1. Introduction
A counting process is nothing but a stochastic process
that counts the number of events that have occurred up to the current time
t, equipped with the following assumptions [
1,
2]:
;
;
for is the number of events occurring in the interval
In recent decades, Poisson processes have found wide application in different research areas [
3,
4], such as medicine and biomedicine, economy, epidemiology, finance, physics and biology [
4,
5,
6,
7,
8,
9]. The exponential decay predicted by the standard Poisson process is used to estimate the inter-arrival distribution of phenomena as phone communication connections even if, recently, a failure of this model has been shown for different complex systems in which the long-term memory effects involve long-tailed properties. Many contributions have been focused on the generalization of the standard Poisson process using fractional calculus and fractional operators providing a fractional version of the Poisson process that allows a power law decay of the counting probabilities to be predicted [
10,
11,
12,
13] A representative example of application of this class of processes is the power-law decay of the duration of network sessions at large session-times. This has led to several important contributions related to fractional Poisson processes [
14], introduced by N. Laskin [
15] as a generalization of the Kolmogorov–Feller equation and recovered by E. Orsingher and L. Beghin [
16], by replacing the time derivative with the fractional Dzerbayshan-Caputo derivative of order
.
This paper is aimed at providing several generalization of counting processes possessing anomalous power-law scaling without applying fractional operators, enforcing the transition structure of Lévy Walks (LW) [
17] described by means of the transitional age-formalism [
18,
19,
20], where the age is defined as the time that has elapsed from the latest transition; i.e., from the latest event.
The article is organized as follows.
Section 2 presents the general structure of simple counting processes, considering Poisson processes and fractional Poisson processes as particular cases. The evolution equations for the probability density with respect to the transition age
are obtained through the age–time dynamics of an LW process, and the boundary conditions are specified and discussed.
Section 3 introduces the concept of generalized counting processes, showing how the stationary assumption in the renewal mechanism can be modified. The case in which the transitional age depends on the actual number of transitions that has occurred is analyzed. In
Section 4, doubly stochastic counting processes [
21,
22], introduced by Cox [
23] and developed by Bartlett [
24], are recovered by assuming that the parameters describing the occurrence of a new event stochastically depend on time. The hierarchical level of stochasticity of these processes can be attributed to environmental fluctuations (environmental stochasticity) that interact with the intrinsic level of stochasticity in the occurrence of events. A scaling analysis is performed, showing that by considering an asymmetrical Poisson–Kac process [
25,
26,
27,
28], the long-term scaling exponent of the counting probability hierarchy can be modulated. Extensions to the model defined in
Section 4 are given and analyzed, focusing on the case in which the stochastic process characterizing the transition rate is associated with the transition mechanism of an LW process. This leads to the presence of two different transition ages related to the occurrence of events and to transitions in the environmental fluctuations.
2. Simple Counting Processes
The age description of LWs [
18,
19] allows for a simple and natural generalization of counting processes. In this section, we consider the general structure of simple counting processes containing Poisson processes and fractional Poisson processes as particular cases. The exact meaning of the concept of “simple counting processes” is given below; see Equation (
13).
Consider the renewal mechanism of an LW for particle dynamics that proceeds via a sequence of events determining a change in the velocity direction. The process is specified by the probability density function for the transition time , corresponding to the time interval between two subsequent events or, equivalently, by the transition rate , related to by the equation . For a Poisson process, , so that .
Let
be the probability density with respect to the transition age
that
k events (corresponding to changes in the velocity direction) have occurred in the time interval
, described by the counting stochastic variable
. Age
corresponds to the state immediately after a transition (event). We indicate with
the stochastic process representing particle transition age at time
t. Then,
Thus,
represents the fraction of particles with an age between
and
that have already performed
k transitions at time
t. The probabilities
of the strict counting process can be obtained as the marginal of this joint density with respect to
; i.e.,
where
represents the probability that
k events have occurred up to time
t.
