Self-similar solution for fractional Laplacian in cones
Krzysztof Bogdan, Piotr Knosalla, {\L}ukasz Le\.zaj, Dominika, Pilarczyk
TL;DR
This paper constructs a self-similar solution for the fractional Laplacian in cones, providing insights into the behavior of stable Lévy processes and solutions to related heat equations in conical domains.
Contribution
It introduces a novel self-similar solution for the fractional Laplacian in cones, with applications to Lévy processes and asymptotic analysis.
Findings
Yaglom limit and entrance law for stable Lévy process derived
Large-time asymptotics for solutions in cones established
Self-similar solutions characterized in conical domains
Abstract
We construct a self-similar solution of the heat equation for the fractional Laplacian with Dirichlet boundary conditions in every fat cone. As applications, we give the Yaglom limit and entrance law for the corresponding killed isotropic stable L\'{e}vy process and precise large-time asymptotics for solutions of the Cauchy problem in the cone.
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Taxonomy
TopicsAdvanced Mathematical Modeling in Engineering · Nonlinear Partial Differential Equations · Spectral Theory in Mathematical Physics