Prob/Stat Seminar

Strategic decision making in games provides a unique and entertaining application of probability theory. In this talk, we will discuss case studies rom a TV game show where knowing more correct answers to trivia questions does not necessarily mean you will win more money. We will present the optimal strategy for this game based on two different utility functions and discuss how probability can help game show contestants make strategic decisions. Note: This talk will be accessible to both undergraduate and graduate students.

We make progress towards showing a necessary and sufficient condition for existence and uniqueness of the Multiplicative Stochastic Heat Equation derived in previous work on specific compact Riemannian manifolds is in fact sharp for some compact Riemannian manifolds. The key observation needed to improve the argument in the previous cases is that different heat kernel upper bounds are needed when estimating the Brownian Bridge density accurately when the endpoints are close or far away from each other. Based on ongoing joint work with Robert Neel (Lehigh) and Cheng Ouyang (UIC). Time permitting, I will talk about some open problems and the potential role of geometry in them.

The logarithmic Sobolev inequality has been first introduced and studied by L. Gross on a Euclidean space, and since then it found many applications. In particular, many existing results concern the question on how the constant in the logarithmic Sobolev inequality depends on the geometry of the underlying space. In this talk, I will review recent results on the study of the constant (and its dimension-independence) in the logarithmic Sobolev inequality on sub-Riemannian manifolds. As for many of such setting curvature bounds (or classical Bakry-Emery estimates) are not available, we use different techniques. Examples in both finite and infinite dimensional settings are provided.

We derive an exact formula for the probability that a Brownian path on an annulus does not disconnect the two boundary components of the annulus. The leading asymptotic behavior of this probability is governed by the disconnection exponent obtained by Lawler-Schramm-Werner (2001) using the connection to Schramm-Loewner evolution (SLE). The derivation of our formula is based on this connection and the coupling with Liouville quantum gravity (LQG), from which we can exactly compute the conformal moduli of random annular domains defined by SLE curves. Using a similar approach, we also derive exact formulas for the non-intersection probabilities of independent Brownian paths on an annulus, as well as extend the result to the case of Brownian loop soup. Based on joint work with X. Fu, X. Sun, and Z. Xie, and upcoming work with Z. Xie.

We study the three-dimensional incompressible stochastic Navier-Stokes equations on the torus with multiplicative noise, a model for viscous Newtonian flows subject to state-dependent random perturbations. The L3 setting is critical because it is invariant under the natural scaling of the equations, placing it at the threshold of regularity where meaningful well-posedness results can be expected. However, critical spaces are delicate: nonlinear interactions and stochastic forcing can counterbalance the dissipative structure, leaving no room for any loss of regularity. To manage the difficulties associated with criticality, we rewrite the system as an infinite sum of differential equations and decompose the initial data into regular components. Under natural smallness assumptions on the noise, we show that sufficiently small L^3 initial data yield pathwise unique, probabilistically strong solutions that remain small in L3 and are global-in-time with probability close to one, with this probability increasing as the initial norm decreases. For arbitrary L3 data, we obtain local-in-time solutions. This talk is based on joint works with Igor Kukavica and Mustafa Aydin that extend Kato’s classical Navier-Stokes theory to the stochastic setting.