Weird Brownian motion

This paper presents and explores a diffusion model that generalizes Brownian motion (BM). On the one hand, as BM: the model’s mean square displacement grows linearly in time, and the model is Gaussian and selfsimilar (with Hurst exponent 12 ). On the other hand, in sharp contrast to BM: the model is not Markov, its increments are not stationary, and its non-overlapping increments are not independent. Moreover, the model exhibits a host of statistical properties that are dramatically different than those of BM: aging and anti-aging, positive and negative momenta, correlated velocities, persistence and anti-persistence, aging Wiener–Khinchin spectra, and more. Conventionally, researchers resort to anomalous-diffusion models—e.g. fractional BM and scaled BM (both with Hurst exponents different than 12 )—to attain such properties. This model establishes that such properties are attainable well within the realm of diffusion. As it is seemingly Brownian yet highly non-Brownian, the model is termed Weird BM.


Introduction
Discovered by Ingen-Housz [1] and Brown [2], diffusion is arguably the most elemental random motion in the physical sciences [3,4]. Pioneered by Bachelier [5,6], by Einstein and Smoluchowski [7,8], and by Wiener [9], Brownian motion (BM)-also known as the Wiener process-is the archetypal quantitative model for diffusion [10]. BM displays an assortment * Author to whom any correspondence should be addressed.
Original Content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. of statistical properties (to be described in detail below) that are of major importance: its mean square displacement (MSD) grows linearly in time; it is a Gaussian process [11]; it is a selfsimilar process [12]; it is a Markov process [13]; and it is a Levy process [14], i.e. its increments are stationary, and its non-overlapping increments are independent.
As stated in the Abstract, this paper showcases Weird BM (WBM), a 'weird' generalization of Brownian motion that: while being a Gaussian and selfsimilar diffusion model (with Hurst exponent 1 2 ), it exhibits a host of properties and features that are commonly associated with anomalous diffusion (with Hurst exponents different than 1 2 ). With WBM at hand, theoreticians and experimentalists can model diffusion-with Gaussian statistics and selfsimilar trajectories-in a highly non-Brownian fashion. Also, WBM provides researches with a critical and rather counter-intuitive 'take-home message': anomalous non-Brownian behaviors are not, necessarily, the characteristics of anomalous diffusion.
The paper is organized as follows. Section 2 constructs WBM, and addresses perspectives via which it is indistinguishable from BM: MSD behavior, positions' statistics, and selfsimilarity. Section 3 unveils profound differences between WBM and BM: a host of statistical properties of WBM that are markedly different than those of BM. Section 4 returns to the MSD behavior, with a 'twist': shifting from real space to Laplace and Fourier spaces, and showing that in these spaces the MSD behaviors of WBM and BM differ dramatically. Section 5 concludes with a recap, with a discussion, and with comparisons of WBM to fractional BM and scaled BM (which are anomalous-diffusion generalizations of BM). Proofs of key results established along the paper are detailed in the Methods (section 6).
Throughout this paper the time axis is t ⩾ 0, and the space axis is −∞ < x < ∞. All the motions considered here initiate at time t = 0 from the spatial origin x = 0. Also, the operation of mathematical expectation is denoted E [·], and the following acronyms are used (three of which were noted above): BM-Brownian motion; WBM-Weird BM; MSD-mean square displacement; GWN-Gaussian white noise.

WBM and perspectives via which it is indistinguishable from BM
The positions, at time t, of BM and of WBM are denoted B (t) and Z (t), respectively. The corresponding velocities of BM and of WBM are denotedḂ (t) andŻ (t). The velocity process of BM is GWN, and it displays properties that will be described and used hereinafter. At time t, the position of WBM is constructed from GWN as follows: where K (t; u) is a kernel function that will be specified below. Namely, the WBM position at time t is: the integral of GWN over the temporal interval [0, t]; and the integration is with respect to the square root of a specific kernel function K (t; u), where u is the integration variable (0 ⩽ u ⩽ t). The WBM kernel K (t; u) (0 ⩽ u ⩽ t) of equation (2). The different shapes of this kernel, as determined by the value of the WBM parameter κ, are illustrated (for the time point t = 1) as follows. Blue curve: uphill shape (0 < κ < 1 2 ). Green curve: flat shape (κ = 1 2 ). Yellow curve: downhill shape ( 1 2 < κ < 1).

