Translation invariant mean field games with common noise

Abstract

This note highlights a special class of mean field games in which the coefficients satisfy a convolution-type structural condition. A mean field game of this type with common noise is related to a certain mean field game without common noise by a simple transformation, which permits a tractable construction of a solution of the problem with common noise from a solution of the problem without. This yields new existence results for common noise problems, notably allowing controlled volatility, as well as new uniqueness results for problems without common noise.