Department of

Mathematics


Seminar Calendar
for Probability Seminar events the next 2 months of Tuesday, September 2, 2014.

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More information on this calendar program is available.
Questions regarding events or the calendar should be directed to Tori Corkery.
     August 2014           September 2014          October 2014    
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Tuesday, September 9, 2014

2:00 pm in Altgeld Hall 347,Tuesday, September 9, 2014

High temperature limits for (1+1)-dimensional directed polymer with heavy-tailed disorder

Partha Dey (UIUC Math)

Abstract: The directed polymer model at intermediate disorder regime was introduced by Alberts-Khanin-Quastel in 2012. It was proved that at inverse temperature $\beta n^{-\gamma}$ with $\gamma=1/4$ the partition function, centered appropriately, converges in distribution and the limit is given in terms of the solution of the stochastic heat equation. This result was obtained under the assumption that the disorder variables posses exponential moments, but its universality was also conjectured under the assumption of six moments. We show that this conjecture is valid and we further extend it by exhibiting the non-universal limiting behavior in the case of less than six moments. We also explain the behavior of the scaling exponent for the log-partition function under different moment assumptions and values of $\gamma$. Based on joint work with Nikos Zygouras.

Tuesday, September 16, 2014

2:00 pm in Altgeld Hall 347,Tuesday, September 16, 2014

Spectral method for half-space kinetic equation

Qin Li (Caltech)

Abstract: Kinetic equation (the Boltzmann equation, the neutron transport equation etc) is known to converge to fluid (the Euler equation or the heat equation) in some certain regimes, but the coupling of the two systems when both regimes coexist is still open. The key is to understand the half-space problem that resembles the boundary layer connecting the two systems. In this talk, I will present a unified proof for the well-posedness of a class of linear half-space equations with general incoming data, and propose a Galerkin method to numerically solve it in a systematic way. The main strategy is to use damping-recovering process for coercivity of the collision term, and the even-odd decomposition for resolving the singularity. Numerical results will be shown to demonstrate the accuracy of the algorithm.

Tuesday, September 23, 2014

2:00 pm in Altgeld Hall 347,Tuesday, September 23, 2014

Parabolic Harnack inequalities for a family of time-dependent non-symmetric Dirichlet forms

Janna Lierl (UIUC Math)

Abstract: Moser iteration is a method that is used to prove mean value estimates, which can then be applied to obtain a parabolic Harnack inequality. Aronson and Serrin applied this technique to a wide class of non-symmetric operators on Euclidean space. On complete Riemannian manifolds, it is known from the works of A. Grigor'yan and L. Saloff-Coste that the parabolic Harnack inequality is equivalent to Poincare inequality together with volume doubling, as well as to two-sided heat kernel bounds. Some part of these results was extended to time-dependent non-symmetric Dirichlet spaces by K.-T. Sturm. I will talk about some recent work on applying parabolic Moser iteration in the context of (non-symmetric) time-dependent forms. This is joint work with L. Saloff-Coste.

Tuesday, September 30, 2014

2:00 pm in Altgeld Hall 347,Tuesday, September 30, 2014

A distributional equality for suprema of spectrally positive Levy processes

Zoran Vondracek (UIUC Math and University of Zagreb)

Abstract: Let Y be a spectrally positive Levy process with strictly negative expectation, C an independent subordinator with finite expectation, and X=Y+C. A curious distributional equality proved some ten years ago states that if the expectation of X is strictly negative, then the overall supremum of Y and the supremum of X just before the first time its new supremum is reached by a jump of C have the same distribution. In this talk I will give an alternative proof of an extension of this result and offer an explanation why it is true.

Wednesday, October 1, 2014

2:00 pm in 443 Altgeld Hall,Wednesday, October 1, 2014

Fundamental solution of kinetic Fokker-Planck operator with anisotropic nonlocal dissipativity

Xicheng Zhang (Wuhan University Math)

Abstract: By using a probability approach (the Malliavin calculus), we prove the existence of smooth fundamental solutions for degenerate kinetic Fokker-Planck equation with anisotropic nonlocal dissipativity, where the dissipative term is the generator of an anisotropic L\'evy process, and the drift term is allowed to be cubic growth.

Tuesday, October 14, 2014

2:00 pm in Altgeld Hall 347,Tuesday, October 14, 2014

Sequential Change Detection for Fractional SDEs

Alexandra Chronopoulou (UIUC IESE)

Abstract: We will consider the problem of sequentially detecting a change in a stochastic process that satisfies a fractional stochastic differential equation with an arbitrary Hurst index, H. For this class of dynamics, we will establish sufficient conditions for the Cumulative Sums (CUSUM) test to be an exact (non-asymptotic) solution to Lorden's minimax optimal stopping problem. In this way, we will extend well-known optimality properties of CUSUM for diffusion processes. The main techniques for these extensions come from fractional calculus and Malliavin calculus.

Tuesday, October 21, 2014

2:00 pm in Altgeld Hall 347,Tuesday, October 21, 2014

Adiabatic and Stable Adiabatic Times

Kyle Bradford (University of Nevada, Reno)

Abstract: This talk will detail the stability of Markov chains. One measure of stability of a time-homogeneous Markov chain is a mixing time. I will define similar measures for special types of time-inhomogeneous Markov chains called the adiabatic and stable adiabatic times. I will discuss the use of these Markov chains and I will discuss how the adiabatic and stable adiabatic times relate to mixing times. This talk is an exploration of linear algebra, analysis and probability.

Tuesday, October 28, 2014

2:00 pm in 347 Altgeld Hall,Tuesday, October 28, 2014

Coalescence in branching trees with application to branching random walks.

Krishna Athreya (Iowa State University)

Abstract: Consider a single type Galton Watson branching tree that is super critical with no extinction. Pick two individuals at random by srswor from the nth generation and trace their lines of descent back in time till they meet.Call that generation number the coalescence time Xn. This talk will address the problem of determining the limit behavior of Xn as n goes to infinity for both supercritical and explosive cases. An application to branching random walks will also be discussed.