Motivated by parallels between mean field games and random matrix theory, we develop stochastic optimal control problems and viscosity solutions to Hamilton-Jacobi equa-tions 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 eld games is replaced by a free semicircular Brownian motion, which describes the large-n limit of Brownian motion on the space of self-adjoint matrices. We introduce a classi-cal common noise from mean eld games into the non-commutative setting as well, allowing the problems to combine both classical and non-commutative randomness. Under certain convexity assumptions, we show that the value of the optimal control problems in the non-commutative setting describes the large-n limit of control problems on tuples of self-adjoint matrices. (This talk is based on works in collaboration with D. Jekel, K. Nam and A. Palmer).
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