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Heat kernel estimates and Harnack inequalities for some Dirichlet forms with non-local part. (English) Zbl 1190.60069
Summary: We consider the Dirichlet form given by
${\mathcal E}(f,f)= \frac12 \int_{\mathbb R^d} \sum_{i,j=1}^d a_{ij}(x) \frac{\partial f(x)}{\partial x_i} \frac{\partial f(x)}{\partial x_j} \,dx +\int_{\mathbb R^d\times\mathbb R^d} (f(y)-f(x))^2 J(x,y)\,dx\,dy.$
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 $${\mathcal E}$$.

MSC:
 60J35 Transition functions, generators and resolvents 60J25 Continuous-time Markov processes on general state spaces 60J75 Jump processes (MSC2010)
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