The evolution equations for
thus follow from the age–time dynamics of an LW process; i.e.,
where
, equipped with the renewal boundary condition
that holds for
, as no boundary condition defines the dynamics of
. The initial condition for
in a counting process can be assumed as
where
is an impulsive Dirac delta distribution centered at
, i.e.,
, and
are the Kronecker symbols, corresponding to the fact that at time
, all the particles possess vanishing transitional age. This corresponds to the classical initial condition used for Continuous Time Random Walks, as discussed in [
20]. Other initial conditions are also possible as thoroughly discussed in [
20], corresponding to a different age preparation of the system. Consider Equation (
3) for
; it simply propagates the initial condition (
5), so that its solution can be expressed as
where
Next, consider a generic
. Using the method of characteristics for first-order linear partial differential equations, the solution of Equation (
3) takes the form
while
for
. That is, the functions
are defined for
and vanish for
. Substituting this equation into the boundary condition (
4), one obtains
where
is the probability density function for the transition times and “∗” indicates the convolution operation. Since for
,
, it follows that
,
, and in general
In terms of the counting probabilities
, this implies
Observe that it is possible to express the counting probability recursively as
with
given by Equation (
8).
If it is possible to represent the system of counting probabilities via the recursive convolutional expression (
13), the counting process is said to be
simple. For simple counting processes, the renewal of the generations is determined by a single function, say
(or
or
); its knowledge defines the process completely. Moreover, giving the expression of the counting probability
for some value
as a function of time, Equation (
13) permits the prediction of the probabilities
for
. Essentially, a simple counting process is characterized by the fact that the renewal equation between subsequent generations (a generation is characterized by a given value of
, say
, so its probabilistic description is given by
) is “autonomous”; i.e., it does not change as either time or generational evolution proceeds.
Examples
The approach developed above encompasses classical counting processes, including also some processes recently studied in connection with fractional calculus [
29,
30,
31,
32,
33]. If
, then one recovers the Poisson process. If
is expressed in terms of the Mittag–Leffler function,
one obtains the fractional Poisson process. Following Laskin [
15],
with
,
, so that
. For
,
, and thus
. Of course, by changing the functional form of
, it is possible to provide a variety of different counting processes that still possess the property (
13) of being simple. For instance, by choosing
as
with
and
, one can recover for
the anomalous long-term scaling properties characterizing fractional Poisson processes. In this case,
, and
.
Figure 1 depicts the first counting probabilities
,
for the simple process defined by Equation (
15) with
and
. Solid lines (a) to (c) represent the theoretical expressions (
12), while symbols represent the results of stochastical simulations. The simulations correspond to an ensemble of
particles. Choosing a time step
(in the simulations
), at each time
, the probability of a new event is given by
, where
is the local value of the age at time
. If no events occur, then
, and otherwise
; i.e., it is reset to zero.
As expected from the theory of fractional Poisson processes, the counting probabilities scale in the long-term regime as
3. Generalized Counting Processes
The stationarity assumption in the renewal mechanism can be generalized in several different ways, and the processes so defined can be referred to as generalized counting processes (GCP). This kind of process can find applications in all fields, such as biology or economy, in which changes occur in the environment determining a progressive variation in the dynamics of events (transitions). The simplest example is taken from the analysis of LW processes addressed in [
19,
20], in which the transition age after any transition is not reset to zero but attains a non-vanishing value that depends on the number of transitions that have occurred. This means that a non decreasing sequence of real numbers
can be introduced,
, such that, while Equation (
3) still holds for the evolution of the densities
, the renewal boundary condition takes the form
Since in most of the cases of interest in connection with LW, the transition rates
are decreasing functions of
, such as for Equation (
15), the boundary condition (
17) determines a slowing down in the occurrence of the transitions, corresponding to a structural aging of the process. The scaling implication of Equation (
17) as regards the dynamics of a LW has been addressed in [
19].
