A trajectorial approach to entropy dissipation for degenerate parabolic equations

TL;DR

This paper introduces a trajectorial method using stochastic calculus to analyze entropy dissipation in degenerate diffusion equations, including the porous medium equation, providing new insights into their gradient flow structure and related inequalities.

Contribution

It develops a trajectorial analogue of entropy dissipation for nonlinear degenerate diffusions using stochastic calculus, offering new proofs of gradient flow properties and inequalities.

Findings

Derivation of a trajectorial entropy dissipation identity

New proof of Wasserstein gradient flow property

Simplified proof of the HWI inequality

Abstract

We consider degenerate diffusion equations of the form $\partial_tp_t = \Delta f(p_t)$ on a bounded domain and subject to no-flux boundary conditions, for a class of nonlinearities $f$ that includes the porous medium equation. We derive for them a trajectorial analogue of the entropy dissipation identity, which describes the rate of entropy dissipation along every path of the diffusion. Our approach is based on applying stochastic calculus to the underlying probabilistic representations, which in our context are stochastic differential equations with normal reflection on the boundary. This trajectorial approach also leads to a new derivation of the Wasserstein gradient flow property for nonlinear diffusions, as well as to a simple proof of the HWI inequality in the present context.