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 so far not covered by the classical variational existence theory (Krylov and Rozovskii in J Sov Math 16:1233–1277, 1981; Liu and Röckner in Stochastic partial differential equations: an introduction, Springer, Berlin, 2015; Pardoux in Equations aux dérivées partielles stochastiques nonlinéaires monotones, Thèse, Orsay, 1972) and, in particular, to stochastic variational inequalities and parabolic stochastic equations with general monotone nonlinearities with low or superfast growth to $+\infty$.