Variational solutions to nonlinear stochastic differential equations in Hilbert spaces

Abstract

One introduces a new variational concept of solution for the stochastic differential equation $dX+A(t)X\,dt+\lambda X\,dt=X\,dW,$ $t\in(0,T)$; $X(0)=x$ in a real Hilbert space where $A(t)=\partial\varphi(t)$, $t\in(0,T)$, is a maximal monotone subpotential operator in $H$ while $W$ is a Wiener process in $H$ on a probability space $\{\Omega,\mathcal{F},\mathbb{P}\}$. In this new context, the solution $X=X(t,x)$ exists for each $x\in H$, is unique, and depends continuously on $x$. This functional scheme applies to a general class of stochastic PDE not covered by the classical variational existence theory ([15], [16], [17]) and, in particular, to stochastic variational inequalities and parabolic stochastic equations with general monotone nonlinearities with low or superfast growth to $+\infty$.