In recent joint work with Wilfrid Gangbo, David Jekel, and Kyeongsik Nam, we developed a theory for optimal control problems involving non-commutative variables, inspired by dynamic random matrix models. As matrix dimensions become large, classical descriptions give way to non-commutative analogues, raising new mathematical challenges closely related to free probability theory and quantum statistical mechanics. Our approach uses infinite-dimensional Hamilton-Jacobi-Bellman equations on the space of non-commutative laws, and we developed a theory of viscosity solutions adapted to this non-commutative setting. This framework naturally handles both classical ("common noise") and non-commutative ("free individual noise") randomness. In this talk, I will summarize our progress, highlight challenges unique to the non-commutative setting, and discuss open questions our work may help illuminate.
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