CS49: Analysis of singular diffusions and related areas

date: 7/15/2025, time: 14:00-15:30, room: ICS 140

Organizer: Adina Oprisan (New Mexico State University)

Chair: Adina Oprisan (New Mexico State University)

A bridge between Random Matrix Theory and Schramm-Loewner Evolutions Theory

Vlad Margarint (University of North Carolina Charlotte)

I will describe a new method that connects two areas of Probability Theory: Schramm-Loewner Evolutions (SLE) and Random Matrix Theory. This machinery opens new avenues of research that allow the use of modern techniques from one field to another. One aspect of this research direction is centered in an interacting particle systems model, namely the Dyson Brownian motion. In the first part of the talk, I will introduce basic ideas of SLE theory, then I will describe the connection with modern techniques from Random Matrix Theory via a first application of our method. I will finish the talk with some open problems that emerge using this newly introduced toolbox. More details available at margarintvlad.com. This is a joint work with A. Campbell and K. Luh.

Bibliography

\([1]\) Campbell, A., Luh, K., Margarint, V. (2025). Rate of Convergence in Multiple SLE using Random Matrix Theory. Random Matrices Theory and Applications.

Nonlinear Stochastic Filtering with Volterra Gaussian noises

Andrea Iannucci (Imperial College London)

We consider a nonlinear filtering problem for a signal–observation system driven by a Volterra-type Gaussian rough path, whose sample paths may exhibit greater roughness than those of Brownian motion. The observation process includes a Volterra-type drift, introducing both memory effects and low regularity in the dynamics. We prove well-posedness of the associated rough differential equations and the Kallianpur–Striebel. We then establish robustenss properties of the filter and study the existence, smoothness, and time regularity of its density using partial Malliavin calculus. Finally, we show that, in the one-dimensional case, the density of the unnormalized filter solves a rough partial differential equation, providing a rough-path analogue of the Zakai equation. This is a joint work with Thomas Cass and Dan Crisan

Asymptotic behavior of the Brownian motion with singular drifts

Adina Oprisan (New Mexico State University)

We discuss the influence of a power law drift on the exit time of the Brownian motion from the half line when the order of the drift at 0 and infinity is different. We find that the first hitting time of 0 is finite almost surely, even when the drift near 0 is positive and unbounded, and study the asymptotic behavior of the process at the time of hitting. We find that the behavior of the process near the origin does not influence the time to hit 0 over long time periods and derive subexponential estimates for the tail distribution of the hitting time. Our results extend to the case in which the drifts are regularly varying functions at 0 respectively infinity. This talk is based on a joint work with Dante DeBlassie and Robert Smits.