Diffusion and memory effects for stochastic

processes and fractional Langevin equations

Armando Bazzani a Gabriele Bassi b Giorgio Turchetti a

aDept. of Physics Univ. of Bologna, INFN Sezione di Bologna

bDept. of Physics Univ. of Bologna and DESY Hamburg

Abstract

We consider the diffusion processes defined by stochastic differential equations whenthe noise is correlated. A functional method based on the Dyson expansion for theevolution operator, associated to the stochastic continuity equation, is proposedto obtain the Fokker-Planck equation, after averaging over the stochastic process.In the white noise limit the standard result, corresponding to the Stratonovich in-terpretation of the non linear Langevin equation, is recovered. When the noise iscorrelated the averaged operator series cannot be summed, unless a family of timedependent operators commutes. In the case of a linear equation, the constraints areeasily worked out. The process defined by a linear Langevin equation with additivenoise is Gaussian and the probability density function of its fluctuating componentsatisfies a Fokker-Planck equation with a time dependent diffusion coefficient. Thesame result holds for a linear Langevin equation with a fractional time derivative(defined according to Caputo[1]). In the generic linear or nonlinear case approxi-mate equations for small noise amplitude are obtained. For small correlation timethe evolution equations further simplify in agreement with some previous alterna-tive derivations. The results are illustrated by the linear oscillator with colourednoise and the fractional Wiener process, where the numerical simulation for theprobability density and its moments is compared with the analytical solution.

Key words: Stochastic processes, correlated noise, functional methods, fractionalderivatives.

1 Corresponding author: A.Bazzani Phone (–39-0512091198, Email baz-zani@bo.infn.it.

Preprint submitted to Elsevier Science 16 October 2004

Introduction

Stochastic equations are a basic tool to model physical and biological systems,since the addition of a noise term in deterministic dynamical models allows totake into account, in a phenomenological way, the coupling of neglected degreesof freedom or more generally the fluctuations of external fields, describing theenvironment. The simplest case to treat is the white noise, since the absenceof correlations allows to simulate the system in a straightforward way and towrite the parabolic Fokker-Planck equation, which governs the time evolutionof probability density function (P.D.F.). The functional method, where theDyson expansion[2] for the evolution operator associated to the continuityequation is averaged, appears to be well suited to derive the Fokker-Planckequation in the case of white noise [3; 4; 5; 6]. In any real process the noiseis correlated and the white noise limit can be justified only if the decay ofcorrelation is very rapid. If the noise has a dynamical origin (chaotic degrees offreedom) the memory effects are usually relevant and an adequate treatment isrequired. Memory effects have also been introduced by replacing the ordinarytime derivative with a fractional derivative and the corresponding linear casewill be considered.

We propose to use the Dyson series [2] to analyze the evolution of the proba-bility density for the process defined by a Langevin equation with correlatednoise. The starting point is the continuity equation, which holds provided thatthe Langevin equation is an ordinary differential equation, for any realizationof the noise. By averaging the Dyson series over the stochastic process weobtain an operator series whose resummation, whenever possible, leads to anevolution equation for the probability density.

In the white noise limit the resummation process of the formal series ex-pansion leads to the ordinary Fokker-Planck equation, which corresponds tothe Stratonovich interpretation of the Langevin equation. When the noise iscorrelated an exact resummation cannot be achieved, unless a commutationcondition is satisfied: in this case the P.D.F. satisfies a generalized Fokker-Planck equation with a drift and a second order integro-differential operator.For a linear Langevin equation with correlated noise the constraints on thecoefficients imposed by the commutation condition are easily worked out. Inthe generic linear or nonlinear case an approximate evolution equation for theP.D.F. based on a small noise expansion is proposed. The additional expansionfor small correlation time leads to a simplified equation, the same found byRisken [7] following a different procedure. The same approximation has beenobtained by various authors using different methods [8; 9; 10; 11].

The proposed method is applicable to any stochastic differential equation forwhich a continuity equation holds and allows to recover in a rather simple way

2

the exact or approximate equations for the corresponding P.D.F, previouslyderived by other methods [7; 12; 13]

Langevin equations with delay or a memory kernel require a different approachbecause the state at time t depends on the previous history of the system,preventing to write the continuity equation as a first order partial differentialequation.

The only case of Langevin equations with a memory kernel, which can betreated is the linear case. Their treatment is the same which applies to anylinear equation with a white or correlated noise. A stochastic process x, drivenby η(t), solution of a Langevin equation with white noise, satisfies itself aLangevin equation with a white noise in an extended phase space (x, η) andthe probability density ρ(x, t) is the marginal distribution of ρ(x, η, t), whichsatisfies the ordinary Fokker-Planck equation. Projection or continued fractionexpansion methods [7; 12] have been proposed to obtain the equation satisfiedby ρ(x, t). The fluctuating part of the process χ = x − 〈 x 〉 is Gaussianand its P.D.F. ρ(χ, t) satisfies a simple Fokker-Planck equation with a timedependent coefficient. The equation satisfied by ρ(x, t) = ρ(x− 〈 x(t) 〉, t) andthe procedure to obtain it are usually more involved.

