Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Long time behaviour of Mean Field Games with fractional diffusion

Olav Ersland, Espen Robstad Jakobsen, Alessio Porretta

TL;DR

The paper analyzes long-time behavior of mean field games driven by non-Gaussian noise via fractional diffusion, establishing an exponential turnpike to a unique stationary ergodic state under Lasry–Lions monotonicity. The approach combines nonlocal HJB and Fokker–Planck analyses with truncation, fixed-point arguments, and duality, yielding uniform-in-time regularity and stability, and culminating in a quantitative turnpike inequality. This provides the first exponential turnpike result for MFGs with Lévy processes lacking Brownian components and highlights the role of a confining drift in ensuring ergodicity. The results have implications for understanding and computing long-horizon equilibria in large populations under jump-type dynamics and nonlocal interactions.

Abstract

In this paper we study the long time behaviour of mean field games systems with fractional diffusion, modeling the case that the individual dynamics of the players is driven by independent jump processes and controlled through the drift term, while being confined by an external field in order to guarantee ergodicity. In the case of globally Lipschitz, locally uniformly convex Hamiltonian, and weakly coupled costs satisfying the Lasry-Lions monotonicity condition, we prove that there is a unique solution $(u_T,m_T)$ to the mean field game problem in $(0,T)$ and we show that, if $T$ is sufficiently large, $(u_T,m_T)$ satisfies the so-called turnpike property, namely it is exponentially close to the (unique) stationary ergodic state for any proportionally long intermediate time.

Long time behaviour of Mean Field Games with fractional diffusion

TL;DR

The paper analyzes long-time behavior of mean field games driven by non-Gaussian noise via fractional diffusion, establishing an exponential turnpike to a unique stationary ergodic state under Lasry–Lions monotonicity. The approach combines nonlocal HJB and Fokker–Planck analyses with truncation, fixed-point arguments, and duality, yielding uniform-in-time regularity and stability, and culminating in a quantitative turnpike inequality. This provides the first exponential turnpike result for MFGs with Lévy processes lacking Brownian components and highlights the role of a confining drift in ensuring ergodicity. The results have implications for understanding and computing long-horizon equilibria in large populations under jump-type dynamics and nonlocal interactions.

Abstract

In this paper we study the long time behaviour of mean field games systems with fractional diffusion, modeling the case that the individual dynamics of the players is driven by independent jump processes and controlled through the drift term, while being confined by an external field in order to guarantee ergodicity. In the case of globally Lipschitz, locally uniformly convex Hamiltonian, and weakly coupled costs satisfying the Lasry-Lions monotonicity condition, we prove that there is a unique solution to the mean field game problem in and we show that, if is sufficiently large, satisfies the so-called turnpike property, namely it is exponentially close to the (unique) stationary ergodic state for any proportionally long intermediate time.
Paper Structure (14 sections, 24 theorems, 265 equations)

This paper contains 14 sections, 24 theorems, 265 equations.

Table of Contents

  1. Introduction
  2. Assumptions and main result
  3. The HJB equation
  4. Parabolic HJB equations
  5. A truncated evolution problem and the proof of Theorem \ref{['thm:uniform_second_derivative_HJB']}.
  6. The stationary HJB equation
  7. The Fokker-Planck equation
  8. Well-posedness for evolutionary and ergodic stationary MFG
  9. Existence for the evolutionary MFG
  10. Stationary ergodic system
  11. The exponential turnpike estimate
  12. Long time decay estimates for drift-diffusion operators
  13. Proof of the main result
  14. Lipschitz bounds for Hamilton-Jacobi equations

Key Result

Theorem 2.2

Assume hypotheses nu', B--B2, H1--H6 and F1--F3. Assume that $m_0\in \mathcal{P}_k$ for some $k\in (0,\sigma)$, and $u_T\in W^{1,\infty}(\mathbb{R}^d)$. Let $(u,m)$ be the solution of (eqn:parabolic_MFG) and let $(\lambda, \bar{u}, \bar{m})$ be the solution of (eqn:ergodic_MFG), such that $\bar{m} \ for every $t\in (0,T)$.

Theorems & Definitions (55)

  • Remark 2.1
  • Theorem 2.2
  • Remark 2.3
  • Definition 3.1: Viscosity solutions
  • Theorem 3.2
  • proof
  • Proposition 3.3: Global Lipschitz bound
  • Lemma 3.4: Local Lipschitz bound
  • Theorem 3.5
  • Proposition 3.6
  • ...and 45 more