Advertisement

Frontiers of Mathematics in China

, Volume 10, Issue 4, pp 753–776 | Cite as

Intrinsic contractivity properties of Feynman-Kac semigroups for symmetric jump processes with infinite range jumps

  • Xin Chen
  • Jian WangEmail author
Research Article

Abstract

Let (X t ) t⩾0 be a symmetric strong Markov process generated by non-local regular Dirichlet form Open image in new window as follows:
where J(x, y) is a strictly positive and symmetric measurable function on ℝ d ×ℝ d . We study the intrinsic hypercontractivity, intrinsic supercontractivity, and intrinsic ultracontractivity for the Feynman-Kac semigroup
In particular, we prove that for Open image in new window with α ∈ (0, 2) and V(x) = |x|λ with λ > 0, (T t V ) t⩾0 is intrinsically ultracontractive if and only if λ > 1; and that for symmetric α-stable process (X t ) t⩾0 with α ∈ (0, 2) and V(x) = logλ (1+|x|) with some λ > 0, (T t V ) t⩾0 is intrinsically ultracontractive (or intrinsically supercontractive) if and only if λ > 1, and (T t V ) t⩾0 is intrinsically hypercontractive if and only if λ ⩾ 1. Besides, we also investigate intrinsic contractivity properties of (T t V ) t⩾0 for the case that lim inf|x|→+∞ V(x) < +∞.

Keywords

Symmetric jump process Lévy process Dirichlet form Feynman-Kac semigroup intrinsic contractivity 

MSC

60G51 60G52 60J25 60J75 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Barlow M T, Bass R F, Chen Z -Q, Kassmann M. Non-local Dirichlet forms and symmetric jump processes. Trans Amer Math Soc, 2009, 361: 1963–1999zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Bass R F, Kassmann M, Kumagai T. Symmetric jump processes: localization, heat kernels, and convergence. Ann Inst Henri Poincaré Probab Stat, 2010, 46: 59–71zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Chen X, Wang J. Functional inequalities for nonlocal Dirichlet forms with finite range jumps or large jumps. Stochastic Process Appl, 2014, 124: 123–153zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Chen X, Wang J. Intrinsic contractivity of Feyman-Kac semigroups for symmetric jump processes. arXiv: 1403.3486Google Scholar
  5. 5.
    Chen Z -Q, Kumagai T. Heat kernel estimates for stable-like processes on d-sets. Stochastic Process Appl, 2003, 108: 27–62zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Chen Z -Q, Kumagai T. Heat kernel estimates for jump processes of mixed types on metric measure spaces. Probab Theory Related Fields, 2008, 140: 277–317zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Chen Z -Q, Kim P, Kumagai T. Weighted Poincaré inequality and heat kernel estimates for finite range jump processes. Math Ann, 2008, 342: 833–883zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Chen Z -Q, Kim P, Kumagai T. Global heat kernel estimates for symmetric jump processes. Trans Amer Math Soc, 2011, 363: 5021–5055zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Chen Z -Q, Kim P, Song R. Heat kernel estimates for Dirichlet fractional Laplacian. J Eur Math Soc, 2010, 12: 1307–1329zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Chung K L, Zhao Z. From Brownian Motion to Schrödinger’s Equation. New York: Springer, 1995zbMATHCrossRefGoogle Scholar
  11. 11.
    Davies E B, Simon B. Ultracontractivity and heat kernels for Schrödinger operators and Dirichlet Laplacians. J Funct Anal, 1984, 59: 335–395zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Kaleta K, Kulczycki T. Intrinsic ultracontractivity for Schrödinger operators based on fractional Laplacians. Potential Anal, 2010, 33: 313–339zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Kaleta K, Lőrinczi J. Pointwise eigenfunction estimates and intrinsic ultracontractivitytype properties of Feynman-Kac semigroups for a class of Lévy processes. Ann Probab (to appear), also see arXiv: 1209.4220Google Scholar
  14. 14.
    Kulczycki T, Siddeja B. Intrinsic ultracontractivity of the Feynman-Kac semigroup for relativistic stable processes. Trans Amer Math Soc, 2006, 358: 5025–5057zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Ouhabaz E M, Wang F -Y. Sharp estimates for intrinsic ultracontractivity on C 1,α-domains. Manuscripta Math, 2007, 112: 229–244MathSciNetCrossRefGoogle Scholar
  16. 16.
    Wang F -Y. Functional inequalities for empty spectrum estimates. J Funct Anal, 2000, 170: 219–245zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Wang F -Y. Functional inequalities and spectrum estimates: the infinite measure case. J Funct Anal, 2002, 194: 288–310zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Wang F -Y. Functional Inequalities, Markov Processes and Spectral Theory. Beijing: Science Press, 2005Google Scholar
  19. 19.
    Wang F -Y. Intrinsic ultracontractivity on Riemannian manifolds with infinite volume measures. Sci China Math, 2010, 53: 895–904zbMATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Wang F -Y, Wang J. Functional inequalities for stable-like Dirichlet forms. J Theoret Probab (to appear), also see arXiv: 1205.4508v3Google Scholar
  21. 21.
    Wang F -Y, Wu J -L. Compactness of Schrödinger semigroups with unbounded below potentials. Bull Sci Math, 2008, 132: 679–689zbMATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Wang J. Symmetric Lévy type operator. Acta Math Sin (Engl Ser), 2009, 25: 39–46zbMATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    Wang J. A simple approach to functional inequalities for non-local Dirichlet forms. ESAIM Probab Stat, 2014, 18: 503–513zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Higher Education Press and Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of MathematicsShanghai Jiao Tong UniversityShanghaiChina
  2. 2.School of Mathematics and Computer ScienceFujian Normal UniversityFuzhouChina

Personalised recommendations