Growth and Holder conditions for the sample paths of Feller processes

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-dif ferential operator with symbol -p(x,xi) satisfying parallel to p(.,xi paral lel 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 a ssociated Feller process {X-t}(t greater than or equal to 0) on R-n is a se mimartingale, even a homogeneous diffusion with jumps tin the sense of [21] ), and characterize the limiting behaviour of its trajectories as t --> 0 a nd infinity. To this end, we introduce various indices, e.g., beta(infinity )(x) := inf{lambda > 0 : lim(parallel to xi parallel to-->infinity) sup(par allel 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(paral lel 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 ob tain 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 t he limit inferior and superior, and also for t --> infinity. Our results ex tend the constant-coefficient (i.e., Levy) case considered by W. Pruitt [27 ].