A Semigroup Approach to Fractional Poisson Processes (original) (raw)
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Fractional Poisson process with random drift
Electronic Journal of Probability, 2014
We study the connection between PDEs and Lévy processes running with clocks given by time-changed Poisson processes with stochastic drifts. The random times we deal with are therefore given by time-changed Poissonian jumps related to some Frobenious-Perron operators K associated to random translations. Moreover, we also consider their hitting times as a random clock. Thus, we study processes driven by equations involving time-fractional operators (modelling memory) and fractional powers of the difference operator I − K (modelling jumps). For this large class of processes we also provide, in some cases, the explicit representation of the transition probability laws. To this aim, we show that a special role is played by the translation operator associated to the representation of the Poisson semigroup.
The fractional Poisson process and the inverse stable subordinator
The fractional Poisson process is a renewal process with Mittag-Leffler waiting times. Its distributions solve a time-fractional analogue of the Kolmogorov forward equation for a Poisson process. This paper shows that a traditional Poisson process, with the time variable replaced by an independent inverse stable subordinator, is also a fractional Poisson process. This result unifies the two main approaches in the stochastic theory of time-fractional diffusion equations. The equivalence extends to a broad class of renewal processes that include models for tempered fractional diffusion, and distributed-order (e.g., ultraslow) fractional diffusion. The paper also discusses the relation between the fractional Poisson process and Brownian time.
Statistics & Probability Letters, 2014
Fractional Poisson processes, a rapidly growing area of non-Markovian stochastic processes, are useful in statistics to describe data from counting processes when waiting times are not exponentially distributed. We show that the fractional Kolmogorov-Feller equations for the probabilities at time t can be representated by an infinite linear system of ordinary differential equations of first order in a transformed time variable. These new equations resemble a linear version of the discrete coagulation-fragmentation equations, wellknown from the non-equilibrium theory of gelation, cluster-dynamics and phase transitions in physics and chemistry.
Time-inhomogeneous fractional Poisson processes defined by the multistable subordinator
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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 non-homogeneous versions of such models, which can be defined by means of the so-called multistable subordinator (a jump process with non-stationary increments), denoted by H := H(t), t ≥ 0. 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 analogue of the time-fractional Poisson process.
General Fractional Calculus, Evolution Equations, and Renewal Processes
Integral Equations and Operator Theory, 2011
We develop a kind of fractional calculus and theory of relaxation and diffusion equations associated with operators in the time variable, of the form (D (k) u)(t) = d dt t 0 k(t − τ)u(τ) dτ − k(t)u(0) where k is a nonnegative locally integrable function. Our results are based on the theory of complete Bernstein functions. The solution of the Cauchy problem for the relaxation equation D (k) u = −λu, λ > 0, proved to be (under some conditions upon k) continuous on [(0, ∞) and completely monotone, appears in the description by Meerschaert, Nane, and Vellaisamy of the process N (E(t)) as a renewal process. Here N (t) is the Poisson process of intensity λ, E(t) is an inverse subordinator.
Poisson-type processes governed by fractional and higher-order recursive differential equations
2009
We consider some fractional extensions of the recursive differential equation governing the Poisson process, by introducing combinations of different fractional time-derivatives. We show that the so-called "Generalized Mittag-Leffler functions" (introduced by Prabhakar [20]) arise as solutions of these equations. The corresponding processes are proved to be renewal, with density of the intearrival times (represented by Mittag-Leffler functions) possessing power, instead of exponential, decay, for t tending to infinite. On the other hand, near the origin the behavior of the law of the interarrival times drastically changes for the parameter fractional parameter varying in the interval (0,1). For integer values of the parameter, these models can be viewed as a higher-order Poisson processes, connected with the standard case by simple and explict relationships.
On the Fractional Poisson Process and the Discretized Stable Subordinator
Axioms, 2015
We consider the renewal counting number process N = N (t) as a forward march over the non-negative integers with independent identically distributed waiting times. We embed the values of the counting numbers N in a "pseudo-spatial" non-negative half-line x ≥ 0 and observe that for physical time likewise we have t ≥ 0. Thus we apply the Laplace transform with respect to both variables x and t. Applying then a modification of the Montroll-Weiss-Cox formalism of continuous time random walk we obtain the essential characteristics of a renewal process in the transform domain and, if we are lucky, also in the physical domain. The process t = t(N ) of accumulation of waiting times is inverse to the counting number process, in honour of the Danish mathematician and telecommunication engineer A.K. Erlang we call it the Erlang process. It yields the probability of exactly n renewal events in the interval (0, t]. We apply our Laplace-Laplace formalism to the fractional Poisson process whose waiting times are of Mittag-Leffler type and to a renewal process whose waiting tmes are of Wright type. The process of Mittag-Leffler type includes as a limiting case the classical Poisson process, the process of Wright type represents the discretized stable subordinator and a re-scaled version of it was used in our method of parametric subordination of time-space fractional diffusion processes. Properly rescaling the counting number process N (t) and the Erlang process t(N ) yields as diffusion limits the inverse stable and the stable subordinator, respectively.
State-dependent fractional point processes
Journal of Applied Probability, 2015
In this paper we analyse the fractional Poisson process where the state probabilities p k ν k (t), t ≥ 0, are governed by time-fractional equations of order 0 < ν k ≤ 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 p k ν k (t) and various representations of state probabilities. We show that the Poisson process with intermediate waiting times depending on ν k differs from that constructed from the fractional state equations (in the case of ν k = 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.
4 Fractional Poisson process with random drift
2014
We study the connection between PDEs and Lévy processes running with clocks given by time-changed Poisson processes with stochastic drifts. The random times we deal with are therefore given by time-changed Poissonian jumps related to some Frobenious-Perron operators K associated to random translations. Moreover, we also consider their hitting times as a random clock. Thus, we study processes driven by equations involving time-fractional operators (modelling memory) and fractional powers of the difference operator I − K (modelling jumps). For this large class of processes we also provide, in some cases, the explicit representation of the transition probability laws. To this aim, we show that a special role is played by the translation operator associated to the representation of the Poisson semigroup.