Volume Growth and Recurrence of Fractional Powers of the Laplace--Beltrami Operator

Abstract

Let $M$ be a connected geodesically complete Riemannian manifold without boundary, write $\mu$ for Riemannian volume, and set $V(o,r)=\mu(B(o,r))$ for geodesic balls centered at $o$. For $0<\alpha<2$, let $X^{(\alpha)}$ be the process obtained by subordinating Brownian motion with an independent $\alpha/2$-stable subordinator; its $L^2$-generator is $-(-\Delta)^{\alpha/2}$. We prove that \[ \int^\infty\frac{dt}{V(o,t^{1/\alpha})}=\infty \] implies that \(X^{(\alpha)}\) is recurrent. The proof uses a spectral trace estimate and radial cutoffs on $M\times(0,\infty)$, where the auxiliary measure is $y^{1-\alpha}\,d\mu\,dy$. It proves the sufficient implication in Grigor'yan's Problem~26.