir}+E[-J^ diuiydt^ + ^ [ [ diuiy-^ht d i j + E I diu'tf-'-^ht dt y\(4y-M\'dt'j+Ef^ {^j\{u\f^-M? dt^ + E The first four terms are unchanged from (4.1.22). For the last two, cT ^ 1 r /-T E + « ? - ' E J / " | a ; P ( i ( | i f 7 < a, and | ( y ^ d ( u D ^ - - ^ k , f d ^ y ^ ' l < E | 3 d ( 4 ) i - i y\{u\ydt^"' sup |^e | | + « ^ / 2 E { S U P | ^ . | 7 | (4.1.29) using conjugate indices 27/(7 ~ 2), 2 and 7. For 7 < Q = ap'/2, E { ^^d{u\y] < c + , i ( i - i ) + , 7 ( 1 - ^ ) + and the result follows. There are several more complications if K is a sphere. Let (j){u) = d{uy as before. For this case /3(u) = d{u){u - a)l\u - a\, so V d ( M ) = (3{u)/d{u) = (u- a)/\u - a\. Thus, using Vlu] = u/\u\ for u^O, V(f>{u) = jd{uy-'^^^^—^ = ^d{uy-^f3{u) \U Q/\ and | u - a | | u - a | l ^ - a l V p - a| \u - a\ J Now (4.1.20) becomes diuiy + j f d{u\y-'-^{A{u\),ul-a) ds + j fd(uiy-'^^^^oB{ul)dws Jo l'"s ~ '^l Jo \"'s " "I + ^J*d{uiyds (4.1.30) Jo - «I \ - «I \ul - a\ I This may be justified as before. Since S is positive, the last term may be expanded and estimated as Jo l^s ~ '^l Recall that d(ti^) = 0 if |tij - a| < p and apply (4.1.3) to the second term on the left hand side of (4.1.30): 7 f d { u i y - ' — ^ {A{ul),{ul - a)) ds (4.1.32) Jo \K - a] > - A T f\{uiy-'j-l—\ui\'ds + j f\(uiy-'j-^\B{ui)\lds Jo Ps — "I Jo l^s ~ C-l fdKy-'\ui\'ds + 'r fd(uiy-'—^{Biui)\lds. P Jo Jo |Ws - a| 81 Combine (4.1.31) and (4.1.32) in (4.1.30) and apply some obvious estimates to get < ci d{u\y-'\u\\^dt+ [ d[u\y-^\\u\ - a\\U\\* dt (4.1.33) f d{u\y-^htdt+ C d{u\y-^htdt+ C d{u\] Jo Jo Jo B{u\) dt + sup t<T + sup fd{uiy-'^^^^oB{ul)dw, Jo l^t - o| Once again the analysis is slightly different for p = 2 dian for p > 2. For p > 2 take expectations and apply Burkholder's inequality and (4.1.5) in (4.1.33) to get /•T E <C2 ^lp{uiy}+^{ll d{n^,ydt^ (4.1.34) | y ' ^ d ( u , T " ' l ^ t l ' r f * } + E | j J ^ d ( u , T " K - « l l l l / t l l * r f * } [ [ d{u\y-^ht d< j + E d { u \ y - ' ' h t dt j + E I j ^ ^ d{u\y-^{i + \\ui\\y di j I dinir^-'ht dt^ ' ^ ' 1 + E I d{uiy^-'(l + E + E + K\\fdt 1/2"! \ 1 For p = 2 use (4.1.6) and die linearity of B rather dian (4.1.5) to get fT E s u P d ( 4 ) ^ } + E | l y diulYdt^ (4.1.35) | y ^ d ( u , T - V , f d i | + E | y ^ d ( u t T " ' l l ^ t - « l l l l / t | | * r f « } d{uiy-'^ht d t j + E d(Ui^)^-2/ii d i j +E d(M^)^-2(l + |Mt'|)2 d t j + E + E ^diuiy-^-^htdt^ ' | + E | ( ^ y ^ d ( u ^ ) 2 ^ - 2 ( l + |u^|)2df' / In either case die last four terms may be dealt with as in (4.1.26) (4.1.27), (4.1.24) and (4.1.25) or as in (4.1.26), (4.1.28) (4.1.24) and (4.1.29). For die odier terms: C2Ey\{u\y~^\u\\^ dt\^ < -^E [ [ d{uiy dt^ + ce'^-^E [ [ \ul\^^ dt^ if 7 < a/2. Since 7 > 2, this requires a > 4. + ce .7-1 For 0 < 2; < 7 - 1, f-x-l C 2 E | ^ d{u'ty-^htdt] < C2E I ^^Pdiuir diu'iy-^-'ht dt] { J \ r d t y < { ^l^diuir] + Y i ^ E diu^y dt] + c - ^ - ^ E Choose 7/(x +1) = t: then 7 - x - 1 = 7(1 - l/t) and C2Ey\(u^tr-'htdt] < lE{^^Pd{uiy] + ^ E y \ { u i y d t ] + ce'^('-'/^) T ^ ^ œ+l • /if"^ dt if 7 < a/2. Finally C 2 E | ^ ^ d K r - i | K - a | | | | / , | | d t } < l E { s u P d ( « J ) ' r | + _ L E | j f ^ d K f dt + C € ^ - " - i E ^ ( / \\ul - a | | ^ | | / t | | r dt For some r > 1, E - 1 . I « t - « l l ^ l l / t l | r ^ dt x+l • < E — a|h+i dt E , ^ x+l 1 l A ' If p > 2 choose 7/(x + 1) = 2 and r = p/2. Then for 7 < a, C2EId(nD^-itx^ - a||||/t|| dt] < 1 E {f^Pdiuiy] + ~^{J^d{uiydt] + 6 ^ / ^ Ifp=2choose7/(a:-i-l) = 3/2andr = 4/3. If7 < 3a/4, C2E1^"^d{u^,y-^ui - a\\\\M\ dt] < 1 E {f^Pdiuiy] + y ^ E j ^^d{u iyd t ] + e /^^ . The pertinent restriction on 7 is 7 < a/2, so l i m E { g P d K r / ^ } = 0. 83 Step 4; {u'), is Cauchy in L«/2(fi; C([0,T]; H)). Proof: For e > 0 and ^ > 0, x't -4 + [ M O - M4) ds + [ Biul) - B{4) dWs + [ -^(iiul) - ^(3(4) ds = 0. From Itô's formula 2 / V « ) - M<),< - <) ds + 2 A< - 4) o - B{4)) dWs Jo Jo + 2 j\-/{<) - ]P{4), <-4)ds = [ \Bi4) - B{4)\i ds. Apply the monotonicity condition (4.1.4) and the estimate (2.3.8) to obtain \u^t~4\' <xj\4-4\Us+2{-^+^)j\iul)d{4)ds-2 j\ul-4)o{Bi4)-B{4))dWs. ° (4.1.36) The stochastic integral in this expression is a martingale so upon taking expectations E where \ u l - 4 \ ' ] < x l \ { \ 4 - 4 \ ' ] ds + 2Em $ = (1 + 1) f d{u\)d{4)dt. ( 0 Jo Apply Gronwall's lemma to get E [ [ \ 4 - 4 \ u t ^ < 2 ( ^ ^ ^ y m . For some K > 2 return to (4.1.36), raise to the power K / 2 , take the supremum over t and take expectations to get E ( E \4~4\^ds + E{ sup I s<t To deal widi the first term on the right hand side: Jo {B{4)- B{4))dw, (4.1.37) + E { t t } E \4 - ds^ ^ I < E j f " ^ [ \4 - 4\' d s j < T ^ E | y ' fl\4-4\'ds],. (4.1.38) Burkholder's