Spring 2016
Tuesdays in BNRG 1254 from 3:304:20pm, unless otherwise noted.
Directions to Purdue by air or by car.
Schedule of talks from Fall 2015
Please send comments and suggestions to the seminar organizer, Samy Tindel.
Directions to Purdue by air or by car.
Schedule of talks from Fall 2015
Please send comments and suggestions to the seminar organizer, Samy Tindel.
Date 
Speaker 
Title 


1/19/16 
Fabrice Baudoin
Purdue University 
Stochastic areas and Hopf fibrations
Abstract
We study the stochastic area processes associated to the Brownian motions on the complex symmetric spaces ${\scriptsize CP^n}$ and ${\scriptsize CH^n}$. Limiting laws are obtained and connections to the geometry of Hopf fibrations are made. This is joint work with Jing Wang (UIUC).


1/26/16 
Yu Gu
Stanford University 
Scaling limits of fluctuations in stochastic homogenization
Abstract
Equations with small scales abound in physics and applied science. When the coefficients vary on microscopic scales, sometimes we expect the local fluctuations to average out and the coefficients have some equivalent homogeneity on large scales. The goal of homogenization is to find an equivalent homogeneous media to replace the heterogeneous one, without much effects on the solutions. In this talk, I will try to explain some probabilistic approaches we use to obtain the first order fluctuations in stochastic homogenization. The main ingredients include the invariance principle of a diffusion in random environment, the HelfferSjöstrand covariance representation and the Stein's method. This is a joint work with JeanChristophe Mourrat (ENS Lyon).


2/2/16 
Xiaoqin Guo
Purdue University 
Einstein relation and steady states for the random conductance model
Abstract
The Einstein relation describes the relation between the response of a system to a perturbation and its diffusivity at equilibrium. It states that the derivative (with respect to the strength of the perturbation) of the velocity equals the diffusivity. In this talk we consider random walks in iid random conductances on the integer lattice ${\scriptsize Z^d}$. We show that when ${\scriptsize d\ge 3}$, the invariant measure for the environment viewed from the particle has a first order expansion in terms of the perturbation. The Einstein relation will follow as a corollary of this expansion. This talk is based on a joint work with N. Gantert and J. Nagel.


2/9/16 
Renming Song
UIUC 
Martin boundary and accessibility of discontinuous Feller processes
Abstract
In this talk, I will present some recent results on the
martin boundary of general open sets with respect to discontinuous
Feller processes in metric measure spaces. The minimality of the
Martin kernel associated with a boundary point is equivalent to
the accessibility of the point. This talk is based on some joint papers
with Panki Kim and Zoran Vondracek.


2/16/16 
Eviatar Procaccia
Texas A&M 
Can one hear the shape of a random walk?
Abstract
We consider a Gibbs distribution over random walk paths on the square lattice, proportional to the cardinality of the path's boundary . We show that in the zero temperature limit, the paths condensate around an asymptotic shape. This limit shape is characterized as the minimizer of the functional, mapping open connected subsets of the plain to the sum of their principle eigenvalue and perimeter (with respect to some norm). A prime novel feature of this limit shape is that it is not in the class of Wulff shapes.


2/23/16 
Daesung Kim
Purdue University 
Martingale transforms and the HardyLittlewoodSobolev inequality
Abstract
We give a representation of the fractional integral for symmetric Markovian semigroups as the projection of martingale transforms and prove the HardyLittlewoodSobolev(HLS) inequality based on this representation. The proof rests on a new inequality for a fractional LittlewoodPaley ${\small g}$function.


3/1/16 
David Nualart
University of Kansas 
The Malliavin calculus and its applications (this will be a Math Colloquium, at 4:30pm) Abstract
The purpose of this talk is to present an elementary introduction to the stochastic calculus of variations or Malliavin calculus.
This is a differential calculus on a Gaussian space introduced by Paul Malliavin in the 70s to provide a probabilistic proof of Hormander's hypoellipticity theorem. We will discuss a recent application of Malliavin calculus, combined with Stein's method, to normal approximations.


