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arXiv · 1108.3191

The Brownian traveller on manifolds

Martin Kolb·David Krejcirik
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Abstract

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.

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Martin Kolb, David Krejcirik. 2011-08-16. The Brownian traveller on manifolds. https://arxiv.org/abs/1108.3191

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