inequality applied to the stochastic integral yields, first for p > 2 and 0 < a; < 1; f\ul-4)oiB{ul)-Bi4))dw. Jo [ \ u l - u l \ \ l + \\ul\\ + \\4\\fdsy^ < cE I j^p|< - 4 r / ' (yj K - +\\u y\i+\\ui\\+\\4\\r/^ds (4.1.39) ••'^ + \\u'j\?ds 2(l-x) ^ KX \ 2 ( 2 - x ) ^ 2 ^ ^ { - ^ {[ 1^^^  - ^* I' 2-x E {i + \\u\\\ + \\4\\fi^dt kI2 \ 2 ^ Choose a; = 2/p and note that 2(1 - a;)/(2 -x) = {p- 2)/{p - 1). If K < a then E 'ii + \\ui\\ + \\ul\\ydt K / 2 -is bounded independently of e and 6, and substituting (4.1.38) and (4.1.39) into (4.1.37) produces t/0 Use Gronwall's lemma to get E{^^Pkt^-4r} < C ( E { * ' = / 2 } + E { $ } ^ ) . Forp= 2, E < cE f\K-4)o(B{ul)-B{4))dw, Jo K / 2 " (4.1.40) < c E | 2 P K - t x ^ r / 2 ( £ K - t i f p d . y I < i ; ^ { ' ^ ^ t \ < - 4 r } ^ c E [ l \ u l - 4 r d s ] whence E { -uir}<c[E{ IfJK d.4- cE and applying Gronwall's lemma again shows diat E [^^P\ul - u'x] < cE [yif^/^] . The result will follow once it is shown that E {'î'"/^ | _^ 0 as e, ^ ^ 0. For some x > I, E { $ f } = E | < cE < cE '1 lY )dt I -d{n\)d{ul) Jo f dt + cE )dt gPd(uf ) t \d{u\) U cE I g P d ( u O ^ ]d{ul) dt < cE [^^Pd{ul)^YEU[\d{u\)dt KX \ — E KX \ — 1 5, ^ ^d{ut)dt If Kx/2 < a/2 (i.e., x < a/k), then E ^ ^jj^ ^d(uj) dtj } bounded independently of e. while E {suPd(uf)- ' /2 s- 0 as ^ 0 i f Kx'/2 < a/2. The latter condition is equivalent to X > a/{a — K ) , SO a choice for x is possible if a/n > a/(a — /c), or K < a/2. This completes Step 4. Step 5; Completion of the Proof. From Step 4, as e ^ 0, u^ ^ -u strongly in L°/2(fi; C([0,T]; H)) . From Step 2 {u'), is bounded in the reflexive space W 2 ( f i ; L P ( 0 , T ; V ) ) , so along a subnet u'^ ^ u weakly in L«P/2(fi;LP(0,T; V)). Since L"P'/2(fi; C([0,T]; H)) w L ^ ( f i ; L i ( 0 , T ; H)) and W 2 ( f i ; LP(0,T; V)) L i ( f i ; L i ( 0 , T ; H)), ^ u strongly and u ' ^ û weakly in L i ( 0 ; L i (0 ,T ; H)), which implies û = u¥ x dt a.e.. Also from Step 2, (u^), is bounded in L'*(fi; C([0,T]; H)) and dius u e L'*(fi; C([0,T]; H)) , as in Theorem 3.2. Define a process T?^  by Jo f Then rj^ is a V-valued process with paths of bounded variation in V if is a halfspace, or a H-valued process with paths of bounded variation in H if is a sphere. If iiT is a halfspace ,a/2 lL«/2(n;BV([0,T];V)) = E Var(77^)"/2| = E = E Jo e udWdt a/2' -d{ul) dt a/2' SO (7?^ )e is bounded in L " / 2 ( f i ; BV([0,T]; V)). But L " / 2 ( Û ; BV([0,T]; V)) = L " / 2 ( f i ; C([0,T]; V*)*) weak*-L«/2(fi; C([0,T]; V*)*) = L ^ ( Î Î ; C ( [ 0 , T ] ; V * ) ) * so along a further subnet ij^ ^ fj weak* in weak*-L"/^(fi; C([0,T]; V*)*). If K is a sphere then r}^ ^ fj weak* in weak*-L'*/^(fi; C([0,T]; H)*). Identify the random measure î) with a process r/ by ri{u>, t) := ^(o;, [0, t\). Then 77 is cadlag and of bounded variation in V if K is a half-space, or in H i f is a sphere. From (4.1.1), < c(i+iiu^i|(p-i)p')=c(i -H i i n^ in , so (A(ti ')), is bounded in L " P ' / 2 ^J^. L P ' ([Q, T ] ; V * ) ^ . This is a reflexive space so, along a fiirther subnet. A{u')^X weakly in V^''"^ {^-U' . Since E ] sup /* Jo Xs ds ap>/2' < E •T X «P72 T , x « / 2 ' llXtll^ dt the linear map x /o Xt dt is continuous from L " P ' / 2 ^çi- L P ' ([O,T]; V * ) ^ into L " P ' / 2 (îî; c ([o,T]; V*)) and thus / A{u',)ds^ f xsds weakly in L ^ ^ ' / ^ (îî; C ([0,T]; V*)) . Jo Jo From (4.1.5), so (5(w'))c is bounded in die reflexive space L " P / ^ ( f i ; LP( [0 ,T] ; A(w))). From Burkholder's inequality E j sup / dWf so the linear map B /g 5 , dw, is continuous from L^^ ' / ^^^ j^ . LP([0, T]; A(t(;))) into L«P/2( f i ;C([0 ,T] ;H)) .Thus 5 ( u ' ) ^ e weakly in L'^P/2(fi; L^'([0,T]; A(u;))) and / B{ul)dws-^ I isdws weakly in E''^I^{Û-X{[{),T];W)). Jo Jo Combining these results, and considering the penalized equation (4.1.15), C weakly in L « / 2 (O ;C( [0 ,T ] ;V*) ) , and ut+ I Xsds+ [ ^sdws + Ct = UQ+ f fsds + mt. (4.1.41) Jo Jo Jo Slight changes to the identification procedure used in Theorem 3.2 are needed to show that C = ^• hi (3.2.31) choose g e C([0, T]; V ) n BV([0, T]; V ), and i f K is a halfspace use die integration by parts formula ^ / {ft,dgt) + / {gt,dft) = ( / T , 5 T ) - (/o,5o), Jo Jo which is valid for / and 5- in C([0,T]; V * ) n BV([0,T]; V ) . The weak* convergence of rj' is sufficient to get convergence in (3.2.32), and replace " / i / ^ " " by " v / " " widi u G V . Now T] is continuous in V * and cadlag in ÏÏ. Then Vcj, supj%| < 00, so if (u„)„ C V and Vn h e E. then {T)t,Vn) {Vt,h.) uniformly in t. Thus 77 is weakly continuous and strongly cadlag in ÏÏ, and so must be strongly continuous in H. Since sup^^^lr/jl < J^\\Piul)\dt, {r]% is bounded in L " / 2 ( f i ; C([0,T]; H ) ) ^ weak*-L«/2(fi; L°°([0 ,T] ; H ) ) , so T]' ^ eweak*. But rj' ^ C weakly in L ^ / ^ ^ f i ; C([0,T]; V * )) ^ weak*-L"/2(n; L°°([0 ,T] ; V ) ) and 7/^  ^ ^ weakly in weak*-L"/2(fi; L°°([0 ,T] ; H ) ) weak*-h''/^{il;h°°{[0,T];Y)), so a.s. ^ = ( = T], and E {supi<j Ir/tl^/^} < 00. Similar arguments when K is â halfspace show that E {supi<j ||77t||"/^} < 00. Note that TJ is not known to be strongly measurable as a map : n ^ C([0,T]; H ) ( or C([0,T]; V ) ) , so 77 ^ L " / 2 ( f i ; C([0,T]; H ) ) . If iiT is a halfspace then 7/ is cadlag in Vand continuous in V*. Let f„ ^ p € V* with Vn G V to see that TJ is weak* continuous in V , and thus strongly continuous in V . It remains to identify x as A{u) and Ç as B{u), which will establish (4.1.9), to prove that a.s. Ut e K \ftmdto estabhsh(4.1.10) and (4.1.12). Note first that d is uniformly continuous from V* or H to M so d{u^) d{u) in L"/2 (fi; C ([0,T]; R)). But from Step 3 d{u^) ^ 0 in L " / ^ (fi; C ([0,T]; E)) so almost surely d{u) = 0 for all t, and thus a.s. ut £ K\lt Equation (4.1.12) will be proved next. Define C:= {veC{[{i,T];V*):vteKyt} and let X be in L ° ° ( f i ; E ) . Then for any 7; G C, Xi'Ku" - v) converges in norm in L^(f i ; C([0,T]; V*)) to X{-KU - V) = X{u - v), where x is the projection onto K. Since T]" converges weak* to 7? in L2(fi; C([0,T]; V* ))*, lim E | x J^TTul - Vt, drjl) j = E | x j \ u t - vt, drjt) j . This holds for any X , so as e ^ 0, / {-Ku^t - Vt, drjl) [ {ut- vt,dr]t) Jo Jo weakly in L^(f i ; R ). Recall that \l3{u) G dlxi-^u), so that {\f3{u), •KU-V)> -IK{V). From the definition of 7/^ it then follows that 1: {•Ku't - vt,dr]l) >0 V w G f i , V V G C . /o Since the set {Y G L^( f i ; M ) : Y > 0 a.s.} is weakly closed (it is norm closed and convex), the last inequality and weak convergence result show that fT V f G C / {ut - Vt,dT]t) > 0 a.s. Jo Since C is separable, the "a.s." and "Vt» G C" may be switched. The entire argument may be repeated on [0,t] rather than [0,T], so from the continuity of t 1-^ /o('"s ~ Vs,dT}s) and the separability of C, one obtains a.s. V t G [ 0 , T ] , f {us-Vs,dris)>0 V U G C , Jo whichis (4.1.12). Toprove(4.1.10), repeat this argument withC := {v G C([0,T]; M) : vt £ K \/t} and use L\Çï; C([0, T]; H)) radier than L2(fi; C([0,T]; V* )). Identifying x and ^  uses the method of monotonicity as in Theorem 3.5 . From (4.1.4), for any t ; eL2( f i ;LP( [0 ,T] ;V) ) , Ey]B{ul) - Bivt)\ldt^ < 2E {J\mO - A{vt),ul - vt) dt^ + AE j ^ V t - ^tf dt^ • {4 A .42) Apply Itô's formula ( Theorem 4.1 ) to the penalized equation (4.1.15); K p + 2 f {A{u\),u\)dt + 2 f u\oB{u\)dwt + 2 f {u\,d7jl) Jo Jo Jo = \uo\^ + 2 I {fuu\)dt + 2 I {u\,dmt) + [m]t+ j \B{u\)\ldt. Jo Jo Jo The stochastic integrals are martingales so upon taking expectations: 2e[J\a{U\), u\) dt^ -e[[\B{u\)\l dt^ = E{\uo\'} - E { | 4 P } + 2E {J\ft,ud dt^+E{[m]t} - 2E [[{u^dvl)^ . Use the previous convergence results on this to see that, along a subnet, lim (2E {y V « ) , dt^ -e[[\B{ul)\i d / } ) = E { K P } - E { | u T p } + 2 E | y (/f,Ui) d t j +E{[m]<} -2e[J (u i ,d7/ i ) | . Apply Theorem 4.1 to (4.1.41) to get k T p + 2 / {xt,Ut)dt + 2 I {uto^t,dwt) + 2 I {ut,dTit) Jo Jo Jo = \uo\^ + 2 I {ft,Ut)dt + 2 I {ut,dmt) + [m]t+ [ \^t\ldt Jo Jo Jo from which E {\uof} - E { + 2E {J\fu Ut) dt^ + E {[m]J - ^^[[{-^t, dvt) = 2E {J\xunt) dt^ - E [ [ 161^ dt^ . Combine the last two results to get lim (2E {J\mO, u\) dt^ - E [[\B{u\)\l dt^j = 2E I j \ x u Ut)dt^-E[ [ 16\ldtY 90 From the weak convergence of B(u^) to ^ in A(u;) it is immediate that lim E [J\b{UI), Bivt))^ df I = E I B{vt))^ dt] . Pass to the limit in (4.1.42) to get B{vt)\ldt] < 2E y \xt-A{vt),ut-vt) dt] + XE S^j^ \ut - vtl''dt] (4.1.43) Set t; = ti to obtain E yj^t-B{vt)\idt]=o and thus ^ = B{u). Now (4.1.43) simplifies to 0 < 2 E | J ^ {Xt-A{vt),ut-Vt)dt] + XEy \ut- vtl'^dt} . Set V = u +ry for r > 0 and any t/ € (fi; 17 ([0,T]; V)) and proceed as in Theorem 3.4 to get X = A{u) P X dt a.e. and a.s. J* Xs ds = A{us) ds V i . Example: Krylov & Rozovskii [2], III.2.1, give the following example of non-linear operators satisfying (4.1.1) to (4.1.4). Let G = E and let to be a standard Brownian motion in E . For O = (0,1), H = h^{0), V = Wo'^{0) for p > 2 and m > 1, and constants a and b, define A(ti) := a{-iy dx' p-2 ax™ and B{u) := b a™ti dx' p/2 Then A and B satisfy conditions (4.1.1) to (4.1.4) i f 2a{p - 1) - b'^p'^/4 > 0. However, B does not satisfy the linear growth condition (4.1.5). To fix this, change B to B{u):= b a™ti p/2 / A C+D \ dx"" J where C and D are any positive constants. While I have only proved existence of solutions to my problem for special constraint sets, a continuity result