August 2024 A trajectorial approach to entropy dissipation for degenerate parabolic equations
Donghan Kim, Lane Chun Yeung
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Bernoulli 30(3): 2253-2274 (August 2024). DOI: 10.3150/23-BEJ1672

Abstract

We consider degenerate diffusion equations of the form tpt=Δf(pt) 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. In line with the recent work (Theory Probab. Appl. 66 (2022) 668–707), our approach is based on applying stochastic analysis 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.

Funding Statement

L.C. Yeung is partially supported by the National Science Foundation (NSF) under grant NSF-DMS-20-04997.

Acknowledgements

The authors are indebted to Ioannis Karatzas for suggesting this problem and for many helpful discussions. The authors also thank the editor, an associate editor, and two anonymous referees for their valuable comments and suggestions, which helped us improved the paper significantly.

Citation

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Donghan Kim. Lane Chun Yeung. "A trajectorial approach to entropy dissipation for degenerate parabolic equations." Bernoulli 30 (3) 2253 - 2274, August 2024. https://doi.org/10.3150/23-BEJ1672

Information

Received: 1 November 2022; Published: August 2024
First available in Project Euclid: 14 May 2024

Digital Object Identifier: 10.3150/23-BEJ1672

Keywords: degenerate diffusion , entropy dissipation , Gradient flow , HWI inequality , porous medium equation

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Vol.30 • No. 3 • August 2024
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