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Fractional Brownian motions: memory, diffusion velocity, and correlation functions

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2017 J. Phys. A: Math. Theor. 50 054002

(http://iopscience.iop.org/1751-8121/50/5/054002)

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Journal of Physics A: Mathematical and Theoretical

Fractional Brownian motions: memory, diffusion velocity, and correlation functions*

A Fuliński

M. Smoluchowski Institute of Physics, Jagiellonian University, Łojasiewicza 11, Kraków, Poland

E-mail: andrzej.fulinski@uj.edu.pl

Received 19 May 2016, revised 18 October 2016Accepted for publication 11 November 2016Published 4 January 2017

AbstractFractional Brownian motions (FBMs) have been observed recently in the measured trajectories of individual molecules or small particles in the cytoplasm of living cells and in other dense composite systems, among others. Various types of FBMs differ in a number of ways, including the strength, range and type of damping of the memory encoded in their definitions, but share several basic characteristics: distributions, non-ergodic properties, and scaling of the second moment, which makes it difficult to determine which type of Brownian motion (fractional or normal) the measured trajectory belongs to. Here, we show, by introducing FBMs with regulated range and strength of memory, that it is the structure of memory which determines their physical properties, including mean velocity of diffusion; therefore, the course and kinetics of several processes (including coagulation and some chemical reactions). We also show that autocorrelation functions possess characteristic features which enable identification of an observed FBM, and of the type of memory governing its trajectory.

Keywords: Brownian motion, memory, correlations

(Some figures may appear in colour only in the online journal)

1. Introduction

Recent measurements [1–10] of single-particle trajectories inside living cells or organelles have shown that the Brownian motions (in the following, ‘Brownian motion’ (BM) without further qualification means every type of BM, including normal BM) in crowded microsize

A Fuliński

Fractional Brownian motions: memory, diffusion velocity, and correlation functions

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Journal of Physics A: Mathematical and Theoretical

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* In memoriam Marian Smoluchowski, on the 100th anniversary of the publication of his seminal papers on Brownian motion and diffusion-limited kinetics.

2017

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1751-8121/17/054002+17$33.00 © 2017 IOP Publishing Ltd Printed in the UK

J. Phys. A: Math. Theor. 50 (2017) 054002 (17pp) doi:10.1088/1751-8121/50/5/054002

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systems are anomalous, i.e. the average displacements of the Brownian particle, defined

as ⟨ ( )⟩X t2 , are proportional to tH with /≠H 1 2. Let us remember that the Hurst exponent H < 1/2 describes the so-called antipersistent, H = 1/2—normal, H > 1/2—persistent diffu-sion, and H = 1—ballistic motion.

One of main aims of recent publications concerning the interpretation of these exper imental data is the identification of the type of motion of the Brownian particles. The majority of this work is limited to the determination of a few basic characteristics: Hurst exponents, probabil-ity distribution functions, and non-ergodic properties of measured trajectories. This, however, is not sufficient [11–13] to find all relevant properties of different models of the motions: in many of these processes the aforementioned characteristics are identical, i.e. in these respects all these models do not differ from each other.

To quote Saxton: ‘Much work is being done on anomalous subdiffusion in the plasma membrane, cytoplasm, and nucleus of cells, and in model systems. The main experimental questions: Is diffusion anomalous or normal, and what are the parameters describing it? The main theoretical questions: What mechanism makes the diffusion anomalous? The main ques-tion linking these: How can the various mechanisms be distinguished experimentally?’ [13].

A few papers devoted to this problem have appeared recently, but the proposed tools for discrimination discussed in these are mainly based on subtle differences in spectral properties, scaling (transient, among others) of dispersion and so-called weakly non-ergodic behaviour [14–19], the latter mainly for confined BMs and continuous-time random walks (CTRWs) [15, 20].

In compound systems, various uncontrollable variables, parameters, and forces, effects of inertia (‘memory’), parametric noises, fluctuations of environment (both thermal and due to other neighbour processes and reactions), and so on, are put together into the effective models of BMs.

It is worth noting that all BMs can be connected with appropriate (generalized) Langevin equation  (and also with appropriate Smoluchowski–Fokker–Planck equations), including fractional ones [21].

There are several types of such models and these can be divided into three main groups, which are very different when the physical picture is considered: diffusion on fractals [20], fractional BMs (FBMs), and CTRWs. Of these, diffusion on fractals [20] is rarely discussed in the context of experimental data. The differentiation between at least some CTRW and fractional types of BM can be done in most cases simply by visual observation of some typical features of measured or simulated two- or three-dimensional paths of the (Brownian) particle: in particular, characteristic groupings of the trajectory in a few regions (traps) with random jumps between these regions, specific to CTRW trajectories, and lack of such behaviour in FBM ones. Such characteristic pictures are shown in figure 1. Corresponding experimentally measured paths can be found, for example, in [22] and [23] (see also [24, 25]).

