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.

Key words： Symmetric stable process; time-change; Hölder continuity; coupling

References

1 Barczy M,Li Z H,Pap G.Yamada-Watanabe results for stochastic differential equations with jumps. Int J Stoch Anal, 2015, Art ID 460472, 23pp

2 Bass R F.Stochastic differential equations driven by symmetric stable process. In:Azéma J,Émery M, Ledoux M, Yor M, eds. Séminaire de Probabilités XXXVI.Lecture Notes in Math, Vol 1801. Berlin: Springer, 2003, 302–313

3 Chen Z Q,Wang J.Ergodicity for time-changed symmetric stable processes. Stochastic Process Appl, 2014, 124: 2799–2823

4 Debussche A,Fournier N.Existence of densities for stable-like driven SDE’s with Höolder continuous coefficients. J Funct Anal, 2013, 264: 1757–1778

5 Fournier N,Printems J.Absolute continuity for some one-dimensional processes. Bernoulli, 2010, 16: 343–369

6 Komatsu T.On the pathwise uniqueness of solutions of one-dimensional stochastic differential equations of jump type. Proc Japan Acad Ser A Math Sci, 1982, 58: 353–356

7 Priola E,Wang F Y.Gradient estimates for diffusion semigroups with singular coefficients. J Funct Anal, 2006, 236: 244–264

8 Stroock D.Diffusion processes associated with Lévy generators. Z Wahrsch Verw Gebiete, 1975, 32: 209–244

9 Wang F Y,Xu L H,Zhang X C.Gradient estimates for SDEs driven by multiplicative Lévy noise. J Funct Anal, 2015, 269: 3195–3219