A cost on paths of measures which induces the Fokker-Planck equation

TL;DR

This paper introduces a new construction of a cost functional on measure paths that characterizes solutions to the Fokker-Planck equation with L^2 drift, building on previous work by Feng and Nguyen.

Contribution

It provides an alternative method to define the cost on measure curves solving the Fokker-Planck equation, inspired by ideas from Gomes and Valdinoci.

Findings

The new cost functional precisely characterizes measure curves solving the Fokker-Planck equation.

The construction offers a different perspective from Feng and Nguyen's original approach.

The approach connects measure path costs with PDE solutions in a novel way.

Abstract

J. Feng and T. Nguyen have defined a cost on curves of measures which is finite exactly on the curves which solve a Fokker-Planck equation with $L^2$ drift. In this paper, using ideas of D. Gomes and E. Valdinoci, we give a different construction of the cost of Feng and Nguyen.