CTRW trajectories depend on underlying long-tail distributions of the jumps. These jumps can be determined directly from experimental data. FBM ones depend on the structure of memory encoded in their definitions, which is not directly measurable. This suggests a more detailed investigation of various possible versions of such memory and a search for their sig-natures in fractional Brownian motions.

The questions arise, in what measurable behaviour are all these possible FBM-type pro-cesses different, how can the differences be observed (both in experimental and simulation data), and what formal properties lead to such differences?

The objective of this paper is to try to answer these questions. It is shown here that details embedded in the definitions of different FBM models, in particular the structure of memory,

A FulińskiJ. Phys. A: Math. Theor. 50 (2017) 054002

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determine several physical quantities, which is a partial answer to the second of Saxton’s main questions [13] mentioned above. The most important of these physical quantities appears to be the mean velocity of diffusion describing the spread of concentration of diffusing species. The latter directly influences the kinetics of various processes, including coagulation and chemical reactions [26–28]. This specific problem, with its implications for chemical kinetics, has never been (to the best of the present author’s knowledge) discussed in literature.

Section 2.1 presents basic models of FBMs, section 2.2—the class of FBMs differing con-tinuously in range and strength of memory, i.e. in the impact of former states on the whole trajectory. In section 2.3 we show how the structure of memory influences the trajectories of Brownian particles and, therefore, determines several physical properties related to diffusional processes. Section 3 is devoted to the kinetics of Brownian motions. Section 4 demonstrates that FBM correlation functions possess characteristic signatures which enable us to differenti-ate between these processes (see also [29]). In section 5 the increments of BMs and fractional noises are discussed. The last section contains conclusions and final remarks.

2. Fractional Brownian motions

Here, we shall use the physical notation (less general, but with a clearer physical meaning), i.e. put

( ) ( )   ( ) ( )  ⟨ ( )⟩  ⟨ ( ) ( )⟩ ( )∫ξ ξ ξ ξ ξ σ δ= = = = −W t t t W t u u t t s t sd d , d , 0, ,t

00

0 0 0 0 02

(1)

where W(t) is the Wiener process (normal Brownian motion), ( )ξ t0 —thermal (white) Gaussian noise, and ⟨ ⟩⋅ —ensemble average.

Figure 1. Two-dimensional simulated tracks of Brownian particles. nBM: normal Brownian motion (Wiener process); CTRW: nBM inside traps, Pareto-distributed jumps between traps.

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2.1. Basic models

There are a few models of FBMs proposed in literature: Kolmogorov’s [30], Lévy’s [31], Mandelbrot and Van Ness’ [32], and scaled Brownian motion (SBM) [33–38].

A centred Gaussian process BH(t) is called (Kolmogorov’s) FBM with a Hurst parameter ( )∈H 0, 1 , if it has the covariance (autocorrelation) function [30, 32]:

⟨ ( ) ( )⟩ ( )κ= | | +| | −| − |B t B s t s t s .H H HH H H2 2 2 (2)

The Kolmogorov FBM (K-FBM) process BH(t) can be also defined as Brownian motion driven by fractional (anticorrelated, antipersistent) Gaussian noise ( )ξ tH [15]:

( ) ( )   ( ) ( )  ⟨ ( )⟩

⟨ ( ) ( )⟩ ( ) ( )

∫ξ ξ ξ

ξ ξ κ κ δ

= = =

= − | − | + | − | −− −

B t t B t u u t

t s H H t s H t s t s

˙ , d , 0

2 2 1 4 ,

H H

t

H H

H H HH

HH

02 2 2 1

(3)

with appropriate initial conditions. Note that for H = 1/2 ( )ξ ξ=tH 0—a more detailed relation between ξH and ξ0 is given in the section 5, below.

There are several integral representations of FBMs. However, these definitions lead to frac-tional processes with nonidentical properties.

The oldest one is Lévy FBM (L-FBM) [31]:

( )( / )

( ) ( )/∫ ξ=Γ +

− −X tH

u t u u1

1 2d .L H

tH

0

1 20 (4)

It will be shown in section 4 that this Lévy FBM (as well as its generalizations, defined below) has a correlation function different, though in some respects similar, to (see figures 6 and 9 below), which is defined by equation (2).

