# HEAT KERNEL ESTIMATES FOR DIRICHLET FRACTIONAL LAPLACIAN WITH GRADIENT PERTURBATION

• Chen, Peng (Department of Mathematics University of Macau) ;
• Song, Renming (Department of Mathematics University of Illinois) ;
• Xie, Longjie (School of Mathematics and Statistics Jiangsu Normal University) ;
• Xie, Yingchao (School of Mathematics and Statistics Jiangsu Normal University)
• Accepted : 2018.06.26
• Published : 2019.01.01

#### Abstract

We give a direct proof of the sharp two-sided estimates, recently established in [4, 9], for the Dirichlet heat kernel of the fractional Laplacian with gradient perturbation in $C^{1,1}$ open sets by using Duhamel's formula. We also obtain a gradient estimate for the Dirichlet heat kernel. Our assumption on the open set is slightly weaker in that we only require D to be $C^{1,{\theta}}$ for some ${\theta}{\in}({\alpha}/2,1]$.

#### Acknowledgement

Supported by : Simons Foundation, NNSF of China, NSF of Jiangsu

#### References

1. K. Bogdan and T. Jakubowski, Estimates of heat kernel of fractional Laplacian perturbed by gradient operators, Comm. Math. Phys. 271 (2007), no. 1, 179-198. https://doi.org/10.1007/s00220-006-0178-y
2. Z.-Q. Chen, E. Hu, L. Xie, and X. Zhang, Heat kernels for non-symmetric diffusion operators with jumps, J. Differential Equations 263 (2017), no. 10, 6576-6634. https://doi.org/10.1016/j.jde.2017.07.023
3. Z.-Q. Chen, P. Kim, and R. Song, Heat kernel estimates for the Dirichlet fractional Laplacian, J. Eur. Math. Soc. (JEMS) 12 (2010), no. 5, 1307-1329.
4. Z.-Q. Chen, P. Kim, and R. Song, Dirichlet heat kernel estimates for fractional Laplacian with gradient perturbation, Ann. Probab. 40 (2012), no. 6, 2483-2538. https://doi.org/10.1214/11-AOP682
5. Z.-Q. Chen, P. Kim, and R. Song, Stability of Dirichlet heat kernel estimates for non-local operators under Feynman-Kac perturbation, Trans. Amer. Math. Soc. 367 (2015), no. 7, 5237-5270. https://doi.org/10.1090/S0002-9947-2014-06190-4
6. Z.-Q. Chen and L. Wang, Uniqueness of stable processes with drift, Proc. Amer. Math. Soc. 144 (2016), no. 6, 2661-2675. https://doi.org/10.1090/proc/12909
7. K. L. Chung, Lectures from Markov Processes to Brownian Motion, Grundlehren der Mathematischen Wissenschaften, 249, Springer-Verlag, New York, 1982.
8. K.-Y. Kim and P. Kim, Two-sided estimates for the transition densities of symmetric Markov processes dominated by stable-like processes in $C^{1,{\eta}}$, open sets, Stochastic Process. Appl. 124 (2014), no. 9, 3055-3083. https://doi.org/10.1016/j.spa.2014.04.004
9. P. Kim and R. Song, Dirichlet heat kernel estimates for stable processes with singular drift in unbounded $C^{1,1}$ open sets, Potential Anal. 41 (2014), no. 2, 555-581. https://doi.org/10.1007/s11118-013-9383-4
10. T. Kulczycki and M. Ryznar, Gradient estimates of Dirichlet heat kernels for unimodal Levy processes, Math. Nachr. 291 (2018), no. 2-3, 374-397. https://doi.org/10.1002/mana.201600443
11. H. Li, D. Luo, and J. Wang, Harnack inequalities for SDEs with multiplicative noise and non-regular drift. Stoch. Dyn. 15 (2015), 1550015. https://doi.org/10.1142/S021949371550015X
12. F.-Y.Wang, Harnack Inequalities for Stochastic Partial Differential Equations, Springer Briefs in Mathematics, Springer, New York, 2013.
13. L. Xie and X. Zhang, Heat kernel estimates for critical fractional diffusion operators, Studia Math. 224 (2014), no. 3, 221-263.