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Analysis of mean field games via Fokker-Planck-Kolmogorov equations: existence of equilibria

Stanislav V. Shaposhnikov, Dmitry V. Shatilovich

Abstract

We study mean field games with unbounded coefficients. The existence of a solution is proved. We propose a new approach based on Fokker-Planck-Kolmogorov equations, the Ambrosio-Figalli-Trevisan superposition principle, the method of doubling variables and a~priory estimates with Lyapunov functions.

Analysis of mean field games via Fokker-Planck-Kolmogorov equations: existence of equilibria

Abstract

We study mean field games with unbounded coefficients. The existence of a solution is proved. We propose a new approach based on Fokker-Planck-Kolmogorov equations, the Ambrosio-Figalli-Trevisan superposition principle, the method of doubling variables and a~priory estimates with Lyapunov functions.

Paper Structure

This paper contains 4 sections, 7 theorems, 336 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Auxiliary results
  4. Proof of Theorem \ref{['th1']} and Corollary \ref{['cor1']}

Key Result

Theorem 2.1

Assume that the conditions $\rm (H1), (H2), (H3)$ are fulfilled and $\nu$ is a probability measure on $\mathbb{R}^d$ with $V\in L^1(\nu)$. Then there exists a mapping $t\mapsto\mu_t$ from $[0, T]$ to the space of probability measures on $\mathbb{R}^d$ that is continuous with respect to the weak topo that is for every $\psi\in C_0^{\infty}(\mathbb{R}^d)$ and for all $t\in[0, T]$ we have (iii) the

Theorems & Definitions (31)

  • Theorem 2.1
  • Corollary 2.1
  • Example 2.1
  • proof
  • Example 2.2
  • proof
  • Example 2.3
  • proof
  • Example 2.4
  • proof
  • ...and 21 more