According to Mandelbrot and Van Ness [32], the (Holmgren–Riemann–Liouville) integral (4) ‘puts too great an importance on the origin for many applications’; therefore, they pro-posed another, now the best-known definition, via the Weyl integral:

( ) ( ) ( )

( )( / )

( ) ( ) ( )/ /∫ ξ

= +

=Γ +

− − −−∞

− −⎡⎣⎢

⎤⎦⎥

X t X t T t

T tH

u t u u u

,

1

1 2d .

M H L H H

HH H

01 2 1 2

0

(5)

The autocorrelation function of ( )X tM H is equal to that of BH(t) [39, 40], i.e. the process ( )X tM H fulfils the definition of K-FBM. Therefore, equation  (5) can be treated as another

integral representation of K-FBM. This implies, among others, that the Kolmogorov diffusion coefficient is equal to that of the Mandelbrot–Van Ness (MVN) one. The results of section 4, equations (10)–(11), give

κ = K a .H H M (6)

Another anomalous diffusion process is SBM [33–38] defined by its integral representation:

( )( / )

( )/∫ ξ=Γ +

−X tH

u u u1

1 2d , .S H

tH

0

1 20 (7)

This process describes, among others, the Brownian motion in systems with variable temper ature and/or density [38].

Comparing kernels defining L-FBM and SBM, i.e. (t − u)H−1/2 versus uH−1/2, one may con-clude that, for symmetry, one may still define another basic model, call it the ‘scaled Wiener process’ (SW process), defined by the ‘kernel’ tH−1/2, i.e.

A FulińskiJ. Phys. A: Math. Theor. 50 (2017) 054002

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( )( / )

( )/∫ ξ=Γ +

−X tH

u t u1

1 2d .W H

tH

0

1 20 (8)

Of these four models, practically the only process discussed in literature is ( )X tM H [32], equation (5), though recently more and more attention has been gained by processes related to SBM[41–46].

2.2. Intermediate models

In the light of the above discussion, and of radical differences between correlation functions of ‘memoryless’ SBM and SW versus memory-containing L-FBM and K-FBM (see section 4), it seems reasonable to look more closely at the role of memory (more precisely, at the bal-ance of the impact of present and past forces) in the behaviour of Brownian trajectories, i.e. to investigate how the various possible versions of such memory change various characteristics of Brownian motions.

Consider the fractional Lévy-like Brownian process with variable memory, defined by

( )( / )

( )/∫ µ ν ξ=Γ +

| − |µ−X t

Hu t u u

1

1 2d .H

tH

0

1 20 (9)

A physical interpretation of such processes is obvious: t denotes the running time, i.e. ( )µX tH is the position of the Brownian particle at time t (trivial), whereas the times [ )∈u t0,

denote the range of past times when the Brownian particle was moving under the action of the stochastic forces ( )ξ u0 . Hence, the parameter ν changes the strength of the influence of the past, μ—of the present on the trajectory of the particle. This description makes sense when both µ≠ 0 and ν≠ 0. The limiting cases (both ‘no memory’) describe: ν≠ 0, µ = 0—process driven by noise ( )ξ u0 scaled by ( ) /ν −u H 1 2 (SBM), and µ≠ 0, ν = 0—process itself scaled by ( ) /µ −t H 1 2 and driven by pure thermal noise (SW). SBM emphasizes the influence of noises (forces) near the end of the trajectory (u near t) for persistent (H > 1/2), and near the beginning of the trajectory (u near 0) for antipersistent (H < 1/2) processes, whereas L-FBM (µ ν= = 1), and SW only do so inversely. In other words, these parameters reduce or enhance—depending on H—the impact of memory on the trajectory.

The parameters μ and ν can be chosen in a few different ways, acting slightly differently on memory properties of the process: (i) μ: any value, ν = 1 implies increasing (µ> 1) or decreasing (µ< 1) role of the present time; (ii) µ = 1, ν: any value—the same for all earlier times; or (iii) [ ]µ∈ 0, 1 , ν µ= −1 changes the balance between present and earlier times.

For the sake of completeness, let us mention that it is possible to define several various linear and non-linear combinations of these processes. Such models might be of use for the description of Brownian motions driven by several random forces ( )ξ t0 , ( )ξ t1 , either Gaussian or non-Gaussian, either correlated between themselves or not, etc.

2.3. Structure of memory

The above-mentioned remark about L-FBM placing too great an importance on origin in numerous applications points towards the role of the different impacts of forces at running time t and past times u (‘memory’) on the properties of different FBMs.

Upopn a closer look, the MVS statement in its original form [32] is true only for H > 1/2, i.e. for persistent processes, when the kernel (t − u)H−1/2 emphasizes the stochastic forces

( )ξ u0 at small values of the past times u along the trajectory in comparison to the late ones near running time t. For H < 1/2 the kernel acts in an opposite manner—late times are favoured, at →u t.

