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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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