Spring 2016

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).

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 Helffer-Sjöstrand covariance representation and the Stein's method. This is a joint work with Jean-Christophe Mourrat (ENS Lyon).

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.

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.

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.

We give a representation of the fractional integral for symmetric Markovian semigroups as the projection of martingale transforms and prove the Hardy-Littlewood-Sobolev(HLS) inequality based on this representation. The proof rests on a new inequality for a fractional Littlewood-Paley ${\small g}$-function.

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.

It is well-known that Riemannian manifolds with uniformly positive Ricci curvature have Bonnet-Myers' 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 Bakry-Emery Ricci tensor, which is naturally associated with a (possibly non-symmetric) 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.

The moment Lyapunov exponent is computed for the solution of the parabolic Anderson equation with an (1+1)-dimensional time-space 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.

Abstract coming soon.

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.

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 Nourdin-Viens approach which allows us to derive further concentration inequalities. We provide an application to Fractional Brownian Motion.

Excited random walks (also called cookie random walks) are model for self-interacting random motion where the transition probabilities are dependent on the local time at the current location. While self-interacting random walks are typically very difficult to study, many results for (one-dimensional) 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, non-zero limiting speed, limiting distributions and more. In this talk I will prove functional limit laws for one-dimensional 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.

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.