This work focuses on numerical algorithms for approximating the ergodic means for suitable functions of solutions to stochastic differential equations with Markov regime switching. Our main effort is devoted to obtaining the convergence and rates of convergence of the approximation algorithms. The study is carried out by obtaining laws of large numbers and laws of iterated logarithms for numerical approximation to long-run averages of suitable functions of solutions to switching diffusions.

We discuss variational formulation of the infinite body problem. To simplify, we assume particles are indistinguishable, weakly interacting (smooth interacting potential), follow Newton’s law. We identify the state of the evolution as a probability measure. We look at the Hamilton-Jacobi theory. Singularities (i.e. mass condensation) arise from such evolution.

It is exactly the understanding formation and destruction of these singularities that brought the use of metric geometry techniques and probabilistic coupling ideas into play. We study an associated Hamilton-Jacobi equation in the space of probability measures. We discover that these are PDEs in infinite dimensional quotient spaces. In the end, we deliver a well posedness result.

This talk will be part of Arturo's comprehensive exam. The talk is open to everyone.

We present results related to the convergence in distribution of the following sequences of random elements: The approximate derivative of the self-intersection local time in [0,T] for the fractional Brownian motion. The approximate self-intersection local time process, for the fractional Brownian motion. The process of weak symmetric Riemann sums for self-similar Gaussian processes. For each of these sequence of random elements, we apply Malliavin calculus techniques to find conditions under which central and non-central limit hold. In this talk, we will emphasize the problem of determining the asymptotic behavior of the process of approximate self-intersection local time for a fractional Brownian motion of Hurst parameter H. We will show that, depending on the value of H, this process either converges to a Brownian motion under the topology of uniform convergence over compact sets, or to a Hermite process. As part of the proof, we present a new approach for proving tightness for processes, based on Meyer's inequality.

In this talk, we first introduce a method of analyzing the asymptotics of a discrete Toeplitz/Hankel determinant by rewriting it as the product of its continuous counterpart and a Fredholm determinant. Then we will discuss a few probability/physics models as applications of this method.

We consider Bernoulli shifts over a countable group. Over an amenable group, it is known that the entropy of a shift is non-increasing under factor maps. However, in the non-amenable case, no such condition holds. We provide some history and background and proceed to construct entropy increasing factors between Bernoulli shifts over the free group of rank at least two that have the additional property of being monotone.

This will be part of Amanda Wilkens' preliminary examination. Everyone is welcome to attend the talk.

This is seminar will be part of Peter Lewis' Oral examination.

We discuss results regarding regularity of stochastic Burgers-type equations with space-time white noise. First, we present some results regarding Holder regularity and moment bounds for the solution of a stochastic Burgers' equation. To establish such properties, we define a process with a Feynman-Kac representation, prove several results about this process, and show that its Hopf-Cole transformation is the solution to Burgers' equation. Next, we present previous and ongoing work regarding the existence and smoothness of densities for solutions to Burgers-type equations via Malliavin calculus. Time permitting, we discuss attempts at global well-posedness for a general class of SPDEs which contains Burgers' equation.

In the past 15 years, a lot of effort went to studying the KPZ equation and other models sharing the same large scale behaviour (Kardar-Parisi-Zhang universality class). The large scale statistics were first obtained for models in infinite volume, and there have been recent progress for models defined on a finite domain with various types of boundary condition, for instance periodic (cf Baik and Liu's work), Neumann, Dirichlet. In this talk, we will explain how to characterize the probability distribution of the solution to the KPZ equation on the positive reals with Neumann type boundary condition. The Laplace transform of the solution can be computed as a remarkably simple functional of the eigenvalues of a large symmetric real matrix, whose statistics are well-known. The proof is not direct but goes through the study of discrete stochastic integrable systems: the stochastic six-vertex model and the asymmetric simple exclusion process (ASEP) on a half-lattice.

Consider the question of finding the maximum of a random polynomial over a manifold or a graph when the dimension of the space tends is large. The spherical Sherrington-Kirkpatrick (SSK) models is a finite temperature version of this problem when the space is a sphere. The free energy of the SSK model is a random variable, and the interest is the limit as the dimension tends to infinity. The convergence of the free energy to a non-random value in the large dimensional limit is a famous result of Parisi and Talagrand. Here we consider the fluctuations when the polynomial is a quadratic function. We use the random matrix theory and describe the law of the fluctuations at all temperature except for the critical temperature. This is a joint work with Ji Oon Li.

In a traffic flow of cars in a single lane, a slowly moving car will have an effect on the following cars. Totally asymmetric simple exclusion (TASEP) process is a simple probabilistic model of a similar feature. It is one of the fundamental models in the interacting particle systems. Furthermore, it is one of the first models for which the law of the fluctuations of the particle locations after large time is determined. Surprisingly, the law is related to random matrix theory. We will discuss the fluctuations of the TASEP and other models which exhibit a similar random matrix behavior.

Consider a d-dimensional branching particle system in random environment with correlated branching mechanism. We show that the empirical measure in the n-th approximation converges weakly to a measure-valued process that solves a martingale problem. Then we show that the process has a density w.r.t. the Lebesgue measure. This density is the weak solution to a SPDE, that is jointly Holder continuous with exponents 1 in space and 1/2 in time.

This talk will also be a part of Panqiu's preliminary examination. Everyone is welcome to attend the seminar.