Centre-of-mass like superposition of Ornstein–Uhlenbeck processes: a pathwayto non-autonomous stochastic differential equationsand to fractional diffusion

# Centre-of-mass like superposition of Ornstein–Uhlenbeck processes: a pathway to non-autonomous stochastic differential equations and to fractional diffusion

Mirko D’Ovidio Silvia Vitali Vittoria Sposini Oleksii Sliusarenko Paolo Paradisi Gastone Castellani Gianni Pagnini DIFA - Department of Physics and Astronomy, University of Bologna, Viale B. Pichat 6/2, I-40127 Bologna, Italy Department of Basic and Applied Science for Engineering, Sapienza University of Rome, Via Antonio Scarpa 16, I-00161 Rome, Italy BCAM – Basque Center for Applied Mathematics, Alameda de Mazarredo 14, E-48009 Bilbao, Basque Country, Spain Ikerbasque – Basque Foundation for Science, Calle de María Díaz de Haro 3, E-48013 Bilbao, Basque Country, Spain ISTI-CNR, Institute of Information Science and Technology ”A. Faedo”, via Moruzzi 1, I-56124, Pisa, Italy Institute for Physics and Astronomy, University of Potsdam, Karl-Liebknecht-Strasse 24/25, D-14476 Potsdam-Golm, Germany
###### Abstract

We consider an ensemble of Ornstein–Uhlenbeck processes featuring a population of relaxation times and a population of noise amplitudes that characterize the heterogeneity of the ensemble. We show that the centre-of-mass like variable corresponding to this ensemble is statistically equivalent to a process driven by a non-autonomous stochastic differential equation with time-dependent drift and a white noise. In particular, the time scaling and the density function of such variable are driven by the population of timescales and of noise amplitudes, respectively. Moreover, we show that this variable is equivalent in distribution to a randomly-scaled Gaussian process, i.e., a process built by the product of a Gaussian process times a non-negative independent random variable. This last result establishes a connection with the so-called generalized gray Brownian motion and suggests application to model fractional anomalous diffusion in biological systems.

###### keywords:
Ornstein–Uhlenbeck process, Heterogeneous ensemble, Superposition, Center of mass, Non-autonomous stochastic differential equation, Randomly-scaled Gaussian process, Generalized grey Brownian motion, Anomalous diffusion, Fractional diffusion.
journal: XXX

## 1 Introduction

Let , be a sequence of Wiener processes with and . As usual, we denote by the expectation with respect to where is the starting point for the process under consideration.

In this paper we consider the ensemble of the Ornstein–Uhlenbeck (OU) processes satisfying the stochastic differential equations (SDE)

 dXκt=−Xκtτκdt+√2σκdWκt,Xκ0=x, (1)

where, , and . We call relaxation times and noise amplitudes. In particular, the following equivalence in law holds true

 Xκt=e−tτκ(x+Wκφ(t)),κ=1,2,…,N, (2)

where

 φ(t)=σκτκ(e2t/τκ−1). (3)

Hence, each is a real-valued Gaussian process with . The correlation function

 Ex[XκtXκs]=e−(t+s)/τκmin(φ(t),φ(s)) (4)

gives the covariance function

 C(t,s)=σκτκ\e−|t−s|/τκ−(σκτκ+x2)\e−(t+s)/τκ. (5)

Without loss of generality, we assume that and therefore

 C(t,s)=σκτκ(\e−|t−s|/τκ−\e−(t+s)/τκ). (6)

For a given , the infinitesimal generator of is therefore

 Au=−xτκdudx+σκd2udx2,u∈Cc(R,R), (7)

where is the set of smooth and compactly supported functions on . In view of the applications in diffusion problems, in analogy with the Langevin equation (for ) we introduce the mass-like variable such that

 1τκ=γκmκ,√σκ=√σ0mκ, (8)

where is a fixed parameter independent of , and, , the pair of independent parameters in the SDE (1) is now replaced by the pair .

In the following, we study the centre-of-mass like process

 Zt=∑κmκMXκt, (9)

with .

