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Time-Inhomogeneous Jump Processes and Variable Order Operators

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Abstract

In this paper we introduce non-decreasing jump processes with independent and time non-homogeneous increments. Although they are not Lévy processes, they somehow generalize subordinators in the sense that their Laplace exponents are possibly different Bernštein functions for each time t. By means of these processes, a generalization of subordinate semigroups in the sense of Bochner is proposed. Because of time-inhomogeneity, two-parameter semigroups (propagators) arise and we provide a Phillips formula which leads to time dependent generators. The inverse processes are also investigated and the corresponding governing equations obtained in the form of generalized variable order fractional equations. An application to a generalized subordinate Brownian motion is also examined.

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Authors and Affiliations

  1. Department of Statistical Sciences, Sapienza - University of Rome, Rome, Italy

    Enzo Orsingher, Costantino Ricciuti & Bruno Toaldo

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  1. Enzo Orsingher
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  2. Costantino Ricciuti
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  3. Bruno Toaldo
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Correspondence to Bruno Toaldo.

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Orsingher, E., Ricciuti, C. & Toaldo, B. Time-Inhomogeneous Jump Processes and Variable Order Operators. Potential Anal 45, 435–461 (2016). https://doi.org/10.1007/s11118-016-9551-4

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  • DOI: https://doi.org/10.1007/s11118-016-9551-4

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