The WBM kernel
The WBM kernel function is (0 ⩽ u ⩽ t), where: the exponent κ is the WBM parameter; and the coefficient c κ is a positive constant that depends on κ. The coefficient c κ is set so that the kernel's total mass be t, i.e. t 0 K (t; u) du = t. Thus, a calculation involving the beta function implies that the coefficient is c κ = Γ (2κ) Γ (2 − 2κ). Consequently, the admissible values of the WBM parameter are 0 < κ < 1.
Depending on the value of the WBM parameter κ, the WBM kernel-as a function of its integration variable u-displays three different shapes (see figure 1).
▶ Uphill shape: when 0 < κ < 1 2 the kernel is monotone increasing from K (t; 0) = 0 to K (t; t) = ∞; in this shape the kernel concentrates its mass 'close to the present', i.e. near the right end of the temporal interval [0, t].
▶ Flat shape: when κ = 1 2 the kernel is constant, K (t; u) ≡ 1; in this shape the kernel distributes its mass evenly over the temporal interval [0, t]. ▶ Downhill shape: when 1 2 < κ < 1 the kernel is monotone decreasing from K (t; 0) = ∞ to K (t; t) = 0; in this shape the kernel concentrates its mass 'at the distant past', i.e. near the left end of the temporal interval [0, t].
When the WBM parameter is κ = 1 2 then WBM yields BM. Indeed, when κ = 1 2 then the WBM kernel is flat, K (t; u) ≡ 1, and hence equation (1) implies that Z(t) = B(t) (for all t ⩾ 0). So, BM is a special case of WBM, and therefore WBM generalizes BM.
When the WBM parameter is κ = 1 2 then the kernel's shape switches between convex and concave (see figure 1), and the switching occurs at the inflection point u * = κ · t. Therefor, the WBM parameter κ has a geometric meaning: it is the calibrated location-the calibration being relative to the length t of the temporal interval [0, t]-of the WBM kernel's inflection point.
As noted above, the total mass of the WBM kernel is t. So, the WBM kernel distributes t units of mass over a temporal interval that spans t units of length, and the overall 'mass-tolength ratio' is t/t = 1. The WBM kernel distributes its t units of mass evenly over the temporal interval [0, t] only when the WBM parameter is κ = 1 2 , i.e. only in the case of BM. A Gini-score quantification of the degree to which the WBM kernel distributes its mass unevenly over the temporal interval [0, t] is described in the methods.
The WBM kernel is a product of two components. The first component, , is similar to the kernel function of the Riemann-Liouville version of Fractional BM [50]. The second component, u 1−2κ (0 ⩽ u ⩽ t), is similar to the kernel function of Scaled BM [51]. Fractional BM and Scaled BM are Gaussian and selfsimilar models that generalize BM so as to generate anomalous diffusion. Here, an 'amalgam' of the two kernel functions is used in order to produce WBM: a Gaussian and selfsimilar diffusion model that is highly non-Brownian.

Diffusion and positions' statistics
First and foremost, our task is to show that WBM is a diffusion model. Specifically, we need to show that E[|Z (t) | 2 ]-the MSD of the WBM position Z (t), relative to the initial WBM position Z (0) = 0-is a linear function of the temporal variable t. To that end we use a key fact regarding integrals of GWN, which is described as follows. Given a square-integrable 'test function' φ (u) (u ⩾ 0), consider its integral with respect to GWN, I =´∞ 0 φ (u)Ḃ (u) du. The statistical distribution of the integral I is Gaussian with: mean zero E [I] = 0; and variance E[I 2 ] =´∞ 0 φ (u) 2 du. Applying this key fact to the integral structure of the WBM positions (equation (1)), and using the fact that the total mass of the WBM kernel was set to be t (i.e.´t 0 K (t; u) du = t), yields the following result. The statistical distribution of the WBM positions is Gaussian with: mean zero E [Z (t)] = 0; and variance Therefore, the MSD of WBM is indeed a linear function of the temporal variable t, and the slope of this linear function-the diffusion coefficient of WBM-is one: d dt E |Z (t) | 2 ≡ 1. So, via the perspective of positions' statistics, and hence via the MSD perspective: BM and WBM are indistinguishable from each other (see figure 2). Specifically, for any time point t, the statistical distributions of the BM position B (t) and of the WBM position Z (t) are identical: Gaussian with mean zero, and with variance t. For all values of the WBM parameter κ, the statistical distribution of this position is 'standard normal': Gaussian with mean zero and with variance one. The panels of the figure depict histograms-of the WBM position Z(1), generated by simulations-for several values of WBM parameter κ; the κ = 0.5 panel correspond to the case of BM. The black 'bell curve' appearing in each panel is the 'standard normal' density function. The match between the histograms and the Gaussian 'bell curve' is self-evident. The algorithm via which the WBM histograms were simulated is described in the methods.

