Construction of Hunt processes by the Lyapunov method and applications to generalized Mehler semigroups
A talk in the Bielefeld Stochastic Afternoon series by
Iulian Cimpean
| Abstract: | It is well known that generalized Mehler semigroups may not
correspond to cadlag (or even cad) Markov processes with values in
the original state space. The problem of characterizing those
generalized Mehler semigroups that correspond to cadlag Markov
processes is highly non-trivial and has remaind open for more than a
decade. The aim of this talk is to present a general approach to tackle
the cadlag problem for generalized Mehler semigroups, by regarding
it as a particular case of the general problem of constructing Hunt
(hence cadlag and quasi-left continuous) processes from a given Markov
semigroup. More precisely, starting from a Markov semigroup on a general
(possibly non-metrizable) state space, we show that 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 cadlag (in most
cases Hunt) Markov process. Based on such general existence results, we derive checkable sufficient conditions for a large class of generalized Mehler semigroups in order to posses an associated Hunt process with values in the original space, in contrast to previous results where an extension of the state space was required. Furthermore, we test these conditions on a stochastic heat equation on $L^2(D)$ whose drift is given by the Dirichlet Laplacian on a bounded domain $D \subset \mathbb{R}^d$, driven by a 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 $L^2(D)$ endowed with the norm topology. Based on a joint work with L. Beznea and M. Röckner. Within the CRC this talk is associated to the project(s): B1 |