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BIRS Workshop Lecture Videos

Stochastic limits for deterministic fast-slow systems Melbourne, Ian


Consider a fast-slow system of ordinary di↵erential equations of the form x ̇ = a ( x , y ) + ✏ \0 1 b ( x , y ) , y ̇ = ✏ \0 2 g ( y ) ,\r\nwhere it is assumed that b averages to zero under the fast flow generated by g. Here x 2 Rd and y lies in a compact manifold. We give conditions under which solutions to the slow equations converge to solutions of a d-dimensional stochastic di↵erential equation as ✏ ! 0. The limiting SDE is given explicitly.\r\nOur theory applies when the fast flow is Anosov or Axiom A, as well as to a large class of nonuniformly hyperbolic fast flows (including the one defined by the well-known Lorenz equa- tions), and our main results do not require any mixing assumptions on the fast flow.\r\nThis is joint work with David Kelly and combines methods from smooth ergodic theory with methods from rough path theory.\r\n

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