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.