We highlight that, because of the center-of-mass like formulation, the process here considered differs from the superposition of OU processes considered for example in Refs. csorgo_etal-cjm-1990 (); lin-spa-1995 () or from the superposition of OU-type processes considered in Refs. leonenko_etal-s-2005 (); barndorffnielsen_etal-mcap-2005 (); grahovac_etal-jsp-2016 (), and also the type of provided results is different. In particular, in this paper we derive the non-autonomous SDE satisfied by a process equivalent in distribution to the process and more we show also that the process is equivalent in distribution to a randomly-scaled Guassian process, i.e., a process built by the product of a non-negative random variable times a Gaussian process. This last result can be understood in the framework of the so-called generalized grey Brownian motion (ggBm), which can be recovered when the Gaussian process is the fractional Brownian motion. The ggBm is a generalization of the theory of the white noise analysis by introducing non-Gaussian measures of Mittag–Leffler type mura-phd-2008 (); mura_etal-jpa-2008 (); mura_etal-itsf-2009 (); grothaus_etal-jfa-2015 (); grothaus_etal-jfa-2016 ().

Under the physical point of view, the considered system can be understood as a heterogeneous ensemble of Brownian particles, i.e., a system composed of non-identical Brownian particles that differ in their density (mass devided by volume). The study of the centre of mass allows for estimating the average concentration and the momentum of inertia of the ensemble by computing the mean and the mean square displacement, respectively. Such a system can be related to the so-called anomalous diffusion which refers to diffusion processes that, in opposition to Brownian motion, do not show a Gaussian density function of particle displacements and neither a linear growth in time of the mean square displacement. Anomalous diffusion is a widespread phenomenon anomaloustransport (); klm () that requires specific statistical tools meroz_etal-phyrep-2015 (), that is sometimes related to fractional diffusion metzler_etal-jpa-2004 (), and that can be generated by the polydispersity when classical thermodynamics holds gheorghiu_etal-pnas-2004 (), i.e., with Gaussian noise as in the present case, or by noises with long-range spatiotemporal correlations with even ”anomalous” thermodynamics gheorghiu_etal-pnas-2004 (), i.e., with non-Gaussian noise. In this respect, the derived link between the process and the ggBm allows for perspective applications of the present results to model fractional diffusion in biological systems in view of the promising application of the ggBm in these systems molina_etal-pre-2016 (); sposini_etal-njp-2018 ().

In the next Section 2, the main theorems are stated and their proofs are given in Section 3. In Section 4, the numerical simulations related to the main results of Section 2 are shown. In Section 5 the conclusions are reported and the application to model anomalous diffusion in biological systems is discussed.

## 2 Main results

In this Section we present the two main results, i.e., Theorem 1 and Theorem 3, that we obtain for the process defined in (9). The first concerns the determination of the non-autonomous SDE satisfied by a process that is equivalent in distribution to , and the second concerns the equivalence in distribution of the process with a randomly-scaled Gaussian process, i.e., a process defined by the product of a non-negative random variable and a Gaussian process.

We introduce the scaled process

 Yκt=Xκt√σκ, (10)

for which

 P0(Yκt∈dy)=∫Ωτpκ(y;t|τ)q(dτ)dy, (11)

where the Gaussian density is conditioned to and

 P(τκ∈Ωτ)=∫Ωτq(dτ)=1. (12)

We assume that , that is has population density . We notice that , , where is distributed according with . We denote by “” the equality in distribution for random variables.

We maintain the superscript in (11) although the sequence (and therefore, the sequence (10)) are identically distributed. Indeed, we have that and where .

###### Theorem 1.

Let be the center-of-mass like stochastic process in (9). Then, where satisfies the following non-autonomous SDE

 dZ∗t=−Z∗tτeff(t)dt+√2σ0MdWefft,Z∗0=z, (13)

with , such that , and where is defined as

 τeff(t)=⎡⎢ ⎢ ⎢ ⎢⎣∫+∞−∞∫Ωτy2pκ(y;t|τ)q(τ)dτdy∫+∞−∞∫Ωτy2τ2pκ(y;t|τ)q(τ)dτdy⎤⎥ ⎥ ⎥ ⎥⎦1/2. (14)

We observe that the SDE (13) is non-autonomous because of the dependence on of the drift coefficient through the function , and that the noise amplitude is not a process but a non-negative random variable. The opposite situation with a drift independent of and the noise amplitude given in terms of a stochastic process has been recently studied in Ref. benth_etal-spa-2018 (). By setting

 Λ=σ0/M2,

from (8), we have that is an independent non-negative random variable.

###### Theorem 2.

Let be the ensemble defined in (1) and (8). Let be the center-of-mass like process defined in (9). Let be the scaled process in (10). Denote by the Gaussian process , . Then we have that , , as ,

 1√NZt\lx@stackreld→√ΛBHt. (15)
###### Theorem 3.

