Time-fractional Poisson Process Research Articles

Fractional Criticality Theory and Its Application in Seismology

To understand how the temporal non-locality («memory») properties of a process affect its critical regimes, the power-law compound and time-fractional Poisson process is presented as a universal hereditary model of criticality. Seismicity is considered as an application of the theory of criticality. On the basis of the proposed hereditarian criticality model, the critical regimes of seismicity are investigated. It is shown that the seismic process has the property of «memory» (non-locality over time) and statistical time-dependence of events. With a decrease in the fractional exponent of the Poisson process, the relaxation slows down, which can be associated with the hardening of the medium and the accumulation of elastic energy. Delayed relaxation is accompanied by an abnormal increase in fluctuations, which is caused by the non-local correlations of random events over time. According to the found criticality indices, the seismic process is in subcritical regimes for the zero and first moments and in supercritical regimes for the second statistical moment of events’ reoccurrence frequencies distribution. The supercritical regimes indicate the instability of the deformation changes that can go into a non-stationary regime of a seismic process.

Renewal processes linked to fractional relaxation equations with variable order

We introduce and study here a renewal process defined by means of a time-fractional relaxation equation with derivative order α(t) varying with time t≥0. In particular, we use the operator introduced by Scarpi in the seventies [23] and later reformulated in the regularized Caputo sense in [5], inside the framework of the so-called general fractional calculus. The obtained model extends the well-known time-fractional Poisson process of fixed order α∈(0,1) and tries to overcome its limitation consisting in the constancy of the derivative order (and therefore of the memory degree of the inter-arrival times) with respect to time. The variable order renewal process is proved to fall outside the usual subordinated representation, since it can not be simply defined as a Poisson process with random time (as happens in the standard fractional case). Finally a related continuous-time random walk model is analyzed and its limiting behavior established.

Squirrels can remember little: A random walk with jump reversals induced by a discrete-time renewal process

We consider a class of discrete-time random walks with directed unit steps on the integer line. The direction of the steps is reversed at the time instants of events in a discrete-time renewal process and is maintained at uneventful time instants. This model represents a discrete-time semi-Markovian generalization of the telegraph process. We derive exact formulae for the propagator using generating functions. We prove that for geometrically distributed waiting times in the diffusive limit, this walk converges to the classical telegraph process. We consider the large-time asymptotics of the expected position: For waiting time densities with finite mean the walker remains in the average localized close to the departure site whereas escapes for fat-tailed waiting-time densities (i.e. densities with infinite mean) by a sublinear power-law. We explore anomalous diffusion features by accounting for the ‘aging effect’ as a hallmark of non-Markovianity where the discrete-time version of the ‘aging renewal process’ comes into play. By deriving pertinent distributions of this process we obtain explicit formulae for the variance when the waiting-times are Sibuya-distributed. In this case and generally for fat-tailed waiting time PDFs a t2-ballistic superdiffusive scaling emerges in the large time limit. In contrast if the waiting time PDF between the step reversals is light-tailed (‘narrow’ with finite mean and variance) the walk exhibits normal diffusion and for ‘broad’ waiting time PDFs (with finite mean and infinite variance) superdiffusive large time scaling. We also consider time-changed versions where the walk is subordinated to a continuous-time point process such as the time-fractional Poisson process. This defines a new class of biased continuous-time random walks exhibiting several regimes of anomalous diffusion.

Extended eigenvalue–eigenvector method

In this paper, we show that the eigenvalue–eigenvector method (EEM) and its extension can be utilized to obtain the state probabilities of Poisson-type counting processes. An extension of the EEM, namely, the extended eigenvalue–eigenvector method (EEEM) is introduced which can be used to solve the linear system of fractional differential equations involving Caputo fractional derivative. As an illustration, we obtain the known state probabilities of the time fractional Poisson process and the fractional birth process using the EEEM.

Convoluted Fractional Poisson Process

In this paper, we introduce and study a convoluted version of the time fractional Poisson process by taking the discrete convolution with respect to space variable in the system of fractional differential equations that governs its state probabilities. We call the introduced process as the convoluted fractional Poisson process (CFPP). The explicit expression for the Laplace transform of its state probabilities are obtained whose inversion yields its one-dimensional distribution. Some of its statistical properties such as probability generating function, moment generating function, moments etc. are obtained. A special case of CFPP, namely, the convoluted Poisson process (CPP) is studied and its time-changed subordination relationships with CFPP are discussed. It is shown that the CPP is a L\'evy process using which the long-range dependence property of CFPP is established. Moreover, we show that the increments of CFPP exhibits short-range dependence property.

Tempered fractional Poisson processes and fractional equations with Z-transform

In this article, we derive the state probabilities of different type of space- and time-fractional Poisson processes using z-transform. We work on tempered versions of time-fractional Poisson process and space-fractional Poisson processes. We also introduce Gegenbauer type fractional differential equations and their solutions using z-transform. Our results generalize and complement the results available on fractional Poisson processes in several directions.

Time-inhomogeneous fractional Poisson processes defined by the multistable subordinator

The space-fractional and the time-fractional Poisson processes are two well-known models of fractional evolution. They can be constructed as standard Poisson processes with the time variable replaced by a stable subordinator and its inverse, respectively. The aim of this paper is to study nonhomogeneous versions of such models, which can be defined by means of the so-called multistable subordinator (a jump process with nonstationary increments), denoted by . Firstly, we consider the Poisson process time-changed by H and we obtain its explicit distribution and governing equation. Then, by using the right-continuous inverse of H, we define an inhomogeneous analog of the time-fractional Poisson process.

