Multi-valued, singular stochastic evolution inclusions

TL;DR

This paper establishes existence and uniqueness results for complex stochastic evolution inclusions with applications to singular diffusion equations, including stochastic 1-Laplacian and fast diffusion equations, in Hilbert spaces.

Contribution

It provides the first abstract variational framework for multi-valued stochastic evolution inclusions with singular diffusions, including new existence and ergodicity results.

Findings

Existence and uniqueness of solutions for stochastic singular diffusion equations.

Existence of a unique weak-* mean ergodic invariant measure under additive noise.

Application to stochastic 1-Laplacian and fast diffusion equations.

Abstract

We provide an abstract variational existence and uniqueness result for multi-valued, monotone, non-coercive stochastic evolution inclusions in Hilbert spaces with general additive and Wiener multiplicative noise. As examples we discuss certain singular diffusion equations such as the stochastic 1-Laplacian evolution (total variation flow) in all space dimensions and the stochastic singular fast diffusion equation. In case of additive Wiener noise we prove the existence of a unique weak-* mean ergodic invariant measure.