Continuity properties of symmetric stable processes

Let [special characters omitted] be a process and [special characters omitted], be a sequence of processes obtained by subordinating an m-symmetric right process Xt on a Lusin space (E; m) via subordinators having Laplace exponents &phis; and [special characters omitted], respectively. In this paper we prove con vergence in probability and convergence almost surely along a subsequence, when limn →∞&phis;n. Under the condition [special characters omitted], where [special characters omitted] is the measure of the subordinator with Laplace exponent &phis;, and d is the dimension of E, we show that the processes obtained by subordinating Brownian motion converge in probability along the entire sequence. The standard symmetric α/2-stable process satisfies this condition.We extend these convergence results to symmetric Lévy processes and subordinated killed Brownian motion. We also explore the Lipschitz continuity of symmetric α-stable processes and prove that it holds for the transition densities of these processes