Purdue Probability Seminar, Fall 2025

Time: Wednesdays from 1:30-2:30pm (EST), unless otherwise noted.

Location:

WALC 3122

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In this talk, I will describe some recent progress on singular stochastic partial differential equations in the setting of non-commutative probability theory - examples will include the stochastic quantization of Fermionic quantum field theories and the setting of free probability. This is based on joint work with Martin Hairer and Martin Peev.

This talk will discuss various notions of boundaries at infinity of random walks in random potentials. Recent results on existence and uniqueness will be presented for a class of models that generalizes first- and last-passage percolation, random walks in random environments, and directed polymers. The resulting boundary structures are related to jointly stationary distributions, geodesic rays, Busemann functions, harmonic functions and the associated Martin boundary, and extremal Gibbs-DLR measures. Based on joint works with Sean Groathouse, Sergazy Nurbavliyev, Firas Rassoul-Agha, and Timo Seppäläinen.

Reinforcement learning (RL) is one of the three main paradigms of machine learning. Traditionally, it has been studied in discrete time and space via Markov decision processes. In 2020, Wang, Zariphopoulou and Zhou [WZZ] formulated a continuous version of RL under the machinery of stochastic control theory and proved results under this formulation. Non-Markovian dynamics in mind, Chakraborty Honnappa and Tindel [CHT] recasted this formulation and introduced rough paths as the "random" driver, instead of Brownian motion. In this talk we will review the [WZZ] construction and interpretation of continuous time RL. Moreover, we will mention the results by [CHT] and present our new results in this direction. This is based on a joint work with Prakash Chakraborty, Harsha Honnappa and Samy Tindel.

This talk will discuss the area process and its asymptotics on partial flag manifolds with blocks of equal size. A Brownian motion on these manifolds can be represented as a matrix valued diffusion obtained from a unitary Brownian motion in block form. To obtain an explicit expression for the characteristic function of the area processes the matrix Jacobi operators on the simplex are introduced and studied. These polynomials simultaneously generalise the Heckman-Opdam polynomials of type BC and the Jacobi polynomials on the simplex. This work generalises the results for the full flag manifold, obtained by Fabrice Baudoin, Nizar Demni, Jing Wang and myself, to a more general class of partial flag manifolds.