SDE In Infinite Dimensions

In this thesis we are concerned with the following two problems.1. The stochastic reflection problem on an infinite dimensional convex set and BVfunctions in a Gelfand triple.We introduce a definition of BV functions in a Gelfand triple which is an extensionof the definition of BV functions in [ADP10] by using Dirichlet form theory with anunderlying Gaussian measure as reference measure. By this definition, we can considerthe stochastic reflection problem associated with a self-adjoint operator and a cylin-drical Wiener process on a convex setΓin a Hilbert space . We prove the existenceand uniqueness of a strong solution of this problem whenΓis a regular convex set. Theresult is also extended to the non-symmetric case. Finally, we extend our results to thecase whenΓ= K_a , where K_a = { f∈L~2(0, 1)∣f≥α},α≥0.We then generalize the above to the case where the Gaussian measure is replacedby a differentiable measure. Again we work in a Gelfand triple and use Dirichlet formtheory. By this definition, we can consider the stochastic reflected quantization problemassociated with a self-adjoint operator and a cylindrical Wiener process on a convexsetΓin a Banach space . We prove the existence of a martingale solution of thisproblem whenΓis a regular convex set.2. The stochastic quasi-geostrophic equation.We study the 2d stochastic quasi-geostrophic equation in T~ 2for general parameterα∈(0, 1) and multiplicative noise. We prove the existence of weak solutions with reg-ular additive noise and the existence of martingale solutions with multiplicative noiseand pathwise uniqueness under some condition in the general case, i.e. for allα∈(0, 1). In the subcritical caseα> 1/2, we prove existence and uniqueness of (probabilisti-cally) strong solutions and construct a Markov family of solutions. The large deviationsprinciple in the subcritical case with multiplicative noise is also obtained.