Construction of Hunt processes by the Lyapunov method and applications to generalized Mehler semigroups

Abstract

It is known that in general, generalized Mehler semigroups defined on a Hilber space H may not correspond to càdlàg (or even càd) Markov processes with values in H endowed with the norm topology. In this paper we deal with the problem of characterizing those generalized Mehler semigroups that do correspond to càdlàg Markov processes, which is highly non-trivial and has remained open for more than a decade. Our approach is to reconsider the càdlàg problem for generalized Mehler semigroups as a particular case of the much broader problem of constructing Hunt (hence càdlàg and quasi-left continuous) processes from a given Markov semigroup. Following this strategy, a consistent part of this work is devoted to prove that starting from a Markov semigroup on a general (possibly non-metrizable) state space, the existence of a suitable Lyapunov function with relatively compact sub/sup-sets in conjunction with a local Feller-type regularity of the resolvent are sufficient to ensure the existence of an associated càdlàg Markov process; if in addition the topology is locally generated by potentials, then the process is in fact Hunt. Other results of fine potential theoretic nature are also pointed out, an important one being the fact that the Hunt property of a process is stable under the change of the topology, as long as it is locally generated by potentials. Based on such general existence results, we derive checkable sufficient conditions for a large class of generalized Mehler semigroups in order to possess an associated Hunt process with values in the original space, in contrast to previous results where an extension of the state space was required; to this end, we first construct explicit Lyapunov functions whose sub-level sets are relatively compact with respect to the (non-metrizable) weak topology, and then we use the above mentioned stability to deduce the Hunt property with respect to the stronger norm topology. As a particular example, we test these conditions on a stochastic heat equation on $L^2\left(D\right)$ whose drift is given by the Dirichlet Laplacian on a bounded domain $D\subset R^d$, driven by a (non-diagonal) Lévy noise whose characteristic exponent is not necessarily Sazonov continuous; in this case, we construct the corresponding Mehler semigroup and we show that it is the transition function of a Hunt process that lives on the original space $L^2\left(D\right)$ endowed with the norm topology.