Stochastic variational inequalities and applications to the total variation flow perturbed by linear multiplicative noise

TL;DR

This paper introduces a novel method to establish existence and uniqueness of solutions for a stochastic total variation flow with multiplicative noise, demonstrating finite time extinction in low dimensions.

Contribution

The work develops a new approach to prove well-posedness of a stochastic variational inequality for highly singular diffusion, extending the stochastic total variation flow theory.

Findings

Proved existence and uniqueness of solutions for the stochastic total variation flow.

Established finite time extinction of solutions in dimensions 1 to 3.

Developed a new method applicable to stochastic variational inequalities with singular diffusivity.

Abstract

In this work, we introduce a new method to prove the existence and uniqueness of a variational solution to the stochastic nonlinear diffusion equation $dX(t)={\rm div} [\frac{\nabla X(t)}{|\nabla X(t)|}]dt+X(t)dW(t) in (0,\infty)\times\mathcal{O},$ where $\mathcal{O}$ is a bounded and open domain in $\mathbb{R}^N$, $N\ge 1$, and $W(t)$ is a Wiener process of the form $W(t)=\sum^\infty_{k=1}\mu_k e_k\beta_k(t)$, $e_k \in C^2(\bar\mathcal{O})\cap H^1_0(\mathcal{O}),$ and $\beta_k$, $k\in\mathbb{N}$, are independent Brownian motions. This is a stochastic diffusion equation with a highly singular diffusivity term and one main result established here is that, for all initial conditions in $L^2(\mathcal{O})$, it is well posed in a class of continuous solutions to the corresponding stochastic variational inequality. Thus one obtains a stochastic version of the (minimal) total variation flow. The new approach developed here also allows to prove the finite time extinction of solutions in dimensions $1\le N\le 3$, which is another main result of this work. Keywords: stochastic diffusion equation, Brownian motion, bounded variation, convex functions, bounded variation flow.