Solutions of stochastic partial differential equations considered as Dirichlet processes

Abstract

We consider the parabolic stochastic partial differential equation $$u(t,x)=\Phi \left(x\right)+{\int }_0^t\mathrm{Lu}(s,x)+f(s,x,u(s,x),\mathrm{Du}(s,x\left)\right)ds$$ $$+{\int }_0^t{g}_i(s,x,u(s,x),\mathrm{Du}(s,x\left)\right)d{B}_s^i,$$ \noindent where f and g are supposed to be Lipschitzian and L is a self-adjoint operator associated with a Dirichlet form defined on a finite- or infinite-dimensional space. We prove that it admits a unique solution which is a Dirichlet process and, thanks to Itô's formula for Dirichlet processes, we prove a comparison theorem.