Geodesic
A probabilistic representation of the fractional differential operator
Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 33, Tome 515 (2022), pp. 5-18

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We consider a class of Lévy processes that includes symmetric $\alpha$-stable processes for $\alpha \in (1,2)$. We obtain a family of stochastic operators using these processes and study the family's properties. We show that constructed stochastic operators approximate the fractional differential operator of order $\alpha$ for the spectral parameter with non-positive real part. 
@article{ZNSL_2022_515_a0,
     author = {T. E. Abildaev},
     title = {A probabilistic representation of the fractional differential operator},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {5--18},
     publisher = {mathdoc},
     volume = {515},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2022_515_a0/}
}
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T. E. Abildaev. A probabilistic representation of the fractional differential operator. Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 33, Tome 515 (2022), pp. 5-18. http://geodesic.mathdoc.fr/item/ZNSL_2022_515_a0/