Probability seminar
 Title: TEMPERED FRACTIONAL CALCULUS Speaker: Mark Meerschaert (Michigan State University) Time: 2:30 p.m., Monday, November 24, 2014 Room: LOW 117 Abstract: Fractional derivatives and integrals are (distributional) convolutions with a power law. Including an exponential term leads to tempered fractional derivatives and integrals. Tempered fractional Brownian motion, the tempered fractional integral or derivative of a Brownian motion, is a new stochastic process whose increments can exhibit semi-long range dependence. The tempered finite difference operator is also useful in time series analysis, where it provides a useful new stochastic model for turbulent velocity data. Tempered stable processes are the limits of random walk models, where the power law probability of long jumps is tempered by an exponential factor. These random walks converge to tempered stable stochastic process limits, whose probability densities solve tempered fractional diffusion equations. Tempered power law waiting times lead to tempered fractional time derivatives. Applications include geophysics and finance.     Title: RANDOM WALKS ON METRIC MEASURE SPACES Speaker: Mathav Murugan (Cornell University) Time: 2:30 p.m., Monday, November 17, 2014 Room: LOW 117 Abstract: A metric space is a length space if the distance between two points equals the infimum of the lengths of curves joining them. For a measured length space, we characterize Gaussian estimates for iterated transition kernel for random walks and parabolic Harnack inequality for solutions of a corresponding discrete time version of heat equation by geometric assumptions (Poincare inequality and Volume doubling property). Such a characterization is well known in the setting of diffusion over Riemannian manifolds (or more generally local Dirichlet spaces) and random walks over graphs (due to the works of A. Grigor'yan, L. Saloff-Coste, K. T. Sturm, T. Delmotte, E. Fabes & D. Stroock). However this random walk over a continuous space raises new difficulties. I will explain some of these difficulties and how to overcome them. We will discuss some motivating examples and applications. This is joint work with Laurent Saloff-Coste (in preparation).     Title: SPECTRAL DYNAMICS OF RANDOM REGULAR GRAPHS AND THE POISSON FREE FIELD Speaker: Soumik Pal (University of Washington) Time: 2:30 p.m., Monday, November 10, 2014 Room: LOW 117 Abstract: A single permutation, seen as union of disjoint cycles, represents a regular graph of degree two. Consider $d$ many independent random permutations and superimpose their graph structures. It is a common model of a random regular (multi-) graph of degree $2d$. We consider the following dynamics. The 'dimension' of each permutation grows by coupled Chinese Restaurant Processes, while in 'time' each permutation evolves according to the random transposition Markov chain. Asymptotically in the size of the graph one observes a remarkable evolution of short cycles and linear eigenvalue statistics in dimension and time. We give a Poisson random surface description in dimension and time of the limiting cycle counts for every $d$. As $d$ grows to infinity, the fluctuation of the limiting cycle counts, across dimension, converges to the Gaussian Free Field. When time is run infinitesimally slowly, this field is preserved by a stationary Gaussian dynamics. The laws of these processes are similar to eigenvalue fluctuations of the minor process of a real symmetric Wigner matrix whose coordinates evolve as i.i.d. stationary stochastic processes. Part of this talk is based on joint work with Toby Johnson and the rest is based on joint work with Shirshendu Ganguly.     Title: A SPATIAL GENERALIZATION OF KINGMAN'S COALESCENT Speaker: Dan Lanoue (University of California, Berkeley) Time: 2:30 p.m., Monday, November 3, 2014 Room: LOW 117 Abstract: The Metric Coalescent (MC) is a measure-valued Markov Process generalizing the classical Kingman Coalescent. We show how the MC arises naturally from a discrete agent based model (the Compulsive Gambler process) of social dynamics and prove an existence and uniqueness theorem extending the MC to the space of all Borel probability measures on any locally compact Polish space. We'll also look in depth at the case of the MC on the unit interval.     Title: OIL AND WATER Speaker: Christopher Hoffman (University of Washington) Time: 2:30 p.m., Monday, October 20, 2014 Room: LOW 117 Abstract: We introduce a two-type internal DLA model which is an example of a non-unary abelian network. Starting with $n$ oil'' and $n$ water'' particles at the origin, the particles diffuse in $\bf Z$ according to the following rule: whenever some site $x \in\bf Z$ has at least 1 oil and at least 1 water particle present, it fires by sending 1 oil particle and 1 water particle each to an independent random neighbor $x \pm 1$. Firing continues until every site has at most one type of particles. We establish the correct order for several statistics of this model and identify the scaling limit under assumption of existence. This is joint work with Elisabetta Candellero, Shirshendu Ganguly and Lionel Levine.     Title: FUNDAMENTAL SOLUTION OF KINETIC FOKKER-PLANCK OPERATOR WITH ANISOTROPIC NONLOCAL DISSIPATIVITY Speaker: Xicheng Zhang (Wuhan University, China) Time: 2:30 p.m., Monday, October 13, 2014 Room: LOW 117 Abstract: By using the probability approach (the Malliavin calculus), we prove the existence of smooth fundamental solutions for degenerate kinetic Fokker-Planck equation with anisotropic nonlocal dissipativity, where the dissipative term is the generator of an anisotropic Levy process, and the drift term is allowed to be cubic growth.     Title: DETERMINANTAL PROBABILITY: SURPRISING RELATIONS Speaker: Russell Lyons (Indiana University) Time: 2:30 p.m., Monday, October 6, 2014 Room: LOW 117 Abstract: (1) For each subset $A$ of the circle with measure $m$, there is a sequence of integers of Beurling-Malliavin density $m$ such that the set of corresponding complex exponentials is complete for $L^2(A)$. (2) Given an infinite graph, simple random walk on each tree in the wired uniform spanning forest is a.s. recurrent. (3) Let $Z$ be the set of zeroes of a random Gaussian power series in the unit disk. Then a.s., the only function in the Bergman space that vanishes on $Z$ is the zero function. (4) In our talk, we explain a theorem that has (1) and (2) as corollaries. We also describe a conjectural extension that has (3) (which is not known) as a corollary. All these depend on determinantal probability measures. All terms above will be explained.     Title: A GAS PARTICLE IN A GRAVITATIONAL FIELD Speaker: Douglas Rizzolo (Univeristy of Washington) Time: 2:30 p.m., Monday, September 29, 2014 Room: LOW 117 Abstract: We will discuss the motion a tagged gas particle in a gravitational field. Our starting point will be a Markov approximation to a Lorentz gas model with variable density. We investigate how the density of the ambient gas impacts the recurrence or transience of the tagged particle. Additionally, we will show that there are multiple scaling regimens leading to nontrivial diffusive limits. This talk is based on joint work with Krzysztof Burdzy.