  Assuming the heat kernel on a doubling Dirichlet metric measure space has a
generalized sub-Gaussian bound, we prove an asymptotically sharp spectral upper
bound on the survival probability of the associated diffusion process. As a
consequence, we can show that the supremum of the mean exit time over all
starting points is finite if and only if the bottom of the spectrum is
positive. Among several applications, we show that the spectral upper bound on
the survival probability implies a bound for the Hot Spots constant for
Riemannian manifolds. Our results apply to many interesting examples including
Carnot groups, sub-Riemannian manifolds with transverse symmetries, and
fractals.
