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Heat kernel estimates for the fractional Laplacian with Dirichlet conditions. (English) Zbl 1204.60074
The main result of the paper gives the following sharp two-sided estimate for the heat kernel of the fractional Laplacian \((-\Delta )^{\alpha /2}\), \(0<\alpha <2\), with Dirichlet condition on the boundary: \[ C^ {-1} P^ {x} (\tau _ {D} > t) P^ {y} (\tau _ {D} > t) \leq \frac{p_ {D} (t, x, y)}{p(t, x, y)} \leq C P^ {x} (\tau _ {D} > t) P^ {y} (\tau _ {D} > t) \] where \(0 < t \leq 1\), \(x, y \in D\), \(C=C(\alpha , D)\) is a constant and \(D\) is a more general type of domain than the case of \(C^{1,1}\) domains, which was treated by Z.-Q. Chen, P. Kim and R. Song, [J. Eur. Math. Soc. (JEMS) 12, No. 5, 1307–1329 (2010; Zbl 1203.60114)].

MSC:
60J35 Transition functions, generators and resolvents
60J50 Boundary theory for Markov processes
60J75 Jump processes (MSC2010)
31B25 Boundary behavior of harmonic functions in higher dimensions
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