Search for: All records

We consider linear and nonlinear transport equations with irregular velocity fields, motivated by models coming from mean field games. The velocity fields are assumed to increase in each coordinate, and the divergence therefore fails to be absolutely continuous with respect to the Lebesgue measure in general. For such velocity fields, the well-posedness of first- and second-order linear transport equations in Lebesgue spaces is established, as well as the existence and uniqueness of regular ODE and SDE Lagrangian flows. These results are then applied to the study of certain nonconservative, nonlinear systems of transport type, which are used to model mean field games in a finite state space. A notion of weak solution is identified for which unique minimal and maximal solutions exist, which do not coincide in general. A selection-by-noise result is established for a relevant example to demonstrate that different types of noise can select any of the admissible solutions in the vanishing noise limit. 

Abstract The goal of this paper is to prove a comparison principle for viscosity solutions of semilinear Hamilton–Jacobi equations in the space of probability measures. The method involves leveraging differentiability properties of the 2-Wasserstein distance in the doubling of variables argument, which is done by introducing a further entropy penalization that ensures that the relevant optima are achieved at positive, Lipschitz continuous densities with finite Fischer information. This allows to prove uniqueness and stability of viscosity solutions in the class of bounded Lipschitz continuous (with respect to the 1-Wasserstein distance) functions. The result does not appeal to a mean field control formulation of the equation, and, as such, applies to equations with nonconvex Hamiltonians and measure-dependent volatility. For convex Hamiltonians that derive from a potential, we prove that the value function associated with a suitable mean-field optimal control problem with nondegenerate idiosyncratic noise is indeed the unique viscosity solution.