# Weak Existence and Uniqueness for McKean-Vlasov SDEs with Common Noise

## TL;DR

This paper develops a framework for weak solutions to McKean-Vlasov SDEs with common noise, establishing existence and uniqueness results under specific regularity and non-degeneracy conditions, linking solutions to associated SPDEs.

## Contribution

It introduces a new definition of weak solutions incorporating compatibility, and proves existence and uniqueness results for McKean-Vlasov SDEs with common noise using compactness and Girsanov transformations.

## Key findings

- Existence of weak solutions under boundedness and continuity of coefficients.
- Weak uniqueness when the private noise is non-degenerate and the drift is regular.
- Connection between weak solutions and solutions to related SPDEs.

## Abstract

This paper concerns the McKean-Vlasov stochastic differential equation (SDE) with common noise. An appropriate definition of a weak solution to such an equation is developed. The importance of the notion of compatibility in this definition is highlighted by a demonstration of its r\^ole in connecting weak solutions to McKean-Vlasov SDEs with common noise and solutions to corresponding stochastic partial differential equations (SPDEs). By keeping track of the dependence structure between all components in a sequence of approximating processes, a compactness argument is employed to prove the existence of a weak solution assuming boundedness and joint continuity of the coefficients (allowing for degenerate diffusions). Weak uniqueness is established when the private (idiosyncratic) noise's diffusion coefficient is non-degenerate and the drift is regular in the total variation distance. This seems sharp when one considers using finite-dimensional noise to regularise an infinite dimensional problem. The proof relies on a suitably tailored cost function in the Monge-Kantorovich problem and representation of weak solutions via Girsanov transformations.