Probability Archives for Fall 2013 to Spring 2014

Strong and weak solutions to stochastic Landau-Lifshitz and applications to ferromagnetism

When: Wed, September 25, 2013 - 2:00pm
Where: 1313
Speaker: Zdzislaw Brzezniak (University of York) -
Abstract: I will speak about the existence of weak solutions (and the existence
and uniqueness of strong solutions)
to the stochastic Landau-Lifshitz equations for multi
(and one)-dimensional
spatial domains. I will also describe the corresponding Large Deviations
principle and it's applications to the ferromagnetic wire.

The talk is based on a joint works with B. Goldys
and T. Jegaraj.

From averaging to homogenization in cellular flows

When: Wed, October 9, 2013 - 11:00am
Where: Math0407
Speaker: Zsolt Pajor-Gyulai (UMD) -
Abstract: Let v be an incompressible periodic vector field on the plane of amplitude A without unbounded flow lines. We assume that the plane is divided periodically into cells, with the motion along v consisting of rotation along the closed flow lines inside each cell.

If we consider an elliptic Dirichlet problem on this background in a domain of size of order R, then for fixed A and large R homogenization methods yield the asymptotic, while for fixed R and large A the classical averaging theory applies.

We obtain a limit theorem for the corresponding diffusion process that will imply the asymptotics of the solution to the corresponding PDE, encompassing both the averaging and the homogenization regimes, as well as the regime where the transition between the averaging and homogenization occurs.

Quasi reflected Brownian motions and Lorentz processes

When: Wed, October 16, 2013 - 11:00am
Where: MATH1308
Speaker: Peter Nandori (Courant Institute) -

Scaling limits for the exit problem for conditioned diffusions via Hamilton-Jacobi equations.

When: Wed, October 30, 2013 - 11:00am
Where: MATH1308
Speaker: Yuri Bakhtin (Georgia Tech) -
Abstract: The classical Freidlin--Wentzell theory on small random
perturbations of dynamical systems operates mainly at the level of
large deviation estimates. It would be interesting and useful to
supplement those with central limit theorem type results. We are able
to describe a class of situations where a Gaussian scaling limit for
the exit point of conditioned diffusions holds. Our main tools are
Doob's h-transform and new gradient estimates for Hamilton--Jacobi
equations. Joint work with Andrzej Swiech.

Stochastic evolutions in ecological models and their scalings

When: Wed, December 4, 2013 - 11:00am
Where: 1108.0
Speaker: Oleksandr Kutovyi (MIT) -
Abstract: We analyze an interacting particle system with a Markov evolution of birth-and-death type in continuum. The corresponding Vlasov-type scaling, which is based on a proper scaling of corresponding Markov generators and has an algorithmic realization in terms of related hierarchical chains of correlation functions equations is studied. The existence of rescaled and limiting evolutions of correlation functions as well as convergence to the limiting evolution are shown.

Recent advances in First Passage Percolation.

When: Wed, December 18, 2013 - 2:00pm
Where: Math 3206
Speaker: Antonio Auffinger (U. Chicago) -
Abstract: First-passage percolation is a model of a random metric on a infinite network. It deals with a collection of points which can be reached within a given time from a fixed starting point, when the network of roads is given, but the passage times of the road are random. It was introduced back in the 60's but most of its fundamental questions are still open. In this talk, we will overview some recent advances in this model focusing on the existence, fluctuation and geometry of its geodesics. Based on joint works with M. Damron and J. Hanson.

Intertwinings, wave equations and growth models

When: Thu, January 9, 2014 - 2:00pm
Where: Math 3206
Speaker: Mykhaylo Shkolnikov (UC Berkeley) -
Abstract: We will discuss a general theory of intertwined diffusion processes of any dimension. Intertwined processes arise in many different contexts in probability theory, most notably in the study of random matrices, random polymers and path decompositions of Brownian motion. Recently, they turned out to be also closely related to hyperbolic partial differential equations, symmetric polynomials and the corresponding random growth models. The talk will be devoted to these recent developments which also shed new light on some beautiful old examples of intertwinings. Based on joint works with Vadim Gorin and Soumik Pal.