may be proved for arbitrary constraint functions whose domain has nonempty interior. o Theorem 4.5. Let ip be a lower semicontmuous proper convex function on H with <l). If continuous solutions u and r] exist to the equation a.s. ut+ [ A{us)ds+ I B{us)dws + r)t = uo+ f fsds + mt V i e [ 0 , T ] (4.1.44) Jo Jo Jo with a.s. Ut G Vt , <p{ut) dt < oo,r] is of bounded variation in H , and a.s. V i G [ 0 , T ] , f {us - Vs,dr)s) > f <p{us) ds - f ip{vs) ds V u G C([0,T]; E ) Jo Jo Jo (4.1.45) for uo G L ° ° ( î î , . F o , P ; H ) and Uo G a.s., f G L°° ( f i x [0,T],P x dt,Y*) and m e M°°''^{E.), then for any a > 2, solutions also exist for UQ G L° ' ( f i , /"ojP; H ) and UQ G a.s., f G L"P'/2(j7.LP'([o ,T]; V*)) and m G A1" '" (H) . The maps uo ^ u, f ^ u and m ^ u are continuous from L ' ^ ( f i , J-o,P; H) , L«p ' /2 (n ;LP' ( [0 ,T]; V*)) and M"'%m) into L"(fi ;C([0,T];H)) ( with the nonn topology ) and L « P / 2 ( 0 ; LP([0 ,T] ; V ) ) ( with the weak topology ), and the maps UQ t-^ T], f T] and m t-^ 7] are continuous Irom L°'(fi , .Fo,P; 11), L«p ' /2 (0 ; LP'([0,T]; V* )) andM^'^U) intoweak*-L«/2(fi; BV([0 ,T] ; H)) . Proof: As in Step 2 of Theorem 4.4, E { 3 K r } ^ E { ( / ^ utrdty''\+E{\r,\r} (4.1.46) < c | l + E { h o n + E ^ ( 1^ WftWïdt^ \ a/2' + E r i« /2 Given solutions (u^, rj^) and (u^, r/^) corresponding to {UQ, f^,m^) and {u^, m^), use Theorem 4.1 and proceed as in Theorem 3.5 to get \u] - 4\'^ < \ul - ul\'^ + X [ \ul-ul\^ds+[m^-m% (4.1.47) Jo + 2 / Wul-ulrds] / \\f]-fl\\l ds \J0 J \J0 1/2 + 2[m'-m']]" (^j\B{u\)-B{ul)\lds + 2 j\u\-ul,d{m^ -m%) Jo + 2 j {u]-ul)o{B{u\)-B{ul))dws. Jo Note that Jo The stochastic integrals in (4.1.47) are martingales, so taking expectations produces + c(E{\ul-ul\'} + E{[m'-m']j} i/p' {Jo hi view ofthe bounds (4.1.46), i f < yoinL2(fi , .Fo,P; H ) , / " ^ / inLP'(f2 x [0,T]; V * ) and m " ^ mmJ^^'%U),thenu'" uinV{Q. x [0 ,T] ;H) . Return to (4.1.47), raise to the power a/2, take the supremum over t, estimate the ''d{m} - rrî^y integral as in (4.1.18) and the " d w " integral as in (4.1.39) or (4.1.40) to get {u^)n Cauchy in L « ( f i ; C ( [ 0 , T ] ; H ) ) i f < UQ in L " ( n , JTQ,?; H ) , ^ / in L « P ' / 2 ( O L P ' ( [ 0 , T ] ; V * ) ) and m " ^ mmj^"'\n). The rest ofthe argument goes as in Step 5 ofTheorem 4.4: along a subsequence vP- u weakly in L«p/2(fi; LP([0,T]; V ) ) and r/" ^ 7/ weak* in weak*-L"/2(fi; BV([0,T]; H ) ) . But the limits are imique, so the entire sequences (w")n and {rf')n converge to u and T/. 4.2 Linear Operators on Sobolev Spaces. Constraint sets other than those used in §4.1 could be allowed if the Itô inequality of §3.1 was used instead of Pardoux's Itô formula, but then the penalized solution would have to be a semimartingale in EL This can be forced, under strong assumptions, by solving the penalized equation in a triplet where V , rather than H , is the pivot space. The necessary assumptions on A and B look very artificial i f this is done abstractly. Instead, I have specialized to the case of linear partial differential operators on Sobolev spaces. Only very basic facts about Sobolev spaces are needed. Unfortunately, in the midst of the proof a rather strange condition ( 4.2.1 below ) is needed. This condition seems to be an artefact of the method of proof, rather than being intrinsic to the problem. The only example I have of a constraint function satisfying (4.2.1) is the indicator of a sphere in H , which has already been covered in greater generality in the previous section, so the results of this section must be regarded as incomplete. A discussion of (4.2.1) is given after the proof of Theorem 4.6. Let O be a bounded open region in and let M = 1?{0), V = Hj((!)) and W = H§(C>). Let io and ii be the imbeddings of V into H and W into V respectively. Then V* = H~^(0), W* = H-2((r)) and The norm and inner product on Vwi l l be denoted by || • | | i and ((•,•))!• The norm and inner product on W will be denoted by || • ||2 and ((•, •))2. Let L : V V* be the canonical isometry of V onto V*; i.e. {Lu, v) = {{u, v))i V u, u e V . If ((•, is chosen to be ((•u, u))i := / u{x)v{x) dx + \^ / diu(x)div(x) dx, Jo Jo where di denotes the distributional derivative with respect to the z*'' coordinate, then L takes the specific form d Lu = u— E ^iiU = {I — A ) ^ . t=l The differential operator 7 - A also defines a map : W ÏÏ related to L hy i^Liu = Li-iu\/ u e W, with i j : M H-^ W*. To avoid complicating the notation, the imbeddings io and ii will usually be omitted, and the maps L, Li and L \ , which are only formally distinct, will all be denoted by L. Let G be a separable Hilbert space and w a Brownian motion in 6 with covariance