Probability Archives for Fall 2012 to Spring 2013

Directed random polymers and Macdonald processes

When: Fri, September 14, 2012 - 2:00pm
Where: Math 1308
Speaker: Ivan Corwin Clay (Mathematics Institute and Massachusetts Institute of Technology) -
Abstract: The goal of the talk is to survey recent progress in understanding statistics of certain exactly solvable growth models, particle systems, directed polymers in one space dimension, and stochastic PDEs. A remarkable connection to representation theory and integrable systems is at the heart of Macdonald processes, which provide an overarching theory for this solvability. This is based off of joint work with Alexei Borodin.

The role of noise in ensembles of nanomagnetic particles - Probability

When: Wed, September 26, 2012 - 2:00pm
Where: Math 1313
Speaker: Misha Neklyudov (Tubingen Universitat)
Abstract: The dynamics of nanomagnetic particles is described by the stochastic Landau-Lifshitz-Gilbert equation. We show that, in the case of finite number of spins, the system relaxes exponentially fast to the unique invariant measure which is described by a Boltzmann distribution. Furthermore, we provide Arrhenius type law for the rate of the convergence to the distribution.
Then, we discuss two implicit discretizations to approximate transition functions both, at finite and infinite times: the first scheme is shown to inherit the geometric `unit-length' property of single spins, as well as the Lyapunov structure, and is shown to be geometrically ergodic; moreover, iterates converge strongly with rate for finite times. The second scheme is computationally more efficient since it is linear; it is shown to converge weakly at optimal rate for all finite times. We use a general result of Shardlow and Stuart to then conclude convergence to the invariant measure of the limiting problem for both discretizations.
At last, we discuss the corresponding SPDE and present construction of the solution through finite elements method. The noise is assumed to be of the trace class. Computational examples will be reported to illustrate the theory. This is a joint work with A. Prohl.

Ergodic states in population dynamics

When: Mon, October 1, 2012 - 11:00am
Where: Math 3206
Speaker: Stanislav Molchanov (University of North Carolina at Charlotte) -

The central problem in the population dynamics is the construction of the stationary in space and time interacting particles ensembles. The corresponding models have to include the underlying random motion, the death and the birth (splitting) of the particles. In addition, such stationary states must be stable with respect to small random perturbations (noses) and demonstrate “patches”.
The talk will present two popular models (contact model by Kondratiev – Skorokhod and logistic model by Bolker – Pacala) and several results in mathematical analysis of these models.

Bridging Scales in Molecular Motor Models: From Single to Multiple Motor Systems

When: Wed, October 10, 2012 - 2:00pm
Where: Math 1313
Speaker: Peter Kramer (Rensselaer Polytechnic Institute) -
Recent years have seen increasing attention to the subtle effects on intracellular transport caused when multiple molecular motors bind to a common cargo. We develop and examine a coarse-grained model which resolves the spatial configuration as well as the thermal fluctuations of the molecular motors and the cargo. This intermediate model can accept as inputs either common experimental quantities or the effective single-motor transport characterizations obtained through systematic analysis of detailed molecular motor models. Through stochastic asymptotic reductions, we derive the effective transport
properties of the multiple-motor-cargo complex, and provide analytical explanations for why a cargo bound to two molecular motors moves more slowly at low applied forces but more rapidly at high applied forces than a cargo bound to a single molecular motor. We also discuss how our theoretical framework can help connect in vitro data with in vivo behavior.