Let be the ensemble defined in (1) and (8). Let be the center-of-mass like process defined in (9). Let be the scaled process in (10). Denote by the Gaussian process , . Then we have that , , ,

 Zt\lx@stackreld=√NΛBHt=ZHt. (16)

We observe that since is a random variable that is different for any realization of the process and it is independent of , the process is not the same process studied in Ref. garet-spa-2000 ().

###### Lemma 1.

The stochastic process has a non-exponential correlation controlled by the population of distributed according to the density . The exponential correlation is recovered in the case .

###### Proof.

Consider the scaled process

 Yκt=Xκt√σκ, (17)

which satisfies the SDE

 dYκt=−Yκtτκdt+√2dWκt, (18)

then it holds

 Zt=√σ0M∑κYκt. (19)

Each stochastic process is an OU process with exponential correlation (4), i.e., , , according to (18). The correlation of the process is

 E[ZtZs] = E[σ0M2E[∑κYκt∑κYκs∣∣∣τκ]] (20) = = E[σ0M2∑κτκ(\e−|t−s|/τκ−\e−(t+s)/τκ)] = E[σ0M2]∑κE[τκ(\e−|t−s|/τκ−\e−(t+s)/τκ)] = NE[σ0M2]∫Ωττ(\e−|t−s|/τ−\e−(t+s)/τ)q(dτ),

where, we recall that is an independent random variable and are independent and identically distributed. The distribution modifies the classical correlation displayed, , by . When , the exponential correlation follows. ∎

From Theorem 3, we have that

 E[ZtZs]=NE[Λ]E[BHtBHs]=NE[Λ]R(t,s), (21)

and then, after re-scaling , the following holds

###### Lemma 2.

Let , , be the density of the non-negative random variable , then the -dimensional density of is

 (22)

where and , , is the covariance matrix of .

###### Corollary 1.

Let , with , and , then the one-point one-time density function of is

 P(z;t)=1√4πλt2H∫Ωλexp{−z24λt2H}f(λ)dλ. (23)
###### Remark 1.

If is the density function of , then from the relation we have that the density function of is

 g(M)=2σ0M2f(σ0M2).
###### Remark 2.

If , , are independent identically distributed variables according to the distribution , then follows the same distribution, i.e.,

 g(M)≡ρ(m).
###### Corollary 2 (See Theorem 2.1 and Corollary 2.1 in Ref. pagnini_etal-fcaa-2016 ()).

Let and , with mainardi_etal-fcaa-2001 ()

 K−θα,β(λ)=12π∫+∞−∞\e−iωλ{12πi∫c+i∞c−i∞\eptpβ−1pβ+Ψθα(ω)dp}dω, (24)

where , , and , then the one-point one-time density function of is pagnini_etal-fcaa-2016 ()

 P(z;t) = 1√4πλt2β/α∫+∞0exp{−z24λt2β/α}K−α/2α/2,β(λ)dλ (25) = 1tβ/αK0α,β(|z|tβ/α).

The density function defined in (25) is the kernel of the following space-time fractional diffusion equation mainardi_etal-fcaa-2001 (); pagnini_etal-fcaa-2016 ()

 ⎧⎪ ⎪⎨⎪ ⎪⎩tDβ∗u=zDα0u,inΩ,u(z,0)=u0(z),inR, (26)

with , , , and where is the Caputo time-fractional derivative of order defined through the Laplace transfor pair

 ∫+∞0\e−st{tDβ∗u(z,t)}dt=sβ˜u(z,s)−m−1∑j=0sβ−1−ju(j)(z,0+),

with , , and is the Riesz–Feller space-fractional derivative of order , and asymmetry parameter , defined through the Fourier transform pair

 ∫+∞−∞\e+iξz{zDαθu(z,t)}dz=−|ξ|α\ei(signξ)θπ/2ˆu(ξ,t).
###### Corollary 3.

The density function in (25) solves the space-time fractional diffusion equation (26), whose special cases are the time-fractional diffusion (), the space-fractional diffusion (), the neutral fractional diffusion () and the classical diffusion (, ).

The stochastic process defined in (16) is built with the same constructive approach adopted by Mura mura-phd-2008 () to build up the ggBm mura-phd-2008 (); mura_etal-jpa-2008 (); mura_etal-itsf-2009 (), i.e., a Gaussian process times a non-negative random variable, and here we call such type processes: randomly-scaled Gaussian processes. The ggBm proposed by Mura is recovered from (16) in the case is the fractional Brownian motion and where , , is the M-Wright/Mainardi function mainardi_etal-ijde-2010 (); pagnini-fcaa-2013 (), i.e., . Then we have the following

###### Corollary 4.

Let and , with and , then the one-point one-time density function of is pagnini-fcaa-2012 ()

 P(z;t) = 1√4πλt2H∫+∞0exp{−z24λt2H}Mβ(λ)dλ (27) = 12tHMβ/2(|z|tH).

