Electronic Journal of Probability

Heat kernel estimates and Harnack inequalities for some Dirichlet forms with non-local part

Mohammud Foondun

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Abstract

We consider the Dirichlet form given by $$ {\cal E}(f,f) = \frac{1}{2}\int_{R^d}\sum_{i,j=1}^d a_{ij}(x)\frac{\partial f(x)}{\partial x_i} \frac{\partial f(x)}{\partial x_j} dx$$ $$ + \int_{R^d \times R^d} (f(y)-f(x))^2J(x,y)dxdy.$$ Under the assumption that the ${a_{ij}}$ are symmetric and uniformly elliptic and with suitable conditions on $J$, the nonlocal part, we obtain upper and lower bounds on the heat kernel of the Dirichlet form. We also prove a Harnack inequality and a regularity theorem for functions that are harmonic with respect to $\cal E$.

Article information

Source
Electron. J. Probab., Volume 14 (2009), paper no. 11, 314-340.

Dates
Accepted: 2 February 2009
First available in Project Euclid: 1 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464819472

Digital Object Identifier
doi:10.1214/EJP.v14-604

Mathematical Reviews number (MathSciNet)
MR2480543

Zentralblatt MATH identifier
1190.60069

Subjects
Primary: 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07]
Secondary: 60J75: Jump processes

Keywords
Integro-differential operators. Harnack inequality. Heat kernel Holder continuity

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Foondun, Mohammud. Heat kernel estimates and Harnack inequalities for some Dirichlet forms with non-local part. Electron. J. Probab. 14 (2009), paper no. 11, 314--340. doi:10.1214/EJP.v14-604. https://projecteuclid.org/euclid.ejp/1464819472


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