TY - CHAP

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

AU - Chen, Zhen Qing

AU - Zhang, Xicheng

N1 - Publisher Copyright:
© 2017, Springer International Publishing AG.

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 -