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arXiv · 2502.17329

Viscosity Solutions in Non-commutative Variables

Wilfrid Gangbo·David Jekel·Kyeongsik Nam·Aaron Z. Palmer
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Abstract

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.

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Wilfrid Gangbo, David Jekel, Kyeongsik Nam, Aaron Z. Palmer. 2025-02-24. Viscosity Solutions in Non-commutative Variables. https://arxiv.org/abs/2502.17329

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