The Probability seminar is held on Tuesday 4:05pm - 5:05pm in Ayres Hall.

Spring 2024:

 

Title: Constrained Hamiltonian Monte Carlo for conditional diffusion, Part III.


Title: Constrained Hamiltonian Monte Carlo for conditional diffusion, Part II.


Title: Constrained Hamiltonian Monte Carlo for conditional diffusion, Part I.

Abstract: The goal of this talk is the exposition and exploration of probability theoretical foundations for a variation of constrained path space sampling as laid out by Beskos et al. (Stoch. Process. Their Appl. 121(10):2201–2230, 2011) which avoids the requirement of closed form expressions for a large class of target processes. To motivate the problem in a simpler setting, I will outline some probability theoretical aspects of constrained Hamiltonian Monte Carlo for sampling a class of singular conditional distributions on finite dimensional state spaces arising from equality constraints with smooth zero locus. Then, I will describe some special considerations which must be made in the infinite dimensional path space setting of an analogous class of conditional diffusions.


Title: Phase transition issue of focusing Gibbs measures for Schrödinger-wave systems.

Abstract: We study the phase transition phenomenon of the singular Gibbs measure associated with the Schrödinger-wave systems, initiated by Lebowitz, Rose, and Speer (1988). In the three-dimensional case, this problem turns out to be critical, exhibiting a phase transition according to the size of the coupling constant. In the weakly coupling region, the Gibbs measure can be constructed as a probability measure, which is singular with respect to the Gaussian free field. On the other hand, in the strong coupling case, the Gibbs measure can not be normalized as a probability measure. In particular, the finite-dimensional truncated Gibbs measures have no weak limit, even up to a subsequence.


Title:  Optimal minimax rate of learning interaction kernels.

Abstract: Nonparametric estimation of nonlocal interaction kernels is crucial in various applications involving interacting particle systems. A central question is whether the optimal minimax rate of convergence for this problem aligns with the rate in classical nonparametric regression. Our study confirms this alignment for systems with a finite number of particles. We introduce a tamed least squares estimator (tLSE) that attains the optimal convergence rate for a broad class of exchangeable distributions. The tLSE bridges the smallest eigenvalue of random matrices and Sobolev embedding. The upper minimax rate relies on nonasymptotic estimates for the left tail probability of the smallest eigenvalue of the normal matrix. The lower minimax rate is derived using the Fano-Tsybakov hypothesis testing method. 


Title: On a class of stochastic fractional heat equations.

Abstract: For the fractional heat equation $\frac{\partial}{\partial t} u(t,x) = -(-\Delta)^{\frac{\alpha}{2}}u(t,x)+  u(t,x)\dot W(t,x)$ where  the covariance function of the Gaussian noise $\dot W$ is defined by the heat kernel, we establish Feynman-Kac formulae for both  Stratonovich and Skorohod solutions, along with their respective moments. In particular, we prove that  $d<2+\alpha$ is a sufficient and necessary condition for the equation to have a unique square-integrable mild Skorohod solution.  One motivation lies in the occurrence of this equation in the study of a random walk in  random environment which is generated by a field of independent random walks starting from a Poisson field.  This is joint work with Meng Wang and Wangjun Yuan. 


Title: Hyperbolic Anderson equations with time-independent Gaussian noise: Stratonovich regime - Part II.


Title: Hyperbolic Anderson equations with time-independent Gaussian noise: Stratonovich regime - Part I.

Abstract: In this talk, we shall report some progress made for the hyperbolic Anderson model in Stratonovich regime. Our approach provides new ingridient on representation and computation for Stratanovich moments. The work is based on a collaborative project with Yaozhong Hu.



Fall 2023


Title: On Stein's Method for Infinitely Divisible Laws.

Abstract: Stein's method is a well-known methodology to quantify the proximity of probability measures, which has been mainly developed in the Gaussian and Poisson settings. In this talk, I will present some relatively recent unifying results on Stein's method for infinitely divisible laws. This talk is based on joint works with Benjamin Arras. 


Title: On the computation of limits of stationary measures of asymmetric simple exclusion processes with open boundary.

Abstract: Recently, several advances have been made on limits of stationary measures of asymmetric simple exclusion processes with open boundary (open ASEP). This talk will present an overview on the method underlying many of these developments. The starting point of the method is a new representation of the probability generating function of stationary measures of open ASEP in terms of the so-called Askey-Wilson Markov processes introduced by Bryc and Wesolowski (2010, 2017). Compared to the well-known Derrida’s matrix ansatz from the 1990s, the advantage of this new representation is that for the Laplace transform of stationary measures of open ASEP it becomes much easier to compute its asymptotics. Computing the limit of Laplace transform then leads to the limit of stationary measures. This conceptually straightforward computation, however, consists of two further delicate steps. One is the computation of the so-called tangent process of Askey-Wilson process, and the other is the derivation of a duality formula of Laplace transforms of certain Markov processes. The talk shall present how the method has been applied and adapted, particularly regarding these two steps, in various setups of open ASEP, and comment on other potential applications. Based on a series of joint works with Wlodek Bryc, Alexey Kuznetsov, Jacek Wesolowski and Zongrui Yang. 


Title: Ergodicity of the generalized Langevin equation with singular potentials.

Abstract: We consider a system of interacting particles governed by the generalized Langevin equation (GLE) in the presence of external confining potentials, singular repulsive forces, as well as memory kernels. Using a Mori-Zwanzig approach, we represent the GLE by a system of SDEs and investigate the large-time stability of the Markov dynamics. Under a general set of conditions on the nonlinearities, we demonstrate that the Markov GLE is always exponentially attractive toward the unique invariant Gibbs measure. Important examples of singular potentials in our results include the Lennard-Jones function and the Coulomb function. This talk is based on a joint work with Manh Hong Duong.


Title:  Short Memory Limit for long-time Statistic in a Stochastic Coleman-Gurtin Model of Heat Conduction.

Abstract: We discuss the stochastic Coleman-Gurtin model, a class of semi-linear differential Volterra equations with polynomial-type potentials that incorporate memory effects while subjected to random perturbations via an additive Gaussian noise. In the first part of the talk, we describe the large-time statistically steady states of the equation. Then, in the short memory limit, i.e., as the memory collapses to a Dirac mass, we establish the system's finite time and time asymptotic validity toward the classical reaction-diffusion equation. The arguments require estimates of suitable Wassernstein distances, which must be uniform regarding the memory. This talk is based on a joint work with Nathan Glatt-Holtz and Vincent Martinez.