Abstract
In this paper, we first establish the sharp two-sided heat kernel estimates and the gradient estimate for the truncated fractional Laplacian under gradient perturbation
where \({\overline {\rm{\Delta }} ^{\alpha /2}}\) is the truncated fractional Laplacian, α ∈ (1, 2) and b ∈ K α−1d . In the second part, for a more general finite range jump process, we present some sufficient conditions to allow that the two sided estimates of the heat kernel are comparable to the Poisson type function for large distance ∣x − y∣ in short time.
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Barlow, M. T., Bass, R. F., Chen, Z. Q. et al.: Non-local Dirichlet forms and symmetric jump processes. Trans. Amer. Math. Soc., 361, 1963–1999 (2009)
Barlow, M. T., Grigor’yan, A., Kumagai, T.: Heat kernel upper bounds for jump processes. J. Reine Angew. Math., 626, 135–157 (2007)
Bass, R. F., Levin, D. A.: Transition probabilities for symmetric jump processes. Trans. Amer. Math. Soc., 354, 2933–2953 (2002)
Bertoin, J.: Lévy Processes. Cambridge University Press, Cambridge, 1996
Bogdan, K., Grzywny, T., Ryznar, M.: Density and tails of unimodal convolution semigroups. J. Funct. Anal., 266(6), 3543–3571 (2014)
Bogdan, K., Jakubowski, T.: Estimates of heat kernel of fractional Laplacian perturbed by gradient operators. Comm. Math. Phys., 271, 179–198 (2007)
Chen, Z. Q., Kim, P., Kumagai, T.: Weighted Poincaré inequality and heat kernel estimates for finite range jump processes. Math. Ann., 342, 833–883 (2008)
Chen, Z. Q., Kim, P., Kumagai, T.: Global heat kernel estimates for symmetric jump processes. Trans. Amer. Math. Soc., 363(9), 5021–5055 (2011)
Chen, Z. Q., Kim, P., Song, R.: Dirichlet heat kernel estimates for fractional Laplacian under gradient perturbation. Ann. Probab., 40, 2483–2538 (2012)
Chen, Z. Q., Kumagai, T.: Heat kernel estimates for stable-like processes on d-sets. Stoch. Process Appl., 108, 27–62 (2003)
Chen, Z. Q., Kumagai, T.: Heat kernel estimates for jump processes of mixed types on metric measure spaces. Probab. Theory and related fields, 140, 277–317 (2008)
Chen, Z. Q., Wang, J. M.: Perturbation by non-local operators. Annal. de l’Institut Henri Poincaré (B) Probabilités et Statistiques, 54(2), 606–639 (2018)
Chen, Z. Q., Wang, L.: Uniqueness of stable processes with drift. Proc. Amer. Math. Soc., 144, 2661–2675 (2016)
Chen, Z. Q., Zhang, X.: Heat kernels and analyticity of non-symmetric jump diffusion semigroups. Probab. Theory Relat. Fields, 165, 267–312 (2016)
Grigor’yan, A., Hu, J., Lau, K.-S.: Estimates of heat kernels for non-local Dirichlet forms. Trans. Amer. Math. Soc., 366(12), 6379–6441 (2014)
Grzywny, T., Szczypkowski, K.: Estimates of heat kernels of non-symmetric Lévy processes. arXiv: 1710.07793v1 [math.AP]
Grzywny, T., Szczypkowski, K.: Heat kernels of non-symmetric Lévy-type processes. J. Differential Equations, 267, 6004–6064 (2019)
Ikeda, N., Nagasawa, N., Watanabe, S.: A construction of Markov processes by piecing out. Proc. Japan Acad., 42, 370–375 (1966)
Jakubowski, T., Szczypkowski, K.: Time-dependent gradient perturbations of fractional Laplacian. J. Evol. Equ., 10, 319–339 (2010)
Kim, P., Lee, J.: Heat kernel of non-symmetric jump processes with exponentially decaying jumping kernel. Stoch. Process Appl., 129, 2130–2173 (2019)
Kim, P., Song, R., Vondracek, Z.: Heat kernels of non-symmetric jump prcesses: beyond the stable case. Potential Anal., 49, 37–90 (2018)
Kolokoltsov, V.: Symmetric stable laws and stable-like jump diffusions. Proc. London Math. Soc., 80, 725–768 (2000)
Meyer, P. A.: Renaissance: recollements, mélanges, raletissement de processus de Markov. Ann. Inst. Fourier, 25, 464–497 (1975)
Szczypkowski, K.: Fundamental solution for super-critical non-symmetric Lévy-type operators. arXiv: 1807.04257v1 [math.AP]
Wang, J. M.: Laplacian perturbed by non-local operators. Math. Z., 279, 521–556 (2015)
Xie, L., Zhang, X.: Heat kernel estimates for critical fractional diffusion operators. Studia Mathematica, 224(3), 221–263 (2014)
Acknowledgements
The author is grateful to Professor Zhen-Qing Chen for valuable comments on an earlier version of this paper.
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Partially supported by NSFC (Grant Nos. 11731009 and 11401025)
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Wang, J.M. Heat Kernel Estimates for Non-symmetric Finite Range Jump Processes. Acta. Math. Sin.-English Ser. 37, 229–248 (2021). https://doi.org/10.1007/s10114-020-9459-1
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DOI: https://doi.org/10.1007/s10114-020-9459-1
Keywords
- Heat kernel
- transition density function
- gradient estimate
- finite range jump process
- truncated fractional Laplacian
- martingale problem