An entropy generation formula on $RCD(K,\infty)$ spaces

TL;DR

This paper extends an entropy generation formula from Euclidean spaces to compact metric measure spaces with the RCD(K,∞) property, enabling analysis of measure curves and stochastic value functions.

Contribution

It proves an entropy generation formula in RCD(K,∞) spaces for bounded measure densities, generalizing previous Euclidean results.

Findings

Entropy generation formula holds in RCD(K,∞) spaces.

Existence of minimal characteristics for stochastic value functions.

Extension of Euclidean Fokker-Planck results to metric measure spaces.

Abstract

J. Feng and T. Nguyen have shown that the solutions of the Fokker-Planck equation in $\R^d$ satisfy an entropy generation formula. We prove that, in compact metric measure spaces with the $RCD(K,\infty)$ property, a similar result holds for curves of measures whose density is bounded away from zero and infinity. We use this fact to show the existence of minimal characteristics for the stochastic value function.