Hölder continuity of semigroups for time changed symmetric stable processes

Abstract

Let{{\displaystyle ({{\displaystyle Z}}_t)}}_{t\ge \mathrm{0}}be a one-dimensional symmetric α-stable process with\alpha \in (\mathrm{0},\mathrm{2}), and let\sigmabe a bounded (from above and from below) and \mathrm{1}/(\alpha \vee \mathrm{1})-Hölder continuous function onℝ. Consider the stochastic differential equationdX=\sigma ({{\displaystyle X}}_{t-})d{{\displaystyle Z}}_t,which admits a unique strong solution. By using thesplitting technique and the coupling method, we derive the Hölder continuity of the associated semigroup.

Keywords

Symmetric stable process / time-change / Hölder continuity / coupling

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2014 Higher Education Press and Springer-Verlag Berlin Heidelberg