Two examples of stochastic processes satisfying a linear Langevin equationare considered: the linear oscillator with a coloured noise and the fractionalWiener process. In the first example the analytical P.D.F. and its momentsare compared with the simulation. The action distribution and its momentsare compared with the results obtained from the equation averaged over theangle [14], showing that they agree asymptotically for large t, according to theaveraging theory [15; 16; 17]. In the second example we consider the P.D.F.and its moments for the fractional Wiener process and the fractional oscillatorwith white noise by comparing analytical results and simulation. In both casesan excellent agreement is found.

The plan of the paper is the following: in section 1 we set up the functionalframework to analyze the stochastic process driven by a white or colored noise,in section 2 we analyze the linear oscillator with correlated noise and in section3 we consider the fractional time linear Langevin equation.

1 Stochastic processes with memory

In any stochastic process the time evolution is determined by a deterministicfield and its fluctuations (noise). The white noise ξ(t), formal time derivativeof a Wiener process w(t), describes a fluctuating field without memory, whosecorrelations have an instantaneous decay. The presence of memory corresponds

3

to a non instantaneous decay of correlations and the decay rate characterizesthe fluctuating field. A simple example is provided by the fluctuating partχ(t) = x(t) − 〈 x(t) 〉 of the process defined by

dx

dt= −α x + ξ(t) (1)

which is asymptotic to the Ornstein-Ulhenbeck process. The correlation ofχ decays exponentially according to

〈χ(t) χ(t′) 〉 =e−α |t−t′| − e−α(t+t′)

2α(2)

and in the α → 0 limit the Wiener correlation 〈 w(t) w(t′) 〉 = min (t, t′) isrecovered.

Replacing ξ(t) with α ξ(t) in equation (1) the correlation function (2), multi-plied by α2, has a limit δ(t− t′) when α → ∞. This formulation is convenientto recover the white noise limit when the correlation time τ = 1/α tends tozero.

The process χ(t) is Gaussian with zero mean, variance σ2(t) ≡ 〈χ2(t) 〉 =(1 − e−2αt)/(2α) and its P.D.F. ρ(χ, t) satisfies the equation

∂ρ

∂t= D(t)

∂2ρ

∂χ2D(t) =

1

2

dσ2

dt(3)

The P.D.F of any correlated noise, defined by a linear equation like (1) satisfies(3) with an appropriate diffusion term D(t). The stochastic process defined byequation (1) for an initial condition x(0) = x0 is given by x(t) = x0 e−αt + χ(t)and its P.D.F. ρ(x, t) = ρ(x − x0 e−αt, t) satisfies the Fokker-Planck equation

∂ρ

∂t= α

∂

∂x(x ρ) +

1

2

∂2ρ

∂x2(4)

In the next subsection 1.2 we show that this elementary result extends to anylinear Langevin equation with additive noise. In subsections 1.1 and 1.3 wederive the equation satisfied by the P.D.F. for a generic process driven by awhite and coloured noise respectively.

1.1 The stochastic continuity equation

The Fokker Planck equation for a linear or nonlinear system with a white noiseis obtained starting from the stochastic continuity equation with an averagingprocedure, which can be carried out explicitly due to the factorization property

4

of the n-points correlation functions. We first consider the following nonlinearequation, where ξ denotes the white noise

dx

dt= Φ(x, t) ≡ a(x, t) + b(x, t) ξ(t) (5)

For a regular field Φ(x, t) (the white noise should be to be regularized tomake to render the procedure rigorously justified) the Jacobian µ(x, t) of thethe transformations from initial coordinates x0 to the coordinates x at time tsatisfies the equation

dµ

dt≡ ∂µ

∂t+ Φ

∂µ

∂x= µ

∂Φ

∂x(6)

Letting ρ0(x0) be the initial density at a point x0, the density at time t atthe point x, solution of equation (5) with initial condition x0 is given by

ρ(x, t) =ρ0(x0)

µ(x, t)(7)

If the divergence of the field Φ is zero, then µ = 1, the density is constantalong any trajectory and satisfies the Liouville equation ∂ρ/∂t+Φ ∂ρ/∂x = 0.If ∂Φ/∂x 6= 0 then µ varies and from (5) and (6) we obtain dρ/dt = −ρ ∂Φ/∂xnamely the continuity equation for the density

∂ρ

∂t= − ∂(Φρ)

∂x(8)

We write the continuity equation (8) in the following operatorial form

∂ρ

∂t= ( A(t) + ξ(t) B(t) ) ρ A = − ∂

∂xa(x, t) B = − ∂

∂xb(x, t) (9)

and its solution as a Dyson series according to

ρ(x, t) = T

exp

t∫

0

((

A(s) + ξ(s) B(s))

ds

ρ0(x) (10)

The stochastic continuity equation (9) describes the evolution of the phasespace density in a fluctuating field Φ(x, t) and becomes the stochastic Liou-ville equation considered by Kubo [18] for a Hamiltonian vector field. Given arealization of Φ(x, t) the density ρ(x, t) describes the time evolution of a clus-ter of particles with initial density ρ(x, 0) all subject to the same realizationof the noise. The average 〈 ρ(x, t) 〉 corresponds to the standard probabilitydensity function for the process defined by the Langevin equation. The sta-tistical properties of the stochastic process ρ(x, t), which might be exploredby computing the moments 〈 ρn(x, t) 〉, will not be investigated here, restrict-ing the analysis to the average value as in [18]. Moving to the interactionrepresentation the average 〈 ρ 〉 with respect to ξ can be computed on the

5

Dyson expansion by taking into account the basic property of the white noisecorrelation functions

〈 ξ(s1) ξ(s2) · · · ξ(s2n−1)ξ(s2n) 〉 =∑

i1,...,i2n

δ(si1 − si2) · · · δ(si2n−1− si2n) (11)

where the sum ranges over the (2n− 1)!! even permutations, the odd correla-tions being zero. Skipping the technical details, which are reported in appendixA, we can prove that the average density satisfies the following equation

∂

∂t〈 ρ 〉 =

(

A +B

2

2

)

〈 ρ 〉 (12)

which agrees with the Fokker-Planck equation for the P.D.F. obtained withthe Stratonovich interpretation of equation (5).

