## Department Colloquium

### Friday, November 8th, 2019

**Time:** 2:30 p.m. **Place:** Jeffery Hall 234

**Speaker:** Ari Arapostathis (UT Austin)

**Title:** Lower bounds on the rate of convergence for heavy-tailed driven SDEs motivated by large scale stochastic networks.

**Abstract: ** We show that heavy-tailed Levy noise can have a dramatic effect on the rate of convergence to the invariant distribution in total variation. This rate deteriorates from the usual exponential to strictly polynomial under the presence of heavy-tailed noise. To establish this, we present a method to compute a lower bound on the rate of convergence. We should keep in mind that standard Foster-Lyapunov theory furnishes only an upper bound on this rate. To motivate the study of such systems, we describe how L\'evy driven stochastic differential equations arise in the study of stochastic queueing networks. This happens when the arrival process is heavy-tailed, or the system suffers asymptotically negligible service interruptions. We identify conditions on the parameters in the drift, the Levy measure and/or covariance function which result in subexponential and/or exponential ergodicity, and we show that these conditions are sharp. In addition, we show that for the queueing models described above with no abandonment, the rate of convergence to the stationary distribution in total variation is polynomial, and we provide a sharp quantitative characterization of this rate via matching upper and lower bounds. We conclude by presenting analogous results on convergence in the Wasserstein distance.

This talk is based on joint work with Hassan Hmedi, Guodong Pang and Nikola Sandric.

**Ari Arapostathis** is a professor in the Department of Electrical and Computer Engineering at The University of Texas at Austin, and holds the Texas Atomic Energy Research Foundation Centennial Fellowship in Electrical Engineering. He received his BS from MIT and his PhD from U.C. Berkeley, in 1982. He is a Fellow of the IEEE, and was a past Associate Editor of the IEEE Transactions on Automatic Control and the Journal of Mathematical Systems and Control. His research has been supported by several grants from the National Science Foundation, the Air-Force Office of Scientific Research, the Army Research Office, the Office of Naval Research, DARPA, the Texas Advanced Research/Technology Program, Samsung, and the Lockheed-Martin Corporation.