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Abstract

In the first part of this thesis, we use the theory of self-duality to provide a variational approach for the resolution of a number of stochastic partial differential equations. We will be able to address the problem of existence of solutions to a class of semilinear stochastic partial differential equations in the form du+A(t,u(t))dt=B(t,u(t)) dW with u(0)=u₀, where for every t∊[0,T], A(t, ‧) is a maximal monotone operator on a reflexive Banach space V, and B is a linear or non-linear operator with values in a Hilbert space H. We use the fact that any maximal monotone operator A can be expressed as a potential of a self-dual Lagrangian L to associate to the equation a (completely) self-dual functional whose minimizer on a suitable path space yields a solution. One particular case of the above equation which already contains a large number of stochastic PDEs is when A is the subdifferential of a convex function φ. More generally, we can deal with equations of the form du(t)= -∂φ(t,u(t))dt + B(t,u(t))dW(t) with u(0)=u₀. We also prove the existence of solutions to SPDEs in divergence form involving a maximal monotone operator β on the n-dimensional Euclidean space which is not necessarily the gradient of a convex function: du= div(β(∇u(t,x)))dt +B(t)dW(t) if u∊[0,T]×D and u(0)=u₀ when u∊∂D where D is a bounded domain in the n-dimensional Euclidean space. In the second part of the thesis, we use methods from optimal transport to address functional inequalities on the n-dimensional sphere. We prove Energy-Entropy duality formulas that yield and improve the celebrated Moser-Onofri inequalities on the 2-dimensional sphere.