Shot-down stable processes
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866909979239251968 |
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| author | Bogdan, Krzysztof Jastrzȩbski, Kajetan Kassmann, Moritz Kijaczko, Michał Popławski, Paweł |
| author_facet | Bogdan, Krzysztof Jastrzȩbski, Kajetan Kassmann, Moritz Kijaczko, Michał Popławski, Paweł |
| contents | The shot-down process is a strong Markov process which is annihilated, or shot down, when jumping over or to the complement of a given open subset of a vector space. Due to specific features of the shot-down time, such processes suggest new type of boundary conditions for nonlocal differential equations. In this work we construct the shot-down process for the fractional Laplacian in Euclidean space. For smooth bounded sets $D$, we study its transition density and characterize Dirichlet form. We show that the corresponding Green function is comparable to that of the fractional Laplacian with Dirichlet conditions on $D$. However, for nonconvex $D$, the transition density of the shot-down stable process is incomparable with the Dirichlet heat kernel of the fractional Laplacian for $D$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_12290 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Shot-down stable processes Bogdan, Krzysztof Jastrzȩbski, Kajetan Kassmann, Moritz Kijaczko, Michał Popławski, Paweł Probability Analysis of PDEs Functional Analysis 35R09, 31C25 (primary), 60J35, 60J75 (secondary) The shot-down process is a strong Markov process which is annihilated, or shot down, when jumping over or to the complement of a given open subset of a vector space. Due to specific features of the shot-down time, such processes suggest new type of boundary conditions for nonlocal differential equations. In this work we construct the shot-down process for the fractional Laplacian in Euclidean space. For smooth bounded sets $D$, we study its transition density and characterize Dirichlet form. We show that the corresponding Green function is comparable to that of the fractional Laplacian with Dirichlet conditions on $D$. However, for nonconvex $D$, the transition density of the shot-down stable process is incomparable with the Dirichlet heat kernel of the fractional Laplacian for $D$. |
| title | Shot-down stable processes |
| topic | Probability Analysis of PDEs Functional Analysis 35R09, 31C25 (primary), 60J35, 60J75 (secondary) |
| url | https://arxiv.org/abs/2301.12290 |