

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 semilong 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:
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 measurevalued 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:
FUNDAMENTAL SOLUTION OF KINETIC FOKKERPLANCK 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 FokkerPlanck 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 BeurlingMalliavin 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.

