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

Conference paper
Part of the Progress in Probability book series (PRPR, volume 72)


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 \(\|\kappa -\tilde{\kappa }\|_{\infty }\).


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

AMS 2010 Mathematics Subject Classification

Primary 60J35 47G20 60J75; Secondary 47D07 



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).


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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

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