Viscosity Solutions in Non-commutative Variables
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866929729286701056 |
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| author | Gangbo, Wilfrid Jekel, David Nam, Kyeongsik Palmer, Aaron Z. |
| author_facet | Gangbo, Wilfrid Jekel, David Nam, Kyeongsik Palmer, Aaron Z. |
| contents | Motivated by parallels between mean field games and random matrix theory, we develop stochastic optimal control problems and viscosity solutions to Hamilton-Jacobi equations in the setting of non-commutative variables. Rather than real vectors, the inputs to the equation are tuples of self-adjoint operators from a tracial von Neumann algebra. The individual noise from mean field games is replaced by a free semi-circular Brownian motion, which describes the large-$n$ limit of Brownian motion on the space of self-adjoint matrices. We introduce a classical common noise from mean field games into the non-commutative setting as well, allowing the problems to combine both classical and non-commutative randomness. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_17329 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Viscosity Solutions in Non-commutative Variables Gangbo, Wilfrid Jekel, David Nam, Kyeongsik Palmer, Aaron Z. Analysis of PDEs Operator Algebras Optimization and Control Probability Primary: 49L12, 46L54, Secondary: 46L52, 49L20, 49L25, 60B20 49L12 49L12, 46L54 Motivated by parallels between mean field games and random matrix theory, we develop stochastic optimal control problems and viscosity solutions to Hamilton-Jacobi equations in the setting of non-commutative variables. Rather than real vectors, the inputs to the equation are tuples of self-adjoint operators from a tracial von Neumann algebra. The individual noise from mean field games is replaced by a free semi-circular Brownian motion, which describes the large-$n$ limit of Brownian motion on the space of self-adjoint matrices. We introduce a classical common noise from mean field games into the non-commutative setting as well, allowing the problems to combine both classical and non-commutative randomness. |
| title | Viscosity Solutions in Non-commutative Variables |
| topic | Analysis of PDEs Operator Algebras Optimization and Control Probability Primary: 49L12, 46L54, Secondary: 46L52, 49L20, 49L25, 60B20 49L12 49L12, 46L54 |
| url | https://arxiv.org/abs/2502.17329 |