Selfsimilarity
The identical MSD behavior of BM and of WBM stems from a profound property that both these motions share: selfsimilarity with Hurst exponent 1 2 . We shall now describe and address the selfsimilarity property.
Given a positive scale parameter s, consider a spatio-temporal scaled version of BM whose positions are B s (t) = B (st) / √ s. Namely, in this version the time axis (t ⩾ 0) is scaled by the factor s, and the space axis (−∞ < x < ∞) is scaled by the factor 1/ √ s. The meaning of the BM selfsimilarity is the following [12]: for any positive scale parameter s, the scaled version of BM is equal in law to the 'original' BM.
With the above scale parameter s, we shift from the spatio-temporal scaled version of BM to a matching spatio-temporal scaled version of WBM: a version whose positions are Z s (t) = Z (st) / √ s. Using the BM selfsimilarity, the integral structure of the WBM positions (equation (1)), and an inherent scale-invariance feature of the WBM kernel (equation (2)), it In each of the four panels: one realization of GWN is generated; and, based on this one realization, the resulting WBM trajectory-Z(t) (0 ⩽ t ⩽ 1)-is simulated and depicted for several values of the WBM parameter κ; the κ = 0.5 trajectories correspond to the case of BM. Marked differences between the κ < 0.5 trajectories and the κ > 0.5 trajectories are selfevident. The algorithm via which the WBM trajectories were simulated is described in the methods.
is proved in the methods that: for any positive scale parameter s, the scaled version of WBM is equal in law to the 'original' WBM. So, via the perspective of selfsimilarity: BM and WBM are also indistinguishable from each other-as both these motions are selfsimilar with Hurst exponent 1 2 . As WBM generalizes BM, the selfsimilarity of WBM generalizes that of BM. In particular, the WBM selfsimilarity implies that the WBM position Z (t) is equal in law to the random variable √ tZ (1). In turn, this equality-in-law implies that E[|Z (t) So, the fact that WBM is a diffusion model stems from the fact that WBM is selfsimilar with Hurst exponent 1 2 . Due to its selfsimilarity, the trajectory of WBM is a fractal object: zooming in and zooming out on the WBM trajectory-doing so via the aforementioned spatio-temporal scaling-yields, statistically, the same 'picture'. Simulations of various WBM trajectories, for different values of the WBM parameter κ, are plotted in figure 3.
As asserted in section 2.2, and as self-evident from figure 2: based on WBM positions' statistics, it is impossible to distinguish between different values of the WBM parameter κ. On the other hand, as self-evident from figure 3, it is possible indeed to distinguish between κ < 1 2 trajectories and κ > 1 2 trajectories of WBM. With this observation we now proceed to section 3.

Unveiling differences between WBM and BM
As established above, the positions of WBM-at any given time point t-are Gaussian with mean zero and with variance t. This fact regarding the WBM positions stems from a deeper property: Gaussianity. Namely, WBM is a Gaussian process: for any given set of time points, 0 < t 1 < · · · < t n < ∞, the statistical distribution of the vector of positions at these time points is multivariate normal. The Gaussianity of WBM is due to the fact that its positions are integrals of GWN [11]. The statistics of a zero-mean Gaussian process are fully characterized by its auto-covariance function. This section will begin with the auto-covariance function of WBM, will then explore various statistical properties of WBM, and will show that these properties differ markedly from those of BM.

The WBM auto-covariance
To derive the auto-covariance function of WBM we use another key fact regarding integrals of GWN, which is described as follows. Given two square-integrable 'test functions' φ n (u) (u ⩾ 0, n = 1, 2), consider their integrals with respect to GWN, I n =´∞ 0 φ n (u)Ḃ (u) du (n = 1, 2). The covariance of the two integrals is E[I 1 I 2 ] =´∞ 0 φ 1 (u) φ 2 (u) du. Namely, this covariance is the inner product-in the Hilbert space of square-integrable test functions-of the functions φ 1 (u) and φ 2 (u).
Applying this key fact to the integral structure of the WBM positions (equation (1)) implies that: the covariance of any two WBM positions, Z (t 1 ) and Z (t 2 ), is where c κ = Γ (2κ) Γ (2 − 2κ) (as set above). Equation (4) is the auto-covariance function of WBM. At its diagonal, t 1 = t 2 , this auto-covariance function coincides with the WBM variance function (equation (3)). In particular, at the parameter value κ = 1 2 , the WBM auto-covariance function yields the BM auto-covariance function: As established in section 2, BM and WBM are indistinguishable from each other via two principal perspectives: the perspective of the statistical distributions of their positions (and hence the MSD perspective); and the perspective of selfsimilarity. Equation (4) sets the spotlight on the dramatic difference between BM and WBM: their markedly different autocovariance functions-which characterize, respectively, the motions' statistics.
Evidently, the auto-covariance function of any random motion is a function of its two temporal variables, t 1 and t 2 . As WBM is selfsimilar, its auto-covariance function can be 'compressed' to a function of a single temporal variable: a non-negative time lag ∆. Indeed, set R (∆) := E[Z (1) Z (1 + ∆)], the covariance of the WBM positions at the time points 1 and 1 + ∆. Then, for any two time points that are separated by the time lag ∆-namely, the time points t and t + ∆-the covariance of the WBM positions at these time points admits the representation Indeed, setting t 1 = t and t 2 = t + ∆ in equation (4), and then applying the change of variables u → v = u/t, yields equation (5). So, in effect, the statistics of WBM are characterized by the function R (∆). Depending on the value of the WBM parameter κ, the function R (∆) displays three different shapes (see figure 4).
The aforementioned properties of the function R (∆) are derived in the methods, as well as the following property: the 'terminal slope' of the function R (∆) is zero, R ′ (∞) = 0. As shall be shown below, the different shapes of the function R (∆) lead to different statistical behaviors of WBM.

Non-stationary increments
Consider the increment of WBM over the time interval (t, t + ∆), i.e.: the displacement Z (t + ∆) − Z (t) of WBM between the time points t and t + ∆ (where t and ∆ are both positive). As the positions of WBM have zero means, so do its increments. And, as WBM is a Gaussian process, the statistical distribution of the WBM increments is Gaussian.