As an initial condition, we can still take Equation (
5). Consequently, with regards to
and
, Equations (
6) and (
8) are still valid. Consider
, in the presence of the boundary condition (
17). The density
for any
is different from zero solely for
and can be expressed in this interval as
where the function
is defined for positive arguments
. The boundary condition (
17) allows the determination of this function
Therefore,
defined for
, while
otherwise. From Equation (
20), the expression for the overall counting probability
follows
in which the function
,
,
has been introduced. The same approach can be applied to all the elements of the system of densities and counting probabilities, and the final result for
is
and vanishes otherwise, where
and
,
are given by
As regards the overall counting probabilities
, one obtains
where we have set
The notation used in Equation (
24) means that
,
and so forth. The proof of this result is developed in
Appendix A. It is important to observe that the counting probabilities
defined by Equation (
26) do not fulfill the requirement (
13) characteristic of a simple counting scheme because of the presence of the factor
that depends explicitly on
k. This result is physically intuitive as the renewal mechanism depends on the generation
k, via the shifts
providing a progressive aging of the process. Observe that the function
as well as
are indeed probabilistically normalized; i.e., they represent density functions,
which follows straightforwardly from their definitions (
24)–(
25).
As an example, consider the process defined by Equation (
15) and considered in the previous section (i.e.,
) and with
where
is a characteristic aging time, so that the aging process depends linearly on the generation number, with
.
Figure 2 depicts some transition functions
defined by Equation (
25) at
and
. For
,
.
Figure 3 compares of the analytical expressions for the counting probabilities Equation (
26) and the results of the stochastic simulation, performed as described in the previous section, with the difference that, at the occurrence of a new event (transition), the age is reset according to the values of
. The first two counting probabilities
and
are not shown as they are identical to the corresponding simple counting problem with
.
Figure 4 depicts the counting probabilities
and
-panel (a) and (b), respectively-for the same process at
, by changing the value of
, from
(simple process) to
. Furthermore, for this class of processes, the asymptotic scaling of the counting probabilities follows Equation (
16), as it is controlled by the functions
, and
,
for any
k.
4. Counting Processes in a Stochastic Environment
It is possible to introduce a further level of complexity (stochasticity) in a counting process by assuming that the parameters describing the occurrence of a new event (such as the transition rate) are not fixed but depend on time in a stochastic way. In other words, they represent a stochastic process.
This generalization could correspond to the case in which the transition mechanism depends on the environmental conditions, and the latter evolve in some random way. Consider for example the statistics of the number of telephone calls within a city, which is a typical phenomenon that can be straightforwardly mapped into a counting process. Its normal statistics can be specified by the function . However, the number of telephone calls can be significantly influenced by the environmental conditions: the sudden occurrence of a calamity (a hurricane, an earthquake, etc.) significantly influences the transition mechanism of the process. Since calamities cannot be easily predicted, it is natural to consider them as stochastic processes. Analogous examples can be provided in biology, especially as regards epidemic spreading or macroevolutionary processes, in which “the event” can be thought of as the origin of a new species (speciation) and the external stochasticity is intrinsic to environmental conditions in geological times.
It is also evident that this kind of counting processes implies a double (hierarchical) level of stochasticity: the intrinsic stochasticity in the occurrence of an event and the environmental stochasticity controlling the variation in the statistical parameters of the process. For these reasons, such processes can be referred to as “doubly stochastic counting processes” or, alternatively, “counting processes in a stochastic environment”. For the sake of brevity, we use the acronym “ES” (environmentally stochastic) to indicate these models. It is assumed that the two sources of stochasticity are independent of each other, and that environmental stochasticity is characterized by a Markovian transition mechanism. This condition could be easily extended to environmental fluctuations possessing semi-Markov properties.
Using the formulation adopted throughout this article, an ES counting process can be characterized by a transition rate
, which is a stochastic process. For instance,
where
is a given function of the transition age and
is a stochastic process, the statistical properties of which are known.
