UNC Probability Seminar

The probability seminar at UNC is organized by the Department of Statistics and Operations Research. There is typically one probabilty seminar every alternate week at UNC (depending on the availability of speakers) while every other week the seminar switches to the Math department at Duke. At UNC the seminar this semester is schedule to be held ateither Hanes 130 or Greenlaw 222 at 4:15 on Thursday. Please see the links on the right for directions on getting here and for other seminars in the triangle area of interest for probabilists. The speakers for this semester and their abstracts are given below.

Note: This is a partial list of speakers for this semester and it will be updated as more speakers confirm their availability.


Fall 2010

  September 2nd: Jim Dai (Georgia Tech)

Title: On the Positive Recurrence of Semimartingale Reflecting Brownian Motions in Three Dimensions

Abstract Let Z be an n-dimensional Brownian motion confined to the non-negative orthant by oblique reflection at the boundary. Such processes arise in applied probability as diffusion approximations for multi-station stochastic processing networks. For dimension n = 2, a simple condition is known to be necessary and sufficient for positive recurrence of Z. The obvious analog of that condition is necessary but not sufficient in three and higher dimensions, where fundamentally new phenomena arise. Building on prior work by Bernard and El Kharroubi (1991) and El Kharroubi et al. (2000, 2002), we provide necessary and sufficient conditions for positive recurrence in dimension n = 3. In this context we find that the fluid-stability criterion of Dupuis and Williams (1994) is not only necessary for positive recurrence but also sufficient; that is, in three dimensions Z is positive recurrent if and only if every path of the associated fluid model is attracted to the origin. I will also discuss recent development for problems in four and higher dimensions. Joint work with Maury Bramson and Michael Harrison.





  September 9th:Rick Durrett (Duke)

Title: An asymptotic theory for randomly-forced discrete heat equations

Location: Duke, Physics 119

Abstract: We consider particle systems that are perturbations of the voter model and show that when space and time are rescaled the system converges to a solution of a reaction diffusion equation in dimensions $d \ge 3$. Combining this result with properties of the PDE and a block construction, we give general, and often asymptotically sharp, conditions for the existence of non-trivial stationary distributions, and for extinction of one type. As applications, we describe the phase diagrams of three systems when the parameters are close to the voter model: (i) a stochastic spatial Lotka-Volterra model of Neuhauser and Pacala, (ii) a model of the evolution of cooperation of Ohtsuki, Hauert, Lieberman, and Nowak, and (iii) a continuous time version of the non-linear voter model of Molofsky, Durrett, Dushoff, Griffeath, and Levin. The first two applications confirm conjectures of Cox and Perkins and Ohtsuki et al.





  September 16th: David Sivakoff (Duke & SAMSI)

Title: Random Site Subgraphs of the Hamming Torus

Location: Duke, Physics 119

Abstract: The critical threshold for the emergence of a giant component in the random site subgraph of a d-dimensional Hamming torus is given by the positive root of a polynomial. This value is distinct from the critical threshold for the random edge subgraph of the Hamming torus. The proof uses an intuitive application of multitype branching processes.





  September 23: Zhen-Qing Chen (Washington)

Title: Stable Processes with Drifts

Location: UNC, Hanes 130

Abstract: A rotationally symmetric stable process in Euclidean space with a drift is a strong Markov process X whose infinitesimal generator L is a fractional Laplacian perturbed by a gradient operator. In this talk, I will present recent results on the sharp estimates on the transition density p_D (t, x, y) of the sub-process of X killed open leaving a bounded open set D. This transition density function p_D(t, x, y) is also the fundamental solution (or heat kernel) of the non-local operator L on D with zero exterior condition.

Based on joint work with P. Kim and R. Song.





  October 28th: Krishna Athreya (Iowa State)

Title: TBA

Location: Duke, Physics 119

Abstract: TBA





  November 18: Steve Shreve (CMU)

Title: TBA

Location: UNC, Hanes 130

Abstract: TBA