S G G G. The space of H valued integrands for w will be denoted by A(H), and the space of V valued integrands by A ( V ) . The norm on A(H) wil l be denoted by | • |w and the norm on A ( V ) by || • ||«,. For B e £ ( G ; H ) , = ?raceo(5(8) 5 , 5 ) , w h e r e / m c e o i s t i i e t r a c e o n H ê i H . For 5 G £ ( G ; V ) , = tracex{B ® B, S), where tracei is die trace on V ê i V . By die Hilbert-Schmidt dieorem there exists an orthonormal basis {gi)i for G and positive numbers CT; such that S = Yll^i '^I9i ® 9i and E S i < oo. Then\B\l = ^((^5/, Bgi), and \\B\\i = Zi cri{{Bg,, Bgi)),. Assume that (1) Uo is an J"o-measurable V-valued random variable. (2) / is a progressively measurable process in ÏÏ. (3) m is a continuous martingale in V , with scalar quadratic variation [m]^ and tensor ( V V valued ) quadratic variation [ m | i . Since V H, m is also a H-valued martingale, with scalar quadratic variation [m] and tensor ( H ®iH valued ) quadratic variation | m | . As before it is also assumed that [m]t = hg ds for some progressively measurable process h. o (4) (/3 : ÏÏ M is lower semicontinuous and convex with ^ 0. As in previous chapters, for € > 0 let TTtU be the imique solution to the inclusion n^u - u + €(p{-K^u) 3 0, and define (3,{u) = l{u - TTew). Relations (3.2.9) to (3.2.16) continue to hold. The restrictive assumptions on UQ, f and m needed to prove existence may be removed by using Theorem 4.5. See the remarks after the proof of Theorem 4.6. The rather obscure assumption mentioned earlier is: BaeWnh^ suchthat {(3,(u), L{u - a)) > 0 V u e W . (4.2.1) Define operators A:V i-*Y* and 5 : V A ( H ) as follows: d / d \ i=l A(u) := — ^ di ^ aijdju + biU + ^ CidiU + du d for aij e C\Ô), hi e 0(0), Ci e C{0) made C{0). Define À := LA Since A{u) € H if Given functions 7A:/ € C^(0)andO G C^(0)for 1 <k<dmdle N , with sup i^.,; IITWII < ° ^ and sup; define for each /, d Bi{u):=J2lkidkU + (iu ,so 5 , € £ ( ¥ ; H) n £(W; V ) oo and B{u){g):=^ai{gi,g)Bi(u) ,so 5 G £ (V; £(<G; H)) n £(W; £(©; V)) . (=1 For later use define B e >C(W; £ (G; V ) ) by B = B The operators  and 5 are used in the proof of Theorem 4.6. To simplify the notation in some stochastic integrals, define h o B{u) := B{u)*h 6 G for / i G H a n d u e V a n d t ; * B{u) := B{u)*v e G for t; G V and u G W. Coercivity: For some 0 > 0, almost everywhere on O, for all (^,),- € R^, d d / oo \ d E E - E ^^mi ^i^j > 2^ E • (4.2.2) j=l j=i \ ;=i / i=i Without the " 5 " terms this is a common condition for ellipticity of the operator ^ • di aij dj . Proposition 4.1. Under condition (4.2.2) there is a real number X such that 2y^{A{u),u)y+X\u\''>e\\u\\i + \B{u)\l yueY and 2w. (Aiu),u)^ + X\\u\\l > e\\u\\l + \\Biu)\\l V u e w . Proof: It may be assumed without loss of generality that u is in Co°(C?). For (4.2.3), 2y^{A{u),u)y-\B{n)\: = J 2 (^ y^  ajjdjU + bju]diu+2u CjdjU + 2du^ dx ^ 3 oo . ~ E ^ ' / ( E '''^•'^'•^ + ^ ' " ) ( E ' ' ' j ' ^ j^ ^'") = / E E - E*^' '>' ' ' ' ' ' j ' diudjudx •^^ i j I + / 2u V 6,- + Ci - V oijiiCi diU dx+ V Jo , J Jo >2e J^J2{diuf -2u u dx dx 2d-^aiCf dx >e\\u\\i-x\u\ For (4.2.4), 2^*{A{u),u)^-\\Biu)\\l = 2{A{u),u)-\B(u)\l + 2 / ^ di ttijdjU + bi^j - ^2 f^idiu - du ^ dkkudx CO -E [^ '^  ( E ^ ^ ' ^ ' ^ + ^ ' " ) ( E ^^'^-^'^ ^ ' ^ ) For / e C j a n d u e C ^ , / difdkkudx = - / fdikkudx= j dkfdikudx. Jo Jo Jo Apply this in (4.2.5) with / = aijdjU + biU and use (4.2.3): dx. 2^^{À{u),u)^-\\B{u)\\i > e\\u\\l-x\u\ •^^ i j i k oo . - E * ^ ' / E ^k{^liidiu + C,iujdk{^^jidju + (^luj dx. 1=1 Jo k i j = e\\u\\l-\\u\ i k j k i k - E ^ ' / ( E T i ' ^ ' W + 0 " ) ( E T j ' ^ i ^ + ^'") + / X / X ^ X / - ^ a/7î/7j7 dikudjku+ "other terms" dx k i j >e\\u\\l-\\u\ + 26 f E E ^ ^ ' ' ^ " ) ^ " ^ "other terms" dx Jo k i where "other terms" involves products of second order derivatives with first order derivatives and lower order terms. The result follows by applying Young's inequality to the "otiier terms". Proposition 4.2. There exist positive numbers \B\ and \È\ such that (1) \uoB{v)\ < \B\\\u\\i\v\ and\uoB(u)\ < \B\\u\'^ ^u,v e V (2) \u*É{v)\ < |5 | | |u | |2 | |« | | i and\u*É{u)\ < \B\\\u\\lyu,ve W (4.2.6) (4.2.7) Proof: It is sufficient to prove these for Bi rather than B, so set G = R and let B{u) = •jidiu+Qu for 7 i , C e C i ( 0 ) . For (4.2.6), uoB{v) = iu,B{v)) = / uy^jidiv + (uvdx Jo Y = — J E di{-jiu)v + Cuv dx = — y -u E lidiu + (vu dx + J uv (2c + E di^i) dx and = -vo B{u) + j uu^2C + E^i'^') dx \u o B{v)\ < c i ( | t ; | | | w | | i + \v\\u\) < Ci\v\\\u\\i \uoB{u)\ = < C2\U\'^. For (4.2.7), u.B(v) = {(u,B{v)))r = j E ( E li^jv + (v)dju dx + uo B{v) = y 'yidijvdju + djjidivdjudx + J dj(vdju +(djvdjudx = - / jidjudijv + dijidjvdju + djjidivdju dx + y ^ / dj^vdju + Ç,djvdjUdx + uo B(v) = - j Y^dj-fidiu + (ujdjvdx-vo B{u) + Y ^ y ^ / dj'Yi{diudjV + divdju)dx + y^ / djC(udjV + vdju) dx + J uv(2C + J2dai^ dx = -vB{u) + y^y^ djji{diudjv +divdju)dx + y^ / dj({udjV + vdju)dx + j w(^2C + E ^ » T ' ) dx ^ \u . B{v)\ < c i ( | | t ; | | i | | t i | | 2 + M l i l l t i l b ) < c i | | t ; | | i | | n | | 2 and \u • B{u)\ < C 2 | | « | | i . + V Theorem 4.6. If for some a>4,uoe L " ( f i , To,P; V ) and (f{uo) G L2(fi , jP"o,P; E) , / € L " ( f i X [0,T]; H) and / i e L ' * ( f i ; L^([0,T]; R)), then there exist unique predictable processes u and rj satisfying a.s. ut+ / A{us)ds+ / B(us)dWs + rjt = UQ+ / fgds + mt V t G [0,T] (4.2.8) Jo Jo Jo with u G L ' ^ ( f i ; C([0,T]; H)) n L ^ ( 0 ; L2( [0 ,T] ; V ) ) and 7/ G L « ( O ; C ( [ 0 , T ] ; H)) n weak*-L"/2(fi; BV([0,T]; H)) . Moreover, a.s. ut elD^^t e [0,T], JQ (p{ut) dt < oo and a.s. V i G [0,T], f\us-Vs,dTj,)> l\{u,)ds- j\{vs)ds V v G C([0,T]; H ) . (4.2.9) Jo Jo Jo Proof: As in Step 1 ofTheorem 4.4, for any e > 0 there exists a unique process ti^ : x [0, T] i - ^ M with paths continuous in H, G V P x dt a.e. and satisfying a.s. u\+ f A{ul)ds-¥ j B{ul)dws+ I li^{u\)ds = UQ+ j fsds + mt V t G [0,T]. Jo Jo Jo Jo (4.2.10) From Step 2 ofTheorem 4.4, {u^)c satisfies E { ? < ? K r } + E | ( y ^ K | | ? d t ) ^ | + E | ( y ^ / 3 , ( 4 ) c i i ) ^ | < c (4.2.11) with c independent of e. These bounds are not adequate for the arguments to follow. The strong assumptions on «o, / and m must be used to drag into W, which will put A{u^) into H and make into a semimartingale in ft This is done by solving a close relative of (4.2.10) in the triplet W ^ V ^ V * < ^ W * . (4.2.12) Recall the definitions À := LA W W* and 5 := B i ^ : W C(G; V ) . Also define / G W* by /( := Lft and P eW^ by $t '•= L(ii{u\). Given these definitions, and using (4.2.4), the equation a.s. ût + I Â{ûs)ds+ I Bsdws+ I $sds = uo+ f fsds + mt Vt G [0,T] (4.2.13) Jo Jo Jo Jo has a unique solution û : fi x [0,T] i-+ V with paths continuous in V and û G W P x dt a.e.. This follows from Theorem II.2.1, Corollary 11.2.1 and Theorem II.2.2 of Krylov & Rozovskii [2]. Note that (4.2.13) is an equation in W*. The imbeddings of one space into another, which have been suppressed in (4.2.13), are as shown in (4.2.12). Thus (4.2.13) translates into a.s. \ft e [0,T] {{ût,v))i + (Â(û , ) ,u ) ds+ (^J^ B{ûs)dws,v^ + 0s,v)ds = ( ( w o , f ) ) i + / Cfs,v)ds+((muv))i V u e W . Jo Using the definitions of L, À, B, / and ^ , this becomes a.s. V / e [ 0 , T ] {ût,Lv)+ I {A{ûs),Lv)ds+ ( f B(ûs) dw^, Lv] + I {(3,{4),Lv)ds Jo \Jo ) Jo = {uo,Lv)+ j {fs,Lv)ds+imt,Lv) V u e W. Jo Since LÇW) is dense in H , a.s. ût+f A{ûs)ds+f B{ûs)dws+I f3^{ul)ds = uo+f fsds+mt V t G [0,T] (4.2.14) Jo Jo Jo Jo where all processes have values in H . Comparing (4.2.14) with (4.2.10), a.s. ul-ût+ f A{ul)-A(ûs)ds+ f B{ul)-B{ûs)dws = 0 V t G [0,T]. Jo Jo A simple application of Itô's formula ( Theorem 4.1 ), (4.2.3) and Gronwall's lemma as in the uniqueness proof of Theorem 4.4 shows that a.s. ût = u\'it e [0,T]. The stronger bounds are obtained from (4.2.13), with tt replaced by u % by proceeding as in Step 2 of Theorem 4.4. It is here that condition (4.2.1) is needed. Applying Itô's formula to (4.2.13) produces aterm 2 JQ{$S, u\ — a)dt = 2 JaiPciO^ ^( '"s ~ "•)) ds, which is non-negative by (4.2.1), and so may be ignored. No changes are required in the rest of Step 2. The bounds obtained are E { f ^ P | K | | ? } + E | ( y ^ | | < | p d . ) ^ I <c . (4.2.15) From (4.2.14) u ' is a semimartingale in H , so one may proceed as in Step 4 of Theorem 3.2 to obtain (3.2.21), which is repeated here for convenience: M4? + 'i fMul)(A{ul),(3,{ul))ds + 2 f^ulMiuDl'ds Jo Jo <ip.{uof + 2 f <p,{ul){fsJ,iul))ds (4.2.16) Jo + 2 I ( v 5 e « ) / 3 . « ) , d m , ) - 2 / < ^ e « ) / 3 e K ) o i ? « ) d « ; , Jo Jo + ] flMOmOllds+l f\ip,{nl)\d[m]s ^ Jo ^ Jo + fmul)\'\B{ul)\lds+ f \(3,{ul)\U[ml. Jo Jo A little more care is needed in treating the "(A(u^), /?e(tij))" term in the absence of a Lipschitz hypothesis on A. Note that for u € W, \A{u)\ < c\\u\\2 and \B(u)\^ < c| |u| |i, use the bounds (3.2.23), (3.2.24) and (3.2.25), take the supremum over t and take expectations to get /•T E g Ç ^ V K ) ' } H - ^ E { y \di\Ut] E + \uim\' dt] + eEy^\ip,{ul)M\\\ul\U dt] E \d'tf\ft\dt] + + I4l)l/tl dt] + E + \uim\'\M dt [J^ Kl'lKWl dt] + 6 E + 141)11411? dt] + E y^ m-i- + 141)11411? dt y^ \di\% dt]+6E y\i+i4i)/it dt]+E j y " ^ +i4i)/it (4.2.17) + E + E + eE + €E <p,{uiy\d't\'htdt 1/2' ^ , ( 4 ) 2 | 4 | 2 | | < | | 2 d / 1/2-eE^ ( ^ % , ( 4 ) 2 | d , f ||4||?di) ^ The last term is dealt with much as in (3.2.28): - { + ^ E { i 4 n i 4 i i ? • The estimates (2), (4) and (6) following (3.2.28) are irrelevant, histead use 6 E | y V e (4 ) l l 4 l l l 4 l | 2 r f < } 4ci + cE " | | ' 4 | | 2 d i g P ^ V e ( 4 ) ' } + c E m'^dt 0 J \ 1/2 / ,T \u\t!