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

Abstract

Abstract. We establish relations between the existence of the L-superharmonic functions that have compact level sets (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 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. Mathematics Subject Classification (2000). Primary 60J45, 31C15; Secondary 47D07, 60J40, 60J35.

Potential-theoretical methods in the construction of measure-valued Markov branching processes

“…In the proof of Theorem 4.9, a main step in obtaining the càdlàg property of the trajectories of the measure-valued (X, )-superprocess was the existence of a nest of weak compact sets on the space of measures, produced by a special excessive function having compact level sets. It turns out that this is an efficient way to obtain the path regularity of a Markov process in other infinitedimensional situations too; a presentation of this method and its applications in relevant examples are given in the survey article [6].…”

Measure-valued discrete branching Markov processes

“…(2.2) The following assertions are equivalent (see Proposition 4.1 in [15] and Proposition 2.1.1 in [14]):…”

Potential theory of infinite dimensional Lévy processes

“…(ii) Subsection 3.2 from [9] presents an informal description of constructing compact Lyapunov functions for the infinite dimensional Lévy processes. Then λ(0) = 0, λ is negative definite and continuous on H. Choosing a Hilbert-Schmidt extension E of H as above there exist probability measures ν t , t > 0, on (E, B(E)) such that…”