Laplacian perturbed by non-local operators

Jie Ming Wang

doi:10.1007/s00209-014-1380-9

Laplacian perturbed by non-local operators

Jie Ming Wang^*

^*Corresponding author for this work

School of Mathematics and Statistics

Research output: Contribution to journal › Article › peer-review

8 Citations (Scopus)

Abstract

Suppose that d ≥ 1 and 0 < β < 2.We establish the existence and uniqueness of the fundamental solution q^b(t, x, y) to the operator (equations found) and b(x,z) is a bounded measurable function on R^d× R^dwith b(x, z) = b(x,−z) for x, z ∈ R^d.We show that if for each x ∈ R^d, b(x, z) ≥ 0 for a.e. z ∈ R^d, then q^b(t, x, y) is a strictly positive continuous function and it uniquely determines a conservative Feller process X^b, which has strong Feller property. Furthermore, sharp two-sided estimates on q^b(t, x, y) are derived.

Original language	English
Pages (from-to)	521-556
Number of pages	36
Journal	Mathematische Zeitschrift
Volume	279
Issue number	1-2
DOIs	https://doi.org/10.1007/s00209-014-1380-9
Publication status	Published - Feb 2015

Keywords

Brownian motion
Feller semigroup
Integral kernel
Laplacian
Lévy system
Non-local operator
Perturbation
Positivity

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10.1007/s00209-014-1380-9

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title = "Laplacian perturbed by non-local operators",

abstract = "Suppose that d ≥ 1 and 0 < β < 2.We establish the existence and uniqueness of the fundamental solution qb(t, x, y) to the operator (equations found) and b(x,z) is a bounded measurable function on Rd× Rdwith b(x, z) = b(x,−z) for x, z ∈ Rd.We show that if for each x ∈ Rd, b(x, z) ≥ 0 for a.e. z ∈ Rd, then qb(t, x, y) is a strictly positive continuous function and it uniquely determines a conservative Feller process Xb, which has strong Feller property. Furthermore, sharp two-sided estimates on qb(t, x, y) are derived.",

keywords = "Brownian motion, Feller semigroup, Integral kernel, Laplacian, L{\'e}vy system, Non-local operator, Perturbation, Positivity",

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language = "English",

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T1 - Laplacian perturbed by non-local operators

AU - Wang, Jie Ming

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N2 - Suppose that d ≥ 1 and 0 < β < 2.We establish the existence and uniqueness of the fundamental solution qb(t, x, y) to the operator (equations found) and b(x,z) is a bounded measurable function on Rd× Rdwith b(x, z) = b(x,−z) for x, z ∈ Rd.We show that if for each x ∈ Rd, b(x, z) ≥ 0 for a.e. z ∈ Rd, then qb(t, x, y) is a strictly positive continuous function and it uniquely determines a conservative Feller process Xb, which has strong Feller property. Furthermore, sharp two-sided estimates on qb(t, x, y) are derived.

AB - Suppose that d ≥ 1 and 0 < β < 2.We establish the existence and uniqueness of the fundamental solution qb(t, x, y) to the operator (equations found) and b(x,z) is a bounded measurable function on Rd× Rdwith b(x, z) = b(x,−z) for x, z ∈ Rd.We show that if for each x ∈ Rd, b(x, z) ≥ 0 for a.e. z ∈ Rd, then qb(t, x, y) is a strictly positive continuous function and it uniquely determines a conservative Feller process Xb, which has strong Feller property. Furthermore, sharp two-sided estimates on qb(t, x, y) are derived.

KW - Brownian motion

KW - Feller semigroup

KW - Integral kernel

KW - Laplacian

KW - Lévy system

KW - Non-local operator

KW - Perturbation

KW - Positivity

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DO - 10.1007/s00209-014-1380-9

M3 - Article

AN - SCOPUS:84927165375

SN - 0025-5874

VL - 279

SP - 521

EP - 556

JO - Mathematische Zeitschrift

JF - Mathematische Zeitschrift

IS - 1-2

ER -

Laplacian perturbed by non-local operators

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Keywords

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