UVA Probability Seminar
Wednesdays, 3:30 - 4:30pm
Kerchof 326
 Organizers: Christian Gromoll & Tai Melcher
 Mathematics Recent Seminars Other Seminars Maps & Directions

# 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) Amber Puha (CSU San Marcos)
* please note time, date, and/or location change

# 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
 Mathematics Recent Seminars Other Seminars Maps & Directions