## Heat kernels for reflected diffusions with jumps on inner uniform domains

HTML articles powered by AMS MathViewer

- by Zhen-Qing Chen, Panki Kim, Takashi Kumagai and Jian Wang PDF
- Trans. Amer. Math. Soc.
**375**(2022), 6797-6841

## Abstract:

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 $(x, y)$.## References

- Sebastian Andres and Martin T. Barlow,
*Energy inequalities for cutoff functions and some applications*, J. Reine Angew. Math.**699**(2015), 183–215. MR**3305925**, DOI 10.1515/crelle-2013-0009 - D. G. Aronson,
*Non-negative solutions of linear parabolic equations*, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3)**22**(1968), 607–694. MR**435594** - Martin T. Barlow, Richard F. Bass, and Takashi Kumagai,
*Parabolic Harnack inequality and heat kernel estimates for random walks with long range jumps*, Math. Z.**261**(2009), no. 2, 297–320. MR**2457301**, DOI 10.1007/s00209-008-0326-5 - A. Bensoussan, J.-L. Lions, and G. Papanicolaou,
*Asymptotic analysis for periodic structures*, AMS Chelsea Publishing, Providence, RI, 2011. Corrected reprint of the 1978 original [MR0503330]. MR**2839402**, DOI 10.1090/chel/374 - Richard F. Bass and Pei Hsu,
*Some potential theory for reflecting Brownian motion in Hölder and Lipschitz domains*, Ann. Probab.**19**(1991), no. 2, 486–508. MR**1106272** - Zhen-Qing Chen and Masatoshi Fukushima,
*Symmetric Markov processes, time change, and boundary theory*, London Mathematical Society Monographs Series, vol. 35, Princeton University Press, Princeton, NJ, 2012. MR**2849840** - Zhen-Qing Chen, Panki Kim, and Takashi Kumagai,
*Weighted Poincaré inequality and heat kernel estimates for finite range jump processes*, Math. Ann.**342**(2008), no. 4, 833–883. MR**2443765**, DOI 10.1007/s00208-008-0258-8 - Zhen-Qing Chen, Panki Kim, and Takashi Kumagai,
*On heat kernel estimates and parabolic Harnack inequality for jump processes on metric measure spaces*, Acta Math. Sin. (Engl. Ser.)**25**(2009), no. 7, 1067–1086. MR**2524930**, DOI 10.1007/s10114-009-8576-7 - Zhen-Qing Chen, Panki Kim, and Takashi Kumagai,
*Global heat kernel estimates for symmetric jump processes*, Trans. Amer. Math. Soc.**363**(2011), no. 9, 5021–5055. MR**2806700**, DOI 10.1090/S0002-9947-2011-05408-5 - Zhen-Qing Chen, Panki Kim, Takashi Kumagai, and Jian Wang,
*Heat kernel upper bounds for symmetric Markov semigroups*, J. Funct. Anal.**281**(2021), no. 4, Paper No. 109074, 40. MR**4249776**, DOI 10.1016/j.jfa.2021.109074 - Zhen-Qing Chen, Panki Kim, and Renming Song,
*Dirichlet heat kernel estimates for rotationally symmetric Lévy processes*, Proc. Lond. Math. Soc. (3)**109**(2014), no. 1, 90–120. MR**3237737**, DOI 10.1112/plms/pdt068 - Zhen-Qing Chen, Panki Kim, and Renming Song,
*Dirichlet heat kernel estimates for subordinate Brownian motions with Gaussian components*, J. Reine Angew. Math.**711**(2016), 111–138. MR**3456760**, DOI 10.1515/crelle-2013-0090 - Zhen-Qing Chen and Takashi Kumagai,
*Heat kernel estimates for stable-like processes on $d$-sets*, Stochastic Process. Appl.**108**(2003), no. 1, 27–62. MR**2008600**, DOI 10.1016/S0304-4149(03)00105-4 - Zhen-Qing Chen and Takashi Kumagai,
*Heat kernel estimates for jump processes of mixed types on metric measure spaces*, Probab. Theory Related Fields**140**(2008), no. 1-2, 277–317. MR**2357678**, DOI 10.1007/s00440-007-0070-5 - Zhen-Qing Chen and Takashi Kumagai,
*A priori Hölder estimate, parabolic Harnack principle and heat kernel estimates for diffusions with jumps*, Rev. Mat. Iberoam.**26**(2010), no. 2, 551–589. MR**2677007**, DOI 10.4171/RMI/609 - Zhen-Qing Chen, Takashi Kumagai, and Jian Wang,
*Stability of heat kernel estimates for symmetric non-local Dirichlet forms*, Mem. Amer. Math. Soc.**271**(2021), no. 1330, v+89. MR**4300221**, DOI 10.1090/memo/1330 - Zhen-Qing Chen, Takashi Kumagai, and Jian Wang,
*Stability of parabolic Harnack inequalities for symmetric non-local Dirichlet forms*, J. Eur. Math. Soc. (JEMS)**22**(2020), no. 11, 3747–3803. MR**4167018**, DOI 10.4171/jems/996 - Zhen-Qing Chen, Takashi Kumagai, and Jian Wang,
*Heat kernel estimates and parabolic Harnack inequalities for symmetric Dirichlet forms*, Adv. Math.**374**(2020), 107269, 71. MR**4157572**, DOI 10.1016/j.aim.2020.107269 - Masatoshi Fukushima, Yoichi Oshima, and Masayoshi Takeda,
