Estimates for MFGs

Diogo A. Gomes*, Edgard A. Pimentel, Vardan Voskanyan

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingChapterpeer-review


In the absence of special transformations or explicit solutions, the analysis of MFGs often relies on a priori bounds. Here, we investigate estimates that are commonly used. We begin by using the maximum principle to obtain one-sided bounds. Next, we consider energy-type estimates that give additional bounds. These two techniques extend to a broad class of mean-field game problems. Equally important are the consequences of these bounds when combined with earlier results. In Sect. 6.3, we develop some of these aspects. In the remainder of this chapter, we discuss other methods that rely on the particular structure of the problems. First, we present a second-order estimate that is used frequently in the periodic setting. Next, we consider a technique that gives Lipschitz bounds for stationary first-order MFGs. Subsequently, we examine energy conservation principles. Finally, we prove estimates for the Fokker–Planck equation that depend on uniform ellipticity or parabolicity of the MFG system.

Original languageEnglish (US)
Title of host publicationSpringerBriefs in Mathematics
PublisherSpringer Science and Business Media B.V.
Number of pages19
StatePublished - 2016

Publication series

NameSpringerBriefs in Mathematics
ISSN (Print)2191-8198
ISSN (Electronic)2191-8201

Bibliographical note

Publisher Copyright:
© 2016, Springer International Publishing Switzerland.


  • Energy Conservation Principle
  • Mean Field Game Problem
  • Mean Field Games
  • Second-order Estimate
  • Uniform Ellipticity

ASJC Scopus subject areas

  • General Mathematics


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