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Wellposedness Of Second Order Master Equations For Mean Field Games With Nonsmooth Data.pdf

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Preview of Wellposedness of Second Order Master Equations for Mean Field Games with Nonsmooth Data
🔗 Source: dornsife.usc.edu
📊 Size: 861 KB
📄 Pages: 94 pages
⬇️ Downloads: 40

Summary

While classical solutions typically demand monotonicity and smooth data, the authors aim to relax these conditions, addressing an open problem in the literature. They propose three weaker solution concepts: good solutions, weak solutions, and weak-viscosity solutions, and prove well-posedness under all three for the master equation.

Key Points:

Relaxes Data Regularity: Moves beyond classical solutions by not requiring differentiability of measures.
 Global Well-Posedness: Establishes global (in time) well-posedness for the master equation using a uniform a priori estimate for Lipschitz continuity of solutions in measures, highlighting the importance of monotonicity.
* Convergence of Nash Systems and Propagation of Chaos: Extends results to high-dimensional systems of partial differential equations (PDEs) arising from N-player games, proving convergence under mild regularity conditions and a propagation of chaos property for optimal trajectories.

Main Contributions:

1. Three Weaker Solution Concepts: Introduce good solutions, weak solutions, and weak-viscosity solutions to accommodate less regular data.
2. Uniform A Priori Estimate: Derive a crucial uniform estimate for the Lipschitz continuity of solutions in measures, enabling global well-posedness even without differentiability of measures.
3. Smooth Mollifier on Wasserstein Space: Develops a new smooth mollifier for functions on Wasserstein space, contributing to the analysis.
4. Convergence and Chaos Propagation in N-Player Games: Demonstrates convergence of the Nash system for MFGs under mild conditions and establishes propagation of chaos for optimal trajectories.

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Description

This paper studies second-order master equations from mean field games with nonsmooth data, relaxing the regularity of the data while maintaining the monotonicity condition.

Technical Information

  • File Format: PDF
  • File Size: 861 KB
  • Pages: 94
  • Language: EN
  • Total Downloads: 40
  • Last Updated: 1 week ago

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