Prob/Stat Seminar

We consider a family of Grushin-type singularities on surfaces and discuss possible extensions of Brownian motion to the singularity. The results make use of the classical theory of one-dimensional diffusions and their boundary conditions as well as conformal mappings. We will introduce the relevant geometric and probabilistic notions at an introductory level.

Stabilizing the state of a dynamical system to a target point is a classical problem in control theory. However, in many physical problems, only part of the state is known. A commonly used idea is to apply a stabilizing feedback to an estimation of the state, relying on a dynamical system called the observer. The observer learns the state of the system from its dynamics and a measured output. This strategy is known as dynamic output feedback stabilization.

However, usual methods break down when there exist inputs that make the state reconstruction impossible, that is, unobservable. Armed with case studies from quantum physics and system engineering, we develop a strategy of embedding of systems. Considering high-dimensional (sometimes infinite-dimensional) extensions of the dynamical system at hands actually allows the introduction of new and better suited observability techniques that help us resolve this issue.

Many nutritional studies focus on the relationship between individuals' diets and resulting health outcomes. When examining these relationships, researchers are generally interested in individuals' long-term, average intake of nutrients; however, typically only 1-2 days of data are collected. If analyses are performed without accounting for the error in estimating usual intake, estimates will be biased.

In this work, we focus on situations where the association between intake and health outcomes is nonlinear. Since we can only obtain noisy measurements of intake, we propose implementing a nonlinear measurement error model which accounts for the nuisance day-to-day variance when estimating long-term average intake. Estimation of the model is performed using maximum likelihood. Applications will be based on analyses of data from the National Health and Nutrition Examination Survey (NHANES).

When there exists a nice decomposition of a manifold into "leaves" of immersed submanifolds, we call the decomposition a foliation. At every point of the manifold this defines a subset of the tangent space (called a distribution) that is tangent to the leaves. In contrast, if one can find a distribution that is nowhere tangent to a submanifold, we call the structure a contact structure. In a natural sense these objects seem to be opposites; however it is possible to consider a structure called a confoliation that interpolates between them, and to thereby recover interesting results that unify the seemingly disjoint theories. In this talk we will provide an introduction to these ideas; the only prerequisite will be an introductory knowledge of topology.

We study a class of backward doubly stochastic differential equations (BDSDEs) involving martingales with spatial parameters, and show that they provide probabilistic interpretations (Feynman-Kac formulae) for certain semilinear stochastic partial differential equations (SPDEs) with space-time noise. As an application of the Feynman-Kac formulae, random periodic solutions and stationary solutions to certain SPDEs are obtained.

We consider a parabolic Anderson model with Gaussian noise whose space time covariance function is singular. We shall give some information about the moments of the solution when the stochastic heat equation is interpreted in the Skorohod sense. Of special interest is the critical case, for which one observes a blowup of moments for large times. The talk is based on a joint work with Xia Chen, Aurelien Deya and Samy Tindel.

In this talk we will construct the discrete log-gamma line ensemble, which is associated with the log-gamma polymers. This log-gamma line ensemble enjoys a random walk Gibbs resampling invariance that follows from the integrable nature of the inverse gamma polymer model via the geometric RSK correspondence. By exploiting such resampling invariance, we show the tightness of this log-gamma line ensemble under the weak noise scaling. Furthermore, a Gibbs property, as enjoyed by the KPZ line ensemble, holds for all subsequential limits.