Consequently, the Gaussian distribution of the increment Z (t + ∆) − Z (t) is determined by its variance.
To compute the variance of the increment Z (t + ∆) − Z (t), recall that the variance of the difference of two random variables is: the sum of their variances, minus twice their covariance. Applying this basic fact to the difference of the WBM positions Z (t + ∆) and Z (t), and using the WBM variance function (equation (3)): equation (5) implies that the increment's variance is In what follows the increment's variance is denoted, in short, V (t; ∆).
Keeping the time-lag variable ∆ fixed, and taking the limit t → ∞, equation (6) implies that Substituting the initial slope R ′ (0) of the function R (∆) (whose calculated values were specified in section 3.1 above) into the limit result of equation (7) implies that: depending on the value of the WBM parameter κ, the Gaussian distribution of the increment Z (t + ∆) − Z (t) displays-in the temporal limit t → ∞-three different asymptotic behaviors.
Thus, WBM has stationary increments only in the BM case.

Propagator and Markov-breaking
Consider two time points: a 'present' time point t, and a 'future' time point t + ∆ (where t and ∆ are both positive). The propagator of a random motion is the conditional statistical distribution of its position at the future time point t + ∆, given its position at the present time point t.
To derive the propagator of WBM we use a key fact regarding the conditional statistical distributions of bivariate normal distributions. Specifically, consider a random vector (X 1 , X 2 ) whose statistical distribution is bivariate normal, and whose components have: zero means; variances v 1 and v 2 ; and covariance c. Then, the conditional statistical distribution of the component X 2 -given the value of the component X 1 -is Gaussian with: conditional mean Applying this key fact to the WBM positions Z (t) and Z (t + ∆), and using equation (5), yields the WBM propagator: a Gaussian propagator that is characterized by the following conditional mean and variance of the future position Z (t + ∆)-given the present position Z (t). The conditional mean is And the conditional variance is With the WBM propagator at hand, it is tempting-yet utterly wrong-to assume that WBM is a Markov process. To elucidate why WBM is not Markovian, consider three time points: the present time t and the future time t + ∆ (as above), and also a past time point u (where 0 < u < t). With these time points fixed, consider the conditional statistical distribution of the future position Z (t + ∆)-given the present and past positions, Z (t) and Z (u). Using the conditional statistical distributions of multivariate normal distributions, it can be shown that: when κ = 1 2 , the conditional mean of the future position Z (t + ∆) depends on the present position Z (t), as well as on the past position Z (u). So, WBM is Markovian only in the BM case.

Momenta
The WBM increment Z (t + ∆) − Z (t) was addressed above (equation (6)). With the WBM propagator at hand, this increment shall now be addressed via a conditional perspective: the conditional statistical distribution of the increment-given the increment's 'initial position' Z (t). As WBM is a Gaussian process, the increment's conditional statistical distribution is Gaussian. Combining this general fact (regarding general Gaussian processes) with the WBM propagator yields the increment's conditional mean and variance.
Information regarding the position Z (t) changes the statistical distribution of the increment Z (t + ∆) − Z (t) significantly. Indeed, with no knowledge of the position Z (t), the increment is governed by one Gaussian statistical distribution. And, knowing the position Z (t), the increment is governed by an altogether different conditional Gaussian statistical distribution. The means and variances of these different Gaussian distributions are specified-side by sidein table 1. The increment's conditional mean and variance (which are specified in the right column of table 1) follow straightforwardly from equations (8) and (9).
In particular, knowing the position Z (t) changes the increment's mean from zero, E[Z (t + ∆) − Z (t)] = 0, to potentially non-zero. Indeed, as specified in table 1, the increment's conditional mean-given the position Z (t)-is: a linear function of the position Z (t), with slope [R (∆/t) − 1]. Thus, knowing the position Z (t) induces a 'momentum'-or, termed alternatively, a 'drift'-that is quantified by the slope [R (∆/t) − 1]. Specifically, the momentum is determined by the sign of the slope [R (∆/t) − 1] as follows. (I) The momentum is negative when the slope is negative, R (∆/t) < 1. (II) The momentum is zero when the slope is zero, R (∆/t) ≡ 1. (III) The momentum is positive when the slope is positive, R (∆/t) > 1.
In turn, the different shapes of the function R (∆)-which were specified in section 3.1 above-yield the following conclusion. Depending on the value of the WBM parameter κ, WBM displays three different momenta.
Thus, WBM has an always-zero momentum only in the BM case.
In the 'unconditional column' no information (regarding the WBM positions and/or trajectory) is disclosed. In the 'conditional column' the WBM position Z (t) is disclosed.

