Time-optimal problem in the space of probabilities measures

This paper focuses on the value function in the time-optimal problem for a continuity equation in the space of probability measures. We derive the dynamic programming principle for this problem. In particular, we prove that the Kruzhkov transform of the value function is a unique discontinuous viscosity solution to the corresponding Dirichlet problem for the Hamilton-Jacobi equation. Finally, we establish the $\Gamma$-convergence of the value function in a perturbed problem to the value function in the unperturbed problem.