1.2 The linear case with memory

In the case of a linear Langevin equation with an additive noise there is aconvenient alternative to equation (12), which holds also when the drivingnoise has a memory. As outlined at the beginning of section 1, we consider theP.D.F of the fluctuating part ρ(χ, t) of the stochastic process, which satisfies anequation like (4). This can be verified by observing that the process is Gaussianand the proof is given at the end of this section. In the case of white noise thisequation is simpler with respect to the Fokker-Planck equation (12), satisfiedby the P.D.F ρ(x, t) = ρ(x − 〈 x(t) 〉, t) of the process x(t). For a correlatednoise the exact extension of equation (12) is possible only for a special classof linear equations. The starting point is the representation of the processaccording to

x(t) = 〈 x(t) 〉 +

t∫

0

K(t, s) ξ(s)ds (13)

If x(t) is defined by the linear equation x = a(t) x + η(t), with fundamentalsolution A(t) (equal to ea t when a is time independent) and η(t) is a correlatednoise defined by the equation η = b(t) η+ξ(t), with fundamental solution B(t),the kernel K in (13) is given by K(t, s) =

∫ ts du A(t) A−1(u) B(u) B−1(s) .

The first step is to compute the variance of the fluctuating part χ = x − 〈 x 〉

σ2(t) ≡ 〈χ2(t) 〉 =

t∫

0

K2(t, s) ds (14)

6

The odd moments vanish whereas the even moments are given by

〈χ2n 〉 =

t∫

0

K(t, s1) ds1 · · ·t∫

0

K(t, s2n) ds2n〈 ξ(s1) · · · ξ(s2n) 〉 = (2n − 1)!! σ2n

(15)as a consequence the process is Gaussian and its P.D.F. is given by

ρ(χ, t) =1

√

2πσ2(t)exp

(

− χ2

2σ2(t)

)

(16)

The corresponding Fokker-Planck equation reads

∂

∂tρ(χ, t) = D(t)

∂2

∂χ2ρ(χ, t) D(t) =

1

2

dσ2

dt(17)

The process we are considering is Gaussian since letting the correlation func-tion be defined by

C(t1, t2) =

t1∫

0

ds1 K(t1, s1)

t2∫

0

ds2 K(t2, s2) δ(s1−s2) =

min(t1,t2)∫

0

K(t1, s1) K(t2, s1) ds1

(18)all the higher (even) order correlation functions (odd orders vanish) are de-termined by the generating functional [19]

G(ζ(t)) = exp

−1

2

t∫

0

dt1

t∫

0

dt2φ(t1, t2)ζ(t1) ζ(t2)

(19)

whose logarithm, generating the connected part i.e. the cumulants, has onlythe quadratic component.

1.3 The nonlinear case with memory

In the nonlinear case with a correlated noise η an equation for the P.D.F.cannot be written in closed form. The standard procedure consists in con-sidering the stochastic process y = (x, η) driven by a white noise in a twodimensional phase space. The Fokker-Planck equation, written according tothe Stratonovich prescription, can be obtained by the functional method asin the one dimensional case. The probability density ρ(x, t) of the stochasticprocess x(t) is the marginal distribution of ρ(x, η, t) from which it is obtainedby integration upon η.

The equation satisfied by ρ(x, t) can be formally written as an operator ex-pansion, after averaging the Dyson series obtained for the continuity equation,

7

for a generic non linear equation with correlated noise. The series can be ex-plicitly summed only when a one parameter family of operators commutes,see appendix B.

If the deterministic field has a delay or a memory kernel, the continuity equa-tion cannot be written as a first order partial differential equation, due to thedependence on the previous history. In the nonlinear case the equation for theP.D.F. cannot be written, not even at the formal level.

We consider the following Langevin equation

dx

dt= a(x, t) + b(x, t) η(t) η(t) =

t∫

0

K(t, s) ξ(s) ds (20)

where the correlation function of the noise η(t) is given by (18) The continuityequation is still given by equation (9) and we can write the solution as ρ(x, t) =U(t) ρ0(x) where the evolution operator satisfies the equation

∂

∂tU(t) = ( A(t) + B(t) η(t) ) U(t) U(0) = I (21)

Using the interaction picture we compute the average with respect to theprocess η, see Appendix B. The expansion can be summed explicitly onlywhen the operators

W(t) = U−1A (t) B(t) UA(t)

at different times do commute and the following result is obtained

∂

∂t〈 U(t) 〉 =

(

A(t)+

t∫

0

ds C(t, s) B(t) UA(t−s) B(s) UA(s− t))

〈 U(t) 〉 (22)

where the operator UA(t) satisfies the following evolution equation

∂

∂tUA(t) = A(t) UA(t) UA(0) = I (23)

