Rate-independent stochastic evolution equations: parametrized solutions

TL;DR

This paper extends the vanishing viscosity method to stochastic rate-independent evolution equations, proving the existence of weak solutions via viscous regularization and time rescaling, accommodating general cylindrical martingale noise.

Contribution

It introduces a novel stochastic framework for rate-independent equations, establishing existence of parametrized martingale solutions with rescaled noise and weak solution concepts.

Findings

Existence of weak solutions via viscous regularization.

Convergence of approximate solutions to parametrized martingale solutions.

Applicability to general cylindrical martingale noise models.

Abstract

By extending to the stochastic setting the classical vanishing viscosity approach we prove the existence of suitably weak solutions of a class of nonlinear stochastic evolution equation of rate-independent type. Approximate solutions are obtained via viscous regularization. Upon properly rescaling time, these approximations converge to a parametrized martingale solution of the problem in rescaled time, where the rescaled noise is given by a general square-integrable cylindrical martingale with absolutely continuous quadratic variation. In absence of jumps, these are strong-in-time martingale solutions of the problem in the original, not rescaled time.