A Linear Equation on the Set of Probability Vectors on Graphs

TL;DR

This paper studies solutions to a linear Hamilton-Jacobi equation in the Wasserstein space of probability vectors on graphs, proving existence under certain smoothness conditions on the initial value function.

Contribution

It establishes the existence of solutions to a linear Hamilton-Jacobi equation on probability measures over graphs, extending analysis in Wasserstein spaces.

Findings

Existence of solutions under differentiability assumptions

Application to probability vectors on finite graphs

Extension of Hamilton-Jacobi theory to graph-based Wasserstein spaces

Abstract

In this paper we investigate solutions to a linear Hamilton-Jacobi equations in the Wasserstein space of probability vectors on a finite simply connected graph. We prove that there exists a solution under the assumption that the initial value function $u_0:\mathcal{P}(G)\to\mathbb{R}$ is Fr\'echet continuously differentiable.