Nonlocal Characterizations of Stochastic Completeness on Complete Riemannian Manifolds

TL;DR

This paper establishes new nonlocal characterizations of stochastic completeness on complete Riemannian manifolds, linking it to fractional Laplacian properties and semigroup behaviors.

Contribution

It introduces several novel nonlocal criteria for stochastic completeness, including fractional Laplacian identities and solution uniqueness, expanding the understanding of the topic.

Findings

Stochastic completeness is equivalent to zero-mean identities involving the fractional Laplacian.

The paper proves L^p-contractivity and smoothing properties of the subordinate semigroup.

It derives short-time and long-time asymptotics for the fractional heat kernel and jump probabilities.

Abstract

In this paper, we first prove that the following generalized conservation principle holds on complete Riemannian manifolds: for every \(0<s<1\) and \(t>0\), \[ T_t^{(s)}\mathbf 1+\int_0^t T_\tau^{(s)}\mathcal R_s\,d\tau=1 \qquad\text{on }M, \] where \(\mathcal R_s\) is the intrinsic killing term measuring the loss of mass of the subordinate semigroup, and the condition \(\mathcal R_s\equiv0\) is equivalent to the stochastic completeness of \(M\). We then provide several new nonlocal characterizations of stochastic completeness. In particular, we show that stochastic completeness is equivalent to genuinely nonlocal conditions, including the zero-mean identity \[ \int_M (-\Delta)^s\varphi\,dV_g=0 \qquad\forall\,\varphi\in C_c^\infty(M), \] as well as the uniqueness of bounded distributional solutions to the associated fractional elliptic and parabolic equations. We also revisit the equivalent \(L^1\)-core characterization for the generator of the heat semigroup, which plays an important role in our approach. In addition, we prove \(L^p\)-contractivity and smoothing properties of the subordinate semigroup, establish both short-time and long-time asymptotic results for the fractional heat kernel, derive the short-time asymptotics of jump probabilities for the associated Markov process, and study the variational characterization and minimality properties of the fractional resolvent. Together, these results provide a unified analytic and probabilistic framework for the fractional Laplacian on complete Riemannian manifolds.