The Brownian traveller on manifolds

  • Martin Kolb

    Universität Paderborn, Germany
  • David Krejčiřík

    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.

Cite this article

Martin Kolb, David Krejčiřík, The Brownian traveller on manifolds. J. Spectr. Theory 4 (2014), no. 2, pp. 235–281

DOI 10.4171/JST/69