Asymptotic behavior of solutions of one-dimensional parabolic SPDE

We investigate the weak convergence as time {dollar}t to infty{dollar} of the solutions of SPDE in one dimension with Dirichlet boundary conditions. We first prove a result for the linear case, which is easy since the solution is Gaussian. Then we get a result for non-linear drift case by proving a new comparison theorem for SPDE.; To investigate an elliptic SPDE with Dirichlet boundary condition, we set up an equivalence between an elliptic SPDE and a system of SODE. From there, we construct a counter example that shows non-uniqueness of the solution for SPDE, and look at how the Lipschitz constant of the drift term affects the uniqueness.