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Similar analysis can be easily done for all other models of FBMs: the late forces ( →u t) are amplified for persistent, and damped for antipersistent, SBM and SW processes, though not identically (see below); for various intermediate processes the effect depends on the interplay of additional parameters, multiplying the past and running time, etc. In all these cases the effect on antipersistent and persistent processes is (roughly) inversed. This effect is clearly visible in the behaviour of the mean velocity of diffusion (figure 2) and of correlation func-tions (figures 3–5 below).

In particular, the main difference between memory-containing L-FBM and memoryless SBM lies in the fact that SBM emphasizes the influence of noises (forces) near the end of the trajectory (u near t) for persistent processes, and near the beginning of the trajectory (u near 0) for antipersistent processes, whereas L-FBM—just inversely. We shall see that this difference influences the shapes of their correlation functions.

The SW process, being the product of normal Brownian motion W(t) and deterministic factor ( ) / ( / )/= Γ +−F t t H 1 2H 1 2 , still has different preferences for the influence of random forces on the trajectory. The presence of F(t) means that the longer the trajectory, the stronger the impact of random impulses is for persistent motion and the weaker it is for antipersistent motion, and that this impact is different in subsequent ranges [ ]∈t t0, 1 , [ ]∈t t0, 2 , etc.

The MVN process, presented as the remedy for the flaw in the Lévy process, introduces an additional memory parameter: reach of influence of far past into the trajectory. Indeed, the greater time t, the more significant the addition from its tail T to the value of ( )X tM H , and in this way the tail compensates the uneven treatment of various parts of the trajectory by XL H, but again it ‘compensates too great an importance on the origin’ for persistent processes, whereas for antipersistent ones it compensates too great an importance on the end of trajectory.

The integral representations of all models of BMs discussed above suggest that the main source of differences between various BMs lies in the memory kernels. However, the above discussion of the behaviour of four main types, i.e. L-FBM, SBM, SW, and MVN-FBM (equation 5), shows that the effect of the ‘memory’ depends not only on the shape of the kernels (interplay between past and present), embedded in the definitions of different types of FMBs, but also on the value of the Hurst exponent H, type of damping, and the reach of memory (L-FBM versus M-FBM) into the past. We shall call this set of mathematical details contained in the definitions, and subtly influential details of the fractal structure of various BM trajectories, the ‘structure of memory’. The sections which fol-low will show that these formal differences translate into observed physical behaviour of Brownian motions, especially, into shapes of autocorrelation functions, and speed of diffusion, which in turn directly determine the behaviour of several other physical and chemical processes.

The following sections will show that these observable quantities also depend on the types of driving stochastic forces (noises)—compare Langevin equations and integral representations of W(t) (normal diffusion) and B(t) (K-FBM), equations (2) and (3), and their diffusion veloci-ties and correlation functions. This point will be elaborated in more detail in section 5 below.

To conclude this section, let us point out that (i) memory- and memoryless processes can be recognized by their Markovian properties (see [47–51]) by application of the Bachelier–Smoluchowski (Chapman–Kolmogorov) functional equation [52, 53] (see also [48]); (ii) differ-ent approaches to the memory problem were presented recently in [50, 54]; and (iii) analogical processes can be defined as memory generalizations of the MVN–FBM (equation (5)) process.

3. Kinetics

The properties of above-defined fractional processes differ in some respects. The most impor-tant of these from a physical point of view appears to be the differences in their velocities, the latter describing (among others) the spread of concentration of diffusing species after time t.

A FulińskiJ. Phys. A: Math. Theor. 50 (2017) 054002

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These differences in velocities of Brownian particles (both mean and first passage times) are embedded in prefactors (mobilities—effective diffusion coefficients) of the dispersion ⟨ ( )⟩X t2 (being the measure of the mean length of the particle’s path), and in the present context can be ascribed to the differences in the memory structure of different FBMs:

⟨ ( )⟩ / ( / )σ= = Γ +X t a K t K H, 1 2 ,i HH

H2 2

02 2 (10)

where ai is the ‘memory coefficient’ distinguishing various types of FBMs. For the types discussed in section 2 (i = L, S, W and μ):

∫

∫ µ ν

µ µ ν µ ν ν ν

µ ν

= = =

= + = + −

= | − |

=− − | − | ≠

=

µ

∞− −

−

−⎪

⎪

⎡⎣⎢

⎤⎦⎥

⎧⎨⎩

a a a H

a a b b x x x

a x x

H

a

1, 1 2 ,

1 , d 1 ,

d

sgn 2 , 0,

, 0.

