Intertwining certain fractional derivatives
The authors provide an intertwining relation involving the Riemann-Liouville operators of order \(\alpha\in(1,2)\) [\textit{J. Bertoin}, Lévy Processes. Cambridge: Cambridge University Press (1996; Zbl 0861.60003), Chapter 2]. For their proof, they use and extend a criterion found in [\textit{P. Carmona}, \textit{F. Petit} and \textit{M. Yor}, Stochastics Stochastics Rep. 47, No. 1--2, 71--101 (1994; Zbl 0830.60072)]. Then, they provide a new proof of some results (from [\textit{V. Bernyk}, \textit{R. C. Dalang} and \textit{G. Peskir}, Ann. Probab. 39, No. 6, 2385--2423 (2011; Zbl 1235.60036)]) by interpreting the stable process reflected at its infimum as the unique self-similar recurrent extension leaving 0 continuously of the stable process killed when it enters the negative half-line. Finally, invoking a closed formula for the densities of exponential functionals of certain spectrally negative Lévy processes, obtained by \textit{P. Patie} [C. R., Math., Acad. Sci. Paris 347, No. 7--8, 407--411 (2009; Zbl 1162.60015); Ann. Probab. 40, No. 2, 765--787 (2012; Zbl 1241.60020)], and based on Mellin inversion, they give another proof of the key argument of their main theorem concerning the intertwining relation.
scientific article; zbMATH DE number 6031716
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