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

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

2009-01-26

arxiv.org/abs/0901.4127v1

Mathematics

Probability

Scientific paper

We consider the Dirichlet form given by \sE(f,f)&=&{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^d\times \bR^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 $\sE$.

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