Journal of Spectral Theory

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

The Brownian traveller on manifolds

Martin Kolb[1] and David Krejčiřík[2]

(1) Institut für Mathematik, Universität Paderborn, Warburger Str. 100, 33098, Paderborn, Germany
(2) Department of Mathematics, Czech Technical University in Prague, Trojanova 13, 120 00, Prague 2, Czechia

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