Weak stability by noise for approximations of doubly nonlinear evolution equations

TL;DR

This paper proves the existence and stability of weak solutions for stochastic doubly nonlinear evolution equations with rough additive noise, highlighting stochastic effects that prevent uniqueness unlike in deterministic cases.

Contribution

It establishes probabilistically weak solutions and their stability for stochastic doubly nonlinear equations under rough noise conditions, a novel result in this area.

Findings

Existence of weak solutions under rough noise conditions

Stability of solutions in a weak probabilistic sense

Non-uniqueness in the deterministic case is overcome by stochastic effects

Abstract

Doubly nonlinear stochastic evolution equations are considered. Upon assuming the additive noise to be rough enough, we prove the existence of probabilistically weak solutions of Friedrichs type and study their uniqueness in law. This entails stability for approximations of stochastic doubly nonlinear equations in a weak probabilistic sense. Such effect is a genuinely stochastic, as doubly nonlinear equations are not even expected to exhibit uniqueness in the deterministic case.