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2009 Heat kernel estimates and Harnack inequalities for some Dirichlet forms with non-local part
Mohammud Foondun
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Electron. J. Probab. 14: 314-340 (2009). DOI: 10.1214/EJP.v14-604

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 $\cal E$.

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Mohammud Foondun. "Heat kernel estimates and Harnack inequalities for some Dirichlet forms with non-local part." Electron. J. Probab. 14 314 - 340, 2009. https://doi.org/10.1214/EJP.v14-604

Information

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

zbMATH: 1190.60069
MathSciNet: MR2480543
Digital Object Identifier: 10.1214/EJP.v14-604

Subjects:
Primary: 60J35
Secondary: 60J75

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

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