Applications of Compact Superharmonic Functions: Path Regularity and Tightness of Capacities

We establish relations between the existence of the L{\mathcal{L}}-superharmonic functions that have compact level sets (L{\mathcal{L}} being the generator of a right Markov process), the path regularity of the process, and the tightness of the induced capacities. We present several examples in infinite dimensional situations, like the case when L{\mathcal{L}} is the Gross–Laplace operator on an abstract Wiener space and a class of measure-valued branching process associated with a nonlinear perturbation of L{\mathcal{L}}.