My research focuses on the ergodic properties (long-time equilibrium behavior) and scaling limits of random dynamical systems with singular interactions. The word "singular" refers to shock-like contact dynamics, such as collisions in a billiard system or local time interactions in a reflected diffusion, that cannot be described simply in terms of a smooth interaction potential. Systems with these types of interactions often defy standard methods of analysis and require new ideas to be dealt with effectively.
Lately, I have been interested in interacting particle models arising in mathematical physics and finance. An important class of models in this area are the rank-based diffusions (also known as "topological diffusions") introduced in the late 90’s and early 2000's by Fernoltz and Karatzas to model equity markets. We can mathematically represent such systems as reflected diffusions in high- (possibly infinite-) dimensional spaces. Understanding the infinite-dimensional versions of these models is not just an academic exercise. Modern data science relies on continuity bounds, convergence rates, and other mathematical results which, roughly speaking, do not depend too much on the dimension of the system in question. Any fact about these infinite-dimensional systems should provide some "dimension-free" insight into the more realistic high-dimensional setting. Work of mine in this area, in collaboration with Amarjit Budhiraja and Sayan Banerjee, includes establishing an SPDE scaling limit for fluctuations of the infinite Atlas model, a prototypical example in the class of rank-based diffusions. More recently, we have obtained results on the strong existence and pathwise uniqueness of solutions to a class of infinite-dimensional singular stochastic differential equations (SDE). For various choices of parameters, this class of SDEs can describe rank-based diffusions, as well as models with proven or conjectured Kardar-Parisi-Zhang scaling behavior, connecting the model to some of the most actively researched areas in modern probability theory.
Prior to this work, my PhD dissertation, written under the supervision of Krzysztof Burdzy, derives stochastic billiard dynamics from physical principles, describing how systems of rigid bodies with random or otherwise non-standard contact dynamics which conserve certain physical quantities may be obtained as the scaling limit of classical rigid body systems with deterministic contact dynamics. (See this cool video on bouncy balls, describing a phenomenon for which the main results of my dissertation provide a statistical mechanical explanation.) Followup work of mine classifies the types of collision dynamics which give rise to ergodic behavior in certain random dynamical systems with locally conserved quantities. Understanding the interaction of probability theory with classic hard problems in dynamical systems continues to be a motivating force in my research.
"Archimedes' principle for a gas of non-spherical particles." With Krzysztof Burdzy and Jacek Małecki. In preparation.
"Strong existence, pathwise uniqueness and chains of collisions in infinite Brownian particle systems." With Sayan Banerjee and Amarjit Budhiraja. Submitted. [arXiv]
"Semi-deterministic processes with applications in random billiards." To appear in Transactions of the AMS. [arXiv]
"Fluctuations of the Atlas model from inhomogeneous stationary profiles." With Sayan Banerjee and Amarjit Budhiraja. Recommended to Annals of Probability, conditional on minor revisions. [arXiv]
"Rough Collisions." Memoirs of the AMS, 304 (2024), no. 1533. [journal] [arXiv]