UVA Probability Seminar
Wednesdays, 3:30 - 4:30pm
Kerchof 326
 Organizers: Christian Gromoll & Tai Melcher
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# Spring 2014

12 Feb Tom Laetsch (U Conn) A random walk through sub-Riemannian geometry Tobias Hurth (Georgia Tech) Invariant densities for dynamical systems with random switching Sneha Subramanian (U Penn) Random zero sets under repeated differentiation of an analytic function Amber Puha (CSU San Marcos) An Unconventional Functional Central Limit Theorem for the Queue Length Process in a Shortest Remaining Processing Time Queue
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# Abstracts

 A random walk through sub-Riemannian geometry Tom Laetsch (U Conn) A sub-Riemannian manifold $M$ is a connected smooth manifold such that the only smooth curves in $M$ which are admissible are those whose tangent vectors at any point are restricted to a subset $\mathcal{H} \subset TM$, called the horizontal distribution. Such spaces have several applications in physics and engineering, as well as in the study of hypo-elliptic operators. In this talk we will construct a family of geometrically natural hypo-elliptic Laplacian-type operators and discuss the trouble with defining one which is canonical. We will also attempt to construct a random walk on a sub-Riemannian manifold which converges weakly to a process whose infinitesimal generator is one of our hypo-elliptic Laplacian-type operators.top of page Invariant densities for dynamical systems with random switching Tobias Hurth (Georgia Tech) Consider a finite family of smooth vector fields on a finite-dimensional smooth manifold $M$. For a fixed starting point on $M$ and an initial vector field, we follow the solution trajectory of the corresponding initial-value problem for an exponentially distributed random time. Then, a new vector field is selected at random from the given family, and we start following the induced trajectory for another exponentially distributed time. Iterating this construction, we obtain a stochastic process $X$ on $M$. To $X$, we adjoin a second process $A$ that records the driving vector field at any given time. The two-component process $(X,A)$ is Markov. In the talk, I will present sufficient conditions for uniqueness and absolute continuity of its invariant measure. These consist of a Hörmander-type hypoellipticity condition that holds at a point on M that can be approached from all other points on the manifold. If $M$ is the real line, one can show that the densities of the invariant measure are smooth away from critical points. For analytic vector fields, we can derive the asymptotically dominant term of the densities at critical points. This is joint work with Yuri Bakhtin.top of page Random zero sets under repeated differentiation of an analytic function Sneha Subramanian (U Penn) We study the result of repeatedly differentiating a random entire function whose zeros are points of a Poisson process of intensity 1 on $\mathbb{R}$. For the first part of the talk, we shall discuss a toy-model based on a random polynomial, whose zeros are i.i.d. Rademacher random variables. For the second, we shall use our results from the first part as a poorism to discuss the main result regarding Poisson process. Based on joint work with Robin Pemantle.top of page An Unconventional Functional Central Limit Theorem for the Queue Length Process in a Shortest Remaining Processing Time Queue Amber Puha (CSU San Marcos) In a shortest remaining processing time (SRPT) queue, the job that requires the least amount of processing time is preemptively served first. One effect of this is that the queue length is small in comparison to the total amount or work in the system (measured in units of processing time). In fact, it is minimized so well that the sequence of queue length processes associated with a sequence of SRPT queues rescaled with standard functional central limit theorem scaling and satisfying standard heavy traffic conditions converges in distribution to the process that is identically equal to zero. This happens despite the fact that under this same regime the rescaled workload processes converge to a non-degenerate reflected Brownian motion. In particular, the queue length process is of smaller order magnitude than the workload process. In the case of processing time distributions that satisfy a rapid variation condition, we implement an alternative, unconventional scaling that leads to a non-trivial limit for the queue length process. This result quantities this order of magnitude difference between queue length and workload processes. We illustrate this result for Weibull processing time distributions.top of page
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