Wellposedness of HamiltonJacobi equations in the Wasserstein space: nonconvex Hamiltonians and common noise
Abstract
We establish the wellposedness of viscosity solutions for a class of semilinear HamiltonJacobi 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 finitedimensional 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 finitedimensional 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 semicontinuous (with respect to the weak topology) and the supersolution is just lower semicontinuous. We then apply these results to meanfield control problems with common noise and zerosum games over the Wasserstein space.
 Publication:

arXiv eprints
 Pub Date:
 December 2023
 DOI:
 10.48550/arXiv.2312.02324
 arXiv:
 arXiv:2312.02324
 Bibcode:
 2023arXiv231202324D
 Keywords:

 Mathematics  Analysis of PDEs;
 Mathematics  Optimization and Control;
 Mathematics  Probability