Generalized Yosida Approximation and Multi-Valued Stochastic Evolution Inclusions

TL;DR

This paper extends the Yosida approximation to stochastic evolution inclusions, proving existence, uniqueness, and extinction properties of solutions under a generalized framework with broad applications.

Contribution

It introduces a nonstandard duality mapping-based Yosida approximation, establishing key solution properties for multi-valued stochastic evolution inclusions.

Findings

Proves existence and uniqueness of solutions.

Demonstrates finite-time extinction with probability one.

Provides explicit bounds on extinction time moments.

Abstract

In this paper, we generalize the classical Yosida approximation by utilizing a nonstandard duality mapping to establish the existence and uniqueness of both (probabilistically) weak and strong solutions and demonstrate the continuous dependence on initial values for a class of multi-valued stochastic evolution inclusions within the variational framework. Furthermore, leveraging this generalized Yosida approximation, we derive the finite-time extinction of solutions with probability one and also provide an explicit upper bound of the moment of extinction time for multi-valued stochastic evolution inclusions perturbed by linear multiplicative noise. The main results are applicable to various examples, including multi-valued stochastic porous media equations, stochastic $\Phi$-Laplace equations and stochastic evolution inclusions involving subdifferentials.