Advertisement

Strong Stability of Heat Kernels of Non-symmetric Stable-Like Operators

  • Zhen-Qing Chen
  • Xicheng Zhang
Conference paper
Part of the Progress in Probability book series (PRPR, volume 72)

Abstract

Let \(d\geqslant 1\) and α ∈ (0, 2). Consider the following non-local and non-symmetric Lévy-type operator on \(\mathbb{R}^{d}\):
$$\displaystyle{\mathcal{L}_{\alpha }^{\kappa }f(x):= \mbox{ p.v.}\int _{\mathbb{R}^{d}}(\,f(x + z) - f(x))\frac{\kappa (x,z)} {\vert z\vert ^{d+\alpha }} \mathrm{d}z,}$$
where \(0 <\kappa _{0}\leqslant \kappa (x,z)\leqslant \kappa _{1}\), κ(x, z) = κ(x, −z), and \(\vert \kappa (x,z) -\kappa (\,y,z)\vert \leqslant \kappa _{2}\vert x - y\vert ^{\beta }\) for some β ∈ (0, 1). In Chen and Zhang (Probab Theory Relat Fields 165:267–312, 2016), we obtained two-sided estimates on the fundamental solution (also called heat kernel) p α κ (t, x, y) of \(\mathcal{L}_{\alpha }^{\kappa }\). In this note, we establish pointwise estimate on \(\vert p_{\alpha }^{\kappa }(t,x,y) - p_{\alpha }^{\tilde{\kappa }}(t,x,y)\vert\) in terms of \(\|\kappa -\tilde{\kappa }\|_{\infty }\).

Keywords

Heat kernel estimate Levi’s method Non-symmetric stable-like operator Strong stability 

AMS 2010 Mathematics Subject Classification

Primary 60J35 47G20 60J75; Secondary 47D07 

Notes

Acknowledgements

The research of ZC is partially supported by NSF Grants DMS-1206276. The research of XZ is partially supported by NNSFC grant of China (Nos. 11271294, 11325105).

References

  1. 1.
    R.F. Bass, D.A. Levin, Transition probabilities for symmetric jump processes. Trans. Am. Math. Soc. 354, 2933–2953 (2002)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    R.F. Bass, H. Ren, Meyers inequality and strong stability for stable-like operators. J. Funct. Anal. 265, 28–48 (2013)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    L. Caffarelli, L. Silvestre, The Evans-Krylov theorem for nonlocal fully nonlinear equations. Ann. Math. 174, 1163–1187 (2011)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Z.-Q. Chen, Symmetric jump processes and their heat kernel estimates. Sci. China Ser. A Math. 52, 1423–1445 (2009)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Z.-Q. Chen, T. Kumagai, Heat kernel estimates for stable-like processes on d-sets. Stoch. Process. Appl. 108, 27–62 (2003)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Z.-Q. Chen, T. Kumagai, Heat kernel estimates for jump processes of mixed types on metric measure spaces. Probab. Theory Relat. Fields 140, 277–317 (2008)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Z.-Q. Chen, X. Zhang, Heat kernels and analyticity of non-symmetric jump diffusion semigroups. Probab. Theory Relat. Fields 165, 267–312 (2016)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Z.-Q. Chen, Y. Hu, Z. Qian, W. Zheng, Stability and approximations of symmetric diffusion semigroups and kernels. J. Funct. Anal. 152, 255–280 (1998)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of WashingtonSeattleUSA
  2. 2.School of Mathematics and StatisticsWuhan UniversityWuhanPeople’s Republic of China

Personalised recommendations