Nonlocal Characterizations of Stochastic Completeness on Complete Riemannian Manifolds

In this paper, we first prove that the following generalized conservation principle holds on complete Riemannian manifolds: for every $0<s<1$ and $t>0$, $$T_t^{(s)}\mathbf{1}+{\int }_0^t{T}_{\tau }^{(s)}{\mathcal{R}}_{s}d\tau =1\text{on }M,$$ where ${\mathcal{R}}_s$ is the intrinsic killing term measuring the loss of mass of the subordinate semigroup, and the condition ${\mathcal{R}}_s\equiv 0$ 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 $${\int }_M(-\mathrm{\Delta }{)}^s\phi d{V}_g=0\mathrm{\forall }\phi \in C_c^{\mathrm{\infty }}(M),$$ 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.