The density function defined in (27) is the kernel of the following Erdélyi–Kober fractional diffusion equation pagnini-fcaa-2012 ()

 ⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩∂u∂t=2Hβt2H−1Dβ−1,1−β2H/β∂2u∂x2,inΩ,u(z,0)=u0(z),inR, (28)

with , , , where is the Erdélyi–Kober fractional derivative defined as follows. Let , , , , the Erdélyi–Kober fractional derivative is defined as

 Dγ,μηφ(t)=n∏j=1(γ+j+1ηtddt)(Iγ+μ,n−μηφ(t)),

where

 Iγ,μηφ(t)=ηΓ(μ)t−η(μ+γ)∫t0τη(γ+1)−1(tη−τη)μ−1φ(τ)dτ.
###### Corollary 5.

The density function in (27) solves a fractional diffusion equation in the Erdélyi–Kober sense (28), whose special cases are the time-fractional diffusion (), the Brownian non-Gaussian motion (), the Gaussian non-Brownian motion () and the classical diffusion ().

## 3 Proofs of Theorems 1, 2 and 3

### 3.1 Proof of Theorem 1

The stochastic dynamics of the process is governed by

 dZt=∑κmκMdXκt.

By remembering the definitions in (8), if we plug (1) into (9), we can derive the following Langevin-type dynamics:

 dZt = 1M∑κ{−γκXκtdt+√2σ0dWκt} = −1M{∑κγκXκt}dt+√2σ0M∑κdWκt.

We introduce now the scaled process , then we have

 ∑κγκXκt = ∑κγκ√σκYκt=√σ0∑κYκtτκ (29) \lx@stackreld= √σ0 ⎷∑κE[(Yκtτκ)2]W1 = √σ0 ⎷∑κE[(Yκtτκ)2]√∑κE[(Yκt)2]√∑κE[(Yκt)2]W1 = 1τeff√∑κE[(√σ0Yκt)2]W1 = 1τeff√∑κE[(mκXκt)2]W1 \lx@stackreld= 1τeff∑κmκXκt=MZtτeff,

where

 (30)

which is a function of time, i.e., , with the Gaussian densities corresponding to the SDE (18).

In the derivation of (29), the first line follows from (8) and the second from the application of the rule of the sum of Gaussian variables reminding that the process is Gaussian, see (18). The third line contains the multiplication and division by and in the fourth is introduced and the parameter is moved below the square root. The fifth line follows from (8) and, finally, in the sixth line the rule of the sum of Gaussian variables is used again in the first equality and (9) is used to have the last equality.

To conclude, by setting , the stochastic process satisfies the following non-autonomous SDE

 dZt=−Ztτeffdt+√2σ0MdWefft.

### 3.2 Proof of Theorem 2

By considering the scaled process , the stochastic process results to be

 Zt=√σ0M∑κYκt.

Let us write

 ¯¯¯¯YNt=1NN∑κ=1Yκt. (31)

Recall that , . For ,

 Ex[¯¯¯¯YNt]=xE0[e−t/τ√σ]=0, (32)

and

 Var0[¯¯¯¯YNt]=1NE0[τ(1−e−2t/τ)]=1NE0[(Y1t)2]. (33)

From the Central Limit Theorem, as

 ¯¯¯¯YNt−E0[¯¯¯¯YNt]√Var0[¯¯¯¯YNt]=√N¯¯¯¯YNt√E0[(Y1t)2]=1√E0[(Y1t)2]1√N∑κYκt\lx@stackreld→W1,

that is, as ,

 1√NN∑κ=1Yκt\lx@stackreld→√E0[(Y1t)2]W1=BHt,∀t>0.

Finally, by multiplying both side by , we obtain the following equivalence in distribution

 ∀t>0,1√NZt\lx@stackreld→√σ0MBHt,N→∞. (34)

Moreover, by setting , the stochastic process is statistically equivalent in the sense of the Central Limit Theorem to the process , i.e.,

 1√NZt→√ΛBHt.

