The Brownian traveller on manifolds

Kolb, M. and Krejcirik, D. (2013) The Brownian traveller on manifolds. Journal of Spectral Theory, 4 (2). pp. 235-281. ISSN 1664-0403 doi: 10.4171/JST/69

Abstract/Summary

We study the inuence 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 at case. As the main result, we establish a sharp extra polynomial-type decay for the heat semigroup on negatively curved manifolds comparing to the at 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.

Item Type	Article
URI	https://reading-clone.eprints-hosting.org/id/eprint/34321
Identification Number/DOI	10.4171/JST/69
Refereed	Yes
Divisions	Science > School of Mathematical, Physical and Computational Sciences > Department of Mathematics and Statistics
Publisher	European Mathematical Society
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