Should i stay or should i go? zero-size jumps in random walks for lévy flights

1 Introduction

This research is motivated by the fact that, in the literature dedicated to random walks for anomalous diffusion, the specific value of the frequency of the jumps with zero-size is disregarded as if it does not affect the motion of the walker, e.g., [50, 65, 6, 70, 62, 46, 2, 47, 63, 44, 16, 18, 23, 33, 45, 71, 36]. Actually, in the literature it is disregarded if the walker does not move in the majority of the iterations because the most frequent jump-size is zero (i.e., the jump-size distribution is unimodal with mode located in zero) or, in opposition, if the walker always moves because the jumps with zero-size never occur (i.e., the jump-size distribution is bi-modal and equal to zero in zero). As a matter of fact, in the large-time limit, this irrelevance holds true for random-walk models of the Brownian motion when the corresponding jump-processes follow a Gaussian law or a coin-flipping rule. On the other side, anomalous diffusion is explained by Lévy flights, rather than by the Brownian motion, and Lévy flights are defined as Markovian random walks that converge to stable densities because of power-law distributed jumps [65, 6].

Before starting, we declare that we are more confident in using terminology, notions and notation adopted in physics. Hence, we do not refer to the considered diffusion processes as random-walk models with Lebesgue measure when they satisfy the Central Limit Theorem or as long-jump processes with Hausdorff measure when they satisfy the generalised Central Limit Theorem in the sense of Lévy, but we term the processes according to the resulting probability density function (pdf), namely we call Brownian motion the processes whose walker’s distribution converges to a Gaussian law and we term Lévy flights the processes whose walker’s distribution converges to a stable law.

The evolution in time of Lévy flights emerges to be governed by a fractional diffusion equation [66, 46, 47, 71]. A number of properties of Lévy flights has been studied, e.g., [10, 11, 13, 57, 55, 54]. However, in the probabilistic derivation of the fractional diffusion equation, the distinctive singularity of the fractional Laplacian is obtained, with a constant time-step, when the distribution of jumps is bi-modal and equal to zero in zero [68], see also Appendix B. Hence, the frequency of jumps with zero-size is expected to play a key role in modelling fractional anomalous diffusion.

Here we analyse this literature inconsistency between the apparent irrelevance of the frequency of zero-size jumps, as promoted by random-walk models for Lévy flights, and the link between the jump distribution and the distinctive singularity of the fractional Laplacian, as established by probability arguments for deriving the fractional diffusion equation [68]. In particular, in this paper we provide an example to show that it is not guaranteed that a Markovian continuous-time random walk (CTRW), with jump-sizes uncoupled from the waiting-times and displaying power-law tails, converges to a stable density when the jumps follow a bi-modal distribution equal to zero in zero, that is the one in agreement with the probabilistic derivation [68], and, moreover, the resulting diffusive process can be non self-similar.

The consequence of this loss of self-similarity is the emergence of a time-scale for realizing the large-time limit. Such time-scale results to be dependent on the stability parameter by spanning from zero to infinity. Hence, the large-time limit could not be reached in real systems.

Even if this can be considered a second order effect, in diffusion processes the maximum of the walker’s distribution stays located in the starting site at all elapsed times and, in the large-time limit, the scaling-law in time of the distribution around its maximum is important for determining the properties of recurrence and transience of the random walk [1]. Therefore, attaining the large-time limit together with the scaling-law in time of the walker’s distribution maximum have a fundamental role on determining the suitability of the CTRW approach for modelling Lévy flights, because of a failing performance or a compatible performance by a CTRW model for reproducing recurrence and transience of the many observed signatures of Lévy flights. Since Lévy flights can be modelled, for example, also through stochastic differential equations driven by Lévy-noise [14, 46, 11], through parametric subordination [26, 24, 25] or through other methods [21, 20, 22, 69], this result establishes a criterion for the selection of proper modelling approaches for Lévy flights.

