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 $t>0$, $$T_t^{(s)}\mathbf 1+\int_0^t T_\tau^{(s)}\mathcal R_s\,d\tau=1 \qquad\text{on }M,$$ 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.