## Electronic Journal of Probability

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

Mohammud Foondun

#### 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

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
Secondary: 60J75: Jump processes

Rights

#### 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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