^dt Zl2 4ci dt \ldt E { f K P l W I I ? « } < ^ E { / ; K r * } . . c E { / | K | i î . < and * E Thus + , 3 ^ , 5 / 3 ^ ^ 1 / 3 ) ^ from which E | ^^^P\d'Xj < ce^/^ as in Theorem 3.2. Only minor changes in Step 3 ofTheorem 3.2 are needed to get {u% Cauchy inV{n; C([0,T]; H)) . Use (4.2.3) in (xrz), use (4.2.6) in (xbs) and proceed as before to get To complete the proof proceed at first as in Step 5 of Theorem 4.4: ^ u strongly in L 2 ( f i ; C ( [ 0 , T ] ; H ) ) and weakly in L«( f i ;L2( [0 ,T] ; V) ) . Using (4.2.15) one may even as-sume diat ^ u weakly in L " ( f i ; L2([0,T]; W)). Since A md B are linear, A{u^) -> A{u) weakly in L " ( 0 ; L2([0,T]; H ) ) and B(u') ^ B(u) weakly in L « ( f i ; L2(([0,T]; A(E[))). Then j/f := /o/?,(u^)ds ^ rj weak* in weak*-L2(n; BV([0,T]; H ) ) and T?' ^ ( weakly in L2(fi; C([0,T]; H ) ) . No change is required in die rest of Step 5 ofTheorem 3.2. By approximating uo, f and m and using Theorem 4.5, the assumptions in Theorem 4.6 may be weakened to a > 2, U Q G L " ( f i , Jb ,P; H) and uo G a.s., f G h"{Q;V([Q,T]; V*)) and m G M^'^iE) with<i[m] < dt. It is clear that any U Q G L°'{Q,,fo,P; M) may be approximated by V-valued simple functions as in Theorem 3.3, and that any progressive / in L " ( f i ; L^([0, T]; V* )) may be approximated by V-valued progressively measurable simple functions. To approximate m G M°'''^(E.), let (e„)„ be an orthonormal basis for H consisting of elements of V . Define V-valued martingales M " by = E ,=i ("^t5 ei)e,. Then so m in M^'^U). Also, [M% = £ Er=i(e. ® ei,Q^)Kds i f [m]t = J^h^ds, so d[M"] < dt. Condition (4.2.1) is not easy to check. To see that it works for the indicator of a sphere, recall that/3e(u) = c{u){u-a), where c{u) = d{u)/{€\u-a\).Then{f3^u,L{u-a)) = c(u)\\u - aW^ > 0. Conditions like (4.2.1) do occur in the literature, in connection with showing that the siun of two maximal monotone operators is maximal monotone: see, for example. Theorems 11.4 and 23.4 in Deimling [1]. If (f is the indicator of a closed convex set K it is possible to find a sufficient condition for (4.2.1) to hold, which may be easier to check, using some ideas of Brézis [1], pl7, and Brézis & Pazy [1]. First, define D •.= {ueE: L{u - a) e M} md F : D Uhy Fu = L(u - a). For any A > 0, / + AL is onto H by the Lax-Milgram lemma, so / + \F is also onto M, and the resolvent Rx := (7 + XF)~^ exists. ( Note this means that F is hypermaximal monotone, and thus maximal monotone. ) Proposition 4.3. Condition (4.2.1) is satisfied if RxK C K \f \ > 0. Also, RxK C K if and only if yuedKnDp, iFu,y)>Q yyedlxiu). (4.2.18) Proof: Let Fx := FRx be the Yosida approximation to F, and note that Rxu + XFxu = u. Then {Fxu - Fxv, u-v) = {FRxu - FRxv, Rxu - Rxv + \Fxu - XFxv) > X\Fxu - Fxv\'^ > 0, so Fx is monotone. If TTu is the projection of u onto K, then a simple consequence of the inclusion u - ir^u € edlxiT^cu) and TT = TT^ Ve is that iru may be characterized as the unique element of K such that (u — TTu,y — TTu) < Oyy e K. From the monotonicity of F A , [FXU - FX'KU,U - TTU) > 0, so {Fxu,(iu) > {FX'KU,U - rru) = (iru - Rx'!ru,u- nu)/X. But i f RxK c then (TTU- Rxnu,u- ir-u) > 0, and thus {Fxu,fiu) > 0 V A > 0. Since Fxu^ FuasX-^ 0+ for u e D, itfolows that {Fu,(5u) > 0 for u e W C £>. A n equivalent condition to RxK C K is given by Brézis & Pazy [1], Corollary 2.1 : If F i s maximal monotone and i i ' C H is closed and convex, then RxK C KMX > 0 iff D/T - n / i ' is dense in DF n K, F+ dix is maximal monotone, ( F + dlxTu = Fu \/u e Dp r\ K, and TT-^^K C K. 1 have stated this for single valued F . Recall that ( F + ÔIKT is the element of smallest norm in F + dÏK. Since I'f = H, it is immediate that DF d K is dense in n i i ' , and that T^^I^ see that F + ÔIK is maximal monotone, note that F may be written as dip for the convex function tp defined on H by 'tp{u) := {Fu, u), so F + dix = dip + dix- But by Corollary 2.5, pl28 of Barbu & Precupanu [1], dip + dix = d{ip + Ix), which is maximal monotone. Thus RxK C K ^ (F + ÔIKTU = FU, "^U e DF n K. li U e K then dlxiu) = {0}, so (F+ ÔIK)"». = FU = {F + ÔIK^U. Thus only u G dK n i ^ F need be considered. Suppose diat {Fu,y) > 0 Vy € dlK{u). Then |Fu + yp = |Fu |2 + 2(Fu,y) + lyp > |Ftip, so {F+dlxfu = Fu. Conversely, suppose tiiat {F+dlxfu = Fu. Set a; = Fu and let y e dlxiu). Then |a; +1/1 > |a;| ^ (x + > |a;p ^ 2(x, r/) + |?