ISRMT seminar: Noncommutative stochastic calculus

Noncommutative or free probability is a branch of mathematics that is useful for describing the large-N limits of many N×N random matrix models. In this theory, classical probability spaces are replaced by pairs (𝒜,𝜏), where 𝒜 is an (operator) algebra and 𝜏 : 𝒜 → ℂ is a certain kind of linear functional. In such a pair, 𝒜 and 𝜏 are conceptualized as the space of ``noncommutative random variables'' and the ``expectation'' functional on 𝒜, respectively. The analogy with classical probability goes much further; indeed, there are notions of distribution, independence, L^p-spaces, conditional expectation, and more. My talk will focus on my recent joint work with David Jekel and Todd Kemp on developing a noncommutative theory of stochastic calculus. I shall frame the discussion around some joint work in progress with Guillaume Cébron and Nicolas Gilliers: applications of the theory to the characterization of large-N limits of solutions to N×N matrix stochastic differential equations.