The Feller Property for Absorption Semigroups

Abstract

Let U=(U(t); t 0) be a substochastic strongly continuous semigroup on L 1 (X, m) where X is locally compact and m a Borel measure on X. We give conditions on absorption rates V implying that the (strong) Feller property carries over from U* to U* V . These conditions are essentially in terms of the Kato class associated with U. Preparing these results we discuss the perturbation theory of strongly continuous semigroups and properties of one-parameter semigroups on L (m). In the symmetric case of Dirichlet forms we generalize the results to measure perturbations. For the case of the heat equation on R d we show that the results are close to optimal.

Classes of time-dependent measures, non-homogeneous Markov processes, and Feynman-Kac propagators

“…Lemmas 6.5 and 6.9 were obtained in [17,21] in the case of the heat semigroup. A similar result concerning time-independent perturbations of semigroups on L 1 was obtained earlier in [32,Lemma 4.2].…”

“…We discuss the strong continuity of propagators on the space L r , the (L r − L q )-smoothing property, and various versions of the Feller property. More results concerning the similarities in the behaviour of semigroups (propagators) and their perturbations by potentials can be found in [1,2,3,4,5,6,7,14,16,17,18,19,20,21,22,32,34,37,38,39,40,42,43,44,45]. In Section 8 we establish that the Feller property, the Feller-Dynkin property, and the BUC-property are inherited by Feynman-Kac propagators from free propagators under additional restrictions on functions and time-dependent measures generating Feynman-Kac propagators.…”

“…for all x ∈ E where c n is defined by (32). Next we choose a sequence of refinements of the partition τ = t 0 < t 1 < · · · < t k = τ + δ on the left-hand side of (36) such that the maximum length of the partition intervals tends to 0, and pass to the limit, using the monotone convergence theorem and the continuity of the functional A V (τ, t) with respect to the variable t. This establishes estimate (31) and completes the proof of Lemma 4.9.…”

The non-autonomous Kato class

“…4.5]. Moreover, K is closely related to the set of potentials which are Miyadera perturbations of the semigroup Ut t^0 on L 1 R n generated by D, see [6,11,12,13]. More precisely, the enlarged Kato class is defined by K X fV 0 X R n 3 C X V 0 measurable, kV 0 k U`I g, where…”

“…(To be precise, in [6,12,13] the class K was introduced for real potentials V 0 . However, the results of these papers which are used below carry over to complex-valued V 0 by considering jV 0 j.)…”