Non-homogeneous space-time fractional Poisson processes

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 process (SFPP). We study the fractional generalization of the non-homogeneous Poisson process and call it the non-homogeneous space-time fractional Poisson process (NHSTFPP). We compute their pmf and generating function and investigate the associated differential equation. The limit theorems for the NHSTFPP process are studied. We study the distributional properties, the asymptotic expansion of the correlation function of the non-homogeneous time fractional Poisson process (NHTFPP) and subsequently investigate the long-range dependence (LRD) property of a special NHTFPP. We investigate the limit theorem for the fractional non-homogeneous Poisson process (FNHPP) studied by Leonenko et al. (2014). Finally, we present some simulated sample paths of the NHSTFPP process.

Studies on generalized Yule models

We present a generalization of the Yule model for macroevolution in which, for the appearance of genera, we consider point processes with the order statistics property, while for the growth of species we use nonlinear time-fractional pure birth processes or a critical birth-death process. Further, in specific cases we derive the explicit form of the distribution of the number of species of a genus chosen uniformly at random for each time. Besides, we introduce a time-changed mixed Poisson process with the same marginal distribution as that of the time-fractional Poisson process.

Saigo space–time fractional Poisson process via Adomian decomposition method

We obtain the state probabilities of various fractional versions of the classical homogeneous Poisson process using an alternate and simpler method known as the Adomian decomposition method (ADM). Generally these state probabilities are obtained by evaluating probability generating function using Laplace transform. A generalization of the space–time fractional Poisson process involving the Caputo type Saigo differential operator is introduced and its state probabilities are obtained using ADM.

Some probabilistic properties of fractional point processes

ABSTRACTIn this article, the first hitting times of generalized Poisson processes Nf(t), related to Bernštein functions f are studied. For the space-fractional Poisson processes, Nα(t), t > 0 (corresponding to f = xα), the hitting probabilities P{Tαk < ∞} are explicitly obtained and analyzed. The processes Nf(t) are time-changed Poisson processes N(Hf(t)) with subordinators Hf(t) and here we study and obtain probabilistic features of these extended counting processes. A section of the paper is devoted to processes of the form where are generalized grey Brownian motions. This involves the theory of time-dependent fractional operators of the McBride form. While the time-fractional Poisson process is a renewal process, we prove that the space–time Poisson process is no longer a renewal process.

Correlated fractional counting processes on a finite-time interval

We present some correlated fractional counting processes on a finite-time interval. This will be done by considering a slight generalization of the processes in Borges et al. (2012). The main case concerns a class of space-time fractional Poisson processes and, when the correlation parameter is equal to 0, the univariate distributions coincide with those of the space-time fractional Poisson process in Orsingher and Polito (2012). On the one hand, when we consider the time fractional Poisson process, the multivariate finite-dimensional distributions are different from those presented for the renewal process in Politi et al. (2011). We also consider a case concerning a class of fractional negative binomial processes.

Correlated fractional counting processes on a finite-time interval

We present some correlated fractional counting processes on a finite-time interval. This will be done by considering a slight generalization of the processes in Borges et al. (2012). The main case concerns a class of space-time fractional Poisson processes and, when the correlation parameter is equal to 0, the univariate distributions coincide with those of the space-time fractional Poisson process in Orsingher and Polito (2012). On the one hand, when we consider the time fractional Poisson process, the multivariate finite-dimensional distributions are different from those presented for the renewal process in Politi et al. (2011). We also consider a case concerning a class of fractional negative binomial processes.

State-Dependent Fractional Point Processes

In this paper we analyse the fractional Poisson process where the state probabilitiespkνk(t),t≥ 0, are governed by time-fractional equations of order 0 &amp;lt; νk≤ 1 depending on the numberkof events that have occurred up to timet. We are able to obtain explicitly the Laplace transform ofpkνk(t) and various representations of state probabilities. We show that the Poisson process with intermediate waiting times depending on νkdiffers from that constructed from the fractional state equations (in the case of νk= ν, for allk, they coincide with the time-fractional Poisson process). We also introduce a different form of fractional state-dependent Poisson process as a weighted sum of homogeneous Poisson processes. Finally, we consider the fractional birth process governed by equations with state-dependent fractionality.

State-Dependent Fractional Point Processes

In this paper we analyse the fractional Poisson process where the state probabilities pkνk(t), t ≥ 0, are governed by time-fractional equations of order 0 &lt; νk ≤ 1 depending on the number k of events that have occurred up to time t. We are able to obtain explicitly the Laplace transform of pkνk(t) and various representations of state probabilities. We show that the Poisson process with intermediate waiting times depending on νk differs from that constructed from the fractional state equations (in the case of νk = ν, for all k, they coincide with the time-fractional Poisson process). We also introduce a different form of fractional state-dependent Poisson process as a weighted sum of homogeneous Poisson processes. Finally, we consider the fractional birth process governed by equations with state-dependent fractionality.

The space-fractional Poisson process

In this paper, we introduce the space-fractional Poisson process whose state probabilities pkα(t), t≥0, α∈(0,1], are governed by the equations (d/dt)pkα(t)=−λα(1−B)αpkα(t), where (1−B)α is the fractional difference operator found in the time series analysis. We explicitly obtain the distributions pkα(t), the probability generating functions Gα(u,t), which are also expressed as distributions of the minimum of i.i.d. uniform random variables. The comparison with the time-fractional Poisson process is investigated and finally, we arrive at the more general space–time-fractional Poisson process of which we give the explicit distribution.