In Spring 2022, we meet on Monday afternoons 12pm (central)/23 pm (eastern) on Zoom. Please contact me for the Zoom link if you are interested to join the seminar.
Talk Schedule
February 21

Speaker: Li Chen (Louisiana State University)
Title: Dirichlet Fractional Gaussian fields on the Sierpinski gasket Abstract: In this talk, we discuss the Dirichlet fractional Gaussian fields on the Sierpinski gasket. We show that they are limits of fractional discrete Gaussian fields defined on the sequence of canonical approximating graphs. This is a joint work with Fabrice Baudoin (University of Connecticut). 
March 21

Speaker: George Yin (University of Connecticut)
Title: Stochastic Kolmogorov Systems: Some Recent Progress Abstract: We present some of our recent work on stochastic Kolmogorov systems. The motivation stems from dealing with important issues of ecological and biological systems. Focusing on environmental noise, we will address such fundamental questions: what are the minimal conditions for longterm persistence of a population, or longterm coexistence of interacting species. [The talk reports some of our joint work with D.H. Nguyen, N.T. Dieu, N.H. Du, and N.N Nguyen.] 
April 4

Speaker: Erkan Nane (Auburn University)
Title: Moments of fractional stochastic heat equations in a bounded domain Abstract: We consider the fractional stochastic heat type equation with nonnegative bounded initial condition, and with noise term that behaves in space like the Riesz kernel and is possibly correlated in time, in the unit open ball centered at the origin in \(\mathbb{R}^d\). When the noise term is white in time, we establish a change in the growth of the solution of these equations depending on the noise level. On the other hand when the noise term behaves in time like the fractional Brownian motion with index \(H\in (1/2,1)\), we also derive explicit bounds leading to a wellknown intermittency property. 
April 11

Speaker: Le Chen (Auburn University)
Title: Exact solvability and moment asymptotics of SPDEs with timeindependent noise Abstract: In this talk, I will report a joint work with Raluca Balan and Xia Chen [BCC22] and a followingup work with Nicholas Eisenberg [CE22]. In this line of research, we first study the stochastic wave equation in dimensions \(d\leq 3\), driven by a Gaussian noise \(\dot{W}\) which does not depend on time. We assume that the spatial noise is either white, or the covariance functional of the noise satisfies a scaling property similar to the Riesz kernel. The solution is interpreted in the Skorohod sense using Malliavin calculus. We obtain the exact asymptotic behaviour of the \(p\)th moment of the solution when either the time or \(p\) goes to infinity. For the critical case, namely, when \(d=3\) and the spatial noise is white, we obtain the exact transition time for the second moment to be finite. The main obstacle for this work is the lack of the FeynmanKac representation for the moment, which has been overcome by a careful analysis of the Wiener chaos expansion. Our methods turn out to be very general and can be applied to a broad class of SPDEs, which include stochastic heat and wave equations as two special cases. [BCC22] Raluca M. Balan, Le Chen, and Xia Chen. “Exact asymptotics of the stochastic wave equation with timeindependent noise”. In: to appear in Ann. Inst. Henri Poincaré Probab. Stat., preprint at arXiv:2007.10203 (2022). [CE22] Le Chen and Nicholas Eisenberg. “Interpolating the Stochastic Heat and Wave Equations with Timeindependent Noise: Solvability and Exact Asymptotics”. In: to appear in Stoch. Partial Differ. Equ. Anal. Comput., preprint at arXiv:2108.11473 (2022). 
April 18

Speaker: Fabrice Baudoin (University of Connecticut)
Title: Asymptotic windings of the block determinants of a unitary Brownian motion and related diffusions Abstract: We study several matrix diffusion processes constructed from a unitary Brownian motion. In particular, we use the Stiefel fibration to lift the Brownian motion of the complex Grassmannian to the complex Stiefel manifold and deduce a skewproduct decomposition of the Stiefel Brownian motion. As an application, we prove asymptotic laws for the determinants of the block entries of the unitary Brownian motion. This is a joint work with Jing Wang (Purdue University). 
April 25

Speaker: Maria Gordina (University of Connecticut)
Title: Limit laws for a hypoelliptic diffusions Abstract: In this talk we will consider several classical problems for hypoelliptic diffusions: the small ball problem (SBP), Chung's laws of iterated logarithm (LIL) , and finding the OnsagerMachlup functional. Namely we will look at hypoelliptic Brownian motion on the Heisenberg group and a Kolmogorov diffusion for the SBP and LIL, and the OnsagerMachlup functional for hypoelliptic Brownian motion in the Heisenberg group. One of these processes is not Gaussian, but it has a spacetime scaling property. Kolmogorov diffusion does not have this property, but it is Gaussian, so one should use a different approach. The OnsagerMachlup functional is used to describe the dynamics of a continuous stochastic process, and it is closely related to the SBP and LIL. Unlike in the Riemannian case we do not rely on the tools from differential geometry such as comparison theorems or curvature bounds as these are not easily available in the subRiemannian setting. The talk is based on the joint work with Marco Carfagnini. 