CS22: Noncommutative Stochastic Processes

Organizer: Anna Wysoczańska-Kula (University of Wrocław)

Stochastic optimization in free probability

David Jekel

This is based on joint work with Wilfrid Gangbo, Kyeongsik Nam, and Aaron Palmer \([1]\), in which we develop an analog of optimal control theory in the free probability setting, motivated in part by connections with mean field games.

Free probability: Free probability, developed in large part by Voiculescu, is a theory of non-commuting random variables which can be used the describe the large-\(n\) behavior of certain \(n \times n\) random matrices. The idea is that the algebra of classical random variables \(L^\infty(\Omega,P)\) is replaced by a non-commutative von Neumann algebra and the expectation is replaced by a tracial state \(\tau: M \to \mathbb{C}\). The role of the normal distribution from classical probability is played by Wigner’s semicircular distribution, and classical independence is replaced by free independence, a condition motivated by free products of groups. In particular, a \(d\)-variable free Brownian motion is a process \(S_t = (S_t^{(1)},\dots,S_t^{(d)})\) where the increments are freely independent and the different coordinates are also freely independent. A free Brownian motion describes the large-\(n\) behavior of a Brownian motion on the space \(M_n(\mathbb{C})_{\operatorname{sa}}^d\) of \(d\)-tuples of self-adjoint matrices.

Stochastic optimization problems: We consider stochastic optimization problems that using both a free Brownian motion \(S_t\) and a classical Brownian \(W_t\); these processes are thus classical random objects taking values in a tracial von Neumann algebra, equipped with both classical and non-commutative filtrations. We consider processes satisfying \[dX_t = \beta_F dS_t + \beta_C \,dW_t + \alpha_t\,dt,\] where \(\alpha_t\) is a control that we want to choose. The stochastic optimization problem is \[\text{minimize } \mathbb{E} [\int_0^1 L(X_t,\alpha_t)\,dt + g(X_1)] \text{ over } \alpha,\] where the running cost \(L\) and the terminal cost \(g\) are given functions that can be evaluated on non-commuting random variables. The inclusion of both \(S_t\) and \(W_t\) is motivated by mean field game theory (see \([2]\)): The \(S_t\) represents individual noise associated to a single player (motivated by the Brownian motion that affects all the matrix entries), and the \(W_t\) represents a common noise that affects all the players (which in the matrix model would be a random scalar multiple of the identity matrix added to each of the \(d\) matrices).

Results: We study the value of the minimization problem as a function of the initial condition \(X_0\), and show that it satisfies a non-commutative version of the dynamic programming principle and Hamilton–Jacobi equation. A key challenge is that the value of the optimization problem may depend on the choice of ambient von Neumann algebra \(M\) and filtration. We thus take the infimum over all choices of \(M\) as well, and analyze how to join together different given choices of \(M\). However, under certain convexity assumptions, the answer is the same regardless of the choice of \(M\), and in this case we can show that the value of the free stochastic optimal problem gives the large-\(n\) limit of matrix optimal control problems. We remark that special cases of this setup relate closely with free entropy theory and the large deviation principle of Biane, Capitaine, and Guionnet \([3]\).

Bibliography

\([1]\) Wilfrid Gangbo, David Jekel, Kyeongsik Nam, and Aaron Z. Palmer. Viscosity solutions in non-commutative variables. Preprint, 2025, arXiv:2502.17329

\([2]\) René Carmona, François Delarue, et al. Probabilistic theory of mean field games with applications I. Springer, 2018.

\([3]\) Philippe Biane, Mireille Capitaine, and Alice Guionnet. Large deviation bounds for matrix Brownian motion. Inventiones mathematicae, 152, 2003, pp. 433–459.