  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.