If the noise is uncorrelated C(t, s) = δ(t − s) the equation for the P.D.F.〈 ρ 〉(x, t) = 〈 U(t) 〉 ρ0(x) obtained from (22) agrees with (12). Equation (22)holds even when the correlated noise η(t) is not the solution of a first orderdifferential equation driven by a white noise or the integral transform of awhite noise as in (20), since this equation and its derivation depend only on theexistence of a correlation function C(t, s). The commutation conditions can beeasily established in the linear case where a(x, t) = a0(t)+x a1(t) and b(x, t) =b0(t)+x b1(t). Supposing for simplicity the coefficients to be time independentand setting a1 = −α < 0, b1 = β the action of the evolution operator UA(t) =eA t where A = ∂

∂x(α x−a0) on a function f(x) can be determined by observing

8

that f(x, t) = UA(t) f(x) satisfies the continuity equation. The solution of∂∂t

f + ∂∂x

(a0 − α x) f = 0, according to equation (7), is given by f(x, t) =f(eαtx + a0 α−1 (1 − eαt) )/µ(t) where µ(t) = e−αt so that finally

UA(t) f(x) = eαt f(

eαt x + a01 − eα t

α

)

(24)

The action of the operator W(t) on the function f(x) is given by

W(t) f(x) = −b0eα t d

dxf(x) − β f(x) − β

(

x + a0eαt − 1

α

) d

dxf(x) (25)

It can be easily checked that the commutation condition occurs in the followingcases: a0 = b0 = 0 or β = 0 or a0 = α = 0. If the coefficients are time dependentthere are some further restrictions: for instance if the drift term vanishes thecommutation condition is satisfied only if b0(t) = b1(t) or b0 = 0 or b1 = 0.Just to give a simple example we consider the special case

dx

dt= −αx + η(t) η(t) = w(t) ≡

t∫

0

ξ(s) ds (26)

so that K(t, s) = 1 and C(t, s) = min(t, s). A straightforward computationshows that the equation for ρ(x, t) is given by

∂ρ

∂t= α

∂

∂x(xρ) +

e−αt − 1 + αt

α2

∂2ρ

∂x2(27)

Since the equation is linear we know that the process is Gaussian. The meantrajectory is given by 〈 x(t) 〉 = x0e

−αt and the variance can be easily computedso that we have

ρ(x, t) =1√

2πσ2exp

(

−( x − x0 e−αt )2

2σ2

)

σ2 =1

α3

(

α t − 2 (1 − e−αt) +1

2(1 − e−2αt)

)

(28)

and it is not hard to check that it satisfies equation (27).

When the operators do not commute it is still possible to write down anapproximate equation by considering a small noise expansion. Replacing η(t)with ǫ η(t) where ǫ is the noise amplitude and η is defined in such a waythat the limit of vanishing correlation time is the white noise 〈 η(t′) η(t) 〉 →δ(t − t′), equation (22) still holds in the non commuting case provided thatthe integral is multiplied by ǫ2. However this equation is correct up to termsof order ǫ4 and consequently is only a small noise approximation. A simplifiedversion of the equation can be obtained if the correlation has an exponentialdecay by considering a further expansion in the correlation time obtained with

9

subsequent integration by parts. For simplicity we assume the correlation todecay exponentially

C(t, s) =α

2e−α |t−s| lim

α→∞C(t, s) = δ(t − s) (29)

Letting Q(t, s) = B(t) UA(t − s) B(s) UA(s − t) and Q(n)(t, s) = ∂n

Q(t, s)/∂sn

the integral I in equation (22) becomes

I =α

2

t∫

0

ds e−α(t−s)Q(t, s) =

N∑

n=0

(−1)n

2αn

[

Q(n)(t, t) − e−α t

Q(n)(t, 0)

]

+

+(−1)n+1

2αn

t∫

0

ds e−α(t−s)Q

(n+1)(t, s) ds (30)

If a, b and the corresponding operators A, B are time independent, we haveQ(t, s) = B e(t−s) A

Be(s−t) A and the integral explicitly reads

I =1

2

(

B2 − e−αt

BeAtBe−At

)

+1

2α

(

B [A, B] + e−αt ( BeAtBAe−At−BAeAt

B e−At ))

+

+ O(

1

α2

)

(31)

To conclude the equation for the probability density in the small noise andshort correlation time limit reads

∂

∂tρ(x, t) =

(

A + ǫ2 B2

2+

ǫ2

2αB [A, B]

)

ρ(x, t) + O

(

ǫ2

α2, ǫ4

)

(32)

2 Oscillator with coloured noise

The case of linear systems with a coloured noise can be easily treated evenwhen the phase space dimension is larger than 1. Letting x ∈ R

d we considerthe following equation

dx

dt= A x + e η(s) (33)

where η(s) =∫ t0 K(t, s) ξ(s) ds is a coloured noise and A, e are a constant

matrix and vector respectively. The fluctuating part of x(t) is given by

x(t) = x(t)−〈 x(t) 〉 =

t∫

0

k(t, s) ξ(s) ds k(t, s) =

t∫

s

eA (t−u) eK(u, s) du

(34)

10

and the variance matrix reads

σ2ij(t) = 〈 xi(t) xj(t)〉 =

t∫

0

ki(t, s) kj(t, s) ds (35)

Computing the higher order moments one recognizes the Gaussian nature ofthe process whose probability density is given by

ρ(x, t) =1

√

det (2π σ2)exp

(

−(x− 〈x(t) 〉 ) · (2σ2)−1 · (x− 〈x(t) 〉 ))

(36)

Another alternative is to solve the linear system for y = (x, η) ∈ Rd+1, to

compute the variance matrix and the corresponding probability density ρ(y, t)which is still Gaussian and recover ρ(x, t) as a marginal distribution afterintegrating over η. The variances 〈 xi(t)xk(t) 〉 obtained from both distributionsare obviously the same.

