Abstract
In this paper, we study the precise behavior of the transition density functions of censored (resurrected) α-stable-like processes in C 1,1 open sets in \({\mathbb R^d}\) , where d ≥ 1 and \({\alpha\in (1, 2)}\) . We first show that the semigroup of the censored α-stable-like process in any bounded Lipschitz open set is intrinsically ultracontractive. We then establish sharp two-sided estimates for the transition density functions of a large class of censored α-stable-like processes in C 1,1 open sets. We further obtain sharp two-sided estimates for the Green functions of these censored α-stable-like processes in bounded C 1,1 open sets.
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Chen, ZQ., Kim, P. & Song, R. Two-sided heat kernel estimates for censored stable-like processes. Probab. Theory Relat. Fields 146, 361–399 (2010). https://doi.org/10.1007/s00440-008-0193-3
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DOI: https://doi.org/10.1007/s00440-008-0193-3
Keywords
- Fractional Laplacian
- Censored stable process
- Censored stable-like process
- Symmetric α-stable process
- Symmetric stable-like process
- Heat kernel
- Transition density
- Transition density function
- Green function
- Exit time
- Lévy system
- Boundary Harnack principle
- Parabolic Harnack principle
- Intrinsic ultracontractivity