Probability Archives for Academic Year 2015

A dynamical system driven by a fast Brownian motion

When: Wed, September 24, 2014 - 2:00pm
Where: MATh1313
Speaker: Mickey Salins (UMD) -
Abstract: We study the limiting process of a stochastic partial differential equation driven by a fast Brownian motion. Interestingly, the limiting process involves a martingale whose quadratic variation is singular with respect to Lebesgue measure. Joint work with Zsolt Pajor-Gyulai.

Stochastic heat equation with rough dependence in space.

When: Wed, October 1, 2014 - 11:00am
Where: MATH3206
Speaker: David Nualart (University of Kansas) -
We will present some new results on the one-dimensional stochastic heat equation driven by a Gaussian noise which is white in time and it has the covariance of a fractional Brownian motion with Hurst parameter H in the space variable, assuming 1/4<H<1/2. For this equation we will discuss the existence of a unique solution if the nonlinear coefficient is Lipschitz. In the linear case, one can derive a Wiener chaos expansion for the solution and establish Feynman-Kac formulas for the moments which lead to sharp lower and upper bounds.

Repeated Out of Sample Fusion in Interval Estimation of Small Tail Probabilities in Food Safety

When: Wed, October 8, 2014 - 2:00pm
Where: MATH1313
Speaker: Benjamin Kedem (UMD) -
Abstract: In food safety and bio-surveillance in many cases it is desired
to estimate the probability that a contaminant such as some
insecticide or pesticide exceeds unsafe very high thresholds.
The probability or chance in question is then very small. To
estimate such a probability we need information about large values.
However, in many cases the data do not contain information about
exceedingly large contamination levels. A solution is suggested
whereby more information about small tail probabilities is
obtained by REPEATED FUSION of the real data with computer generated
"fake" data. The method provides short, yet reliable, interval
estimates from moderately large samples. An illustration is
provided using exposure data from the
National Health and Nutrition Examination Survey (NHANES).

Periodic homogenization with an interface (after M. Hairer and C. Manson)

When: Wed, October 15, 2014 - 2:00pm
Where: MATH1313
Speaker: Zsolt Pajor-Gyulai (UMD) -
Abstract: The diffusive limit of diffusion processes with periodic coefficients is well known to be a Brownian motion with an explicitly computable effective diffusion matrix. The topic of this talk is a situation when the coefficients are periodic only outside of an interface region of finite thickness. It turns out that the limit process is a semimartingale whose bounded variation part is proportional to the local time spent on the interface.

Graph-based Polya's urns

When: Wed, November 12, 2014 - 11:00am
Where: MATH3206
Speaker: Yuri Lima (UMD) -
Abstract: Given a finite connected graph, place a bin at each vertex. Two bins are called a pair if they share an edge of the graph. At discrete times, a ball is added to each pair of bins. In a pair of bins, one of the bins gets the ball with probability proportional to its current number of balls. In this talk we discuss the limiting behavior of the proportion of balls in the bins. The results build on works of Benaim, Benjamini, Chen, Lima, and Lucas.

The supremum of L^2 normalized random holomorphic fields

When: Wed, December 3, 2014 - 11:00am
Where: MATH3206
Speaker: Renjie Feng (UMD) -
Abstract: In this talk, I will define random polynomials and their generalization to complex manifolds. The main result is regarding the landscape of such random holomorphic fields:
I will show that the expected value of the supremum of L^2 normalized random holomorphic fields of degree n on m-dimensional Kahler manifolds are asymptotically of order \sqrt{m\log n}. This is the joint work with S. Zelditch.

A local limit theorem for random walks in balanced environments

When: Wed, December 3, 2014 - 2:00pm
Where: MATH1313
Speaker: Mikko Stenlund (University of Helsinki) -
Abstract: Central limit theorems for random walks in quenched random environments have attracted plenty of attention in the past years. Recently, finer local limit theorems - yielding a Gaussian density multiplied by a highly oscillatory modulating factor - for such models have been obtained. In the one-dimensional nearest-neighbor case with i.i.d. transition probabilities, local limits of uniformly elliptic ballistic walks are now well understood. We complete the picture by proving a similar result for the only recurrent case, the balanced one, in which such a walk is diffusive. The method of proof is entirely different from the ballistic case.


Local Limit Theorem for sums of independent bounded random variables

When: Wed, February 11, 2015 - 11:00am
Where: Math3206
Speaker: Dmitry Dolgopyat (UMD) -
Abstract: We prove Local Limit Theorem for sums of independent random variables
satisfying appropriate tightness conditions. In particular we show that
the Local Limit Theorem holds if summands are uniformly bounded.

Branching models with interaction and in random media

When: Fri, March 6, 2015 - 3:00pm
Where: MATH3206
Speaker: Janos Englander (University of Colorado Boulder) -
Abstract: We will discuss some spatial branching models where the particles interact with each other and some other models where the particles are put into a random environment, affecting the branching.
The main questions concern the large time behavior of these systems.

Moderate Deviation Principles for Stochastic Dynamical Systems

When: Mon, April 6, 2015 - 3:00pm
Where: MATH3206
Speaker: Amarjit Budhiraja (UNC, Chapel Hill) -
Abstract: We present Moderate Deviation Principles for stochastic dynamical systems with small noise that is given in terms of Poisson random measures (PRM) and finite or infinite dimensional Brownian motions (BM). As applications we consider a family of stochastic partial differential equations with jumps and large collections of weakly interacting countable state Markov processes. Proofs rely on certain variational representations for expected values of nonnegative functionals of PRM and BM.

Local thermal equilibrium for certain stochastic models of heat transport

When: Fri, April 10, 2015 - 3:00pm
Where: MATH3206
Speaker: Peter Nandori (Courant) -
Abstract: This talk is about nonequilibrium steady states (NESS) of a class of
stochastic models in which particles exchange energy with their 'local
environments' rather than directly with one another. The physical domain of the system can be a bounded region of R^d for any dimension d. We assume that the temperature at the boundary of the domain is prescribed and is nonconstant, so that the system is forced out of equilibrium. Our main result is local thermal equilibrium in the infinite volume limit. We also prove that the mean energy profile of NESS satisfies Laplace's equation for the prescribed boundary condition.
Our method of proof is duality: by reversing the sample paths of particle
movements, we convert the problem of studying local marginal energy
distributions at x to that of joint hitting distributions of certain random walks starting from x. This is a joint work with Yao Li and Lai-Sang Young.