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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Journal ArticleDOI
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.
Journal ArticleDOI
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 paper, the authors present a self-contained and unified approach to Bernstein functions and closely related function classes, bringing together old and establishing new connections, and an extensive list of complete Bernstein functions with their representations is provided.
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Limit theorems for continuous-time random walks with infinite mean waiting times
TL;DR: In this article, the scaling limit of a continuous-time random walk is shown to be an operator Levy motion subordinated to the hitting time process of a classical stable subordinator.
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Sur les fonctions absolument monotones
TL;DR: Les fonctions absolument monotones jouent le m~me rble fondamenta l duns la th6orie des f o n c t i o n s analytiques d 'une variable r6elle que les fonsctions (simplement) monotone for la classe g6n6rale des fonction £ variat ion born6e as mentioned in this paper.
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TL;DR: The notion of long range dependence is discussed from a variety of points of view, and a new approach is suggested, including connections with non-stationary processes.