The Brownian traveller on manifolds

TL;DR

This paper investigates how intrinsic curvature affects the long-term decay of heat in tubular neighborhoods of geodesics on two-dimensional manifolds, revealing slower decay in positively curved cases and a sharp polynomial decay in negatively curved cases.

Contribution

It establishes a precise decay rate difference for heat semigroups based on manifold curvature, using Hardy inequalities and self-similar analysis.

Findings

Exponential decay is slower in positively curved manifolds.

Sharp polynomial decay is proven for negatively curved manifolds.

Methodology combines Hardy inequalities with self-similar variables.

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.