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

Abstract

Abstract. We study the inheritance of properties of free backward propagators associated with transition probability functions by backward FeynmanKac propagators corresponding to functions and time-dependent measures from non-autonomous Kato classes. The inheritance of the following properties is discussed: the strong continuity of backward propagators on the space L r , the (L r − L q )-smoothing property of backward propagators, and various generalizations of the Feller property. We also prove that a propagator on a Banach space is strongly continuous if and only if it is separately strongly continuous and locally uniformly bounded.

Time-dependent Schrödinger perturbations of transition densities

“…We point out that our main estimate, Theorem 3 below, is more precise and explicit than those mentioned above. Accordingly, it also strengthens in the present context the celebrated Khas'minski's lemma, one of the main tools in the probabilistic theory of Schrödinger perturbations of generators of Markov processes (see, e.g., [11], [16]). This strengthening is of independent interest-the estimate is valid in the full range of times, rather than only in small time intervals, and the proof gives a deeper insight into the interplay between individual terms of the series involved.…”

“…It is possible to extend the present results to more general integral kernels or to measures (rather than functions q; see [20], [26], [16] for a related study). In fact, considering q(dudz) = ηδ u 0 (du)dz, where η ≥ 1 and δ u 0 is the probability measure concentrated at u 0 , shows that p q may explode in finite time u 0 .…”

“…Following [30], [16], we will say that q : R × X → R belongs to the parabolic (space-time) Kato class for p if We say that p is probabilistic if (31) X p(s, x, t, y) dy = 1 for s < t, x ∈ X.…”

Localization and Schrödinger Perturbations of Kernels

“…We also wish to mention recent results [4,8] for non-local Schrödinger-type perturbations (see [18] and [21], too). Schrödinger perturbations of the Gaussian transition density are studied in [19,22], see also [11]. We refer to [3,5,13,14,16] for further instances, applications and forms of the 3P (or 3G) inequality 4.24.…”

On Combinatorics of Schrödinger Perturbations

“…In this casep is also a transition density, one generated by a time-dependent additive (Schrödinger) perturbation of the generator of p, see Example 3 and [3]. For recent developments in Schrödinger perturbations of time-nonhomogeneous transition probabilities we refer the reader to [3], and [14], [15], [21]. We will assume that there are 0 ≤ η < ∞ and a function Q : R × R → [0, ∞) satisfying the following condition of super-additivity:…”