The equivalence between discrete spectrum and stochastic incompleteness

A complete noncompact Riemannian manifold is stochastically incomplete if its heat kernel $p(x,y,t)$ satisfies

$$\int_M p(x,y,t)\,dy<1$$

for some, equivalently any, $(x,t)\in M\times(0,+\infty)$; otherwise it is stochastically complete. The discrete-spectrum–stochastic-incompleteness conjecture. A complete noncompact Riemannian manifold has discrete spectrum if and only if it is stochastically incomplete. The conjecture proposes an equivalence between a spectral property and a probabilistic property of complete noncompact Riemannian manifolds; the supplied text gives supporting results in particular classes but does not establish the equivalence in general.