Deterministic and Stochastic Differential Equations in Hilbert Spaces Involving Multivalued Maximal Monotone Operators

Abstract

This work deals with a Skorokhod problem driven by a maximal operator: \begin{aligned} &du(t)+Au(t)(dt)\ni f(t)dt+dM(t), \; 0<t<T,\\ &u(0)=u_{0}, \end{aligned} which is a multivalued deterministic differential equation with a singular inputs $dM(t)$, where $t\rightarrow M(t)$ is a continuous function. The existence and uniqueness result is used to study an It\^{o}'s stochastic differential equation \begin{aligned} &du(t)+Au(t)(dt)\ni f(t,u(t))dt+B(t,u(t))dW(t),\; 0<t<T,\\ &u(0)=u_{0}, \end{aligned} in a real Hilbert space $H$, where $A$ is a multivalued ($\alpha$-)maximal monotone operator on $H$, and $f(t,u)$ and $B(t,u)$ are Lipschitz continuous with respect to $u$. Some asymptotic properties in the stochastic case are also found.