scispace - formally typeset
Search or ask a question
Journal ArticleDOI

Non-homogeneous space-time fractional Poisson processes

TL;DR: The space-time fractional Poisson process (STFPP) as mentioned in this paper is a generalization of the TFPP and the space fractional poisson process, defined by Orsingher and Poilto (2012).
Abstract: The space-time fractional Poisson process (STFPP), defined by Orsingher and Poilto (2012), is a generalization of the time fractional Poisson process (TFPP) and the space fractional Poisson...
Citations
More filters
Posted Content
TL;DR: In this article, a model of anomalous diffusion in anisotropic medium, which is obtained as a weak limit of suitable continuous-time random walks, is presented as a possible application.
Abstract: It is well-known that compositions of Markov processes with inverse subordinators are governed by integro-differential equations of generalized fractional type. This kind of processes are of wide interest in statistical physics as they are connected to anomalous diffusions. In this paper we consider a generalization; more precisely we mean componentwise compositions of $\mathbb{R}^d$-valued Markov processes with the components of an independent multivariate inverse subordinator. As a possible application, we present a model of anomalous diffusion in anisotropic medium, which is obtained as a weak limit of suitable continuous-time random walks.

8 citations

Journal ArticleDOI
TL;DR: In this article, a model of anomalous diffusion in anisotropic medium, which is obtained as a weak limit of suitable continuous-time random walks, is presented as a possible application.

8 citations

Journal ArticleDOI
01 May 2020
TL;DR: In this article, the compound Poisson processes of order $k$ (CPPoK) were introduced and its properties were discussed, using mixture of tempered stable subordinator and its right continuous inverse, the two subordinated CPPoK with various distributional properties were studied.
Abstract: In this article, the compound Poisson processes of order $k$ (CPPoK) is introduced and its properties are discussed. Further, using mixture of tempered stable subordinator (MTSS) and its right continuous inverse, the two subordinated CPPoK with various distributional properties are studied. It is also shown that space and tempered space fractional versions of CPPoK and PPoK can be obtained, which generalize the results in the literature.

3 citations

Posted Content
TL;DR: In this article, the authors introduced fractional Poisson felds of order k in n-dimensional Euclidean space and derived the marginal probabilities, governing difference-differential equations of the introduced processes.
Abstract: In this article, we introduce fractional Poisson felds of order k in n-dimensional Euclidean space $R_n^+$. We also work on time-fractional Poisson process of order k, space-fractional Poisson process of order k and tempered version of time-space fractional Poisson process of order k in one dimensional Euclidean space $R_1^+$. These processes are defined in terms of fractional compound Poisson processes. Time-fractional Poisson process of order k naturally generalizes the Poisson process and Poisson process of order k to a heavy tailed waiting times counting process. The space-fractional Poisson process of order k, allows on average infinite number of arrivals in any interval. We derive the marginal probabilities, governing difference-differential equations of the introduced processes. We also provide Watanabe martingale characterization for some time-changed Poisson processes.

2 citations

References
More filters
Book
01 Jan 2004
TL;DR: In this paper, the authors present a general theory of Levy processes and a stochastic calculus for Levy processes in a direct and accessible way, including necessary and sufficient conditions for Levy process to have finite moments.
Abstract: Levy processes form a wide and rich class of random process, and have many applications ranging from physics to finance Stochastic calculus is the mathematics of systems interacting with random noise Here, the author ties these two subjects together, beginning with an introduction to the general theory of Levy processes, then leading on to develop the stochastic calculus for Levy processes in a direct and accessible way This fully revised edition now features a number of new topics These include: regular variation and subexponential distributions; necessary and sufficient conditions for Levy processes to have finite moments; characterisation of Levy processes with finite variation; Kunita's estimates for moments of Levy type stochastic integrals; new proofs of Ito representation and martingale representation theorems for general Levy processes; multiple Wiener-Levy integrals and chaos decomposition; an introduction to Malliavin calculus; an introduction to stability theory for Levy-driven SDEs

2,908 citations

Book
31 May 2010
TL;DR: The Eulerian Functions The Bessel Functions The Error Functions The Exponential Integral Functions The Mittag-Leffler Functions The Wright Functions as mentioned in this paper The Eulerians Functions
Abstract: Essentials of Fractional Calculus Essentials of Linear Viscoelasticity Fractional Viscoelastic Media Waves in Linear Viscoelastic Media: Dispersion and Dissipation Waves in Linear Viscoelastic Media: Asymptotic Methods Pulse Evolution in Fractional Viscoelastic Media The Eulerian Functions The Bessel Functions The Error Functions The Exponential Integral Functions The Mittag-Leffler Functions The Wright Functions.

1,593 citations

MonographDOI
01 Jan 2009

902 citations

01 Jan 1971
TL;DR: In this article, a linear operator of order functions of order (1.2) is defined and an operator of fractional integration is employed to prove results on the solutions of the integral equation.
Abstract: is an entire function of order $({\rm Re}\alpha)^{-1}$ and contains several well-known special functions as particular cases. We define a linear operator $\mathfrak{C}(\alpha, \beta;\rho;\lambda)$ on a space $L$ of functions by the integral in (1.2) and employ an operator of fractional integration $I^{\mu}$ : $L\rightarrow L$ to prove results on $\mathfrak{C}(\alpha, \beta;\rho;\lambda)$ ; these results are subsequently used to discuss theorems on the solutions of (1.2). The technique used can be apPlied to obtain analogous results on the integral equation

822 citations