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Hölder continuity of semigroups for time changed symmetric stable processes

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Abstract

Let (Z t ) t⩾0 be a one-dimensional symmetric α-stable process with α ∈ (0, 2), and let σ be a bounded (from above and from below) and 1/(α ∨ 1)-Hölder continuous function on ℝ. Consider the stochastic differential equation dX t = σ(X t−)dZ t , which admits a unique strong solution. By using the splitting technique and the coupling method, we derive the Hölder continuity of the associated semigroup.

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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, 23 pp

    Google Scholar 

  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

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  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

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Dejun Luo

  2. School of Mathematics and Computer Science, Fujian Normal University, Fuzhou, 350007, China

    Jian Wang

Authors
  1. Dejun Luo
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  2. Jian Wang
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Correspondence to Jian Wang.

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Luo, D., Wang, J. Hölder continuity of semigroups for time changed symmetric stable processes. Front. Math. China 11, 109–121 (2016). https://doi.org/10.1007/s11464-015-0501-z

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  • DOI: https://doi.org/10.1007/s11464-015-0501-z

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