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Compound kernel estimates for the transition probability density of a Lévy process in $ \mathbb{R}^n$


Author: Victoria Knopova
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 89 (2013).
Journal: Theor. Probability and Math. Statist. 89 (2014), 57-70
MSC (2010): Primary 60G51; Secondary 60J75, 41A60
DOI: https://doi.org/10.1090/S0094-9000-2015-00935-2
Published electronically: January 26, 2015
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Abstract | References | Similar Articles | Additional Information

Abstract: We construct in the small-time setting the upper and lower estimates for the transition probability density of a Lévy process in $ \mathbb{R}^n$. Our approach relies on the complex analysis technique and the asymptotic analysis of the inverse Fourier transform of the characteristic function of the process.


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  • 1. E. T. Copson, Asymptotic Expansions, Cambridge University Press, Cambridge, 1965. MR 0168979 (29:6234)
  • 2. M. T. Barlow, R. B. Bass, Z.-Q. Chen, and M. Kassmann, Non-local Dirichlet forms and symmetric jump processes, Trans. Amer. Math. Soc. 361 (2009), 1963-1999. MR 2465826 (2010e:60163)
  • 3. M. T. Barlow, A. Grigoryan, and T. Kumagai, Heat kernel upper bounds for jump processes and the first exit time, J. Reine Angew. Math. 626 (2009), 135-157. MR 2492992 (2009m:58077)
  • 4. Z.-Q. Chen, Symmetric jump processes and their heat kernel estimates, Sci. China Ser. A. 52 (2009), 1423-1445 MR 2520585 (2010i:60220)
  • 5. Z.-Q. Chen and T. Kumagai, Heat kernel estimates for stable-like processes on $ d$-sets, Stoch. Proc. Appl. 108 (2003), 27-62. MR 2008600 (2005d:60135)
  • 6. Z.-Q. Chen and T. Kumagai, Heat kernel estimates for jump processes of mixed types on metric measure spaces, Probab. Theory Related Fields 140(1-2) (2008), 277-317. MR 2357678 (2009e:60186)
  • 7. Z.-Q. Chen, P. Kim, and T. Kumagai, On Heat kernel estimates and parabolic Harnack inequality for jump processes on metric measure spaces, Acta Math. Sin. (Engl. Ser.) 25 (2009), 1067-1086. MR 2524930 (2011b:60334)
  • 8. Z.-Q. Chen, P. Kim, and T. Kumagai, Global heat kernel estimates for symmetric jump processes, Trans. Amer. Math. Soc. 363(9) (2011), 5021-5055. MR 2806700
  • 9. Y. Ishikawa, Asymptotic behavior of the transitions density for jump type processes in small time, Tôhoku Math. J. 46 (1994), 443-456. MR 1301283 (95j:60132)
  • 10. Y. Ishikawa, Density estimate in small time for jump processes with singular Lévy measures, Tôhoku Math. J. 53(2) (2001), 183-202. MR 1829978 (2002g:60124)
  • 11. K. Kaleta and P. Sztonyk, Upper estimates of transition densities for stable-dominated semigroups, J. Evol. Equ. 13 (2013), 633-650 MR 3089797
  • 12. K. Kaleta and P. Sztonyk, Estimates of Transition Densities and their Derivatives for Jump Lévy Processes, Preprint, 2013.
  • 13. C. Klüppelberg, Subexponential distributions and characterizations of related classes, Probab. Theory Related Fields 82 (1989), 259-269. MR 998934 (90j:60020)
  • 14. V. Knopova and R. L. Schilling, Transition density estimates for a class of Lévy and Lévy-type processes, J. Theor. Probab. 25 (2012), 144-170. MR 2886383
  • 15. V. Knopova and A. Kulik, Exact asymptotic for distribution densities of Lévy functionals, Electronic J. Probab. 16 (2011), 1394-1433. MR 2827465 (2012g:60157)
  • 16. V. Knopova and A. Kulik, Intrinsic small time estimates for distribution densities of Lévy processes, Rand. Oper. Stoch. Eq. 21 (4) (2013), 321-344. MR 3139314
  • 17. V. Knopova, Asymptotic behaviour of the distribution density some Lévy functionals in $ \mathbb{R}^n$ Theory Stoch. Process. 17 (33) (2011), no. 2, 35-54. MR 2934558
  • 18. V. Knopova and A. Kulik, Parametrix Construction for Certain Lévy-type Processes and Applications, Preprint, 2013.
  • 19. V. Knopova and A. Kulik, Intrinsic Compound Kernel Estimates for the Transition Probability Density of a Lévy Type Processes and their Applications, Preprint, 2013.
  • 20. R. Léandre, Densité en temps petit d'un processus de sant, Seminaire de Probabilitts XXI (J. Azéma, P. A. Meyer, and M. Yor, eds.), Lecture Notes in Math., vol. 1247, Springer, Berlin, 1987, pp. 81-99. MR 941977 (89g:60179)
  • 21. E. Omey, F. Mallor, and J. Santos, Multivariate subexponential distributions and random sums of random vectors, Adv. Appl. Probab. 38(4) (2006), 1028-1046. MR 2285692 (2008b:60037)
  • 22. E. Omey, Subexponential distribution functions in $ \mathbb{R}^d$, J. Math. Sci. 138(1) (2006), 5434-5449. MR 2261594 (2008c:62016)
  • 23. J. Picard, On the existence of smooth densities for jump processes, Probab. Theory Related Fields 105 (1996), 481-511. MR 1402654 (97h:60056)
  • 24. J. Picard, Density in small time for Lévy processes, ESAIM Probab. Statist. 1 (1997), 358-389. MR 1486930 (98i:60068)
  • 25. J. Picard, Density in small time at accessible points for jump processes, Stoch. Process. Appl. 67 (1997), 251-279. MR 1449834 (98f:60163)

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Additional Information

Victoria Knopova
Affiliation: V. M. Glushkov Institute of Cybernetics, National Academy of Science of Ukraine, 40, Academician Glushkov Avenue, 03187, Kyiv, Ukraine
Email: vic_knopova@gmx.de

DOI: https://doi.org/10.1090/S0094-9000-2015-00935-2
Keywords: Transition probability density, transition density estimates, L\'evy processes, Laplace method
Received by editor(s): June 1, 2013
Published electronically: January 26, 2015
Article copyright: © Copyright 2015 American Mathematical Society

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