Positive recurrence of piecewise OrnsteinUhlenbeck processes and common quadratic Lyapunov functions 

Ton Dieker (Georgia Tech)
We study the positive recurrence of piecewise OrnsteinUhlenbeck (OU) diffusion processes, which arise from manyserver queueing systems with phasetype service requirements. These diffusion processes exhibit different behavior in two regions of the state space, corresponding to `overload' and `underload'. The resulting switching behavior cause standard techniques for proving positive recurrence to fail. Using and extending the framework of common quadratic Lyapunov functions from the theory of control, we construct Lyapunov functions for the diffusion approximations corresponding to systems with and without abandonment. Using these Lyapunov functions, we prove that piecewise OU processes have a unique stationary distribution. top of page

ClarkOcone formula and central limit theorems for Brownian local time increments 

David Nualart (U Kansas) *
The purpose of this talk is to discuss some applications of the ClarkOcone representation formula. This formula provides an explicit expression for the stochastic integral representation of functionals of the Brownian motion in terms of the derivative in the sense of Malliavin calculus. We will compare this formula with the classical Ito formula and we will discuss its application to derive a central limit theorem for the modulus of continuity in the space variable of the Brownian local time increments. top of page

A subelliptic Taylor isomorphism theorem in infinite dimensions, I 

Tai Melcher
We will discuss recent results for subelliptic heat kernel measures on a particular class of infinite dimensional Lie groups. We will begin with the background for the Taylor isometry in the flat case and in finite dimensions, as well as discussing how we define Brownian motion on these infinite dimensional Lie groups. Then we will state the Taylor theorem in this setting, which says that there is a unitary mapping from the space of holomorphic functions which are square integrable with respect to the law of the Brownian motion onto the space of derivatives of those functions at the identity. We will discuss elements of the proof time permitting. This is joint work with M. Gordina. top of page

Computing in Probability 

Larry Leemis (William & Mary)*
Statistical languages such as SAS and R have been used for decades to analyze large data sets. Relatively little work has been done, however, on using symbolic languages for the manipulation of random variables. This talk introduces a prototype probability language named APPL (A Probability Programming Language) developed by the speaker and his colleagues, and highlights some recent applications.
top of page

 
