Mathematics and Physics Seminar Series

AnnouncingAnnouncing

A Seminar Presentation

on Friday

February 9, 2018

at 1:45 pm - 2.30 pm in

Maxcy 212

at The University of New Haven

Coupling Brownian motion on the

Heisenberg group and its applications to

gradient estimates

— Phanuel Mariano

PhD Candidate

Department of Mathematics

University of Connecticut

Abstract: Brownian motion is a mathematical model for the random movement of

a particle. It was ﬁrst observed by Robert Brown in 1827 while looking at pollen grains

through a microscope. Shizuo Kakutani, in 1944, was the ﬁrst to show a connection be-

tween Brownian motion and harmonic functions (which are solutions to Laplace’s equa-

tion). We construct a successful non-Markovian coupling (where the coupled processes

meet almost surely) of Brownian motions on the Heisenberg group, which is the simplest

nontrivial example of a sub-Riemannian manifold. Sub-Riemannian manifolds often oc-

cur in the study of constrained systems, such as the motion of a car on a surface. We

use this coupling to furnish purely analytic gradient estimates for harmonic functions on

the Heisenberg group by purely probabilistic means. This talk is based on joint work

with Sayan Banerjee and Maria Gordina

Fur ther Information

For further information, please contact Dr. Yasanthi Kottegoda at the Department of Mathematics and Physics,

Ofﬁce: Maxcy 315, 203-932-1206, YKottegoda@newhaven.edu.