Preprint
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Inserted: 11 nov 2026

Year: 2026

Abstract:

This paper is devoted to the well-posedness of a class of infinite horizon discounted Mean Field Games (MFGs) problems, where both an exponential discount and a so-called cheap control factors are present simultaneously, and scale in a possibly different way. We introduce a new class of transformations of variables, at the level of both the MFG PDE and the corresponding FBSDE systems. We then show how these transformations reveal a new mechanism that generates semiconvexity estimates for the value function and ensures the existence of mean field equilibria, as long as the parameters land in a suitable regime, that we quantitatively describe. This new mechanism demonstrates furthermore how discount factors lead to a source of strong displacement monotonicity for the Lagrangian, without any initial monotonicity assumptions on the running costs. Consequently, in such regimes of parameters uniqueness of solutions manifests. Our approach uses a Lagrangian point of view, which lets us treat general non-separable Lagrangians and both deterministic and idiosyncratic noise driven models in a transparent, unified way. For a specific class of quadratic Lagrangians, using turnpike-type methods, we further obtain an exponential contractivity estimate for the mean field equilibria.