Stochastic Spectral Theory for Selfadjoint Feller Operators

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

“…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.…”

Nonautonomous Kato classes of measures and Feynman-Kac propagators

“…Section 2 is devoted to the study of various classes of time-dependent measures. These classes generalize the Kato class of measures (see [AS,BM1,DvC,F,Gu1,JL,S,V1,V2,Z] for more information on the Kato classes of functions and measures), the enlarged Kato class (see [V1, V2], see also [Gu1,GK]), and the nonautonomous Kato classes of functions (see [Gu3,LS,N,QZ1,QZ2,RRSV,SV]). Our main result in Section 2 is a characterization of classes of time-dependent measures in terms of the corresponding potential operators (see Theorem 1).…”

Schwartz operators

“…Later on an important connection with SPDE's was established, see, e.g. [1,122,12,32,59,60,61,63,64,65,92,126,83,85,33,35,141,6]. In all these constructions and applications, probability measures on infinite dimensional spaces are constructed by starting from projective systems of finite dimensional probability measures.…”