Journal of Spectral Theory

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Volume 4, Issue 2, 2014, pp. 235–281
DOI: 10.4171/JST/69

Published online: 2014-07-13

The Brownian traveller on manifolds

Martin Kolb[1] and David Krejčiřík[2]

(1) Universität Paderborn, Germany
(2) Czech Technical University in Prague, Czech Republic

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.

Keywords: Heat equation and curvature, Hardy inequality

Kolb Martin, Krejčiřík David: The Brownian traveller on manifolds. J. Spectr. Theory 4 (2014), 235-281. doi: 10.4171/JST/69