### 3.3 Proof of Theorem 3

The proof of Theorem 3 follows by considering the previous theorem. Let us consider the Fourier transforms of both densities. We have that

 E0[eiξZt]= ∫eiξzddzP0(Zt

## 4 Numerical simulations

In this Section we show by numerical simulations the equivalence in distribution of the three processes (9), (13) and (16) as proved in Theorems 1 and 3. The results of the simulations are shown in Fig. 1 by plotting the density functions of the three processes at different times. Moreover, the insets show the corresponding variances.

The centre-of-mass like process is simulated by using its definition (9) through the process , which obeys Eq. (1). We remind, that in (1) the parameters and , see Eq. (8), follow the corresponding distributions and . In particular, in Panel a), these distributions are

 g(σ)=−38σ−3/4Γ(−2/3)exp[−σ−3/8],q(τ)=exp(−τ), (35)

where we remind that , and in Panel b)

 g(σ)=ησν−1Γ(ν/η)exp(−ση),q(τ)=αΓ(1/α)1τL−αα(τ), (36)

where is the Lévy extremal density and , , , .

The process is simulated by

 ZHt=√NΛBHt=√NΛ∑κYκt=√Nσ0M∑κYκt, (37)

where obeys Eq. (10).

Finally, the process is simulated by the SDE

 dZ∗t=−Z∗tτeff(t)dt+√2Nσ0MdWt,

where, in analogy with previous cases, we use the properties of the sum of Gaussian variables to set . Since we already have the trajectories that we simulated for the process , for numerical convenience, the computation of is performed according to definition in terms of series given in Eq. (30). Moreover, we set because in the simulations we sum trajectories.

Simulations support the equivalence in distribution among the three processes, at least for the chosen distributions of the parameters.

## 5 Conclusions and perspectives for applications

In this paper we have considered an ensemble of OU processes characterized by a population of relaxation times and of noise amplitudes. In the framework of the Brownian motion, the OU process can be viewed as a stochastic dynamical equation and is called Langevin equation. In this sense, we have considered a heterogeneous ensemble of Brownian particles.

In particular, we have studied the stochastic dynamics of the centre-of-mass like process of this ensemble and proved that this process is equivalent in distribution to a process generated by a non-autonomous SDE with time-dependent drift term and uncorrelated white noise, i.e., no-memory effects emerge in opposition to approaches based on the generalized Langevin equation or on Langevin equations with coloured noises, and it is equivalent in distribution also to a randomly-scaled Gaussian process, i.e., the product of a Gaussian process times a non-negative random variable.

Moreover we showed that anomalous diffusion emerges as a consequence of the heterogeneity. In fact, we found that the population of the relaxation times contributes to the emergence of an anomalous scaling in time, as shown in (20), and the population of the noise amplitudes contributes to have a non-Gaussian density function (22).

The present study is a further step towards the interpretation of anomalous diffusion as a consequence of a complex environment pagnini-pa-2014 (); molina_etal-pre-2016 (); pagnini_etal-fcaa-2016 () that is here represented by the heterogeneity of the ensemble of particles. The improvement provided with this paper lays in the fact by using the OU process we establish a relation between the anomalous diffusion and the Brownian motion modelled by the Langevin equation, which is a dynamical equation, so put the emergence of anomalous diffusion on physical basis. Then, future developments will concern the study of the motion and the diffusion properties of a test-particle immersed in a complex environment embodied by such heterogeneous ensemble. This has a direct application for modelling anomalous diffusion hofling_etal-rpp-2013 () and collective motion mayor_etal-nr-2016 () in biological systems. In fact, anomalous diffusion in biological systems is largely observed in experiments and it is due to macromolecular crowding in the interior of cells and in cellular membranes, because of their densely packed and heterogeneous structures hofling_etal-rpp-2013 (); regner_etal-bj-2013 (). The proposed modelling approach can be applied for modelling diffusion of meso-scopic test-particle in an environment also composed of meso-scopic particles embodying the heterogeneous and crowded environment where the test-particle moves. The study of diffusion of a test-particle in a heterogeneous and crowed environment motivates future developments of the present research.

## Acknowledgments

This research is supported by the Basque Government through the BERC 2014-2017 program and the BERC 2018-2021 program, and by the Spanish Ministry of Economy and Competitiveness MINECO through BCAM Severo Ochoa excellence accreditation SEV-2013-0323 and through project MTM2016-76016-R ”MIP”. VS acknowledges BCAM, Bilbao, for the financial support to her internship research period, and SV acknowledges the University of Bologna for the financial support through the ”Marco Polo Programme” for funding her PhD research period abroad spent at BCAM, Bilbao.

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