The stochastic value function on metric measure spaces

TL;DR

This paper proves that in $RCD(K, olinebreak \infty)$ metric measure spaces, the stochastic value function obeys the viscous Hamilton-Jacobi equation, extending Fleming's theorem from Euclidean spaces to more general settings.

Contribution

It establishes the viscous Hamilton-Jacobi equation for the stochastic value function on $RCD(K, olinebreak \infty)$ spaces, generalizing classical results to metric measure spaces.

Findings

Stochastic value function satisfies viscous Hamilton-Jacobi equation on $RCD(K, olinebreak \infty)$ spaces.

Extension of Fleming's theorem from Euclidean spaces to metric measure spaces.

Provides a link between stochastic analysis and geometric analysis in metric measure spaces.

Abstract

Let $(S,d)$ be a compact metric space and let $m$ be a Borel probability measure on $(S,d)$. We shall prove that, if $(S,d,m)$ is a $RCD(K,\infty)$ space, then the stochastic value function satisfies the viscous Hamilton-Jacobi equation, exactly as in Fleming's theorem on ${\bf R}^d$.