3/8/16 
Kazumasa Kuwada
Tokyo Institute of Technology 
Maximal Diameter Theorem and Heat Flow (this will be a Math Colloquium, at 4:30pm) Abstract
It is wellknown that Riemannian manifolds with uniformly positive Ricci curvature have BonnetMyers' diameter upper bound, and that the equality is attained if and only if the space is isometric to a sphere. The latter assertion is called S. Y. Cheng's maximal diameter theorem.
In this talk, we give two generalizations of the maximal diameter theorem. The first one is concerned with the BakryEmery Ricci tensor, which is naturally associated with a (possibly nonsymmetric) diffusion generator on manifold. The second one is in the framework of (possibly singular) metric measure spaces with 'uniform positive lower Ricci curvature bound' in a certain sense. In each theorem, a canonical heat flow on these spaces plays a fundamental role.


3/22/16 
Xia Chen
University of Tenessee 
Precise intermittency for the parabolic Anderson equation with a
$(1+1)$dimensional timespace white noise
Abstract
The moment Lyapunov exponent is computed for the solution of the
parabolic Anderson equation with an (1+1)dimensional timespace white
noise. Our main result positively confirms an open problem left by
Bertini and Cancrini (1995). The results for other related settings
will be posted for comparison. Some conjectures and problems will be
given.


3/29/16 
No seminar
Special Event 
We are going to the Math is Key Lecture, by S. James Gates Jr 4:30, Fowler Hall
Abstract coming soon.


4/5/16 
Tal Orenshtein
Université Lyon 1 
Router walks on regular trees
Abstract
Router Walk is a generalization of the Rotor Router model.
Initially there is a router configuration on the graph: at each vertex
there is an infinite sequence of routers pointing to one of the
neighbors. Given a router configuration, the walk is defined
deterministically: at each visit to a vertex the walker follows the
next unused router in its current location and jumps to a neighbor. In
this talk, which is based on a recent joint work with Sebastian Mueller, we
shall discuss the problem of recurrence and transience of router walks
on regular trees with i.i.d. router sequences.


4/12/16 
AbdolReza Mansouri
Queen's University 
Concentration inequalities via Malliavin calculus: An extension of the NourdinViens approach
Abstract
In their recent seminal work (Density estimates and concentration inequalities via Malliavin calculus, EJP, 2009), Nourdin and Viens have derived novel concentration inequalities using the Malliavin calculus. In this talk, which is based on joint work with John Treilhard (Yale), we present an extension to the original NourdinViens approach which allows us to derive further concentration inequalities. We provide an application to Fractional Brownian Motion.


4/19/16 
Jon Peterson
Purdue University 
Functional limit laws for recurrent excited random walks
Abstract
Excited random walks (also called cookie random walks) are model for selfinteracting random motion where the transition probabilities are dependent on the local time at the current location. While selfinteracting random walks are typically very difficult to study, many results for (onedimensional) excited random walks are remarkably explicit. In particular, one can easily (by hand) calculate a parameter of the model that will determine many features of the random walk: recurrence/transience, nonzero limiting speed, limiting distributions and more.
In this talk I will prove functional limit laws for onedimensional excited random walks that are recurrent. For certain values of the parameters in the model the random walks under diffusive scaling converge to a Brownian motion perturbed at its extremum. This was known previously for the case of excited random walks with boundedly many cookies per site, but we are able to generalize this to excited random walks with periodic cookie stacks. In this more general case, it is much less clear why perturbed Brownian motion should be the correct scaling limit.
This is joint work with Elena Kosygina.


4/26/16 
Alex Fribergh
Université de Montreal 
The ant in 'a' labyrinth
Abstract
One of the most famous open problem in random walks in random environments is to understand the behavior of a simple random walk on a critical percolation cluster, a model known as 'the ant in the labyrinth'. I will present new results on the scaling limit for the simple random walk on the critical branching random walk in high dimension. In the light of lace expansion, we believe that the limiting behavior of this model should be universal for simple random walks on critical structures in high dimensions. This is a joint work with G. Ben Arous and M. Cabezas.
