Solutions and stochastic averaging for delay-path-dependent stochastic variational inequalities in infinite dimensions

TL;DR

This paper establishes well-posedness and a stochastic averaging principle for complex delay-path-dependent stochastic variational inequalities in infinite dimensions, with applications to various particle systems and SPDEs.

Contribution

It introduces a general framework for SVIs with jumps, delays, and path dependence in infinite dimensions, proving existence, uniqueness, and averaging results under non-Lipschitz conditions.

Findings

Proved well-posedness of delay-path-dependent SVIs in infinite dimensions.

Established stochastic averaging principle for strong convergence.

Applied results to particle systems and stochastic PDEs with jumps and delays.

Abstract

In this paper, we study a very general stochastic variational inequality(SVI) having jumps, random coefficients, delay, and path dependence, in infinite dimensions. Well-posedness in terms of the existence and uniqueness of a solution is established, and a stochastic averaging principle on strong convergence of a time-explosion SVI to an averaged equation is obtained, both under non-Lipschitz conditions. We illustrate our results on general but concrete examples of finite dimension and infinite dimension respectively, which cover large classes of particle systems with electro-static repulsion, nonlinear stochastic partial differential equations with jumps, semilinear stochastic partial differential equations (especially stochastic reaction-diffusion equations) with delays, and others.