In particular, in the spirit of Pólya’s theorem [60, 52], recurrence and transience are of paramount importance on the way home. The motion of wild animals has been associated many times to power-law distributions both in the view of the celebrated, and criticised, Lévy flights foraging hypothesis [15, 32, 58, 72, 4, 61, 34], and also in the view of the concept of site fidelity [19, 17]: the recurrent visit of an animal to a previously occupied location [28, 3, 7, 5]. The fact that the large-time limit for determining the recurrence or transience of the process could not be realistically reached clashes against the concept of site fidelity, which is straightforwardly related to recurrence, and this provides a further weakness of the power-law hypothesis for animal behaviour.

To conclude, our result highlights the need to investigate more deeply the role of zero-size jumps in random walks with power-law distributed jumps.

In Section 2, we call the attention to the small wavelength expansion of the characteristic function of jumps that are power-law distributed and we derive the conditions for the loss of self-similarity in the resulting process. In Section 3, we discuss the significance of the derived result both i) in the framework of the probabilistic derivation of the fractional diffusion equation, as far as the relation between zero-size jumps and the distinctive singularity of the fractional Laplacian is concerned, and ii) in the framework of animal behaviour, as far as the concept of site fidelity and the Lévy flights foraging hypothesis are concerned, and we furtherly highlight the effect due to zero-size jumps for reaching the large-time limit. In Section 4, we provide summary and conclusions in the perspective of future research.

2 Power-law tails, zero-size jumps and self-similarity

We denote by ρ(x; t) the distribution of the walker’s displacement x at time t, with x = (x₁, …, x_N) ∈ ℝ^N and t > 0, such that

Moreover, we assume as initial datum ρ(x; 0) = δ(x). In a CTRW model, the distribution ρ(x; t) is governed by the Montroll–Weiss equation [50, 64] that, in the Markovian case with jump-sizes uncoupled from waiting-times, reads

where ρ̂(κ; t) and φ̂(κ) = φ̂(ℓκ) are the characteristic functions of ρ(x; t) and of the jump pdfφ(x) = φ(x/ℓ)/ℓ^N, respectively, with ℓ as the length-unit of the jumps and τ as the time-unit - and also the mean value - of the waiting-times that are exponentially distributed.

Here, we are interested in establishing an observable that allows for discriminating between the case “Should I stay?”: when the walker does not move in the majority of the iterations because the most frequent jump-size is zero, i.e., φ(x) is an unimodal jump pdf such that φ(0) = sup{φ(x) : x ∈ ℝ^N}; and the opposite case “Should I go?”: when the walker always moves because the jumps with zero-size never occur, i.e., φ(x) is a bi-modal jump pdf such that φ(0) = inf{φ(x) : x ∈ ℝ^N} = 0. We say that jump-sizes in the “Should I go?” condition follow a rule à la coin-flipping.

By applying in one single step the analog of the Kramers–Moyal expansion and of the Pawula theorem, whatever the jump pdfφ(x) is such that it holds [46, 47, 45, 71, 36]

then, if in the small wavelength expansion (2.3) we set α = 2, from equation (2.2) we obtain that ρ(x; t) solves the evolution problem

where 𝓓 = ℓ²/τ is the diffusion coefficient and, actually, ρ(x; t) is a Gaussian density:

so we say that this CTRW is a model for the Brownian motion, with variance

On the contrary, if in the small wavelength expansion (2.3) we consider the interval 0 < α < 2, from equation (2.2) we obtain that ρ(x; t) solves the fractional evolution problem

where 𝓓_α = ℓ^α/τ is the fractional diffusion coefficient, therefore 𝓓₂ = 𝓓, and (−Δ)^α/2, α ∈ (0, 2), is the fractional Laplacian [8, 37, 38] such that, actually, ρ(x; t) is a Lévy stable density [29], i.e.,

and we say that this CTRW is a model for Lévy flights [71], with fractional absolute moments [46, 48]

Here we do not specify any particular definition of the fractional Laplacian, because we consider only processes in an unbounded domain and in this case there are at least ten equivalent definitions [37]. In bounded domains, the spectral representation results to be favourite, with respect others, for diffusion problems because it is based on the heat kernel, namely the Brownian motion [12]. Mathematical and physical interpretations of the fractional Laplacian are provided by Hilfer [30, 31] and a number of physical systems governed by space fractional kinetics are reported, for example, by Uchaikin & Sibatov [67]. A noteworthy case of diffusion problem (2.7) is the special case α = 1 that leads to the Cauchy (Lorentz) distribution [29, 1]

where 𝓝 is the normalization factor.

