Abstract
We construct a self-similar solution of the heat equation for the fractional Laplacian with Dirichlet boundary conditions in every fat cone. Furthermore, we give the entrance law from the vertex and the Yaglom limit for the corresponding killed isotropic stable Lévy process and precise large-time asymptotics for solutions of the Cauchy problem in the cone.
Funding Statement
Krzysztof Bogdan was partially supported by the National Science Centre (Poland): grant 2017/27/B/ST1/01339. Łukasz Leżaj was partially supported by the National Science Centre (Poland): grant 2021/41/N/ST1/04139.
Acknowledgments
Part of the research for this work was conducted during Łukasz Leżaj’s post-doctoral stay at the University of Jyväskylä from January to June 2022. He expresses his gratitude to the University for their hospitality and to Professor Stefan Geiss for his warm welcome, guidance, and support during the stay. We thank the referee for very helpful questions and suggestions.
Citation
Krzysztof Bogdan. Piotr Knosalla. Łukasz Leżaj. Dominika Pilarczyk. "Self-similar solution for fractional Laplacian in cones." Electron. J. Probab. 29 1 - 24, 2024. https://doi.org/10.1214/24-EJP1111
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