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

A Linear Equation on the Set of Probability Vectors on Graphs