As an example we consider the harmonic oscillator with a Ornstein-Uhlembecknoise.

x = p

p = −ω2x + ǫ η(t)

η = −α η + α ξ(t) (37)

Introducing the complex variable z =√

ω x − ip/√

ω the first two equationsare replaced by

z = iω z − iǫ√ω

η (38)

and the solution for its fluctuating part is given by

z(t) = z(t) − 〈 z(t) 〉 = − i ǫ√ω

t∫

0

eiω(t−s) − e−β (t−s)

1 + iω/βξ(s) ds (39)

The variance matrix is obtained from the mean of z2 and of |z|2

〈 z2(t) 〉 ≡ ω σ2xx(t) − σ2

pp(t)

ω− 2iσ2

xp(t) 〈 |z|2(t) 〉 ≡ ω σ2xx(t) +

σ2pp(t)

ω(40)

In the limit α → ∞ we recover the limit for the stochastic oscillator withwhite noise. In this case 〈 z2(t) 〉 = −ǫ2ω−2 eiωt sin(ωt) and 〈 |z(t)|2 〉 = ǫ2ω−1 tso that

σ2xx(t) =

ǫ2

2ω2

(

t − sin(2ωt)

2ω

)

σ2xp(t) =

ǫ2

2 ω2sin2(ωt)

σ2pp(t) =

ǫ2

2

(

t +sin(2ωt)

2ω

)

(41)

11

0

5

10

15

20

25

0 0.2 0.4 0.6 0.8 1

F(J

)

J

0

0.05

0.1

0.15

0.2

0 1 2 3 4 5

<J>

T

Fig. 1. Harmonic oscillator with coloured noise for an initial distribution with sup-port at the origin in phase space ρ(x, y, 0) = δ(x)δ(y). The parameters chosen areω = 1, ǫ = 0.2 and N = 105 particles are used in the simulation. The left frameshows the action density at t = 5 for different correlation times: τ = 2 (red) τ = 1(blue) τ = 0 (green). The right frame shows the mean value of the action for thesame values of τ . We recall that for t large σpp ≃ ω2σxx ≃ 1

2ǫ2 (1 + ω2τ2)−1 t

Another relevant variable for the oscillator is the action J = 12|z|2 = E/ω.

From the previous solution z(t) = z0eiωt + z(t) we obtain the mean value

J(t) = J0 + ǫ2 t/(2ω). Choosing for simplicity z0 = 0 it is easy to compute thevariance σ2

J = 〈 (J − 〈 J 〉)2 〉. Letting z(t) =∫ t0 K(ts) dt we have

σ2J =

1

4

t∫

0

|K(t, s)|2 ds

2

+1

4

∣

∣

∣

∣

∣

∣

t∫

0

K(t, s)2 ds

∣

∣

∣

∣

∣

∣

2

=ǫ4

4ω2

(

t2 +sin2(ωt)

ω2

)

(42)

The leading term for ωt → ∞ is the same one obtains from the Fokker-Planckequation, written for the action angle variables, after averaging over the angle.The P.D.F. for the action distribution reads

∂

∂tρ(J, t) =

∂

∂J

ǫ2J

2ω

∂

∂Jρ(J, t) (43)

and equation for the moments can be easily obtained by integration by parts,assuming ρ vanishes sufficiently fast at infinity. The results is the same forthe mean 〈 J 〉 , whereas the variance is given by σ2

J = ǫ2 t2/(4ω2). The actiondistribution for a correlated noise and the first moments are shown in figure1.

3 Fractional time Langevin equation

We consider a dynamical system defined by Dx = Φ(x) where D = d/dt and a

is the vector field in a finite dimensional phase space Rd. By replacing the or-

12

dinary time derivative with the fractional time Caputo derivative D∗αx = Φ(x)

of order α [20], we obtain a system whose memory increases as we decrease αstarting from 1. The derivative Dα is defined for 1 ≥ α > 0 by

D∗α = J1−α D Jαf(t) =

1

Γ(α)

t∫

0

(t − s)α−1 f(s) ds (44)

where J0 = I is the identity, J1 = J is the ordinary integral. The one parameterfamily of integral operators enjoys the group property from which it followsthat J−1

1−α = DJα. The equation with fractional derivative is equivalent toD(x − JαΦ(x)) = 0 namely the integral equation [21].

x(t) = x(0) +1

Γ(α)

t∫

0

(t − s)α−1Φ(x(s)) ds (45)

defined for any α > 0. As a dynamical system this equation has an infinite

dimensional phase space. This is evident from the discrete time version (Eulerintegrator) D∗

α(∆)xk = Φ(xk), which explicitly reads [21]

xn = x0 +(∆t)α

Γ(α + 1)

n−1∑

k=0

Φ(xk) [(n − k)α − (n − k − 1)α] (46)

The extension to the Langevin equation [22; 23] consists in replacing Φ(x)dtwith a(x)dt + b(x)dw. In the discretization scheme (46) Φ(xk) is replacedby a(xk) + b(xk) ξk (∆t)−1/2 where ξk are random variables with zero meanand unit variance. In the Hamiltonian case it is possible to replace the Eulerscheme with a symplectic scheme. The derivation of a Fokker-Planck equationfaces the difficulty of writing a continuity equation for the density when α 6= 1.