W L S

M L H HH H

H

H H

HW

0

1 2 1 22

0

12 1

2 2

2 1

/

( )   ( )

( ) /  

/ /

(11)

Figure 2. Changes in the mean velocity of diffusion as functions of memory parameters, in three different parametrizations. Note the differences in scales.

A FulińskiJ. Phys. A: Math. Theor. 50 (2017) 054002

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Mean distance travelled by the Brownian particle is given by the square root of the

dispersion ⟨ ( )⟩X t2 . Hence the mean velocity of diffusion is ( ) ⟨ ( )⟩ /=v t X t t2 .For the four main types of FBMs described above we have:

( )   ( ) ( ) ( )/( ) ( ) ( )   ( ) ( )

/ ( / ) /

= = =

= + =

=µ µ

−v t K t v t v t v t H

v t b v t v t a v t

v v a a

, 2 ,

1 , ,

.

W HH

L S W

M H L W

i j i j

2 2 2 2 2 2

2 2 2 2

1 2

(12)

This means that the Kolmogorov–Mandelbrot–Van Ness process is faster than the Lévy and scaled ones, which, in turn, due to the lack of denominator 2H in ( )v tW

2 , are slower than the SW process for persistent (2H > 1) and faster for antipersistent (2H < 1) processes.

Figure 2 shows how the speed of diffusion v(t) is changed by the changes in the structure of memory. Note that in all these cases the changes of v(t) of persistent and antipersistent processes are (roughly) inversed.

4. Autocorrelation functions

Direct analytic calculations of the (non-normalized C and normalized Cn) autocorrelation function:

( ) ⟨ ( ) ( )⟩

( ) ( )/ ⟨ ( )⟩⟨ ( )⟩

=

=

C t s X t X s

C t s C t s X t X s

, ,

, , ,n2 2 (13)

from the integral formulae above, are possible only for SBM and SW:

τ τ τθτ θ τ θ

= == =

−C t s a K C a K

C t s a K C a K

, , ,

, , ,S S H

HL H

H

n S S HH

n SW L H

2SW

1 2

, ,1 2

( )   ( )( ) ( / )   ( / )

/

/ (14)

where ( )τ = t smin , , ( )θ = t smax , .The last relation shows that the normalized autocorrelation functions of the normal

(H = 1/2) and scaled (any H > 0) Wiener processes behave identically as functions of time, and differ only in the magnitude of diffusion coefficients.

For Lévy-like FBMs, we get the relatively simple form:

( ) ( ) /∫ µ ν µ ν= | − || − |µ µτ

−C t s a K u t u s u, d .HH

0

1 2 (15)

( ( )µ ν= = =µC C 1L ) which can be computed numerically. MVN-FBM cannot be computed easily from the integral formula (5), however, =C CM K, equation (2) [39, 40]. All these pro-cesses, though, have the same time dependence of dispersion, but differ in the time-behaviour of C(t,s).

The best visualization of similarities and differences between all these memory models is given by non-normalized autocorrelation functions C(t,s) drawn as functions of s for constant t. Such a comparison of the main models is given in figure 3.

Figures 4 and 5 present more details for various μ-type models of memory.These results show that the correlation functions for various models of Brownian motions

differ in their asymptotic behaviour: ( )∼ αC t s s, 0 for →s 0, ∼ α∞s for →∞s , and in behaviour near the turning point s = t.

The asymptotic behaviour is summarized in table 1.

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For normalized Cn(t,s) α α= − Hn i i, , = ∞i 0, . Note that for all these memory models α α+ =∞ H20 , α α+ =∞ 0n n,0 , .

The results presented in table  1 show that the asymptotic behaviour of autocorrelation functions alone are insufficient to discern between SBM and antipersistent K-FBM models, and between various L-descendent (L-FBM, μ-FBM, SW) models. Further information in this respect can be gained from autocorrelation functions of finite increments of these processes (next section), and the behaviour near the turning point s = t. The latter can be characterized by the inclination along the (non-normalized) correlation functions:

( ) ( )α =∂∂

ss

C t sln

ln , .t

(16)

The differences described by these characteristics are shown in figure 6 for K-FBM and L-FBM, and in figures 7 and 8 for various μ-models. ( )α s of K-FBM and L-FBM shown in figure 6, for persistent processes, differ very little near s = t. Note, however, that K-FBM and L-FBM models also differ in the values of α0 and α∞ (table 1).

