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Kinetic-type Mean Field Games with Non-separable Local Hamiltonians

David M. Ambrose, Megan Griffin-Pickering, Alpár R. Mészáros

Abstract

We prove well-posedness of a class of kinetic-type Mean Field Games, which typically arise when agents control their acceleration. Such systems include independent variables representing the spatial position as well as velocity. We consider non-separable Hamiltonians without any structural conditions, which depend locally on the density variable. Our analysis is based on two main ingredients: an energy method for the forward-backward system in Sobolev spaces, on the one hand and on a suitable vector field method to control derivatives with respect to the velocity variable, on the other hand. The careful combination of these two techniques reveals interesting phenomena applicable for Mean Field Games involving general classes of drift-diffusion operators and nonlinearities. While many prior existence theories for general Mean Field Games systems take the final datum function to be smoothing, we can allow this function to be non-smoothing, i.e. also depending locally on the final measure. Our well-posedness results hold under an appropriate smallness condition, assumed jointly on the data.

Kinetic-type Mean Field Games with Non-separable Local Hamiltonians

Abstract

We prove well-posedness of a class of kinetic-type Mean Field Games, which typically arise when agents control their acceleration. Such systems include independent variables representing the spatial position as well as velocity. We consider non-separable Hamiltonians without any structural conditions, which depend locally on the density variable. Our analysis is based on two main ingredients: an energy method for the forward-backward system in Sobolev spaces, on the one hand and on a suitable vector field method to control derivatives with respect to the velocity variable, on the other hand. The careful combination of these two techniques reveals interesting phenomena applicable for Mean Field Games involving general classes of drift-diffusion operators and nonlinearities. While many prior existence theories for general Mean Field Games systems take the final datum function to be smoothing, we can allow this function to be non-smoothing, i.e. also depending locally on the final measure. Our well-posedness results hold under an appropriate smallness condition, assumed jointly on the data.
Paper Structure (10 sections, 17 theorems, 158 equations)

This paper contains 10 sections, 17 theorems, 158 equations.

Table of Contents

  1. Introduction
  2. Standing assumptions and preliminaries
  3. Function spaces
  4. Tools for the Transport Operator
  5. Sobolev Estimates for Composite Functions
  6. Linear estimates
  7. Estimates for solutions of a Kolmogorov equation
  8. Estimates for the MFG equations
  9. Local Well-posedness
  10. Approximation argument for a key estimate

Key Result

Theorem 1.1

Let $H$ and $g$ satisfy suitable regularity assumptions. Let $m^{0}\in H^{s}_z({\mathbb{M} \times \mathbb{R}^d})$ be given, where $s$ is an integer such that $s>d+1$. Then there exist $\epsilon_\ast>0, \delta_\ast>0, T_\ast>0\newline$ such that, for all $\epsilon\in(0, \epsilon_\ast), \delta\in(0,\d

Theorems & Definitions (49)

  • Theorem 1.1
  • Remark 1
  • Remark 2
  • Remark 3
  • Definition 1: Vector fields
  • Remark 4
  • Remark 5
  • Lemma 2.1
  • proof
  • Definition 2
  • ...and 39 more