3.1 The fractional linear case

The fractional linear Langevin equation D∗αx = a0(t) + x a1(t) + b0(t) ξ(t) is

equivalent to the following integral stochastic equation

x(t) = x(0)+1

Γ(α)

t∫

0

(t−s)α−1(

a0(s)+a1(s)x(s)+ b0(s) ξ(s))

ds 0 < α ≤ 1

(47)whose fluctuating part χ = x(t) − 〈 x(t) 〉 can be written as a linear integraltransform of the white noise according to (13). The process χ(t) is Gaussian

13

-1 10

1 α= 1.25

x

ρ

-1 10

1 α= 1

x

ρ

-1 10

1 α= 0.75

x

ρ

Fig. 2. Probability density for a fractional Wiener process at t = 0, 20, 100 forǫ = 0.05. Comparison of the simulation (histograms) with the analytic solution(curves) for α = 1.25, 1, 0.75

0 1000

10

t

µ

µ2 µ4 α=1.25

0 1000

0.25

t

µ

µ2 µ4 α=1

0 1000

0.05

t

µ

µ2 µ4 α=1

Fig. 3. Moments of the probability density for a fractional Wiener process µ2, µ4

with ǫ = 0.05 and α = 1.25, 1, 0.75

and its variance is given by (14). The simplest case a0 = a1 = 0 and b0 = ǫcorresponds to the fractional Wiener noise [4,5,6] and reads

x(t) = x0 +

t∫

0

K(t − s)dw(s) K(t) =ǫ

Γ(α)tα−1 (48)

The P.D.F. ρ = (2πσ2(t))−1/2 exp(−(x−x0)2/2σ2(t) ) of the fractional Wiener

process satisfies equation (17) and the variance σ2 computed according to (13)is given in appendix D. As a consequence its P.D.F. ρ(x, t) is not related to thesolutions of the fractional diffusion equation D∗

αρ = ∂2ρ/∂x2, whose solutionis a stable distribution of order α/2 in the t variable (the P.D.F. of a Wienerprocess is normal of order 2 in x and of order 1/2 in t). The fractional diffusionequation has been considered by various authors [20; 24; 25] as a continuum

14

0 0.50

1 α= 1.25

J

ρ

0 0.5 0

1 α= 1

J

ρ

0 0.5 0

1 α= 0.75

J

ρ

0 1000

1

t

< J

>

α=1.25, 1, 0.75

Fig. 4. Density ρ(J, t) and 〈J〉 for ǫ = 0.05, ω = 2π/100 and α = 1.25, 1, 0.75 inthe case of a fractional harmonic oscillator with noise. The times for the densityhistograms are t = 0, 1, 10 for α = 1, 0.75 and t = 0, 1, 2 for α = 1, 25. The initialaction is around J = 1/8.

limit of the fractional master equation

D∗α(∆t) ρ(xk, ti) =

ρ(xk + ∆x, ti) − 2ρ(xk, ti) + ρ(xk + ∆x, ti)

(∆x)2(49)

which is different from the fractional random walk corresponding to the dis-crete version of the fractional Langevin equation considered here, see AppendixC.

In figure 2 we have compared the results of a simulation with N = 104 particlesobtained from the discrete version of (47) given by (46) where Φ(xk) =ǫξk/

√∆t and ξk are random numbers with zero mean and unit variance. We

have compared the normal Wiener process α = 1 with the fractional processα = 0.75, whose diffusion is slower and the process with α = 1.25, whosediffusion is faster. The process with α > 1 does not correspond to a fractionalderivative (0 < α < 1), but it is still a solution of the integral equation (45),which defines the analytic continuation of the Wiener process for any real αlarger than 0.

In figure 3 we have compared the second and fourth moments of the simulationwith µ2 = σ2 and µ4 = 3σ4.

We have considered also the fractional Langevin equation for the harmonicoscillator [26]

z(t) = z(0) +1

Γ(α)

t∫

0

(t − s)α−1 (iωz(s)ds + ǫdw(s)) z = x − ip (50)

The mean part is 〈z〉 = z0Eα(iωtα), where Eα(z) denotes the Mittag Lofflerfunction. The fluctuating part is given by the convolution of the Wiener pro-

15

cess with the Green’s function.

z − 〈z(t)〉 = ǫ

t∫

0

α(t − s)α−1E ′α(iω(t − s)α) dw(s) (51)

where The results for the distribution in the action J = 12|z|2 and its mean

are shown in figure 4.