Results presented in figure 6 demonstrate, among others, strong changes in inclinations of C(t,s) for K-FBMs and L-FBMs near s = t. On the other hand, it is easy to find that for SBM and for SW processes ( )α s consist of flat straight lines with discontinuity at s = t. In general, all correlation functions behave singularly at t = s, though the character of this singularity is different for different models and for persistent and antipersistent processes (see inset in figure 6). These facts, together with the results shown in figure 6, enable differentia-tion between antipersistent K-FBMs and SBMs, as well as between L-FBMs and SW, which exhibit the same asymptotic behaviour (table 1).

The behaviour of ( )α s for μ-FBMs, shown in figures 7 and 8 enables, in particular, the identification of the values of memory parameters μ and ν and the differentiation between L- and μ- models, for which the asymptotic behaviour is identical. Moreover, comparison of these figures shows clearly the role of the structure of memory: the effect of enhancement of running time t (parameter μ, figure 7) is much stronger than that of enhancement of past times u (parameter ν, figure 8)—running time and past times play different roles in the behaviour of Brownian motions.

Comparison of all these characteristics is sufficient for identification of the memory model of a given Brownian motion.

Dependence of C(t,s) on time t with constant ∆ = −s t does not give interesting results, in the sense that the differences between various FBMs are too weak to be characteristic enough.

5. Increments

Brownian motions (BMs) discussed in this text, although continuous (strictly speaking, Hölder-continuous) are (almost) nowhere differentiable in a normal sense, although there are suggestions that fractional derivatives of some at least nowhere differentiable functions do exist [55]. Nevertheless, it is possible to define ‘normal’ increments of these fractal processes.

5.1. Infinitesimal increments—fractional noises

Denote

( ) ( ) ( )∫ ξ=X t u K t u ud , ,J Ht

t

J H 00

(17)

A FulińskiJ. Phys. A: Math. Theor. 50 (2017) 054002

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where ( )X tJ H is one of processes defined in the section 2. The infinitesimal increment ( )δX tJ H of this process can be defined:

( ) [ ( ) ( )]/δ δ δ= + −δ↓

X t X t X tlim .J H J H J H0 (18)

Figure 3. Calculated autocorrelation functions C(t, s) at constant t = 100 of Brownian motions: K–Kolmogorov’s, L–Lévy’s, S–SBm, W–scaled Wiener, two μ-Bm. Not normalized.

Figure 4. Calculated autocorrelation functions C(t, s) at constant t = 5000 of X t( )µ β , for 1ν = , 0.2...0.8µ = . Not normalized.

A FulińskiJ. Phys. A: Math. Theor. 50 (2017) 054002

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Such quantities do exist, but are (almost) nowhere continuous and can be treated as fractional noises. Infinitesimal increments of BMs discussed here are:

( ) ( )   ( ) ( )

( ) ( ) ( ) ( )( / )

( ) ( ) /( / )

( )

( ) ( ) /( / )

( ) ( ) ( ) ( )

( ) ( ) /( / )

( ) ( )

/

/

/

/ /

∫

∫

∫

δ ξ δ ξ

δ δ ξ ξ ξ

δ ξ ξ

δ ξ µ ν ξ

δ ξ ξ ξ

δ ξ ξ

≡ ≡

= ≡ = =Γ +

≡ =−

Γ +−

≡ =−

Γ +| − |

≡ = +

≡ =−

Γ +− − −

µ µ

−

−

−

−∞

− −⎡⎣⎢

⎤⎦⎥

W t t B t t

X t X t t tt

H

X t tH

Hu t u

X t tH

Hu t u

X t t t t

X t tH

Hu t u u

, ,

1 2,

1 2

1 2d ,

1 2

1 2d ,

,

1 2

1 2d ,

H H

S H W H S W

H

L H Lt

tH

Ht

tH

M H M L T

T H TH H

01 2

0

3 20

3 20

03 2 3 2

0

0

0

(19)

where [ / ( / )]σ= Γ +Q H 1 2H 02.

The MVN process, equation (5), is one of integral representations of K-FBM. Then, incre-ments of both processes are equivalent. This leads to the relation ( ) ( )ξ ξ≡t tH M , which expresses fractional noise ( )ξ tH by a memory integral containing white Gaussian noise (thermal noise)

( )ξ t0 and, therefore, enables numerical simulations of ( )ξ tH :

Figure 5. Calculated autocorrelation functions C(t, s) at constant t = 5000 of X t( )µ β , for 1µ = , 0.2...0.8ν = . Not normalized.

Table 1. C(t,s) for the main types of memory.

Process 0α α∞ Comments

K–FBM–a 2H 0 Equations (2) and (3) H 0.5⩽K–FBM–p 1 2H − 1 Equations (2) and (3) H 0.5⩾SBM 2H 0 Equation (7)μ–FBM H + 0.5 H − 0.5 Equation (9) , 0, 1( ) ( ]µ ν∀ ∈L–FBM, SW H + 0.5 H − 0.5 Equations (4) and (8)

Note: p—persistent, a—antipersistent.