*Dirichlet forms and symmetric Markov processes*, Second revised and extended edition, De Gruyter Studies in Mathematics, vol. 19, Walter de Gruyter & Co., Berlin, 2011. MR**2778606** - Alexander Grigor′yan,
*Heat kernel upper bounds on a complete non-compact manifold*, Rev. Mat. Iberoamericana**10**(1994), no. 2, 395–452. MR**1286481**, DOI 10.4171/RMI/157 - Alexander Grigor’yan,
*Heat kernel and analysis on manifolds*, AMS/IP Studies in Advanced Mathematics, vol. 47, American Mathematical Society, Providence, RI; International Press, Boston, MA, 2009. MR**2569498**, DOI 10.1090/amsip/047 - Alexander Grigor’yan and Jiaxin Hu,
*Upper bounds of heat kernels on doubling spaces*, Mosc. Math. J.**14**(2014), no. 3, 505–563, 641–642 (English, with English and Russian summaries). MR**3241758**, DOI 10.17323/1609-4514-2014-14-3-505-563 - Alexander Grigor’yan, Jiaxin Hu, and Ka-Sing Lau,
*Generalized capacity, Harnack inequality and heat kernels of Dirichlet forms on metric measure spaces*, J. Math. Soc. Japan**67**(2015), no. 4, 1485–1549. MR**3417504**, DOI 10.2969/jmsj/06741485 - Alexander Grigor’yan, Eryan Hu, and Jiaxin Hu,
*Two-sided estimates of heat kernels of jump type Dirichlet forms*, Adv. Math.**330**(2018), 433–515. MR**3787551**, DOI 10.1016/j.aim.2018.03.025 - Pavel Gyrya and Laurent Saloff-Coste,
*Neumann and Dirichlet heat kernels in inner uniform domains*, Astérisque**336**(2011), viii+144 (English, with English and French summaries). MR**2807275** - Rosario N. Mantegna and H. Eugene Stanley,
*Stochastic process with ultraslow convergence to a Gaussian: the truncated Lévy flight*, Phys. Rev. Lett.**73**(1994), no. 22, 2946–2949. MR**1303317**, DOI 10.1103/PhysRevLett.73.2946 - Jun Masamune, Toshihiro Uemura, and Jian Wang,
*On the conservativeness and the recurrence of symmetric jump-diffusions*, J. Funct. Anal.**263**(2012), no. 12, 3984–4008. MR**2990064**, DOI 10.1016/j.jfa.2012.09.014 - P. A. Meyer,
*Renaissance, recollements, mélanges, ralentissement de processus de Markov*, Ann. Inst. Fourier (Grenoble)**25**(1975), no. 3-4, xxiii, 465–497 (French, with English summary). MR**415784**, DOI 10.5802/aif.593 - Hiroyuki Ôkura,
*Recurrence and transience criteria for subordinated symmetric Markov processes*, Forum Math.**14**(2002), no. 1, 121–146. MR**1880197**, DOI 10.1515/form.2002.001 - L. Saloff-Coste,
*A note on Poincaré, Sobolev, and Harnack inequalities*, Internat. Math. Res. Notices**2**(1992), 27–38. MR**1150597**, DOI 10.1155/S1073792892000047 - Laurent Saloff-Coste,
*Aspects of Sobolev-type inequalities*, London Mathematical Society Lecture Note Series, vol. 289, Cambridge University Press, Cambridge, 2002. MR**1872526** - Laurent Saloff-Coste,
*The heat kernel and its estimates*, Probabilistic approach to geometry, Adv. Stud. Pure Math., vol. 57, Math. Soc. Japan, Tokyo, 2010, pp. 405–436. MR**2648271**, DOI 10.2969/aspm/05710405 - Karl-Theodor Sturm,
*Analysis on local Dirichlet spaces. I. Recurrence, conservativeness and $L^p$-Liouville properties*, J. Reine Angew. Math.**456**(1994), 173–196. MR**1301456**, DOI 10.1515/crll.1994.456.173 - Karl-Theodor Sturm,
*Analysis on local Dirichlet spaces. II. Upper Gaussian estimates for the fundamental solutions of parabolic equations*, Osaka J. Math.**32**(1995), no. 2, 275–312. MR**1355744** - K. T. Sturm,
*Analysis on local Dirichlet spaces. III. The parabolic Harnack inequality*, J. Math. Pures Appl. (9)**75**(1996), no. 3, 273–297. MR**1387522** - Dachun Yang and Yuan Zhou,
*New properties of Besov and Triebel-Lizorkin spaces on RD-spaces*, Manuscripta Math.**134**(2011), no. 1-2, 59–90. MR**2745254**, DOI 10.1007/s00229-010-0384-y - L. Wu, Modeling financial security returns using Lévy processes, In:
*Handbooks in Operations Research and Management Science: Financial Engineering*, Vol. 15, edited by J. Birge and V. Linetsky, North-Holland, 2008, pp. 117–162.

## 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: zqchen@uw.edu
**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: pkim@snu.ac.kr
**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: t-kumagai@waseda.jp
**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: jianwang@fjnu.edu.cn
- 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: https://doi.org/10.1090/tran/8678