Whenever the derivation of CTRW models for the Brownian motion and for Lévy flights is strictly based on the small wavelength expansion of the characteristic function of jumps (2.3), the difference between the conditions “Should I stay?” and “Should I go?” is neglected because this limit provides the behaviour of the tails of the resulting walker’s distribution ρ(x; t) and then, in this respect, the distribution of small jump-sizes is irrelevant. Actually, the application of this method could mislead to the undeclared statement - on the back of the mind - that the small wavelength expansion of the characteristic function of the jump pdf should be a series with alternating signs, namely an alternating series, like the Taylor expansion of a completely monotonic function is, but this is not always true for stable densities. Here we investigate the effect of this non-alternation of signs in the resulting random walk.

In the case of the CTRW model for the Brownian motion, this undeclared statement is true both in the “Should I stay?” and “Should I go?” conditions, see Figure 1, in fact, in the isotropic case, it holds

Figure 1

Left: plots of the one-dimensional (N = 1) jump pdfs(2.11a) and (2.12a) corresponding to the Should I stay? and Should I go? conditions, respectively, for the generation of the Brownian motion from CTRW models. Right: plots of the Gaussian walker’s distribution ρ(x; t) (2.5) of the CTRW models for the Brownian motion as generated by using the one-dimensional (N = 1) jump pdfs(2.11a) (filled symbols) and (2.12a) (empty symbols) at t = 10τ, 100τ, 1000τ represented by squares, diamons and triangles, respectively: the short-time effects of the coin-flipping rule (2.12a) is visible.

and also

where ê is the basis vector (ê ⋅ ê = 1) with isotropic symmetry so x = |x| ê = 2ℓê and κ = |κ| ê such that

Conversely, in the case of CTRW models for Lévy flights, although this undeclared statement is true in the “Should I stay?” condition and in fact, with 0 < α < 2, it holds

and ρ(x; t) solves (2.7), unfortunately, the alternating sign expansion (2.13b) is not always true in the “Should I go?” condition.

For mathematical convenience, we provide an example, in the one-dimensional (N = 1) case, of a jump pdf whose small wavelength expansion of the characteristic function is not an alternating series.

In order to arrange the “Should I go?” condition within the framework of power-law distributed jumps, we consider the one-sided (extremal) Lévy densities Lα−α(x) [59], with x ∈ ℝ, i.e., Lα−α(x) > 0 when x > 0 and Lα−α(x) = 0 when x ≤ 0, with 0 < α < 1. Thus one-sided Lévy densities can be used for defining a jump rule à la coin-flipping by taking into account also the remarkable limit L1−1(x) = δ(x − 1). The power-law of the tails of the jump pdfφ(x) is spanned inside the range of the stable parameter (0, 1) ∪ (1, 2) as follows:

We observe that the Brownian coin-flipping rule (2.12a) is recovered from both the Lévy coin-flipping rules (2.14a) and (2.14b) when α = 1, while the special case of the Cauchy distribution (2.10) is not achievable. A study of the considered Lévy coin-flipping rules (2.14a, 2.14b) is moved in Appendix A where formulae useful for the following analysis are derived.

See plots of the one-dimensional (N = 1) jump pdfs(2.13a) and (2.14a, 2.14b) in Figure 2.

Figure 2

Plots of the one-dimensional (N = 1) jump pdfs(2.13a) - left column - and (2.14a, 2.14b) - right column - corresponding to the Should I stay? and Should I go? conditions, respectively. Left column: solid lines decrease as |x|^−(α+1) and dashed lines as |x|^−(2α+1) with 0 < α < 1. Right column: solid lines decrease as |x|^−(α+1) and dashed lines as |x|^{−[(α+1)+1]} with 0 < α < 1.

From the Lévy coin-flipping rule (2.14a), we have that for κ ∈ ℝ the characteristic function is

hence expansion (2.15) is an alternating series if 0 < α ≤ 1/2 such that ρ(x; t) solves (2.7), but if 1/2 < α < 1 then we have

and expansion (2.15) is not a series with alternating signs. In this case, from equation (2.2) it follows that

with

and therefore ρ(x; t) solves the fractional evolution problem

where

and

such that ρ(x; t) is a convolution of Lévy stable densities:

with fractional absolute moments [9, 56]

From the characteristic function (2.17), we have that the tails of the distribution ρ(x; t) (2.22) follow the same power-law of the tails of a stable density of stability parameter α, see Figures 3 and 4, namely

Figure 3

Plots of the tails of the walker’s distribution ρ(x; t) obtained with jump pdf(2.14a) at times t = 10 τ, 100τ, 1000τ marked by squares, triangles and diamonds, respectively. The dashed lines represent the power-law decaying |x|^−(α+1).

Figure 4

Plots of the central part of the walker’s distribution ρ(x; t) obtained with jump pdf(2.14a) at times t = 10 τ, 100τ, 1000τ marked by squares, triangles and diamonds, respectively. The dotted lines represent Lévy stable densities of index α and the dashed lines the power-law decaying |x|^−(α+1). The loss of self-similarity in the interval 1/2 < α < 1 is evident.

Convolution integral (2.22) has been studied in a number of papers as fundamental solution of double-order space-fractional diffusion equation [9], as generalised Voigt function [40, 56], or as a sum of two independent stable random variables [53, 51].

From the Lévy coin-flipping rule (2.14b), we have that

since 0 < α < 1, expansion (2.25) is an alternating series and ρ(x; t) solves (2.7) by replacing α → (α + 1) and Dα→Dα=2(α+1)/2Γ(1/α)α1+αsinπ2αℓα+1τ,, see Figure 5.

Figure 5

Plots of the walker’s distribution ρ(x; t) obtained with jump pdf(2.14b) at times t = 10 τ, 100τ, 1000τ marked by squares, triangles and diamonds, respectively. The dotted lines represent Lévy stable densities of index (α +1) and the dashed lines the power-law decaying |x|^{−[(α+1)+1]}.

Before ending this section, we want to highlight that both jump pdfs (2.14a) and (2.14b) tend to the coin-flipping rule (2.12a) when α → 1 and so both the resulting processes tend to the Brownian motion. However, from series expansions (2.15) and (2.25) it emerges that they tend to the Brownian motion in a very different way. In fact, we observe that series expansion (2.15) reduces to (2.12b) through the third term ∝ |κ|^2α because the coefficient of the second term goes to 0, while series expansion (2.25) reduces to (2.12b) through the second term ∝ |κ|^α+1 because the coefficient of the third term goes to 0.

3 Discussion and significance

3.2 Indetermined homecoming: the effect of the Lévy coin-flipping rule on transience (and recurrence) of anomalous diffusion processes

Our second remark on the above results concerns the problem of transience (and recurrence) for power-law processes and its relation with the concept of site fidelity in animal behaviour [28, 3, 7, 5, 19, 17] and with the Lévy flights foraging hypothesis [15, 32, 58, 72, 4, 61, 34]. Affili, Dipierro & Valdinoci [1] developed an approach for deriving the conditions for transience and recurrence of Markovian random processes that is based on the decaying in time of the walker’s distribution in the starting site, i.e., ρ(0;t). We briefly report that approach [1] in Appendix C, where we have re-arranged it accordingly to the present aims.

Actually, diffusive processes, whose walker’s distribution converges to a stable density with stable parameter 0 < α < 2, are always transient except in the one-dimensional (N = 1) case when 1 ≤ α < 2 [1, 49], see Appendix C.

