The total variation flow perturbed by gradient linear multiplicative noise

TL;DR

This paper investigates stochastic non-linear diffusion equations with singular diffusivity and gradient noise, establishing existence, uniqueness, and finite-time extinction of solutions, relevant for image restoration and turbulence modeling.

Contribution

It introduces a framework for analyzing solutions to highly singular stochastic diffusion equations with gradient noise, including existence, uniqueness, and extinction properties.

Findings

Proved existence and uniqueness of solutions in stochastic variational inequality framework.

Established finite-time extinction of solutions with probability one.

Applied results to image restoration and turbulence modeling contexts.

Abstract

We consider stochastic non-linear diffusion equations with a highly singular diffusivity term and multiplicative gradient-type noise. We study existence and uniqueness of non-negative variational solutions in terms of stochastic variational inequalities. We also show extinction in finite time with probability one. These kind of equations arise, e.g. in the use for simulation of image restoring techniques or for modeling turbulence.