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.
"Translation invariant mean field games with common noise." Electron. Commun. Probab. 20 1 - 13, 2015. https://doi.org/10.1214/ECP.v20-3822