A Probabilistic Approach to Small Noise Limit for PDEs on the Wasserstein Space

In this work we prove an analogue, for partial differential equations on the space of probability measures, of the classical vanishing noise result known for equations on the Euclidean space. Our result allows in particular to show that the value function arising in various problems of classical mechanics and games can be obtained as the limiting case of second-order PDEs. The method of proof builds on stochastic analysis arguments via a variational representation for functionals of McKean-Vlasov equations, and allows us to deduce the small noise limit result as a consequence of a Freindlin-Wentzell large deviation theorem for McKean-Vlasov equations in the Laplace principle form.