October 24 Dmitry Dolgopyat (UMD)
Motion in a random force field
October 30 3PM (Note special time) Yuri Bakhtin (Georgia Tech)
Noisy heteroclinic networks: small noise asymptotics
I will start with the deterministic dynamics generated by a vector
field that has several unstable critical points connected by
heteroclinic orbits. A perturbation of this system by white noise will be considered. I
will study the limit of the resulting stochastic system in distribution (under appropriate
time rescaling) as the noise intensity vanishes. It is possible to describe the limiting
process in detail, and, in particular, interesting non-Markov effects arise. There are
situations where this result provides more precise exit asymptotics than the classical
Wentzell-Freidlin theory.
November 14 Leonid Koralov (UMD)
Mathematical model for polymers
We consider the distribution of a long homopolymer in a potential field. The typical
shape of the homopolymer depends on the size of the
potential. We shall discuss various phenomena when the
temperature parameter is at or near the critical value. Mathematically, the problems
concern the study of spectral properties
of certain elliptic differential operators.
November 30 (NOTE SPECIAL DATE) 11 AM Elena Kosygina (CUNY)
Excited random walks on integers
We consider a class of non-markovian nearest neighbor random walk
models on one dimensional integer lattice in deterministic and random
i.i.d. environments. Transition probability from x to the nearest neighbors
depends on the environment and on the number of previous visits to x. We
discuss the recurrence and transience, the law of large numbers, positivity
of speed, and the central limit theorem. The proofs are based on the
connection between these models and branching processes with migration.
This is a joint work with Martin Zerner (University of Tuebingen, Germany).
December 5 4:30 PM (NOTE SPECIAL TIME) Leonid Ryzhik (Chicago)
Reaction-diffusion fronts in random media
Abstract. It is well known that scalar reaction-diffusion equations
in homogeneous media admit special planar front solutions which propagate with a constant
speed. They are stable for a large class of nonlinearities. Similar results have been
established in periodic media by Berestycki and Hamel, and Weinberger. On the other hand,
M. Freidlin has shown for KPP nonlinearities in 1970's that all solutions propagate with
the same speed in random media. I will discuss generalizations of the notion of a traveling
front to special solutions in general heterogeneous and, in particular, to random media, as
well as stability of those fronts when the nonlinearity is of the ignition type. This is a
joint work with J. Nolen, J.-M. Roquejoffe and A. Mellet.
February 6 Sandra Cerrai (Universita di Firenze)
On some asymptotic results for systems of stochastic partial differential
equations
In my talk I try to give an overview of a series of results which I have
obtained in the last years and which concern some asymptotic problems for a
class of PDEs (mainly of reaction-diffusion type) perturbed by a noisy term.
I start from the analysis of their long-time behavior and I investigate the
existence and uniqueness of invariant measures and their ergodic properties.
I also try to explain what is the role played by the stochastic perturbation
in the stabilization of solutions.
Concerning the role played by the stochastic perturbation of PDEs, I will
introduce some results on large deviations, both for the path and for the
invariant measures of solutions, in the case of systems of reaction
diffusion equations of Ginzburg-Landau type, perturbed by a noise of
multiplicative type.
I will conclude my talk describing some other limiting results for SPDEs,
such as the Smoluchowski-Kramers approximation for the stochastic damped
wave equation and the averaging principle.
February 13 Mark Freidlin (UMD)
Smoluchowski-Kramers approximation and
small mass asymptotics for the motion of
a charged particle
I will consider the Langewin equation for the motion of
a small mass particle and the motion of the particle when
the friction term is replaced by the Bio-Savart force.
February 20 Jeremy Quastel (Toronto)
Wiener meets Korteweg and deVries
In joint work with Benedek Valko we noticed
that Gaussian white noise is invariant for periodic KdV.
That this has any meaning is perhaps already a surprise.
We will explain what is means, and how it can be proved
using recent results in pde, and perhaps speculate a little
on the physical meaning.
March 5 Stanislav Molchanov (North Carolina)
On random analytic functions
In 1930's, H. Cramer proposed several models of the "random pseudoprimes" to
explain and predict
some features in the distribution of the true prime numbers.
My talk will present several results on the analytic properties of the
"Cramer's type" random "zeta"-functions and the problem of analytic
continuation for more general classes of random analytic functions.
March 12 Alexei Novikov (Penn State) Correlations of phase transformations in polycrystals.
A polycrystal is a composite of crystalline grains. A crystalline grain is a
periodic lattice of atoms. Each grain may undergo a different
(martensitic) phase transformation, that changes its lattice structure.
We will discuss a one-dimensional model of phase transformations in random
polycrystals and will
show how phase transformations in different grains can be correlated.
April 2 Yuri Suhov (Cambridge)
Localisation in the Anderson tight binding model
with several particles.
The Anderson model (which will celebrate
its 50th anniversary in 2008) is among most popular topics in the
random matrix and operator theory.
However, so far the attention here was concentrated on
single-particle models, where the random external potential
is either IID or has a rapid decay of spatial correlations.
Multi-particle models remained out of scope in mathematical
(and, surprisingly, physical) literature. Recently, Chulaevsky
and Suhov (2007) proposed a version of the multi-scale analysis (MSA)
scheme tackling the multi-particle case. I'll discuss one
of results in this direction: localisation in the lattice (tight
binding)
multi-particle models for large values of the amplitude
(coupling) constant.
April 9 Ilya Goldsheid (Queen Mary College, London) Random walks in random environment on a strip and Lyapunov exponents
April 16 Dmitry Dolgopyat (UMD) Central Limit Theorem for random walk in Markov environment
April 23 Kostya Khanin (Toronto)
Localization and pinning in systems with random potentials
I'll discuss directed polymers and first passage percolation
problem for two types of the random media. The first one
corresponds to "white", or independent, random potentials.
In the second case the random potential is of the product type,
where one term depends on the space variable, while the other
depends only on time. I'll show that two models belong to
different universality classes although scaling exponents for
transversal fluctuations are the same in both cases.