Mean Field Games in Hilbert Spaces with Degenerate Diffusion: A Viscosity Solution Approach
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866911677227728896 |
|---|---|
| author | Święch, Andrzej Wessels, Lukas |
| author_facet | Święch, Andrzej Wessels, Lukas |
| contents | We study a degenerate second order mean field game (MFG) system in a Hilbert space $H$ which couples a Fokker--Planck equation describing the evolution of probability measures on $H$ with a Hamilton--Jacobi--Bellman (HJB) equation for the value function. Our main result establishes existence and uniqueness of solutions to this coupled system. Solutions of the HJB equation are interpreted in the viscosity sense. For existence, we extend the classical fixed-point approach based on Tikhonov's theorem to our setting. A central difficulty in this approach is proving uniqueness for the corresponding linear degenerate Fokker--Planck equation. To address this issue, we introduce a class of suitable adjoint equations and employ viscosity solution techniques to construct sufficiently regular solutions. Uniqueness for the full MFG system is then obtained via an adaptation of the Lasry--Lions monotonicity method. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_12690 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Mean Field Games in Hilbert Spaces with Degenerate Diffusion: A Viscosity Solution Approach Święch, Andrzej Wessels, Lukas Analysis of PDEs Optimization and Control Probability We study a degenerate second order mean field game (MFG) system in a Hilbert space $H$ which couples a Fokker--Planck equation describing the evolution of probability measures on $H$ with a Hamilton--Jacobi--Bellman (HJB) equation for the value function. Our main result establishes existence and uniqueness of solutions to this coupled system. Solutions of the HJB equation are interpreted in the viscosity sense. For existence, we extend the classical fixed-point approach based on Tikhonov's theorem to our setting. A central difficulty in this approach is proving uniqueness for the corresponding linear degenerate Fokker--Planck equation. To address this issue, we introduce a class of suitable adjoint equations and employ viscosity solution techniques to construct sufficiently regular solutions. Uniqueness for the full MFG system is then obtained via an adaptation of the Lasry--Lions monotonicity method. |
| title | Mean Field Games in Hilbert Spaces with Degenerate Diffusion: A Viscosity Solution Approach |
| topic | Analysis of PDEs Optimization and Control Probability |
| url | https://arxiv.org/abs/2605.12690 |