## Electronic Journal of Probability

### Fractional Poisson process with random drift

#### Abstract

We study the connection between PDEs and Lévy processes running with clocks given by time-changed Poisson processes with stochastic drifts. The random times we deal with are therefore given by time-changed Poissonian jumps related to some Frobenius-Perron operators $K$ associated to random translations. Moreover, we also consider their hitting times as a random clock. Thus, we study processes driven by equations involving time-fractional operators (modelling memory) and fractional powers of the difference operator $I-K$ (modelling jumps). For this large class of processes we also provide, in some cases, the explicit representation of the transition probability laws. To this aim, we show that a special role is played by the translation operator associated to the representation of the Poisson semigroup.

#### Article information

Source
Electron. J. Probab., Volume 19 (2014), paper no. 122, 26 pp.

Dates
Accepted: 30 December 2014
First available in Project Euclid: 4 June 2016

https://projecteuclid.org/euclid.ejp/1465065764

Digital Object Identifier
doi:10.1214/EJP.v19-3258

Mathematical Reviews number (MathSciNet)
MR3304182

Zentralblatt MATH identifier
1334.60053

Rights

#### Citation

Beghin, Luisa; D'Ovidio, Mirko. Fractional Poisson process with random drift. Electron. J. Probab. 19 (2014), paper no. 122, 26 pp. doi:10.1214/EJP.v19-3258. https://projecteuclid.org/euclid.ejp/1465065764

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