In irreducible Dirichlet metric measure spaces, we establish that the corresponding Laplacian has a discrete spectrum and the associated diffusion killed upon exiting connected open sets is irreducible. These results are obtained under minimal assumptions on heat kernels and without requiring regularity of the boudary. The general framework has applications to various problems including small deviations of diffusions and generalized heat content. Examples include Riemannian and sub-Riemannian manifolds, as well as non-smooth and fractal spaces.
This is joint work with Marco Carfagnini and Masha Gordina.