We observe that distribution ρ(x; t) (2.22) in the Should I go? condition (2.14a) with 1/2 < α < 1 is not self-similar, on the contrary ρ(x; t) is self-similar in the same Should I go? condition (2.14a) but with 0 < α ≤ 1/2 or in the Should I go? condition (2.14b) with 0 < α < 1, and in the Should I stay? condition (2.13a) with 0 < α < 2. As a consequence, in the considered case study, the loss of self-similarity introduces a time-scale T necessary for defining the large-time limit and so for attaining the scaling law in time of ρ(0; t) and determining transience and recurrence. This time-scale T emerges to be dependent on α and tending to infinity when α → 1 and this makes the large-time limit unattainable in real systems. In fact, by starting from the formula

ρ(0;t)=1π∫0∞ρ^(κ;t)dκ=1π∫0∞e−(1−φ^(κ))t/τdκ,(3.2)

we have that when ℓ |κ| ≪ 1 the expansion (2.17) holds and then (3.2) can be approximated by

ρ(0;t)≃1π∫01/ℓe−(ℓακα+12ℓ2ακ2α)t/τdκ+e−t/τ∫1/ℓ∞eφ^(κ)t/τdκ,(3.3)

and, after the change of variable ℓ_αk^αt/τ = ξ, it becomes

ρ(0;t)≃t−1/ααπKα∫0ℓαtℓατe−ξ−12ℓ2αℓα2τtξ2αξ1/α−1dξ+e−t/τ∫ℓαtℓατ∞expφ^ξ1/αtτξ1/α−1dξ,(3.4)

that in the limit t/τ → ∞ reduces to

ρ(0;t)≃t−1/ααπKα1/α∫0∞e−ξ−12ℓ2αℓα2τtξ2αξ1/α−1dξ=Γ(1/α)t−1/ααπKα1/αℓ2αℓατt−12αeℓα2t4ℓ2ατD−1αℓα2tℓ2ατ,(3.5)

where D_p(z), with Re p < 0, is the parabolic cylinder function, see [27, 3.462(1) p. 365 and 9.24-9.25 (9.246) p. 1028] with asymptotic behaviour

Dp(z)∼ez2/4zp,|z|≫1and|z|≫|p|.(3.6)

To conclude, we obtain that, when the following large-time limit is reached

t≫T=τα2sinπ2(1+2α)sinπ2(1+α)2,(3.7)

it holds

ρ(0;t)∼Γ(1/α)απKα1/αt−1/α.(3.8)

However T → ∞ when α → 1, see Figure 6, that poses an issue on the attainability of such large-time limit in real systems.

Figure 6

Plot of the time-scale T as defined in formula (3.7).

We have simulated the process that converges to ρ(x; t) (2.22) and the decaying in time of ρ(0; t) is shown in Figures 7 and 8. By using a fitting procedure, we have estimated the scaling-law in the transient regime τ ≪ t ≪ T and we found that it is a not power-law, see Figure 9. In particular, we have approximated it with the easy-to-read formula

Figure 7

Plots of the decreasing in time of the maximum of the walker’s distribution ρ(0; t) generated through the jump pdf(2.14a) with α = 0.6, 0.7, 0.8, 0.9. The solid line represent the decaying-law t^{−[α+f(α)]}(3.9) and the dashed line is the large-time decaying-law t^−α(3.8). The plots show the duration of the intermediate regime τ ≪ t ≪ T and its enlarging as α → 1.

ρ(0;t)∼t−1/[α+f(α)],f(α)=1α2Γ(2πα)−Γ(π)Γ(2π)−Γ(π),τ≪t≪T,(3.9)

that meets the constraints f(1/2) = 0 and f(1) = 1 in order to recover the limit scaling-laws ρ(0; t) ∼ t⁻², when α = 1/2, and ρ(0; t) ∼ t^−1/2, when α = 1, as expected from (2.15).

Simulations show that, during the intermediate regime τ ≪ t ≪ T, when α → 1 it holds α + f(α) > 1 (see also Figures 7, 8 and 9). Moreover, if α → 1 then T → ∞ and this makes unattainable the large-time limit t ≫ T whenever the studied Markovian CTRW model with jumps following a rule à la coin-flipping corresponds to a real system. In particular, this means that, a recurrence-like scaling, i.e., ρ(0; t) ∼ t^−β with 1 ≤ β < 2, could be observed for a very extended temporal interval because T → ∞, in spite of the transient theoretical scaling ρ(0; t) ∼ t^−1/α, with 1/2 < α < 1.