Let us assume that, in the absence of stochasticity in
, the basic counting process is simple. In the presence of Equation (
30), Equations (
3) and (
4) attain the form
, and
where now
are stochastic processes controlled by the statistics of
.
Throughout this article, we consider for
stochastic processes attaining a finite numbers of realizations (states), and the transitions amongst the different states follow Markovian dynamics [
27]. For simplicity, we assume here that
may attain only two values, letting
be a modulation of a Poisson–Kac process [
25,
26], so that Equation (
30) can be explained as
where
is a Poisson process characterized by the transition rate
. For the sake of clarity, we assume for
the more general initial conditions,
,
, and
,
, where
are probability weights,
. Under this assumption, the transition rates vanish (blocking conditions) whenever the parity of
is odd and attain the value
when even. A realization of this process is depicted in
Figure 5 panel (b). For
, the functional form (
15) has been chosen, with
,
. Panel (a) depicts the evolution of
for the corresponding simple counting process (i.e., in the absence of environmental stochasticity,
). The transition rate
over the realization of the process changes with time, because the age
depends on time, and returns to zero after each transition. In panel (c), the corresponding behavior of
—i.e., the number of events up to time
t—is depicted. Panel (b) refers to Equation (
33) for
, with
. The realizations depicted in panels (a) and (b) refer to the same initial seed in the use of the quasi-random number algorithm implemented.
The average transition time in the environmental conditions is , and the corresponding behavior of is depicted in panel (d). The initial dynamics of these two processes are almost identical (only because, by chance, ). Subsequently, around , the ES realization (panel b) undergoes a series of blocking conditions, and correspondingly, experiences a long time-interval of stagnation for . This indicates that the two processes, and the blocking-effects in the choice of , may deeply influence the statistics of the counting process.
The main quantity of interests are the mean field partial counting probability densities
, where
and
refers to the average with respect to the probability measure of the stochastic process
. Following [
19], these quantities satisfy the evolution equations
equipped with the boundary condition, solely on
, as the “-”-component does not perform any transition,
and with the initial conditions
For this process, the counting probabilities
are expressed by
Figure 6 depicts the counting probabilities (for low
k) associated with this process (lines (a) to (c)) obtained by solving Equations (
35)–(
37) for
given by Equation (
15) with
for two values of
, while the environmental stochasticity is characterized by
and
. Symbols refer to the values of
obtained from the stochastic simulation of the process using an ensemble of
elements. The agreement between mean field probabilities and stochastic simulations is excellent. Lines (d) correspond to the behavior of
for the bare simple stochastic process (
), for which, at
,
It can be observed that the hierarchy of counting probabilities
for the ES counting process possesses a different asymptotic scaling than the corresponding bare process, and specifically
This result is remarkable and underlines how a modulation of the transition rate by means of a simple Poisson–Kac mechanism changes the long-term “anomalous” properties of the counting process, determining the occurrence of a long-term effective scaling exponent
The proof of this result is derived in the next paragraph, considering the scaling analysis of Equation (
35).
Next, consider the influence of the other parameters. The transition rate
of the environmental fluctuations does not modify the asymptotic scaling properties but solely influences quantitatively the counting probabilities. This phenomenon is depicted in
Figure 7: the higher the value of
, the faster the event generation. This phenomenon is evident both as regards the relaxation of
(panel a) as the occurrence of the first event (panel b) expressed by the probability
.
Another interesting property can be observed from the decay of
at intermediate time scales. For small
(in the case of the data depicted in
Figure 7, this means
), the probability
displays a relaxational transition from the initial scaling
for
, pertaining to the bare counting process in the absence of environmental fluctuations to the long-term scaling
.
Figure 8 refers to the influence of
; i.e., of the initial preparation of the system. At short time scales, the influence of
is significant, as the initial transient of
can be expressed as the weighted sum
of the behavior of the bare counting process in the absence of environmental fluctuations
and of the unrelaxed dynamics pertaining to the case where
.
