Probability Archives for Fall 2016 to Spring 2017

Branching random walks and related models

When: Wed, November 25, 2015 - 3:00pm
Where: MATH 1308
Speaker: Boris Vainberg (UNCC) -
Abstract: We'll discuss two topics: 1) Intermittency for branching random walks with heavy tails, 2) Spectal analysis of contact models.

Stochastic quantum integrable systems

When: Mon, December 7, 2015 - 3:00pm
Where: MATH 1308
Speaker: Ivan Corwin (Columbia) -
Abstract: We describe recent work involving stochastic interacting particle systems related to quantum integrable systems. The theory we develop unites all known exactly solvable models in the Kardar-Parisi-Zhang universality class (including also random growth models, directed polymers, random matrices, random tilings, random walks in random environments), as well as provides new examples of solvable systems, and new tools in their analysis.

The Stochastic Boussinesq Equation and applications is Turbolent Convection

When: Wed, February 10, 2016 - 11:00am
Where: Math 3206
Speaker: Nathan Glatt-Holtz (Department of Mathematics Virginia Tech) -
Abstract: Buoyancy driven convection plays a ubiquitous role in physical applications: from cloud formation to large scale oceanic and atmospheric circulation pro- cesses to the internal dynamics of stars. Typically such fluid systems are driven by heat fluxes acting both through the boundaries (i.e. heating from below) and from the bulk (i.e. internal ’volumic’ heating) both of which can have an essentially stochastic nature in practice.

In this talk we will review some recent results on invariant measures for the stochastic Boussinesq equations. These measures may be regarded as canonical objects containing important statistics associated with convection: mean heat transfer, small scale properties of the flow and pattern formation. We discuss ergodicity, uniqueness and singular parameter limits in this class of measures. Connections to the hypo-ellipticity theory of parabolic equations and to Wasser- stein metrics will be highlighted.

On eigenvalue repulsion of Wigner matrices

When: Wed, April 6, 2016 - 11:00am
Where: MATH2400
Speaker: Hoi Nguyen (Ohio State and IAS) -
Abstract: I will address certain repulsion behavior of roots of random polynomials and of eigenvalues of Wigner matrices, and their applications. Among other things, we show a Wegner-type estimate for the number of eigenvalues inside an extremely small interval for quite general matrix ensembles.

Large deviations and Monte Carlo methods for multiscale problems.

When: Wed, April 27, 2016 - 11:00am
Where: MATH3206
Speaker: Kostas Spiliopoulos (Boston University) -
Abstract: Rare events, metastability and Monte Carlo methods for stochastic dynamical systems have been of central scientific interest for many years now. In this talk we focus on rough energy landscapes, that are modeled as multiscale stochastic dynamical systems perturbed by small noise. Large deviations deals with the estimation of rare events. Depending on the type of interaction of the fast scales with the strength of the noise we get different behavior, both for the large deviations and for the corresponding Monte Carlo methods.I will describe how to design asymptotically provably efficient importance sampling schemes for the estimation of associated rare event probabilities, such as exit probabilities,hitting probabilities, hitting times, and expectations of functionals of interest. Standard Monte Carlo methods perform poorly in these kind of problems in the small noise limit. In the presence of multiple scales one faces additional difficulties and straightforward adaptation of importance sampling schemes for standard small noise diffusions will not produce efficient schemes. We resolve this issue and demonstrate the theoretical results by examples and simulation studies. Time permitting we will discuss the case of the second order hypoelliptic Langevin equation as well as construction of efficient Monte Carlo methods for the related problem of escape from a stable equilibrium point.