Dependent velocities and increments
Equation (5) provided a 'compressed representation' of the WBM auto-covariance function.
As shall now be shown, equation (5) also provides a 'compressed representation' of the autocovariance function of the WBM velocities. Consider two time points, t 1 and t 2 , that are separated by a time lag ∆, i.e. t 2 − t 1 = ∆. Equation (5) implies that the covariance of the WBM positions at these time points is . Differentiation with respect to the temporal variables t 1 and t 2 further implies that the covariance of the WBM velocities at these time points is Equation (10) is the auto-covariance function of the WBM velocities. Substituting the specific time points t 1 = 1 and And, in terms of the function R * (∆), equation (10) implies that: the covariance of the WBM velocities at the time points t and t + ∆ admits the representation Equation (11) is the 'velocity counterpart' of equation (5). It follows from equation (11)  In turn, the relationships regarding the correlations of the WBM velocities, together with the shapes of the function R (∆)-which were specified in section 3.1 above-yield the following conclusion. Depending on the value of the WBM parameter κ, the non-overlapping increments of WBM display three different behaviors.
A general fact regarding Gaussian random variables is: the random variables are independent if and only if they are uncorrelated. Thus, WBM has independent non-overlapping increments only in the BM case.

Aggregated correlations
As noted above, the auto-covariance function of any random motion is a function of its two temporal variables, t 1 and t 2 . When a random motion is stationary [52], its auto-covariance function 'compresses' to a function of the time-lag variable ∆ = t 2 − t 1 . Given a stationary process of interest, the integral of its 'compressed' auto-covariance function-the integral being with respect to the time-lag variable ∆-plays a key role in the context of quantifying the process' 'range of dependence' [53][54][55].
It is evident from equation (11) that the auto-covariance function of the WBM velocity process is not a stationary auto-covariance function; namely, the right-hand side of equation (11) is a function of the two time points t and t + ∆ (rather than a function of the time-lag variable ∆ alone). Nonetheless, integrating the auto-covariance function of equation (11) with respect to the time-lag variable ∆ yields an integral that does not depend on the time point t. Specifically, The integral of equation (12) In turn, the properties of the function R (∆)-which were specified in section 3.1 above-lead to the following conclusion. Depending on the value of the WBM parameter κ, the integrals of equations (12) and (13)  Thus, WBM has a zero aggregate of future correlations only in the BM case. As noted above, the integral of equation (12) manifests standing at the present time point t, and summing up all future correlations. Analogously, standing at the present time point t, one can also sum up all 'past correlations'. Doing so yields the aggregate of past correlationś t 0 E[Ż (t − ∆)Ż (t)]d∆. In turn, a calculation (which is similar to the above calculation) implies that this aggregate is equal to R ′ (0), the initial slope of the function R (∆) (whose calculated values were specified in section 3.1 above). So, as with the aggregate of future correlations: depending on the value of the WBM parameter κ, also the aggregate of past correlations yields three different outcomes; and WBM has a zero aggregate of past correlations only in the BM case.

Back to MSD, with a twist
Our analysis of WBM began with the statistics of its positions-which were shown to be Gaussian with mean zero, and with variance t. Consequently, via the perspective of the positions' statistics-and hence via the MSD perspective-BM and WBM turned out to be indistinguishable from each other. This section shall return to the MSD perspective, and do so with a 'twist': shifting from the real WBM positions to corresponding positions in Laplace and Fourier spaces. As shall be shown in this section, the shifts from real space to Laplace and Fourier spaces will enable the MSD perspective to distinguish between BM and WBM.

Laplace approach
Consider the following weighted average of the WBM positions: where τ is a positive parameter. Namely,Z (τ ) averages the WBM positions with respect to the density function of an exponential distribution with mean τ . Hence, the parameter τ manifests the characteristic time-scale of the weighted averageZ (τ ).
As the positions of WBM have zero means, so does their weighted averageZ (τ ). And, as WBM is a Gaussian process, the statistical distribution of the weighted averageZ (τ ) is Gaussian. Consequently, the Gaussian distribution of the weighted averageZ (τ ) is determined by its variance.
To compute the variance of the weighted averageZ (τ ), we make two observations. The first observation is that the integral structure of the WBM positions is, in effect, a convolutional structure. The second observation is that the weighted averageZ (τ ) is, in effect, a Laplace transform of the WBM positions (see section 4.3 below). As proved in the Methods, a derivation using these observations establishes that the weighted-average's variance is: where D(κ) = Γ( 1 2 + κ) 2 /[4 1−κ Γ (2κ)]. As noted above, a random motion is a diffusion if its MSD grows linearly in time. And, when a random motion is a diffusion then: its diffusion coefficient is the positive slope of its temporally linear MSD. In real space the underlying temporal variable is t, the WBM positions are Z (t), and equation (3) asserts that: WBM is a diffusion, and its diffusion coefficient is 1 for all values of the WBM parameter κ. Hence, in particular, the diffusion coefficient in real space cannot distinguish between BM and WBM.
Shifting from real space to Laplace space: the underlying temporal variable switches from t to τ ; the WBM positions switch from Z (t) toZ (τ ); and the diffusion coefficient switches from 1 to D(κ). Indeed, with regard to Laplace space, equation (15) asserts that: WBM is a diffusion, and its diffusion coefficient is D(κ). So, shifting from real space to Laplace space renders BM and WBM-via the Laplace diffusion coefficient D(κ)-distinguishable from each other (see figure 5).

