Electronic Journal of Probability

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

Mohammud Foondun

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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

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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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