Invariant Densities for Dynamical Systems with Random Switching

When: Wed, February 5, 2014 - 11:00am
Where: MTH 2300
Speaker: Tobias Hurth (Georgia Tech) -
Abstract: Consider a finite family of smooth vector fields on a finite-dimensional smooth manifold M. For a fixed starting point on M and an initial vector field, we follow the solution trajectory of the corresponding initial-value problem for an exponentially distributed random time. Then, a new vector field is selected at random from the given family, and we start following the induced trajectory for another exponentially distributed time. Iterating this construction, we obtain a stochastic process X on M. To X, we adjoin a second process A that records the driving vector field at any given time. The two-component process (X,A) is Markov. In the talk, I will present sufficient conditions for uniqueness and absolute continuity of its invariant measure. These consist of a Hoermander-type hypoellipticity condition that holds at a point on M that can be approached from all other points on the manifold. If M is the real line, one can show that the densities of the invariant measure are smooth away from critical points. For analytic vector fields, we can derive the asymptotically dominant term of the densities at critical points. This is joint work with Yuri Bakhtin.

Dynamics of Nanomagnetic Particle Systems

When: Wed, March 19, 2014 - 11:00am
Where: Room 1308
Speaker: Prof. M. Neklyudov (University of Sydney ) -
Abstract: Abstract: The dynamics of nanomagnetic particles is described by the
stochastic Landau-Lifshitz-Gilbert (SLLG) equation.

In the first part of the talk we will discuss the long time behaviour of the finite-dimensional SLLG equation. Firstly, we explain how statistical mechanics argument defines the form of the noise of the equation. Then we will consider different approximations of the equation such as structure preserving discretisation and penalisation approximation. We discuss the convergence of approximations and their consistency with the long time behaviour of the system.

In the second part of the talk we will look at the infinite dimensional case. Firstly, we present a numerical scheme convergent to the solution of SLLG equation. Then we show some numerical results and discuss open problems, such as existence of invariant measure, existence of solution in the case of space-time white noise, etc. In particular we will explain why Krylov-Bogoliubov Theorem is not directly applicable to the proof of the existence of invariant measure even in the case of the coloured noise. In the end we will present certain transformation of SLLG equation which allows to represent the noise as the sum of additive noise and energy conservative noise.

Computational examples will be reported to illustrate the theory.

The talk is based on the recently published book (jointly with L. Banas, Z. Brzezniak, A. Prohl) and on the work in progress of the author.

An Overview of an Analytic Approach for Branching Processes

When: Wed, April 9, 2014 - 11:00am
Where: MTH 1308
Speaker: Mark Daniel Ward (Purdue University, (Sabbatical at UMD)) -
Abstract: One approach to solving some questions in probability
theory--especially questions about asymptotic properties of algorithms and
data structures--is to take an analytic approach, i.e., to utilize
complex-valued methods of attack. These methods are especially useful with
several types of branching processes, leader election algorithms, pattern
matching in trees, data compression, etc. This talk will focus on some of
the highlights of this approach. I endeavor to keep it at a level that is
accessible for graduate students.

Stochastic transport in periodic channels.

When: Wed, May 7, 2014 - 11:00am
Where: MATH1308
Speaker: Leonid Koralov (UMD) -
Abstract: We'll discuss stochastic transport in periodic channels. The problems to be considered are motivated by the study of Brownian motors (mechanisms that create directed motion out of fluctuations of
an external field), which are of interest in biological and industrial applications. We'll study the effective properties of the flow with respect to parameters that describe the geometry of the channel.

Asymptotic behavior and summability improving in non-autonomous Kolmogorov equations

When: Wed, May 14, 2014 - 11:00am
Where: MATH 1308
Speaker: Alessandra Lunardi (University of Parma) -