Heat kernels for non-symmetric diffusion operators with jumps

Zhen Qing Chen; Eryan Hu; Longjie Xie; Xicheng Zhang

doi:10.1016/j.jde.2017.07.023

Heat kernels for non-symmetric diffusion operators with jumps

Zhen Qing Chen, Eryan Hu^*, Longjie Xie, Xicheng Zhang

^*Corresponding author for this work

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

42 Citations (Scopus)

Abstract

For d⩾2, we prove the existence and uniqueness of heat kernels to the following time-dependent second order diffusion operator with jumps: L_t:=[Formula presented]∑i,j=1da_ij(t,x)∂_ij ²+∑i=1db_i(t,x)∂_i+L_t ^κ, where a=(a_ij) is a uniformly bounded, elliptic, and Hölder continuous matrix-valued function, b belongs to some suitable Kato's class, and L_t ^κ is a non-local α-stable-type operator with bounded kernel κ. Moreover, we establish sharp two-sided estimates, gradient estimate and fractional derivative estimate for the heat kernel under some mild conditions.

Original language	English
Pages (from-to)	6576-6634
Number of pages	59
Journal	Journal of Differential Equations
Volume	263
Issue number	10
DOIs	https://doi.org/10.1016/j.jde.2017.07.023
Publication status	Published - 15 Nov 2017
Externally published	Yes

Keywords

Gradient estimate
Heat kernel
Kato class
Lévy system
Non-local operator
Transition density

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10.1016/j.jde.2017.07.023

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Chen, Z. Q., Hu, E., Xie, L., & Zhang, X. (2017). Heat kernels for non-symmetric diffusion operators with jumps. Journal of Differential Equations, 263(10), 6576-6634. https://doi.org/10.1016/j.jde.2017.07.023

@article{6b535a47ced84bec9e9460e130d19ddc,

title = "Heat kernels for non-symmetric diffusion operators with jumps",

abstract = "For d⩾2, we prove the existence and uniqueness of heat kernels to the following time-dependent second order diffusion operator with jumps: Lt:=[Formula presented]∑i,j=1daij(t,x)∂ij 2+∑i=1dbi(t,x)∂i+Lt κ, where a=(aij) is a uniformly bounded, elliptic, and H{\"o}lder continuous matrix-valued function, b belongs to some suitable Kato's class, and Lt κ is a non-local α-stable-type operator with bounded kernel κ. Moreover, we establish sharp two-sided estimates, gradient estimate and fractional derivative estimate for the heat kernel under some mild conditions.",

keywords = "Gradient estimate, Heat kernel, Kato class, L{\'e}vy system, Non-local operator, Transition density",

author = "Chen, {Zhen Qing} and Eryan Hu and Longjie Xie and Xicheng Zhang",

note = "Publisher Copyright: {\textcopyright} 2017 Elsevier Inc.",

year = "2017",

month = nov,

day = "15",

doi = "10.1016/j.jde.2017.07.023",

language = "English",

volume = "263",

pages = "6576--6634",

journal = "Journal of Differential Equations",

issn = "0022-0396",

publisher = "Academic Press Inc.",

number = "10",

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TY - JOUR

T1 - Heat kernels for non-symmetric diffusion operators with jumps

AU - Chen, Zhen Qing

AU - Hu, Eryan

AU - Xie, Longjie

AU - Zhang, Xicheng

PY - 2017/11/15

Y1 - 2017/11/15

N2 - For d⩾2, we prove the existence and uniqueness of heat kernels to the following time-dependent second order diffusion operator with jumps: Lt:=[Formula presented]∑i,j=1daij(t,x)∂ij 2+∑i=1dbi(t,x)∂i+Lt κ, where a=(aij) is a uniformly bounded, elliptic, and Hölder continuous matrix-valued function, b belongs to some suitable Kato's class, and Lt κ is a non-local α-stable-type operator with bounded kernel κ. Moreover, we establish sharp two-sided estimates, gradient estimate and fractional derivative estimate for the heat kernel under some mild conditions.

AB - For d⩾2, we prove the existence and uniqueness of heat kernels to the following time-dependent second order diffusion operator with jumps: Lt:=[Formula presented]∑i,j=1daij(t,x)∂ij 2+∑i=1dbi(t,x)∂i+Lt κ, where a=(aij) is a uniformly bounded, elliptic, and Hölder continuous matrix-valued function, b belongs to some suitable Kato's class, and Lt κ is a non-local α-stable-type operator with bounded kernel κ. Moreover, we establish sharp two-sided estimates, gradient estimate and fractional derivative estimate for the heat kernel under some mild conditions.

KW - Gradient estimate

KW - Heat kernel

KW - Kato class

KW - Lévy system

KW - Non-local operator

KW - Transition density

UR - http://www.scopus.com/inward/record.url?scp=85026756353&partnerID=8YFLogxK

U2 - 10.1016/j.jde.2017.07.023

DO - 10.1016/j.jde.2017.07.023

M3 - Article

AN - SCOPUS:85026756353

SN - 0022-0396

VL - 263

SP - 6576

EP - 6634

JO - Journal of Differential Equations

JF - Journal of Differential Equations

IS - 10

ER -

Heat kernels for non-symmetric diffusion operators with jumps

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Keywords

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