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

Authors
Publisher
Electronic Journal of Probability
Keywords
• 60J75
• 60J35
• HöLder Continuity
• Integro-Differential Operators. Harnack Inequality. Heat Kernel

## 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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