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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Heat kernels for reflected diffusions with jumps on inner uniform domains
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by Zhen-Qing Chen, Panki Kim, Takashi Kumagai and Jian Wang PDF
Trans. Amer. Math. Soc. 375 (2022), 6797-6841


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 $\mathcal {L}$ on $D$ of the form \[ \mathcal {L} u(x)\! =\!\frac 12 \!\sum _{i, j=1}^d\! \frac {\partial }{\partial x_i}\! \left (\!\!a_{ij}(x) \frac {\partial u(x)}{\partial x_j}\!\right ) \!+ \lim _{\varepsilon \downarrow 0}\! \int _{\{y\in D: \, \rho _D(y, x)>\varepsilon \}}\!\! (u(y)-u(x)) J(x, y)\, dy \] satisfying “Neumann boundary condition”. Here, $\rho _D(x,y)$ is the length metric on $D$, $A(x)=(a_{ij}(x))_{1\leq i,j\leq d}$ is a measurable $d\times d$ matrix-valued function on $D$ that is uniformly elliptic and bounded, and \[ J(x,y)≔\frac {1}{\Phi (\rho _D(x,y))} \int _{[\alpha _1, \alpha _2]} \frac {c(\alpha , x,y)} {\rho _D(x,y)^{d+\alpha }} \,\nu (d\alpha ), \] where $\nu$ is a finite measure on $[\alpha _1, \alpha _2] \subset (0, 2)$, $\Phi$ is an increasing function on $[ 0, \infty )$ with $c_1e^{c_2r^{\beta }} \le \Phi (r) \le c_3 e^{c_4r^{\beta }}$ for some $\beta \in [0,\infty ]$, and $c(\alpha , x, y)$ is a jointly measurable function that is bounded between two positive constants and is symmetric in $(x, y)$.
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Additional Information
  • Zhen-Qing Chen
  • Affiliation: Department of Mathematics, University of Washington, Seattle, Washington 98195
  • MR Author ID: 242576
  • ORCID: 0000-0001-7037-4030
  • Email:
  • Panki Kim
  • Affiliation: Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 08826, Republic of Korea
  • MR Author ID: 705385
  • Email:
  • Takashi Kumagai
  • Affiliation: Department of Mathematics, Faculty of Science and Engineering, Waseda University, Tokyo 169-8555, Japan
  • MR Author ID: 338696
  • ORCID: 0000-0001-7515-1055
  • Email:
  • Jian Wang
  • Affiliation: College of Mathematics and Informatics, Fujian Key Laboratory of Mathematical Analysis and Applications (FJKLMAA), Center for Applied Mathematics of Fujian Province (FJNU), Fujian Normal University, 350007 Fuzhou, People’s Republic of China
  • Email:
  • Received by editor(s): April 8, 2021
  • Published electronically: August 11, 2022
  • Additional Notes: 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
  • © Copyright 2022 by the authors
  • Journal: Trans. Amer. Math. Soc. 375 (2022), 6797-6841
  • MSC (2020): Primary 60J35, 60J76; Secondary 31C25, 35K08
  • DOI: