18/07/2013, 16:30 — 17:30 — Sala P3.10, Pavilhão de Matemática
David Krejcirik, Nuclear Physics Institute ASCR, Czech Republic
The Brownian traveller on manifolds
We study the influence of the intrinsic curvature on the large
time behaviour of the heat equation in a tubular neighbourhood of
an unbounded geodesic in a two-dimensional Riemannian manifold.
Since we consider killing boundary conditions, there is always an
exponential-type decay for the heat semigroup. We show that this
exponential-type decay is slower for positively curved manifolds
comparing to the flat case. As the main result, we establish a
sharp extra polynomial-type decay for the heat semigroup on
negatively curved manifolds comparing to the flat case. The proof
employs the existence of Hardy-type inequalities for the Dirichlet
Laplacian in the tubular neighbourhoods on negatively curved
manifolds and the method of self-similar variables and weighted
Sobolev spaces for the heat equation.
- Martin Kolb and David Krejcirik: The Brownian traveller on
manifolds, J. Spectr. Theory, to appear; preprint on arXiv:1108.3191
[math.AP].