Stochastic Nonlinear Parabolic Equations with Stratonovich Gradient Noise

Abstract

Existence and uniqueness of solutions to stochastic differential equation $dX-\text {div}\,a(\nabla X)\,dt=\sum _{j=1}^N(b_j\cdot \nabla X)\circ d\beta _j$ in $(0,T)\times \mathcal O$; $X(0,\xi )=x(\xi )$, $\xi \in \mathcal O$, $X=0$ on $(0,T)\times \partial \mathcal O$ is studied. Here $\mathcal O$ is a bounded and open domain of $\mathbb R^d$, $d\ge 1$, $\{b_j\}$ is a divergence free vector field, $a:[0,T]\times \mathcal O\times \mathbb R^d\rightarrow \mathbb R^d$ is a continuous and monotone mapping of subgradient type and $\{\beta _j\}$ are independent Brownian motions in a probability space $(\Omega ,\mathcal F,\mathbb P)$. The weak solution is defined via stochastic optimal control problem.