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

Zhen Qing Chen; Xicheng Zhang

doi:10.1007/978-3-319-59671-6_2

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

Zhen Qing Chen^*, Xicheng Zhang

^*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceeding › Chapter › peer-review

1 Citation (Scopus)

Abstract

Let d≥ 1 and α ∈ (0, 2). Consider the following non-local and non-symmetric Lévy-type operator on ℝ^d : Lακf(x):=p.v.∫ℝd(f(x+z)−f(x))κ(x,z)|z|d+αdz, where 0 < κ₀≤ κ(x, z) ≤ κ₁, κ(x, z) = κ(x, −z), and |κ(x,z)−κ(y,z)|≤κ2|x−y|β 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 Lακ. In this note, we establish pointwise estimate on |pακ(t,x,y)−pακ̃(t,x,y)| in terms of ∥ κ− κ̃ ∥_∞.

Original language	English
Title of host publication	Progress in Probability
Publisher	Birkhauser
Pages	57-65
Number of pages	9
DOIs	https://doi.org/10.1007/978-3-319-59671-6_2
Publication status	Published - 2017
Externally published	Yes

Publication series

Name	Progress in Probability
Volume	72
ISSN (Print)	1050-6977
ISSN (Electronic)	2297-0428

Keywords

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

Access to Document

10.1007/978-3-319-59671-6_2

Cite this

Chen, Z. Q., & Zhang, X. (2017). Strong Stability of Heat Kernels of Non-symmetric Stable-Like Operators. In Progress in Probability (pp. 57-65). (Progress in Probability; Vol. 72). Birkhauser. https://doi.org/10.1007/978-3-319-59671-6_2

@inbook{3f30543c26a749eda65f93f8e6bef0b2,

title = "Strong Stability of Heat Kernels of Non-symmetric Stable-Like Operators",

abstract = "Let d≥ 1 and α ∈ (0, 2). Consider the following non-local and non-symmetric L{\'e}vy-type operator on ℝd : Lακf(x):=p.v.∫ℝd(f(x+z)−f(x))κ(x,z)|z|d+αdz, where 0 < κ0≤ κ(x, z) ≤ κ1, κ(x, z) = κ(x, −z), and |κ(x,z)−κ(y,z)|≤κ2|x−y|β 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) pα κ(t, x, y) of Lακ. In this note, we establish pointwise estimate on |pακ(t,x,y)−pα{\~κ}(t,x,y)| in terms of ∥ κ− {\~κ} ∥∞.",

keywords = "Heat kernel estimate, Levi{\textquoteright}s method, Non-symmetric stable-like operator, Strong stability",

author = "Chen, {Zhen Qing} and Xicheng Zhang",

note = "Publisher Copyright: {\textcopyright} 2017, Springer International Publishing AG.",

year = "2017",

doi = "10.1007/978-3-319-59671-6_2",

language = "English",

series = "Progress in Probability",

publisher = "Birkhauser",

pages = "57--65",

booktitle = "Progress in Probability",

address = "Switzerland",

}

TY - CHAP

T1 - Strong Stability of Heat Kernels of Non-symmetric Stable-Like Operators

AU - Chen, Zhen Qing

AU - Zhang, Xicheng

PY - 2017

Y1 - 2017

N2 - Let d≥ 1 and α ∈ (0, 2). Consider the following non-local and non-symmetric Lévy-type operator on ℝd : Lακf(x):=p.v.∫ℝd(f(x+z)−f(x))κ(x,z)|z|d+αdz, where 0 < κ0≤ κ(x, z) ≤ κ1, κ(x, z) = κ(x, −z), and |κ(x,z)−κ(y,z)|≤κ2|x−y|β 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) pα κ(t, x, y) of Lακ. In this note, we establish pointwise estimate on |pακ(t,x,y)−pακ̃(t,x,y)| in terms of ∥ κ− κ̃ ∥∞.

AB - Let d≥ 1 and α ∈ (0, 2). Consider the following non-local and non-symmetric Lévy-type operator on ℝd : Lακf(x):=p.v.∫ℝd(f(x+z)−f(x))κ(x,z)|z|d+αdz, where 0 < κ0≤ κ(x, z) ≤ κ1, κ(x, z) = κ(x, −z), and |κ(x,z)−κ(y,z)|≤κ2|x−y|β 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) pα κ(t, x, y) of Lακ. In this note, we establish pointwise estimate on |pακ(t,x,y)−pακ̃(t,x,y)| in terms of ∥ κ− κ̃ ∥∞.

KW - Heat kernel estimate

KW - Levi’s method

KW - Non-symmetric stable-like operator

KW - Strong stability

UR - http://www.scopus.com/inward/record.url?scp=85118472519&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-59671-6_2

DO - 10.1007/978-3-319-59671-6_2

M3 - Chapter

AN - SCOPUS:85118472519

T3 - Progress in Probability

SP - 57

EP - 65

BT - Progress in Probability

PB - Birkhauser

ER -

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

Abstract

Publication series

Keywords

Access to Document

Other files and links

Fingerprint

Cite this