Fourier approach
Consider the following complex-valued integral of the WBM positions: where θ is a real non-zero parameter, and where T is a positive parameter. Namely,Ẑ θ (T) is the Fourier transform of the WBM positions over the temporal window [0, T], and θ is the Fourier variable. See section 4.3 below for why the Fourier transform of the WBM positions is restricted to a finite temporal window. The integralẐ θ (T) is a complex-valued random variable. As argued with regard to the weighted averageZ (τ ) of equation (14), the real and imaginary components of the random variableẐ θ (T) are Gaussian with zero mean. The variance of the random variableẐ θ (T) is E[|Ẑ θ (T) | 2 ], where the squared absolute value |·| 2 is now in the sense of complex numbers. As proved in the Methods, the asymptotic behavior of this variance-with respect to the parameter T-is: where ≈ denotes asymptotic equality (in the limit T → ∞), and where C(κ) = Γ(κ + In other words, equation (17) means that: As noted and re-iterated above with regard to the MSD of WBM in real space (equation (3)): the underlying temporal variable is t; the temporal growth of the MSD is linear; and this growth characterizes diffusion. Shifting from real space to Fourier space: the underlying temporal Figure 6. The WBM anomalous-diffusion coefficient in Fourier space, C(κ) (0 < κ < 1) of equation (17). Specific values of this anomalous-diffusion coefficient include: the left-end limit value lim κ→0 C(κ) = 0; the 'mid-value' C( 1 2 ) = 1; and the right-end limit value lim κ→1 C(κ) = 1 4 π .
▶ Super diffusion: when 0 < κ < 1 2 the temporal growth of the MSD is, asymptotically, a super-linear power law. ▶ Diffusion: when κ = 1 2 the temporal growth of the MSD is, asymptotically, linear. ▶ Sub diffusion: when 1 2 < κ < 1 the temporal growth of the MSD is, asymptotically, a sublinear power law. So, shifting from real space to Fourier space renders BM and WBM-via the MSD perspective-highly distinguishable from each other. Also, while the diffusion coefficient 1 of equation (3) does not depend on the values of the WBM parameter κ, the anomalous-diffusion coefficient C(κ) of equation (17) does (see figure 6).
The Laplace transform happens to fit WBM quite naturally. This natural fit is lost when shifting from the Laplace transform to the Fourier transform, due to three reasons. Firstly, the natural time axis of the Fourier transform is the real temporal line −∞ < t < ∞ (rather than the non-negative temporal half-line t ⩾ 0, which is the natural time axis of the Laplace transform, and the time axis of WBM). Secondly, the aforementioned Laplace relationship Ż (λ) = λZ (λ) does not have a similar Fourier relationship. Thirdly, and most importantly, while the Laplace exponential function exp (−λt) is square integrable, the Fourier geometric function exp (iθt) is not. Consequently, the Fourier transform of GWN is not a 'legitimate' random object, and this 'illegitimacy' is induced to the Fourier transform of the WBM positions. Restricting the Fourier transform of the WBM positions to the finite temporal window [0, T]-as done in equation (16)-renders the Fourier geometric function exp (iθt) square integrable, and hence resolves the Fourier 'illegitimacy' issue.
Consider a general random signal whose value at time t is X (t) (the random-signal's values can be either real or complex). Similarly to equation (16), setX θ (T) =´T 0 exp (iθt) X (t) dt to be Fourier transform of the random signal over the temporal window [0, T], where θ is a real non-zero Fourier variable. The random-signal's sample spectrum over the temporal window is 1 T |X θ (T) | 2 , and the mean of the sample spectrum is 1 In turn, the random-signal's spectral density is defined as the temporal limit of the sample-spectrum's mean: lim T→∞ 1 T E[|X θ (T) | 2 ], provided that this limit exists [56]. When the random signal is a stationary process then the Wiener-Khinchin theorem asserts that the spectral density is well-defined, and that it coincides with the Fourier transform of the random-signal's auto-covariance function [57,58]. When the random signal is a nonstationary process then one has to shift from the Wiener-Khinchin theorem to, so called, 'aging Wiener-Khinchin theorems' [59][60][61][62][63][64]. These theorems replace-in the random-signal's sample spectrum-the temporal harmonic 'diffusion scaling' 1/T by temporal power-law 'anomalousdiffusion scalings' (as well as by more elaborate temporal scalings).
flicker noise. (A universal Poisson mechanism that generates 'anomalous generalizations' of flicker noise was established in [82].) Consider the following ratio of the MSDs in Fourier space: that of the Fourier transform of WBMẐ θ (T), to that of the Fourier transform of BMB θ (T). Equation (17) implies that the asymptotic behavior of this ratio-with respect to the parameter T-is: where ≈ denotes asymptotic equality (in the limit T → ∞). The right-hand side of equation (18) can be re-formulated as C(κ) · F(θT), where F (θ) = |θ| 1−2κ ( −∞ < θ < ∞) is a 'spectral profile'. Evidently, depending on the value of the WBM parameter κ, the spectral profile F (θ) displays three different shapes (see figure 7).
Equation (18) provides a 'ratio re-formulation' of equation (17). Shifting from real space to Fourier space renders BM and WBM-also via the MSD-ratio perspective of equation (18)highly distinguishable from each other.