4.1. Scaling Analysis
The anomalous scaling behavior observed in the hierarchy of counting probabilities
can be theoretically predicted by considering
, which satisfies the Equation (
35) for
, and no boundary conditions should be specified, meaning that Equation (
35) propagates solely the initial condition. The general solution of these equations can be expressed as
defined for
, where function
entering Equation (
43) can be expressed as the primitive of an auxiliary function
; i.e.,
where
, with
, defined for
, and
are constants independent of
and
t. Substituting Equation (
43) with the position (
44) into the balance Equation (
35), one obtains for
and
the linear homogeneous system
which admits a solution provided that the determinant of the coefficient matrix is equal to zero; i.e.,
the solutions (two) of which are expressed by
Consider the case of
given by Equation (
15) or of any
monotonically decaying to zero for
. For large
, the term
is small, meaning that a Taylor expansion to the first-order provides
There are two independent solutions for
, depending on the determination of the square root, which can be labelled as
and
. From Equation (
48), one obtains that one solution is given, for a large
, by
corresponding to the fast decay mode, since
is a constant. The other solution (slow mode) is expressed as
since for large
,
.
Figure 9 depicts the behavior of the two functions
,
, for
expressed by Equation (
15) with
and
, enhancing the two long-term asymptotes expressed by Equations (
49) and (
50).
It follows from Equation (
50) that if
is given by Equation (
15), the long-term slowest transition rate is expressed by
, where
and consequently, the long-term scaling exponent of the counting probability hierarchy is
, which is consistent with the numerical results and with Equation (
41).
The above analysis suggests that it would be possible to modulate the long-term scaling exponent of the counting probability hierarchy by considering an asymmetrical Poisson–Kac process [
25], characterized by unequal transition rates
and
from the two states. In this case, the balance equations for the mean field partial densities are expressed by the equations
Proceeding as above
Figure 10, the system of solutions for
can still be expressed by Equations (
43) and (
44), where the resulting linear system replacing Equation (
45) is now given by
Equation (
53) provides for the (two) functions
,
, the expressions
which, for a large
(and
), behave as
In the large
-limit, for transition rates
decaying to zero Equation (
54) yields the effective transition rate of the slowest decaying mode:
Equation (
56) implies for the effective scaling exponent
the expression
Stochastic simulation results support quantitatively the validity of Equation (
57), as depicted in
Figure 11. These data have been derived by considering the long-term relaxation of
obtained from an ensemble of
elements with
given by Equation (
15) at
.
4.2. Extensions
The ES model analyzed in the previous paragraph can be extended in different ways. Particularly interesting is the case where the process
entering the constitutive Equation (
33) is associated with the transition mechanism of an LW process, so that [
19]
where
represents the transitional age of the environmental fluctuations, returning to zero at each transition, and
is the transition rate of the environmental fluctuations.
For this class of processes, the mean field statistical description involves the partial densities
, which are parametrized with respect to the transitional ages of the counting process
; i.e., of the microstochasticity, and of the environmental fluctuations (macrostochasticity). For this problem, the evolution equations for the partial counting densities
read
equipped with the boundary conditions at
and
for
, and
for
. As an initial condition one may consider the simplest case,
where
are probability weights.
This case is interesting for several reasons. Conceptually, the presence of two transition ages
and
accounts for the duality in the stochastic nature of the fluctuations, related to the occurrence of events (
) and to the transition in the environmental conditions (
). Suppose that the transition rate of the environmental fluctuations
depends on some parameter
. The degrees of freedom in the choice of
suggest that this model could in principle display a phase transition. For a counting process characterized by a transition rate given by Equation (
15), a phase transition at a critical value
of
would correspond to the break-down of the long-term power-law scaling of the counting probabilities
, such that, either above or below
, the counting probabilities would decay asymptotically slower for any power of
t. The analysis of this model will be developed in forthcoming communications.