/p > 0. But 9 /K (U ) is a cone, so let y = r^ r for z e dlxiu) and r > 0. Then 2(x, z) + rl^:^ > 0, and letting r ^ Q shows that {x, z) > 0 for any z G dlK{u). References. Baiocchi, C. & A. Capelo [1]: Variational and Quasi-variational Inequalities. Wiley 1984. Barbu, V . & Th. Precupanu [1]: Convexity and Optimization in Banach Spaces. Reidel 1986. Bensoussan, A . & J.L. Lions [1]: Contrôle Impulsionnel et Inéquations Quasi-variationelles. Dimod 1982. Brézis, H . [1]: Problèmes Unilatéraux. J. Madi. Pures et Appl. 1972,v51,pp 1-168. [2]: Operateurs Maximaux Monotones. North-Holland 1973. Brézis, H . & A. Pazy [1]: Semigroups of Nonlinear Contractions on Convex Sets. J. Func. Anal. 1970,v6,pp 237-281. Dellacherie, C. [1]: Capacités et processus stochastiques. Springer-Verlag 1972. Dellacherie, C. & P.A. Meyer [1]: Probabilités et potentiel. Vols 1 & 2, Herman 1975,1980. Deimling, K . [1]: Nonlinear Functional Analysis. Springer-Verlag 1985. Diestel J. & J.J. Uhl [1]: Vector Measures. A M S Mathematical Surveys # 15,1977. Donati-Martin, C. & E . Pardoux [1]: White noise driven SPDE's with reflection. Preprint. Duvaut, G. & J.L. Lions. [1]: Inequalities in Mechanics and Physics. Springer-Verlag 1976. Glowinski, R., J.L. Lions & R. Trémolières [1]: Numerical Analysis of Variational Inequalities. North-Holland 1981. Gyongy, I. & N .V . Krylov. [1]: On Stochastic Equations with respect to Semimartingales I., Stochastics, 1980,v4,pp 1-21. [2]: On Stochastic Equations with respect to Semimartingales II. Itô Formula in Banach Spaces. Stochastics, 1982,v6,pp 153-173. [3]: On Stochastic Equations with respect to Semimartingales III. Stochastics, 1982,v7,pp 231-254. Haussmann, U.G. [1]: Stochastic P.D.E.s with Unilateral Constraints in Higher Dimensions. Proceedings, Stochastic PDE's and Applications 3, Trento, January 1990. To appear in Pitman Research Notes in Mathematics. Haussmaim, U.G. & E. Pardoux [1]: Stochastic Variational Inequalities of Parabolic Type. Appl. Math. Optim., 1989,v20,pp 163-192. Mille, E. & R.S. Phillips [1]: Functional Analysis and Semi-Groups. A M S Colloquium Pub. #31, 1957. lonescu-Tulcea, A . & C . [1]: Topics in the Theory of Lifting. Springer-Verlag 1969. Jacod J. [1]: Une Condition d'existence et d'Unicité pour les Solutions Fortes d'Equations Différentielles Stochastiques. Stochastics, 1980,v4,pp 23-38. Kinderlehrer, D. & G. Stampacchia. [1]: A n Introduction to Variational hiequalities and their Applications. Academic Press 1980. Krylov, N .V . & B.L. Rozovskii [1]: On the Cauchy Problem for Linear Stochastic Partial Differential Equations. Math. USSR Izvestiya,vll,#6,1977,ppl267-1284. [2]: Stochastic Evolution Equations. Journal of Soviet Mathematics, vl6,1981,pp 1233-1277. Lions, P.L. & A.S. Sznitman. [1]: Stochastic Differential Equations with Reflecting Boundary Conditions. Comm. Pure & Appl. Math. v37,1984,pp511-537. Menaldi, J-L. [1]: Stochastic Variational Inequality for Reflected Diffusion. Indiana Univ. Math. Jour. v32#5,1983. Menaldi, J-L. & M . Robin. [1]: Reflected Diffusion Processes with Jumps. Annals of Probability, 1985,vl3#2,pp319-341. Métivier, M . [1]: Semimartingales. de Gruyter 1982. Métivier, M . & J. Pellaumail [1]: Stochastic Integration. Academic Press 1980. Nualart, D. & E. Pardoux [1]: White noise driven quasilinear SPDE's with reflection. P*reprint. Pardoux, E. [1]: Équations aux dérivées partielles stochastique non-lin'eaires monotones. These, Université Paris Sud, 1975. Rascanu, A . [1]: Existence for a Class of Stochastic Parabolic Variational Inequalities. Stochastics 1981, v5,pp201-239. [2]: On some Stochastic Parabolic Variational Inequalities. Nonlinear Analysis, Theory, Methods and Applications. 1982 v6 #l,pp75-94 Reed, M . & B. Simon [1]: Methods of Modem Mathematical Physics 1: Functional Analysis. Academic Press 1980. Rudin, W. [1]: Real & Complex Analysis, 3rd Edition, M<=Graw-Hill 1987. Saisho, Y [1]: Stochastic Differential Equations for Multi-Dimensional Domain with Reflecting Boundary. Probability Theory and Related Fields v74, 1987, pp455-477. Schaeffer, H.H. [1]: Topological Vector Spaces. Fifdi printing. Springer-Verlag 1986. Tanaka, H. [1]: Stochastic Differential Equations with Reflecting Boimdary Condition in Convex Regions. Hiroshima Matii. J. v9, 1979, pp 163-177. Troyanski, S.L. [1]: On locally uniformly convex and differentiable norms in certain non-separable Banach spaces. StudiaMatii. 37, 1971, ppl73-180. Y o r M . [1]: Existence et imicité de diffusions à valeurs dans un espace de Hilbert. Ann. Inst. Henri Poincaré. vX#l,1974,pp55-88. Yosida K . [1]: Functional Analysis. Fourth Edition, Springer-Verlag 1974. 

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