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Figure 6. Comparison of inclination measure s( )α of autocorrelation functions C(t, s) for K-FBM (red) and L-FBM (blue). Turning point t = 5000. Inset: two examples of singularities at s = t for H = 0.2 and 0.6.

Figure 7. Comparison of inclination measure s( )α of autocorrelation functions C(t, s) for μ-FBMs, dependence on μ for 1ν = , Turning point t = 5000.

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It is easy to find that correlation functions of infinitesimal increments of memoryless processes W(t), ( )X tS H and ( )X tW H are δ-correlated. Autocorrelation functions of remain-ing fractional noises, ( )δX tJ H contain a coloured (non-δ-correlated) part and contribution

( )δ∼ −t s -compensating divergencies appearing during calculations of the average values of dispersions, correlation functions, etc. This is similar to, but more complicated than those appearing in the Kolmogorov fractional noise ( )ξ tH , equation (3). Additionally, the coloured parts are rather complicated. The simplest one is that of the increments of the Lévy process

( )∼ −H t smin2 2 , .These results also suggest that infinitesimal increments of BMs driven by memoryless

kernels and white (δ-correlated) noises are also δ-correlated, whereas fractional noises, being infinitesimal increments of FBMs defined by memory kernels, are coloured.

The above relations establish direct links between Gaussian white noise ( )ξ t0 (thermal noise), and various white and coloured noise ( )ξ tH (FGNs). Identical relations hold for FGNs originating from other white noises, e.g. from white α-stable i.i.d. (independent and identi-cally distributed) processes ( )ξα t,0 .

5.2. Finite increments—sample paths

Infinitesimal increments are not experimentally measurable. What are directly measurable (observable) [56] are only the sample paths { ( )X tJ H n } of all these processes and their finite increments ( )D X tJ k H :

( ) [ ( ) ( )]= + −D X tkDT

X t kDT X t1

,J k H J H J H (20)

Figure 8. Comparison of inclination measure s( )α of autocorrelation functions C(t, s) for μ-FBMs, dependence on ν for 1µ = . Turning point t = 5000.

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= +t t nDTn 0 , n = 0,1,..., DT being the time resolution.Then, the autocorrelation functions of finite increments can be calculated directly from

autocorrelation functions of respective BMs:

( ) ( ) ( ) ( ) ( )= + + − + − + +C t s C t kDT s kDT C t kDT s C t s kDT C t s, , , , , .D X X X X Xk

(21)

Similar relations hold for finite increments of other observables.Figure 9 shows these functions of various BMs, normalized to the same size

( )/ ( ( ) )| |C t s C t s, max , to enable direct comparison:Changes of t = const. lead only to shifts of curves, whereas changes of the range k of incre-

ments kDT lead to narrowing/widening of the peaks. Width that is too narrow may result in the camouflage of characteristic differences of the shapes, disabling the differentiation between BMs, the more that pairs C t s,DW( ) and CDS(t,s), and CL(t,s) and CB(t,s) differ little. On the other hand, if k is too large it may also result in deformation of the main characteristics.

Characteristic features of C(t,s) of increments of Wiener and scaled BM are the flat wings, visible for small enough k. Moreover, these functions are very similar. The only difference lies in the fact that the CDW consists of two straight lines (inclinations α =± 1) for CDW, and of two concave or convex lines for CDS. A characteristic feature of CL(t,s) and CB(t,s) is the presence of anticorrelated regions for antipersistent processes. ( )C t s,SW are different to C(t,s) of all other BMs, and can be recognized by characteristic wings.

6. Final remarks and conclusions

Anomalous fractional Brownian motions are useful universal models for many nonstationary fractional processes observed in reality.

The investigations presented above emphasize the role of memory (or rather, of the bal-ance of the impact of present and past forces) in the behaviour of Brownian motions. It was shown here that it is possible to introduce memory effects gradually, from memoryless forms to strong memory, with SBM, equation (7), and SW processes, equation (8), being the limiting cases of standard fractional Brownian motions. Introduction of μ-FBM processes, with gradu-ally changed roles of past and present times, though perhaps not so important from the point of view of analysis of experimental data, has enabled a better understanding of the role of the structure of memory: the effect of enhancement of running time t versus that of enhancement of past times u is very different (figures 7 and 8). Moreover, analysis of the behaviour of the mean velocity of diffusion and of autocorrelation functions shows that the memory effects are different for antipersistent and persistent processes (figures 2–5, 7 and 8).