Conclusion
BM is the running integral of GWN. Specifically, the position of BM at time t is the integral of GWN over the temporal interval [0, t]. So, underpinning the generation of BM from GWN is a flat kernel function that 'distributes' evenly t units of 'mass' over the t units of length (of the temporal interval [0, t]). Shifting from this flat kernel function to the kernel function of equation (2) maintains the overall 'mass-to-length ratio' t/t = 1, but yet it 'distributes' the 'mass' unevenly over the temporal interval [0, t]. In turn, this uneven distribution of 'mass' shifts BM to a profoundly different motion generated from GWN: WBM, which was introduced and explored in this paper.
The fact that the kernel function of equation (2) has an overall 'mass-to-length ratio' t/t = 1 implies that the statistics of the WBM positions are indistinguishable from those of BM: both these positions are Gaussian with mean zero and with variance t. Consequently, both WBM and BM are diffusion models with diffusion coefficient one, i.e.: the MSD of both these models grows linearly in time, with slope one. Moreover, the kernel function of equation (2) is invariant with respect to changes of scale, and this scale-invariance implies that WBM-just as BM-is selfsimilar with Hurst exponent 1 2 . So, WBM and BM are indistinguishable from each other via three principal perspectives: position's statistics, diffusion and diffusion coefficient, and selfsimilarity. However, the shift from the flat and even kernel function of BM to the non-linear and uneven kernel function of WBM induces a 'motion memory'. In turn, this memory further induces marked differences between BM and WBM. Specifically, the kernel function of equation (2) has a parameter κ that takes values in the unit interval. And, determined by the parameter κ, WBM displays dramatically different behaviors in the three following regimes: the 'sub-Brownian' regime 0 < κ < 1 2 ; the 'super-Brownian' regime 1 2 < κ < 1; and the Brownian regime κ = 1 2 , in which WBM yields BM. The behaviors of WBM in its three regimes are summarized in table 2. Table 2 showcases properties that are commonly associated with anomalous diffusion (rather than with diffusion). Arguably, the two most prominent Gaussian and selfsimilar models that generalized BM so as to generate anomalous diffusion are: FBM [50,[83][84][85][86][87]; and SBM [51,[88][89][90][91][92]. On the one hand, FBM maintains the stationary-increments property of BM, and its Hurst exponent H takes values in the unit interval. On the other hand, SBM maintains the independent-increments property of BM, and its Hurst exponent H is positive. Sub-diffusion and super-diffusion comparisons of WBM to FBM and to SBM are presented, respectively, in tables 3 and 4. These tables specify sub-diffusion and super-diffusion properties of FBM and of SBM that are displayed by WBM-which is a diffusion (rather than an anomalous diffusion).
A key 'take-home message' of this paper is the following: as anomalous-diffusion behaviors were shown here to arise well within the realm of diffusion-such behaviors do not, necessarily, characterize anomalous diffusion. Due to its weird behaviors, WBM is a rather surprising diffusion model that defies common 'Brownian intuition'.   Random motions that emerge-via scaling limits-over macroscopic time scales are selfsimilar processes [93]. Thus, from a macroscopic perspective of random motions, only selfsimilar such processes should be considered. WBM, which was introduced and explored in this paper, is a Gaussian and selfsimilar model of diffusion that is not Markovian. Switching the Gaussian property with the Markovian property leads to 'diffusion in a logarithmic potential' (DLP): a random motion whose evolution is governed by Langevin dynamics with a logarithmic potential [94]. Namely, DLP is a Markovian and selfsimilar model of diffusion that is not Gaussian. In fact, DLP is the only selfsimilar model of diffusion whose evolution is governed by an Ito stochastic differential equation [93,95]. DLP attracted vast statistical-physics interest in recent years [45,[96][97][98][99][100][101][102][103][104][105]. In the context of selfsimilar models of diffusion, the following facts are underscored.
▶ WBM is a Gaussian (non-Markovian) generalization of BM that is produced by the mechanism of equation (1). ▶ DLP is a Markovian (non-Gaussian) generalization of BM that is produced by the Langevin mechanism. ▶ The intersection of WBM and DLP is BM, which is both Gaussian and Markovian.
The WBM generalization of BM points out empirical, engineering, and theoretical directions for future research. Empirically-wise: discover real-world diffusion processes with WBM statistics. Engineering-wise: devise real experiments that generate diffusion processes with WBM statistics. Theoretical directions for future research include: representation of WBM dynamics via fractional equations [24]; stochastic calculus [106] for WBM, as well as for stochastic differential equations driven by WBM; and first-passage times [107] of WBM.

Gini score of the WBM kernel
As noted in section 2.1, the WBM kernel function (equation (2)) distributes t units of mass over the temporal interval [0, t]. The degree to which the WBM kernel distributes it mass unevenly can be quantified by socioeconomic inequality indices [108][109][110]. Given a human society, an inequality index quantifies how unequal the distribution of wealth among the society members is, and it does so via a score that takes values in the unit interval. The score zero characterizes a perfectly equal distribution of wealth in which all the society members have the same (positive) individual wealth. The score one characterizes a perfectly unequal distribution of wealth in which a zero percent of the society members possess all the wealth, and all other society members are completely impoverished. Arguably, the most popular inequality index is the Gini index [111][112][113]. Deeming the kernel's mass to be wealth, it can be shown that the corresponding Gini index is |2κ − 1| (see figure 8), and hence: the closer the WBM parameter is to its mid-value, κ = 1 2 , the more even the distribution of mass; and the closer the WBM parameter is to its boundary values, κ → 0 and κ → 1, the more uneven the distribution of mass.

