Viscosity Solutions in Non-commutative Variables

TL;DR

This paper extends viscosity solutions and stochastic control problems to non-commutative variables, using operator algebras and free probability to connect mean field games with random matrix theory.

Contribution

It introduces a framework for viscosity solutions in non-commutative variables, incorporating both classical and free stochastic processes, bridging mean field games and operator algebra theory.

Findings

Developed viscosity solutions for Hamilton-Jacobi equations in non-commutative variables.

Established connections between mean field games and free probability theory.

Integrated classical and free noise models in the non-commutative setting.

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.