Growth and Holder conditions for the sample paths of Feller processes

• 成果类型: Article
• 署名作者: Schilling, RL
• 署名单位: Nottingham Trent University; Universite Paris Saclay
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s004400050201
• 发表日期: 1998
• 页码: 565-611
• 关键词: pseudo differential-operators; pseudodifferential-operators; variable order; semigroups

摘要

Let (A,D(A)) be the infinitesimal generator of a Feller semigroup such that C-c(infinity)(R-n) subset of D(A) and A\C-c(infinity)(R-n) is a pseudo-differential operator with symbol -p(x,xi) satisfying parallel to p(.,xi parallel to(infinity) less than or equal to c(1 + parallel to xi parallel to(2)) and \Im p(x,xi)\ less than or equal to c(0) Re p(x,xi). We show that the associated Feller process {X-t}(t greater than or equal to 0) on R-n is a semimartingale, even a homogeneous diffusion with jumps tin the sense of [21]), and characterize the limiting behaviour of its trajectories as t --> 0 and infinity. To this end, we introduce various indices, e.g., beta(infinity)(x) := inf{lambda > 0 : lim(parallel to xi parallel to-->infinity) sup(parallel to x-y parallel to less than or equal to 2/parallel to xi parallel to) \p(y,xi)\/parallel to xi parallel to(lambda) = 0} or delta(infinity)(x) := inf{lambda > 0 : lim inf(parallel to xi parallel to-->infinity) inf(parallel to x-y parallel to less than or equal to 2/)parallel to xi parallel to sup(parallel to epsilon parallel to less than or equal to 1) \p(y,parallel to xi parallel to epsilon)\/parallel to xi parallel to(lambda) = 0}, and obtain a.s. (P-x) that lim(t-->0) t(-1/lambda) sup(s less than or equal to t) parallel to X-s - x parallel to = 0 or infinity according to lambda > beta(infinity)(x) or lambda < delta(infinity)(x). Similar statements hold for the limit inferior and superior, and also for t --> infinity. Our results extend the constant-coefficient (i.e., Levy) case considered by W. Pruitt [27].