Spectral theory for one-dimensional (non-symmetric) stable processes\n killed upon hitting the origin

2019/10/28 by Jacek Mucha, Mucha, Jacek
Mathematics · Physics and Astronomy · #Cold Atom Physics and Bose-Einstein Condensates #FOS: Mathematics #Probability (math.PR) #Quantum chaos and dynamical systems #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1910.12821

openalex publication_date 2019/10/28 · openalex created_date 2022/07/28 · openalex updated_date 2026/07/28

Abstract

We obtain an integral formula for the distribution of the first hitting time\nof the origin for one-dimensional \α-stable processes X_t, where\n\α\∈(1,2). We also find a spectral-type integral formula for the\ntransition operators P₀^t of X_t killed upon hitting the origin. Both\nexpressions involve exponentially growing oscillating functions, which play a\nrole of generalised eigenfunctions for P₀^t.\n

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