Stochastic Nonlinear Parabolic Equations with Stratonovich Gradient Noise.

Existence and uniqueness of solutions to stochastic differential equation dX-diva(∇X)dt=∑j=1N(bj·∇X)∘dβj<inline-graphic></inline-graphic> in (0,T)×O<inline-graphic></inline-graphic>; X(0,ξ)=x(ξ)<inline-graphic></inline-graphic>, ξ∈O<inline-graphic></inline-graphic>, X=0<inline-graphic></inline-graphic> on (0,T)×∂O<inline-graphic></inline-graphic> is studied. Here O<inline-graphic></inline-graphic> is a bounded and open domain of Rd<inline-graphic></inline-graphic>, d≥1<inline-graphic></inline-graphic>, {bj}<inline-graphic></inline-graphic> is a divergence free vector field, a:[0,T]×O×Rd→Rd<inline-graphic></inline-graphic> is a continuous and monotone mapping of subgradient type and {βj}<inline-graphic></inline-graphic> are independent Brownian motions in a probability space (Ω,F,P)<inline-graphic></inline-graphic>. The weak solution is defined via stochastic optimal control problem. [ABSTRACT FROM AUTHOR]