Well-posedness of Hamilton-Jacobi equations in the Wasserstein space: non-convex Hamiltonians and common noise

Abstract

We establish the well-posedness of viscosity solutions for a class of semi-linear Hamilton-Jacobi equations set on the space of probability measures on the torus. In particular, we focus on equations with both common and idiosyncratic noise, and with Hamiltonians which are not necessarily convex in the momentum variable. Our main results show (i) existence, (ii) the comparison principle (and hence uniqueness), and (iii) the convergence of finite-dimensional approximations for such equations. Our proof strategy for the comparison principle is to first use a mix of existing techniques (especially a change of variables inspired by \cite{bayraktar2023} to deal with the common noise) to prove a ``partial comparison" result, i.e. a comparison principle which holds when either the subsolution or the supersolution is Lipschitz with respect to a certain very weak metric. Our main innovation is then to develop a strategy for removing this regularity assumption. In particular, we use some delicate estimates for a sequence of finite-dimensional PDEs to show that under certain conditions there \textit{exists} a viscosity solution which is Lipschitz with respect to the relevant weak metric. We then use this existence result together with a mollification procedure to establish a full comparison principle, i.e. a comparison principle which holds even when the subsolution under consideration is just upper semi-continuous (with respect to the weak topology) and the supersolution is just lower semi-continuous. We then apply these results to mean-field control problems with common noise and zero-sum games over the Wasserstein space.