Recurrence for Fractional Powers of Diffusion Operators in Terms of Volume-Growth

Abstract

This contribution summarizes the contents of [11]. There are now sharp conditions available for determining the recurrence of the Laplace-Beltrami operator on a complete Riemannian manifold [2][10] which carry over to the setting of strongly local Dirichlet forms [14]; roughly speaking, quadratic volume growth or less implies recurrence. Let L be the negative-definite self-adjoint diffusion operator associated to a strongly local Dirichlet form. The subordinated semigroup of index α, 0 < α < 2 is the strongly continuous submarkovian semigroup generated by -(-L)α/2. The problem of finding recurrence conditions for subordinated diffusion operators has not received much attention — although see [13], [12]. If L is the Laplace-Beltrami operator on a complete Riemannian manifold with non-negative Ricci curvature, the heat kernel lower bounds [3] entail that -(-L)α/2 is recurrent if the volume growth function v(r) satisfies v(r) ≤ cra for large r. We present here a recurrence condition for subordinated diffusion operators in terms of the volume growth function. For one-dimensional diffusion operators our hypotheses are weak, but in the general case we need a strong “radial symmetry” condition. An outcome of our result is that if the derivative \( \dot{v} \)(r) of the volume growth function satisfies \( \dot{v} \)(r) ≤ cr α-1-€ for some 0 < € ≤ α -1 and 1 < α < 2 then -(-L)α/2 is recurrent.