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Stochastic Analysis and Related TopicsStrong Stability of Heat Kernels of Non-symmetric Stable-Like Operators

Stochastic Analysis and Related Topics: Strong Stability of Heat Kernels of Non-symmetric... [Let d≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$d\geqslant 1$$ \end{document} and α ∈ (0, 2). Consider the following non-local and non-symmetric Lévy-type operator on ℝd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathbb{R}^{d}$$ \end{document}: Lακf(x):=p.v.∫ℝd(f(x+z)−f(x))κ(x,z)|z|d+αdz,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\displaystyle{\mathcal{L}_{\alpha }^{\kappa }f(x):= \mbox{ p.v.}\int _{\mathbb{R}^{d}}(\,f(x + z) - f(x))\frac{\kappa (x,z)} {\vert z\vert ^{d+\alpha }} \mathrm{d}z,}$$ \end{document} where 0<κ0≤κ(x,z)≤κ1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$0 <\kappa _{0}\leqslant \kappa (x,z)\leqslant \kappa _{1}$$ \end{document}, κ(x, z) = κ(x, −z), and |κ(x,z)−κ(y,z)|≤κ2|x−y|β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\vert \kappa (x,z) -\kappa (\,y,z)\vert \leqslant \kappa _{2}\vert x - y\vert ^{\beta }$$ \end{document} 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) pακ(t, x, y) of Lακ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal{L}_{\alpha }^{\kappa }$$ \end{document}. In this note, we establish pointwise estimate on |pακ(t,x,y)−pακ̃(t,x,y)|\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\vert p_{\alpha }^{\kappa }(t,x,y) - p_{\alpha }^{\tilde{\kappa }}(t,x,y)\vert$$ \end{document} in terms of ∥κ−κ̃∥∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\|\kappa -\tilde{\kappa }\|_{\infty }$$ \end{document}.] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

Stochastic Analysis and Related TopicsStrong Stability of Heat Kernels of Non-symmetric Stable-Like Operators

Part of the Progress in Probability Book Series (volume 72)
Editors: Baudoin, Fabrice; Peterson, Jonathon
Chen, Zhen-Qing; Zhang, Xicheng
Stochastic Analysis and Related Topics — Sep 27, 2017
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References (8)

Publisher
Springer International Publishing
Copyright
© Springer International Publishing AG 2017
ISBN
978-3-319-59670-9
Pages
57 –65
DOI
10.1007/978-3-319-59671-6_2
Publisher site
See Chapter on Publisher Site

Abstract

[Let d≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$d\geqslant 1$$ \end{document} and α ∈ (0, 2). Consider the following non-local and non-symmetric Lévy-type operator on ℝd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathbb{R}^{d}$$ \end{document}: Lακf(x):=p.v.∫ℝd(f(x+z)−f(x))κ(x,z)|z|d+αdz,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\displaystyle{\mathcal{L}_{\alpha }^{\kappa }f(x):= \mbox{ p.v.}\int _{\mathbb{R}^{d}}(\,f(x + z) - f(x))\frac{\kappa (x,z)} {\vert z\vert ^{d+\alpha }} \mathrm{d}z,}$$ \end{document} where 0<κ0≤κ(x,z)≤κ1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$0 <\kappa _{0}\leqslant \kappa (x,z)\leqslant \kappa _{1}$$ \end{document}, κ(x, z) = κ(x, −z), and |κ(x,z)−κ(y,z)|≤κ2|x−y|β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\vert \kappa (x,z) -\kappa (\,y,z)\vert \leqslant \kappa _{2}\vert x - y\vert ^{\beta }$$ \end{document} 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) pακ(t, x, y) of Lακ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal{L}_{\alpha }^{\kappa }$$ \end{document}. In this note, we establish pointwise estimate on |pακ(t,x,y)−pακ̃(t,x,y)|\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\vert p_{\alpha }^{\kappa }(t,x,y) - p_{\alpha }^{\tilde{\kappa }}(t,x,y)\vert$$ \end{document} in terms of ∥κ−κ̃∥∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\|\kappa -\tilde{\kappa }\|_{\infty }$$ \end{document}.]

Published: Sep 27, 2017

Keywords: Heat kernel estimate; Levi’s method; Non-symmetric stable-like operator; Strong stability; Primary 60J35; 47G20; 60J75; Secondary 47D07

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