4 Summary and conclusions

We have provided a general functional framework to derive the evolution equa-tion for the probability density corresponding to the Langevin equation withwhite and coloured noise. For the white noise the standard Fokker-Planckequation corresponding to the Stratonovich interpretation of the Langevinequation is recovered. For the correlated noise an exact parabolic equationcan be written only if a commutation condition of a family of time dependentoperators is satisfied. For the linear case satisfying the commutation relationa Fokker-Planck equation holds. In the generic case an approximate equationholds for small noise amplitude, and it takes a simpler form for small correla-tion time. The fluctuating part of the solution of a linear Langevin equationwith additive noise is Gaussian also when the ordinary time derivative is re-placed by a fractional derivative and the distribution of the process includingthe mean component can be compared with the solution of the above men-tioned equations. To this end we have considered the linear oscillator with aUhlembeck noise and the fractional Wiener process.

To conclude we can assess that the functional framework based on field the-oretical techniques like the Dyson series, is quite general and easy to workout. The stochastic continuity equation, related to the previously consideredstochastic Liouville equation [18], has a nontrivial probabilistic meaning onits own, which might be explored before the averaging process is carried out.Given a family of initial distributions with known weights, the stochastic con-tinuity equation gives all the information on their time evolution, whereas thestandard approach concerns only the mean.

16

5 Acknowledgments

We wish to thank H. Mais for stimulating discussions and DESY for hospitalityduring the completion of the present work.

17

6 Appendix A

The density ρ(x, t), which satisfies stochastic continuity equation (1), can bewritten as ρ(x, t) = U(t) ρ0(x) where the operator U(t) satisfies the linearoperator stochastic equation

∂

∂tU(t) =

(

A(t) + ξ(t) B(t))

U(t) U(0) = I (A1)

whose solution, obtained by iterating the corresponding integral equation, isgiven by (10). The presence of the time ordering operator T is required if A

and B do not commute. We define the deterministic evolution operator UA(t)such that U(0) = I according to

∂

∂tUA(t) = A(t) UA(t) UA(t) = T exp

t∫

0

A(s) ds

(A2)

and introduce the interaction representation according to UI(t) = U−1A (t) U(t),

which satisfies the following evolution equation

∂

∂tUI(t) = U

−1A (t) ξ(t) B(t) UA(t) UI(t) (A3)

Our goal is to evaluate 〈 U(t) 〉 and to this end we first evaluate 〈 UI(t) 〉. Thetime evolution of the former averaged operator is then immediate since A isa deterministic operator and 〈 U(t) 〉 = UA(t) 〈 UI(t) 〉. The mean value withrespect to ξ is evaluated on the Dyson series by using (12)

〈 UI(t) 〉 =∞∑

n=0

1

n!

⟨

T

n∏

j=1

t∫

0

dsj U−1A (sj) ξ(sj) B(sj) UA(sj)

⟩

=

=∞∑

n=0

1

(2n)!T

( t∫

0

ds1,

t∫

0

ds2 · · ·t∫

0

ds2n−1

t∫

0

ds2n U−1A (s1)B(s1) UA(s1)

U−1A (s2)B(s2) UA(s2) · · · U

−1A (s2n)B(s2n) UA(s2n)

⟨

ξ(s1) ξ(s2) · · · ξ(s2n−1)ξ(s2n)⟩

)

=

=∞∑

n=0

(2n − 1)!!

(2n)!T

( t∫

0

ds1 · · ·t∫

0

dsnU−1A (s1) B2(s1) UA(s1) · · ·U−1

A (sn) B2(sn)UA(sn)

)

(A4)

18

As a consequence observing that (2n − 1)!!/(2n)! = 2−n/n! we see that

〈 UI(t) 〉ρ(x, t) = T

exp

t∫

0

(U−1A (s)

B2(s)

2UA(s)ds

(A5)

which satisfies the following equation

∂

∂t〈 UI(t) 〉 = U

−1A (t)

B2(t)

2UA(t) 〈 UI(t) 〉 (A6)

The mean value of U(t) is finally obtained observing that

∂

∂t〈 U(t) 〉 =

∂

∂tUA(t) 〈 UI(t) 〉 = AUA(t) 〈 UI(t) 〉+

B2(t)

2UA(t) 〈 UI(t) 〉 = (A+

B2

2) 〈 U(t) 〉

(A7)

19

7 Appendix B

If the noise correlation function is C(t1, t2), moving to the interaction repre-sentation, the average of the operator UI(t) = U

−1A U(t) is given by

〈 UI(t) 〉 =∞∑

n=0

(2n − 1)!!

(2n)!T

t∫

0

ds1

t∫

0

ds2 C(s1, s2) W(s1) W(s2) · · ·

· · ·t∫

0

ds2n−1

t∫

0

ds2n C(s2n−1, s2n) W(s2n−1) W(s2n)

=∞∑

n=0

1

n!T

t∫

0

ds1

s1∫

0

ds2 C(s1, s2) W(s1) W(s2) · · ·

· · ·t∫

0

ds2n−1

s2n−1∫

0

ds2n C(s2n−1, s2n) W (s2n−1) W(s2n)

(B1)

where the operators W(t) are defined by

W(t) = U−1A (t) B(t) UA(t) (B2)

and we have taken into account the symmetry of the product T(W(s1) · · ·W(s2n)with respect to the permutations of the indices 1, . . . 2n. If the operators W(t)at different times do not commute it is not possible to cast the results intoa simpler form which allows the series to be resummed. When the operatorsW(t) at different times commute, the time ordering can be dropped and theseries can be written as

〈 UI(t) 〉 =∞∑

n=0

,1

n!

