We obtain an integral formula for the distribution of the first hitting time of the origin for one-dimensional α-stable processes , where . We also find a spectral-type integral formula for the transition operators of killed upon hitting the origin. Both expressions involve exponentially growing oscillating functions, which play a role of generalised eigenfunctions for .
Work supported by National Science Centre (NCN), Poland, under grant 2015/19/B/ST1/01457
"Spectral theory for one-dimensional (non-symmetric) stable processes killed upon hitting the origin." Electron. J. Probab. 26 1 - 33, 18 February 2021. https://doi.org/10.1214/21-EJP594