Convolution-type derivatives, hitting-times of subordinators and time-changed $C_0$-semigroups
TLDR
In this paper, the authors take under consideration subordinators and their inverse processes (hitting-times) and present the governing equations of such processes by means of convolution-type integro-differential operators similar to the fractional derivatives.Abstract:
In this paper we will take under consideration subordinators and their inverse processes (hitting-times). We will present in general the governing equations of such processes by means of convolution-type integro-differential operators similar to the fractional derivatives. Furthermore we will discuss the concept of time-changed $C_0$-semigroup in case the time-change is performed by means of the hitting-time of a subordinator. We will show that such time-change give rise to bounded linear operators not preserving the semigroup property and we will present their governing equations by using again integro-differential operators. Such operators are non-local and therefore we will investigate the presence of long-range dependence.read more
Citations
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Stable non-Gaussian random processes , by G. Samorodnitsky and M. S. Taqqu. Pp. 632. £49.50. 1994. ISBN 0-412-05171-0 (Chapman and Hall).
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Lévy processes and infinitely divisible distributions
TL;DR: In this paper, the authors consider the distributional properties of Levy processes and propose a potential theory for Levy processes, which is based on the Wiener-Hopf factorization.
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Time fractional equations and probabilistic representation
TL;DR: In this article, the existence and uniqueness of solutions for general fractional-time parabolic equations of mixture type, and their probabilistic representations in terms of the corresponding inverse subordinators with or without drifts, were studied.
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Relaxation patterns and semi-Markov dynamics
Mark M. Meerschaert,Bruno Toaldo +1 more
TL;DR: In this paper, a method based on Bernstein functions was proposed to unify three different approaches in the literature, including power law relaxation, semi-Markov process and semi-Maximax relaxation.
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On fractional tempered stable processes and their governing differential equations
TL;DR: The governing equation of the Tempered Stable Subordinator is derived, which generalizes the space-fractional differential equation satisfied by the law of the α-stable subordinator itself and is expressed in terms of the shifted fractional derivative of order α ?
References
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TL;DR: In this article, the traditional diffusion model was extended to the vector fractional diffusion model, which is the state-of-the-art diffusion model for the problem of diffusion.
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Tempering stable processes
TL;DR: In this paper, the authors consider a general and robust class of multivariate tempered stable distributions and establish their identifiable parametrization and prove short and long time behavior of tempered stable Levy processes and investigate their absolute continuity with respect to the underlying α -stable processes.