TY - JOUR

T1 - HEAT KERNELS FOR REFLECTED DIFFUSIONS WITH JUMPS ON INNER UNIFORM DOMAINS

AU - Chen, Zhen Qing

AU - Kim, Panki

AU - Kumagai, Takashi

AU - Wang, Jian

N1 - Funding Information:
Received by the editors April 8, 2021. 2020 Mathematics Subject Classification. Primary 60J35, 60J76; Secondary 31C25, 35K08. Key words and phrases. Reflected diffusions with jumps, symmetric Dirichlet form, inner uniform domain; heat kernel. The research of the first author was partially supported by Simons Foundation Grant 520542. The research of the second author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (No. 2016R1E1A1A01941893). The research of the third author was supported by JSPS KAKENHI Grant Number JP17H01093 and JP22H00099, and by the Alexander von Humboldt Foundation . The research of the fourth author was supported by the National Natural Science Foundation of China (Nos. 11831014 and 12071076), and the Education and Research Support Program for Fujian Provincial Agencies.
Publisher Copyright:
© 2022 by the authors.

PY - 2022/10/1

Y1 - 2022/10/1

N2 - In this paper, we study sharp two-sided heat kernel estimates for a large class of symmetric reflected diffusions with jumps on the closure of an inner uniform domain D in a length metric space. The length metric is the intrinsic metric of a strongly local Dirichlet form. When D is an inner uniform domain in the Euclidean space, a prototype for a special case of the processes under consideration is symmetric reflected diffusions with jumps on D, whose infinitesimal generators are non-local (pseudo-differential) operators L on D of the form Lu(x) = 1/2 d ∑ i,j=1 ∂/∂xi (aij(x) ∂u(x)/∂xj)) +lim ε↓0 ∫ {y∈D: ρD(y,x)>ε}(u(y)−u(x))J(x, y) dy satisfying “Neumann boundary condition”. Here, ρD(x, y) is the length metric on D, A(x) = (aij(x))1≤i,j≤d is a measurable d×d matrix-valued function on D that is uniformly elliptic and bounded, and J(x, y) := 1/Φ(ρD 1 (x, y)) ∫ [α1,α2] c(α, x, y)/ρD(x, y)d+α ν(dα) where ν is a finite measure on [α1, α2] ⊂ (0, 2), Φ is an increasing function on [0, ∞) with c1ec2rβ ≤ Φ(r) ≤ c3ec4rβ for some β ∈ [0, ∞], and c(α, x, y) is a jointly measurable function that is bounded between two positive constants and is symmetric in (x, y).

AB - In this paper, we study sharp two-sided heat kernel estimates for a large class of symmetric reflected diffusions with jumps on the closure of an inner uniform domain D in a length metric space. The length metric is the intrinsic metric of a strongly local Dirichlet form. When D is an inner uniform domain in the Euclidean space, a prototype for a special case of the processes under consideration is symmetric reflected diffusions with jumps on D, whose infinitesimal generators are non-local (pseudo-differential) operators L on D of the form Lu(x) = 1/2 d ∑ i,j=1 ∂/∂xi (aij(x) ∂u(x)/∂xj)) +lim ε↓0 ∫ {y∈D: ρD(y,x)>ε}(u(y)−u(x))J(x, y) dy satisfying “Neumann boundary condition”. Here, ρD(x, y) is the length metric on D, A(x) = (aij(x))1≤i,j≤d is a measurable d×d matrix-valued function on D that is uniformly elliptic and bounded, and J(x, y) := 1/Φ(ρD 1 (x, y)) ∫ [α1,α2] c(α, x, y)/ρD(x, y)d+α ν(dα) where ν is a finite measure on [α1, α2] ⊂ (0, 2), Φ is an increasing function on [0, ∞) with c1ec2rβ ≤ Φ(r) ≤ c3ec4rβ for some β ∈ [0, ∞], and c(α, x, y) is a jointly measurable function that is bounded between two positive constants and is symmetric in (x, y).

KW - Reflected diffusions with jumps

KW - inner uniform domain; heat kernel

KW - symmetric Dirichlet form

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U2 - 10.1090/tran/8678

DO - 10.1090/tran/8678

M3 - Article

AN - SCOPUS:85139556031

SN - 0002-9947

VL - 375

SP - 6797

EP - 6841

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

IS - 10

ER -