Path-by-path well-posedness of nonlinear diffusion equations with multiplicative noise

TL;DR

This paper establishes the path-by-path well-posedness of stochastic nonlinear diffusion equations with multiplicative noise, leading to the existence of a random dynamical system, and addresses an open problem in the field.

Contribution

It proves the well-posedness of stochastic porous media and fast diffusion equations driven by linear multiplicative noise, solving a previously open problem.

Findings

Proves path-by-path well-posedness of stochastic diffusion equations.

Establishes existence of a random dynamical system for these equations.

Addresses and solves an open problem from prior literature.

Abstract

We prove the path-by-path well-posedness of stochastic porous media and fast diffusion equations driven by linear, multiplicative noise. As a consequence, we obtain the existence of a random dynamical system. This solves an open problem raised in [Barbu, R\"ockner; JDE, 2011], [Barbu, R\"ockner; JDE, 2018+], and [Gess, AoP, 2014].