Journal
FRACTIONAL CALCULUS AND APPLIED ANALYSIS
Volume 25, Issue 2, Pages 488-519Publisher
SPRINGERNATURE
DOI: 10.1007/s13540-022-00025-3
Keywords
Time-fractional evolution equations (primary); Fractional calculus; Randomly scaled Gaussian processes; Randomly scaled Levy processes; Randomely slowed-down; speeded-up Levy processes; Linear fractional Levy motion; Generalized grey Brownian motion; Inverse subordinators; Marichev-Saigo-Maeda operators of fractional calculus; Appell functions; Three parameter Mittag-Leffler function; Feynman-Kac formulae; Anomalous diffusion
Funding
- Projekt DEAL
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In this article, we consider a general class of integro-differential evolution equations that includes the governing equation of the generalized grey Brownian motion and the time- and space-fractional heat equation. We present a general relation between the parameters of the equation and the distribution of the underlying stochastic processes, as well as discuss different classes of processes providing stochastic solutions of these equations. For a subclass of evolution equations containing Marichev-Saigo-Maeda time-fractional operators, we determine the parameters of the corresponding processes explicitly. Moreover, we explain how self-similar stochastic solutions with stationary increments can be obtained via linear fractional Levy motion for suitable pseudo-differential operators in space.
We consider a general class of integro-differential evolution equations which includes the governing equation of the generalized grey Brownian motion and the time- and space-fractional heat equation. We present a general relation between the parameters of the equation and the distribution of the underlying stochastic processes, as well as discuss different classes of processes providing stochastic solutions of these equations. For a subclass of evolution equations, containing Marichev-Saigo-Maeda time-fractional operators, we determine the parameters of the corresponding processes explicitly. Moreover, we explain how self-similar stochastic solutions with stationary increments can be obtained via linear fractional Levy motion for suitable pseudo-differential operators in space.
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