Stochastic modeling of the accumulation of passenger and driver mutations in cancer

When: Wed, October 24, 2012 - 2:00pm
Where: Math 1313
Speaker: Cristian Tomasetti (Harvard School of Public Health) -
Important progress has been made in our understanding of cancer thanks to the ever growing
amount of data originated by sequencing technologies. One useful approach for better under-
standing the process of accumulation of somatic mutations in cancer is given by the integration of
mathematical modeling with sequencing data of cancer tissues.
While it has been hypothesized that some of the somatic mutations found in tumors may occur
prior to tumor initiation, there is little experimental or conceptual data on this topic. To gain
insights into this fundamental issue, we formulated a new mathematical model for the evolution
of somatic mutations in which all relevant phases of a tissue's history are considered. The model
provides a way to estimate the in-vivo tissue-speci c somatic mutation rates from the sequencing
data of tumors. The model also makes novel predictions, validated by our empirical ndings, on
the expected number of somatic mutations found in tumors of self-renewing tissues. Importantly,
our analysis indicates that half or more of the somatic mutations in tumors of self-renewing tissues
occur prior to the onset of neoplasia. Furthermore, a general principle for improving the detection
of driver mutations by reducing the amount of \noise" caused by the passenger mutations will be
Our results have substantial implications for the interpretation of the large number of genome-
wide cancer studies now being undertake

Traps for 1 dimensional random walks in random environment

When: Wed, October 31, 2012 - 2:00pm
Where: Math 1313
Speaker: Dmitry Dolgopyat (UMD) -
Abstract: It is well known that one dimensional random walk in a random environment moves slower than a simple random walk due to presence of traps. We review the results about the distributions of traps for random walks in random environment and describe its implications in different asymptotic regimes.

Space-time stationary solutions of Burgers equation with random forcing

When: Wed, November 7, 2012 - 2:00pm
Where: 1313
Speaker: Yuri Bakhtin (Georgia Tech) -
The Burgers equation is a basic hydrodynamic model
describing the evolution of the velocity field of sticky dust
particles. When supplied with random forcing it turns into an
infinite-dimensional random dynamical system that has been studied since late 1990's. The variational approach to Burgers equation allows to study the system by analyzing optimal paths in the random landscape generated by the random force potential. For a long time only compact cases of Burgers dynamics on the circle or a torus were understood
well. In this talk I discuss the Burgers dynamics on the entire real
line with no compactness or periodicity assumption. The main result is the description of the ergodic components and One Force One Solution principle on each component.
Joint work with Eric Cator and Kostya Khanin.

Control and Random Dynamical Systems in Reproducing Kernel Hilbert Spaces

When: Wed, November 28, 2012 - 2:00pm
Where: 1313.0
Speaker: Boumediene Hamzi (Imperial College, London) -
Abstract: See

A characterization of the topological support of the law of a stochastic wave equation

When: Wed, March 13, 2013 - 11:00am
Where: Math 3206
Speaker: Marta Sanz-Sole' (University of Barcelona) -
Abstract: We consider a non-linear stochastic wave equation driven by a Gaussian noise, white in time and with a spatial stationary covariance. Under suitable conditions, it is known that the sample paths of the random field solution are Holder continuous, jointly in time and in space (see Dalang and Sanz-Sole (2009)). In this talk, we will establish a characterization of the topological support of the law of the solution to this equation in Holder norm. This follows from an approximation theorem, in the convergence of probability, for a sequence of evolution equations driven by a family of regularizations of the driving noise. The implementation of the method depends on whether the initial conditions vanishes or not.

Large deviations for random walks in a random environment on a strip

When: Wed, March 27, 2013 - 2:00pm
Where: 1313.0
Speaker: Jonathon Peterson (Purdue University) -
Abstract: We consider large deviations of random walks in a random environment on the strip $\mathbb{Z} \times \{1,2,\ldots,d\}$. Large deviations for random walks in random environments have been studied in a variety of different types of graphs, but only in the one-dimensional nearest-neighbor case is there a known variational formula relating the quenched and averaged rate functions. We will generalize the argument for the one-dimensional case to that of a strip of finite width and prove quenched and averaged large deviation principles with a variational formula relating the two rate functions. The main novelty in our approach will be to use an idea of Furstenburg and Kesten to obtain probabilistic formulas for the limits of certain products of random matrices.