Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2605.06589 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866918488475435008 |
|---|---|
| author | Gangbo, Wilfrid Munoz, Sebastian Wu, Jeremy Zhang, Zhaoyu |
| author_facet | Gangbo, Wilfrid Munoz, Sebastian Wu, Jeremy Zhang, Zhaoyu |
| contents | We develop a classical well-posedness and regularity theory on a finite connected weighted graph for an extended mean field game system, its associated master equation, and a Hamilton-Jacobi- Bellman equation on the probability simplex, all in the presence of an individual noise operator. The geometric structure is inherited from the logarithmic-mean activation functional of discrete optimal transport, under which the entropic Fokker-Planck equation appears as a gradient flow on the graph and the individual noise operator is a bilinear form in the probability vector and the Wasserstein gradient. A central technical step is a quantitative preservation-of-positivity estimate for the discrete continuity equation, which rules out finite-time boundary degeneracy and yields a classical solution theory for the master equation on the open simplex without imposing any boundary condition. As an application, we recover a Nash equilibrium interpretation of the discrete system in terms of Markov chains on the graph. Our setup is inspired by the computational algorithms for optimal mass transport of [10, 11] and provides a rigorous well-posedness theory for several of the equations derived in [25]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_06589 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Master equations with an individual noise on finite state graphs Gangbo, Wilfrid Munoz, Sebastian Wu, Jeremy Zhang, Zhaoyu Analysis of PDEs Optimization and Control 35F21, 35R02, 49L12, 49N80, 49Q22, 60H10, 60H30, 91A43 We develop a classical well-posedness and regularity theory on a finite connected weighted graph for an extended mean field game system, its associated master equation, and a Hamilton-Jacobi- Bellman equation on the probability simplex, all in the presence of an individual noise operator. The geometric structure is inherited from the logarithmic-mean activation functional of discrete optimal transport, under which the entropic Fokker-Planck equation appears as a gradient flow on the graph and the individual noise operator is a bilinear form in the probability vector and the Wasserstein gradient. A central technical step is a quantitative preservation-of-positivity estimate for the discrete continuity equation, which rules out finite-time boundary degeneracy and yields a classical solution theory for the master equation on the open simplex without imposing any boundary condition. As an application, we recover a Nash equilibrium interpretation of the discrete system in terms of Markov chains on the graph. Our setup is inspired by the computational algorithms for optimal mass transport of [10, 11] and provides a rigorous well-posedness theory for several of the equations derived in [25]. |
| title | Master equations with an individual noise on finite state graphs |
| topic | Analysis of PDEs Optimization and Control 35F21, 35R02, 49L12, 49N80, 49Q22, 60H10, 60H30, 91A43 |
| url | https://arxiv.org/abs/2605.06589 |