A probabilistic approach to vanishing viscosity for PDEs on the Wasserstein space

TL;DR

This paper establishes a vanishing viscosity result for PDEs on the Wasserstein space, connecting second order PDEs with value functions in mechanics and game theory, using stochastic analysis techniques.

Contribution

It extends classical vanishing viscosity results to PDEs on probability measure spaces, enabling new links between stochastic processes and PDE solutions.

Findings

Proves a vanishing viscosity theorem for PDEs on Wasserstein space.

Shows value functions in mechanics and games as limits of second order PDEs.

Establishes a large deviation principle for McKean-Vlasov equations.

Abstract

In this work we prove an analogue, for partial differential equations on the space of probability measures, of the classical vanishing viscosity 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 and allows for instance to prove a Freindlin-Wentzell large deviation theorem for McKean-Vlasov equations.