Growth and Hölder conditions for the sample paths of Feller processes

Abstract

Let eY he be the in®nitesimal generator of a Feller semigroup such that g I R n & he and ejg I R n is a pseudo-differential operator with symbol ÀpxY n satisfying kpY nk I 1 knk 2 and jIm pxY nj 0 Re pxY n. We show that the associated Feller process f t g t!0 on R n is a semimartingale, even a homogeneous diusion with jumps (in the sense of [21]), and characterize the limiting behaviour of its trajectories as t 3 0 and I. To this end, we introduce various indices, e.g., b x I X inffk b 0 X lim knk3I sup kxÀyk 2aknk jpyY njaknk k 0g or d x I X inffk b 0 X lim inf knk3I inf kxÀyk 2aknk sup kk 1 jpyY knkjaknk k 0g, and obtain a.s. P x that lim t30 t À1ak sup s t k s À xk 0 or I according to k b b x I or k`d x I . Similar statements hold for the limit inferior and superior, and also for t 3 I. Our results extend the constant-coecient (i.e., Le vy) case considered by W. Pruitt [27].Mathematics Subject Classi®cation (1991): 60F15, 60J75, 60G17, 35S99, 60J35 Probab. Theory Relat. Fields 112, 565 ± 611 (1998) *Part of this work was done while the author was visiting the Universite d'Evry±Val d'Essonne, France. He would like to thank the sta of the maths department and, in particular, Prof. F. Hirsch, for their support and the good working conditions. Financial support by DFG post-doctoral fellowship Schi 419/1-1 is gratefully acknowledged.

Nonlinear Markov Semigroups and Interacting Lévy Type Processes

“…books [3], [4]) and their spatially nontrivial extensions -Lévy type processes, the latter being basically just the Markov processes with pseudo-differential generators of Lévy-Khinchin type, see [28] for a comprehensive analytic study of these processes, [29] for a probabilistic analysis, where these processes are called jump-diffusions, and also [53] for some connections between these approaches. Lévy and Lévy type processes enjoy quite different properties (e.g.…”

“…Follows from Theorem 4.1 (iv) and the duality arguments. The basic properties of the solution to the Cauchy problem of equations (52), (53) are collected in the following theorem.…”

Long-time behavior for a class of Feller processes

“…Next, the concept of the indices of stability can be generalized to general Lévy processes through the Pruitt indices (see [15]). The Pruitt indices for nice Feller processes have been introduced in [25]. In Theorems 2.12 and 2.13, we also discuss the recurrence and transience property of nice Feller processes, as well as of Lévy processes, in terms of the Pruitt indices.…”

“…Let us remark that, according to [25,Lemma 2.1], condition (C2) is equivalent with the boundedness of the coefficients of the symbol q(x, ξ), that is,…”

Topics on Hamiltonian Dynamics Related to Symbols of Certain Schrodinger Operators Associated with Generators of Levy Processes

“…admits for every fixed x ∈ R n a Lévy-Khinchine representation. Such operators were studied systematically since the late 1980ties, we refer to the monographs [43]- [45] and [7] for overviews in which in particular the seminal existences results W.Hoh [34]- [36] and the equally important papers [57]- [59] of R.Schilling on the probabilistic relevance of the symbol are discussed. The basic problem in the theory is that in general these symbols do not fit into any of the framework discussed above.…”