The Most Visited Sites of Certain Levy Processes

Let X be a symmetric Levy process with Ee^(i(lambda)Xt) = e ^ (-t(psi)(lambda)) Let (phi)(x)= 2/(pi) (integral) (1- cos (lambda(x)) /(psi)(lambda) d(lambda)) Assume that (bullet operator) (psi) (lambda) is regularly varying at zero with index 1<(alpha) <= 2 and (1/(psi)(lambda)) I [(lambda) => 1] (element of) L 1(R). (bullet operator) (phi)(x) is increasing on [0, (infinity) Let L x t denote the local time of X at x up to time t. Following The most visited sites of symmetric stable processes, by Bass, Eisenbaum, and Shi, let V(t) be such that L V(t) t =sup x(element of) R L x t . We call V(t) the most visited site of X up to time t. We show that under the above conditions on X,V(t) is transient. In particular, for all (gamma)>9 (limit)(|V(t)|/ (phi)^(-1)(t(psi)^-1(1/t) (log t) ^(gamma))) This result is obtained for symmetric stable processes in the above reference. We use their approach and many of their methods.