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

Jacek Mucha

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arXiv · 1910.12821

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

Jacek Mucha

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Abstract

We obtain an integral formula for the distribution of the first hitting time of the origin for one-dimensional $\alpha$-stable processes $X_t$, where $\alpha\in(1,2)$. We also find a spectral-type integral formula for the transition operators $P_0^t$ of $X_t$ killed upon hitting the origin. Both expressions involve exponentially growing oscillating functions, which play a role of generalised eigenfunctions for $P_0^t$.

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Jacek Mucha. 2019-10-28. Spectral theory for one-dimensional (non-symmetric) stable processes killed upon hitting the origin. https://arxiv.org/abs/1910.12821

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