Therefore, since recurrence can be understood as homecoming probability and power-law distributions are used for explaining animal behaviour, the significance of this apparent recurrence - in spite of the actual transience - lays into an indetermined homecoming. This indetermination provides a further weakness of the hypothesis of Lévy-like motions for animal behaviour that can be overcome, for example, in the framework of truncated Lévy flights [42, 35].

Figure 8

The same as in Figure 7 but with α = 0.95, 0.97, 0.99, 0.999 for highlithing the delay in attaining the large-time limit t ≫ T.

Figure 9

Plot of the decaying-law of ρ(0; t) as estimated by simulations (black squares). The dotted line corresponds to the formula α + c₂α² + … + c₆α⁶ as provided by the fitting routine scipy.optimize.curve_fit while the solid line corresponds to the formula α + f(α) (3.9) and the dashed line is the reference-line indicating the transient-to-recurrence conversion at α+f(α) = 1.

4 Summary and conclusions

In this paper we have analysed random walks and we have discussed the role of a jump rule à la coin-flipping, namely jumps with a bi-modal distribution that is equal to zero in zero. In particular, we have studied an example of jump process that displays tails decaying with a power-law and we found that, within the framework of Markovian CTRW models for Lévy flights, i.e., the walkers’s distribution converges to a stable density, the self-similarity of the diffusive process is lost for a certain interval of the stability parameter 0 < α < 2: in the particular case of our example the self-similarity is lost when 1/2 < α < 1.

In the derivation of Lévy flights from the CTRW, the key role is played by the asymptotic limit for small wavelength of the characteristic function of jumps and this leaves open the specific form of the full characteristic function of the jump pdf. This asymptotic limit re-phrases in a single step the Kramers–Moyal expansion and the Pawula theorem by reducing the so-called Montroll–Weiss equation, that governs the walker’s distribution in the CTRW approach, to the fractional diffusion equation in the domain of the wavelength. Actually, this procedure is an example for showing the Central Limit Theorem in the sense of Lévy. As a matter of fact, when the jump-sizes follow a rule à la coin-flipping, the small wavelength expansion of their characteristic function is not always a series with alternating signs and this fact causes the loss of self-similarity. In the framework of the studied example, the resulting diffusive process converges to a generalised Voigt profile that is given by the convolution of two stable densities.

We have highlighted that this loss of self-similarity has a double significance. At the mathematical level, the use of a jump pdf corresponding to a rule à la coin-flipping makes the model for Lévy flights consistent with the probabilistic derivation of the fractional diffusion equation where the distinctive singularity of the fractional Laplacian is a consequence of the jump-rule à la coin-flipping, but the resulting evolution equation is indeed a double fractional-order equation in the stability interval 1/2 < α < 1. At the application level, the loss of self-similarity generates an intermediate temporal regime which defines a time-scale for large-time limit. This time-scale results to be depended on α and it tends to infinite when α tends to 1: this means that, whenever the studied process is a reliable model for a physical system, the large-time limit could not be observed in real measurements. This unattainability of the large-time limit has an effect on the transience and recurrence of the process: actually, in spite of the expected transience of the process, the long-extended intermediate regime could display a recurrence-like scaling that leads to an indetermined situation in real cases. This apparent recurrence because of the unattainability of the large-time limit is a property of the studied CTRW model that deserves attention in the future. In fact, if animal movement is modelled through Lévy-like motions then the searching for food, and also the searching for home, can be affected by the adopted jump rule: the searching for food could lead to a double-order equation and the searching for home to an indetermined homecoming in real systems.

To conclude, we state that the research on the derivation of random walks models for Lévy flights and fractional diffusion is not concluded yet, that a further deep investigation on the role of jump-rules à la coin-flipping is necessary. This calls for a generalisation of the present results both in terms of the choice of the jump pdf and in terms of the considered random walk model characterised by power-law tails as, for example, Lévy walks. Moreover, this distinguishible effect due to the jump-rule, i.e., it is à la coin-flipping or not, turns into a distinguishible feature of the motion of animals: namely if they stand in the majority of the iterations or if they always move.

Definitively, the difference between the “Should I stay?” and “Should I go?” conditions cannot be disregarded.