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)].
Reviewer: Liliana Popa (Iaşi)
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 |
Keywords:
fractional Laplacian; Dirichlet problem; heat kernel estimate; Lipschitz domain; boundary Harnack principleCitations:
Zbl 1203.60114References:
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