The structure of memory of a given model of Brownian motion strongly influences several physical properties of trajectories. Of these, the most important seems to be the mean diffu-sion velocity v(t) (section 3). v(t) characterizes the spread of the concentration of diffusing species, and the latter directly influences the kinetics of various processes (including chemical reactions), depending on the speed of providing specific molecules at specific sites [26–28].

Another important quantity is the autocorrelation function, both of the process {X(t)} itself (section 4), and of its finite increments (section 5.2), properties of which can be obtained from measured single-particle tracks. This enables the realization of the secondary purpose of this paper: to find how to differentiate between various theoretically defined models of Brownian motions, and in particular, to find whether the structure of memory can be determined exper-imentally. We have shown that the autocorrelation function C(t,s) makes a good tool for this purpose and that the behaviour of C(t,s) for t = const., with →s 0, s near the turning point s = t, and →∞s are sufficiently characteristic to determine the model of the (fractional)

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Brownian motion in the specific case considered. Additional information, enabling differen-tiation between various BMs, can be obtained from autocorrelation functions of increments of measured sample paths, described at the end of the section 5 (see figure 9 and the discussion therein). In this respect, the calculation of these functions for several increasing widths of increments seems to be especially effective.

The results presented in this investigation also shed some light on the above-mentioned remark that Lévy FBM, equation  (4), ‘puts too great an importance on the origin (of the Brownian trajectory) for many applications’. Our analysis (section 2.3) has shown that, whereas this is true for H > 1/2, i.e. for persistent processes, for antipersistent (H < 1/2) ones the kernel (t − u)H−1/2 acts in an opposite manner—favoured are stochastic forces ( )ξ u0 at late times, i.e. at →u t. Similar analysis for all other models of FBMs shows that the impact of early and late stochastic forces (‘memory’) depends strongly on the value of the Hurst exponent H. In general, for antipersistent processes, various effects related to the structure of memory are reversed with respect to those for persistent ones. This effect is clearly seen in figures 2–5, 7 and 8. From the physical point of view, most important in this respect, are the changes in mean diffusion velocity (figure 2): the FBM models which lead to the fastest diffu-sion for persistent processes, are slowest for antipersistent.

It is also worth noting that correlation functions of increments of Kolmogorov and Lévy FBMs are very similar.

When analyzing the role of the memory in the Brownian motion models, it is worth noting that for the Lévy-like μ-FBMs, equation (9), the changes of the parameter ν (i.e. strength of the memory of the past) with µ = 1 influence the shape of C(t,s) much less than the changes of

Figure 9. Autocorrelation functions CDX(t, s), for constant t = 500, of finite increments of various BMs. Increment kDT = 250.

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the parameter μ (strength of the impact of recent forces), and that the effect is much stronger for antipersistent processes than for the persistent ones (figures 4 and 5).

For all models of Brownian motion discussed here, the normalized autocorrelation func-tions ( ) →C t s, 0n for →∞s and t = const., i.e. all FBM’s become asymptotically uncor-related, though their correlation functions vanish with different speeds: fastest is SBM, slowest—L-FBM, μ-FBM, and SW. Moreover, Kolmogorov’s correlations, equation (2), van-ish faster for antipersistent (subdiffusional) than for persistent (superdiffusional) processes. These differences are related to differences in the diffusion velocities, equations (11), (12) and figure 2, and may also help the recognition whose process is represented by a given cor-relation function.

Other results which seem to be of some interest are the relation presented at the end of the preceding section  relating directly the fractional noises ( )ξ tH with thermal noise ( )ξ t0 . The observation noted after equation  (19) that correlation functions of infinitesimal increments of all those considered here memoryless processes: W(t), ( )X tS H , and ( )X tW H are δ-correlated, whereas those of memory-sensitive processes: B(t), ( )X tL H , and ( )µX tH are coloured. This fact suggests the supposition that this is the general rule: memory embedded in the process {X(t)} is the source of correlations of its infinitesimal increments ≡ fractional noises.

Finally, let us repeat that most of the results presented above hold for any i.i.d. δ-correlated (‘white’) noise ( )ξ t0 , not only for Gaussian-distributed ones. Let us note, however, that the drawback of the use of non-Gaussian noises is that the direct simple relation between ( )ξ t0 and the Maxwell distribution of momenta, i.e. with the temperature of the medium (fluctuation-dissipation theorem) is lost.

Acknowledgments

I wish to express my gratitude to one of the anonymous referees for the suggestion to discuss not only Brownian motions, but also their increments.

This work is partially supported by an NCN Maestro Grant No.2012/06/A/ST1/00258.

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