Integrable stochastic particle systems in one space dimension, like the Totally Asymmetric Simple Exclusion Process (TASEP), have been studied for over 50 years (introduced simultaneously in biology and mathematics in 1969-70). They strike a balance between being simple enough to be mathematically tractable and complicated enough to describe many interesting phenomena. Many natural questions about these systems can be generalized by introducing multiple parameters. The interplay between these parameters is powered by the Yang-Baxter equation, which brings new intriguing results to the well-traveled territory. In particular, I will discuss new Lax-type equations for the Markov semigroups of the TASEP and its relatives. Based on a joint work with Axel Saenz.
This talk will be about the convergence problem in mean field control (MFC), i.e. the challenge of rigorously justifying the convergence of certain "symmetric" $N$-particle stochastic control problems towards their mean field counterparts. On the one hand, this convergence problem is already well-understood from a qualitative perspective, thanks to powerful probabilistic techniques based on compactness. On the other hand, quantitative results (i.e. rates of convergence) are more difficult to obtain, in large part because the value function of the mean field problem (which is also the solution to a certain Hamilton-Jacobi equation on the Wasserstein space) may fail to be $C^1$, even if all the data is smooth. After giving an overview of the convergence problem, I will discuss the results of two recent joint works with Cardaliaguet, Daudin, Delarue, and Souganidis, in which we use some ideas from the theory of viscosity solutions to overcome this lack of regularity and obtain rates of convergence of the $N$-particle value functions towards the value function of the corresponding MFC problem.