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.

TBA