Mathematics and Physics Seminar Series
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,