Solutions of stochastic partial differential equations considered as Dirichlet processes (Q1769782)

The author considers solutions \(u=u(t,x)\) of the parabolic stochastic partial differential equation \[ du= [Lu+ f(t,x,u,Du)]dt +\sum_{i=1}^d g_i(t,x,u,Du) dB_t^i\;, \] on a bounded domain with Lipschitz functions \(f\) and \(g\) and finite-dimensional noise. The operator \(L\) is self-adjoint and generates a Dirichlet form defined on a finite- or infinite-dimensional space. The main result is the existence of unique solutions in a class of Dirichlet processes, where the author considers mild and weak solutions, which he shows are equivalent concepts under his assumptions. Relying on Itô's formula for Dirichlet processes a comparison principle is established.