Proof of the selfsimilarity of WBM
Fix a positive scale parameter s. The BM selfsimilarity implies that BM is equal in law to its spatio-temporal scaled version, whose positions are B s (t) = B (st) / √ s. In turn, the BM The closer the score is to the value zero, the more equal the distribution of mass; perfect equality is attained at the mid-value of the WBM parameter, κ = 1 2 . The closer the score is to the value one, the more unequal the distribution of mass; perfect inequality is attained as the WBM parameter approaches its boundary values, κ → 0 and κ → 1. velocity processḂ (t) (t ⩾ 0) is equal in law to the velocity process of the scaled versionḂ s (t) = B (st) · √ s (t ⩾ 0). Consider the spatio-temporal scaled version of WBM, whose positions are Z s (t) = Z (st) / √ s. For a given set of time points, 0 < t 1 < · · · < t n < ∞, note that: (using equation (1) (noting that equation (2) (using the BM selfsimilarity, as induced to its velocity process) (using equation (1) yet again) Equations (19) ]. This equality-in-law holds for any positive scale parameter s, and for any set of time points 0 < t 1 < · · · < t n < ∞. So, the finite-dimensional distributions of the scaled version of WBM coincide with the finite-dimensional distributions of the 'original' WBM, and hence [12]: WBM is selfsimilar with Hurst exponent 1/2.

Simulation algorithm for WBM
To simulate WBM trajectories over the temporal interval [0, 1] the following discretization approach is taken. A positive integer N is set, and the continuous temporal interval [0, 1] is replaced by a discrete temporal grid comprising of the discrete time points t = n/N (n = 0, 1, 2, . . . , N). The discretized version of GWN over the discrete 'time steps' of the grid is: 1, 2, . . . , N), where η n are independent and identically distributed copies of a 'standard normal' random variable (i.e. a Gaussian random variable with mean zero and with variance one). Namely, the 'noise' that corresponds to the temporal sub-interval ( n−1 N , n N ) is η n / √ N (n = 1, 2, . . . , N). In turn, the discretized version of the WBM trajectory over the time points of the discrete temporal grid is given by the positions: Z N (0) = 0, and (n = 1, 2, . . . , N), where K (t; u) is the WBM kernel function of equation (2).
Differentiating both sides of equation (26) with respect to the variable ∆ implies that So, when the WBM parameter is in the range 0 < κ < 1 2 then the function R (∆) is monotone decreasing, R ′ (∆) < 0. And, when the WBM parameter is in the range 1 2 < κ < 1 then the function R (∆) is monotone increasing, R ′ (∆) > 0.
6.5. Laplace approach: proof of equation ( 15) The kernel function of equation (2) implies that the WBM positions admit the following convolution form: Namely, the random function Z (t) is the convolution of two functions: the determinist function f (t) = t κ− 1 2 / √ c κ ; and the random function ζ (t) = t 1 2 −κḂ (t)-which is GWN that is modulated by a time-dependent 'magnitude'.
Given a temporal function φ (t), its Laplace transform is:φ (λ) =´∞ 0 exp (−λt) φ (t) dt, where λ is a non-negative Laplace variable. A key property of the Laplace transform is that it maps convolutions to products. Thus, applying the Laplace transform to the convolution of equation (33)  . (34) The Laplace transform of the determinist function f (t) = t κ− 1 2 / √ c κ is Being an integral of GWN, the Laplace transform of the random function ζ (t) = t Combining equations (34)- (36) together implies that the Laplace transform of the random function Z (t) is a Gaussian random variable with: mean E[Z (λ)] = 0; and variance As c κ = Γ (2κ) Γ (2 − 2κ), equation (37) implies that In terms of the Laplace transform of the random function Z (t), the weighted average of the WBM positions is Consequently, setting λ = 1/τ and using equations (38) and (39), we conclude that the weighted averageZ (τ ) is a Gaussian random variable with: mean E[Z (τ )] = 0; and variance Equation (40) proves equation (15).
6.6. Fourier approach: proof of equation ( 17) With regard to the variance of the integralẐ θ (T) =´T 0 exp (iθt) Z (t) dt, note that: (using equation (4)) Also, using the change-of-variables t → τ = t − u, note that: and hence Combined together, equations (41), (42) and (44) imply that (the transition from the second to the third line of equation (45) used the change-of-variables u → v = u/T). In turn, equation (45) yields the limit Consider a Gamma distribution with scale parameter s and with Gamma parameter γ (both parameters are positive). The density function of this Gamma distribution is (0 < τ < ∞); and the Fourier transform of this density function iŝ  (47) and (48) imply that Taking the limit s → 0, equation (49) implies that Setting γ = κ + 1 2 in equation (50) Finally, substituting equation (51) into equation (46) yields Equation (52) proves equation (17).

Data availability statement
No new data were created or analysed in this study.