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 Electronic Journal of Probability > Vol. 14 (2009) > Paper 11 open journal systems 


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

Mohammud Foondun, University of Utah


Abstract
We consider the Dirichlet form given by begin{eqnarray*} sE(f,f)&=&frac{1}{2}int_{bR^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_{bR^dtimes bR^d} (f(y)-f(x))^2J(x,y)dxdy. end{eqnarray*} 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 $sE$.


Full text: PDF

Pages: 314-340

Published on: February 2, 2009
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Electronic Journal of Probability. ISSN: 1083-6489