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

TL;DR

This paper investigates deterministic and stochastic differential equations involving multivalued maximal monotone operators in Hilbert spaces, establishing existence, uniqueness, and asymptotic properties for solutions driven by singular and stochastic inputs.

Contribution

It introduces new results on existence and uniqueness for multivalued differential equations with singular inputs and extends these to stochastic equations with Lipschitz continuous coefficients.

Findings

Proved existence and uniqueness of solutions for the deterministic Skorokhod problem.

Extended results to stochastic differential equations with maximal monotone operators.

Analyzed asymptotic behavior of solutions in the stochastic setting.

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.