t∫

0

ds1 O(s1) · · ·t∫

0

dsn O(sn) = exp

t∫

0

O(s) ds

(B3)

where the operator O(t) is defined by

O(t) =

t∫

0

ds C(t, s)W(t) W(s) (B4)

We notice that in the limit of a white noise where C(t, s) = δ(t − s) theoperator becomes O(t) = 1

2W 2(t) ≡ 1

2U−1A (t) B

2(t) UA(t). In this limit thecommutativity condition is no longer necessary and the result is given by (B3)with a time ordering operator in front of the exponential. In the commutingcase the equation satisfied by the operator 〈 UI(t) 〉 can be written as

∂

∂t〈 UI(t) 〉 = O(t) 〈 UI(t) 〉 (B5)

20

Finally the equation for the mean value of U(t) is given by

∂

∂t〈 U(t) 〉 =

∂

∂t

(

UA(t) 〈 U(t) 〉)

=[

A(t) + UA(t) O(t) U−1A (t)

]

〈 U(t) 〉 (B6)

In more explicit form the last equation reads

∂

∂t〈 U(t) 〉 =

[

A(t) +

t∫

0

ds C(t, s) UA(t) W (t) W(s) U−1A (t)+

]

〈 U(t) 〉 (B7)

which is exactly equation (22).

Approximate equations In the non commuting case we can write approxi-mate evolution equations. We notice that in this case

〈 UI(t) 〉 =∞∑

n=0

,1

n!T

t∫

0

ds1 O(s1) · · ·t∫

0

dsn O(sn) 6= T exp

t∫

0

O(s) ds

(B8)

since the time ordering is applied not to the operators O(si) but to the opera-tors W entering in its definition. We also remark that with the definition (B4),where the order of the product cannot be changed in the non commuting case,we have TO(t) = O(t). As a consequence we consider the following series

〈 UI(t) 〉1 = I + ǫ2

t∫

0

ds1 O(s1) + ǫ4

t∫

0

ds1

s1∫

0

ds2 O(s1) O(s2) +

+ ǫ6

t∫

0

ds1

s1∫

0

ds2

s3∫

0

ds3 O(s1) O(s2) O(s3) + . . .

=∞∑

n=0

ǫ2n

n!TO

t∫

0

ds1 · · ·t∫

0

dsn O(s1) · · ·O(sn) = TO exp

t∫

0

O(s) ds

(B9)

where TO is the time ordering referred to the operators O(t). More explicitlyletting t1 < t2 we have

TO ( O(t1)O(t2) ) = O(t2)O(t1) =

t2∫

0

ds2 C(t2, s2) W(t2)W(s2)

t1∫

0

ds1 C(t1, s1) W(t1)W(s1)

T ( O(t1)O(t2) ) =

t2∫

0

ds2

t1∫

0

ds1 C(t2, s2) C(t1, s1) T ( W(t2)W(s2)W(t1)W(s1) )

(B10)

21

From its definition it is evident that 〈 UI(t) 〉1 = 〈 UI(t) 〉 + O(ǫ4) and thatletting 〈 U(t) 〉1 = UA(t) 〈 UI(t) 〉1 which also differs from 〈 U(t) 〉 by orders of ǫ4

the equation it satisfies is given by (B7).

We show also how to write down the equation at the next order. Letting

〈 UI(t) 〉2 = I+ǫ2

t∫

0

ds1 O(s1)+ǫ4

t∫

0

ds1

s1∫

0

ds2

[

T ( O(s1) O(s2) )−O(s1)O(s2)]

+. . . ≡

≡ I +

t∫

0

ds1O2(s1) +

t∫

0

ds1

s1∫

0

ds2 O2(s1) O2(s2) + ...

=∞∑

n=0

1

n!TO

t∫

0

ds1 · · ·t∫

0

dsn O2(s1) · · ·O2(sn) = TO exp

t∫

0

O2(s) ds

(B11)

where we have defined the operator O2(t) according to

O2(t) = ǫ2O(t) + ǫ4

t∫

0

ds [T ( O(t) O(s) ) − O(t) O(s) ] (B12)

It is not difficult to see that 〈 UI(t) 〉2 differs from 〈 UI(t) 〉 by orders of ǫ6.Finally letting 〈 U(t) 〉2 = UA(t) 〈 UI(t) 〉1 be the second order approximationto the evolution operator the equation it satisfies is given by

∂

∂t〈 U(t) 〉2 =

[

A(t) + UA(t) O2(t) U−1A (t)

]

〈 U(t) 〉2 (B13)

22

8 Appendix C

We quote here the correlation matrix for the harmonic oscillator with a colourednoise defined by equation (36)

σ2xx(t) =

ǫ2

2ω2

1(

1 + ω2

β2

)2

[

(

1 +ω2

β2

)

t +ω2

β3

(

1 − e−2βt)

− 4ω

β2e−βt sin(ωt)

−(

1 − ω2

β2

)

sin(2ωt)

2ω− 2

βsin2(ωt)

]

σ2pp(t) =

ǫ2

2

1(

1 + ω2

β2

)2

[

(

1 +ω2

β2

)

t +1

β

(

1 − e−2βt)

+4

β(e−βt cos(ωt) − 1)

+

(

1 − ω2

β2

)

sin(2ωt)

2ω+

2

βsin2(ωt)

]

σ2xp(t) =

ǫ2

2ω

1(

1 + ω2

β2

)2

[

(

1 − ω2

β2

)

sin2(ωt)

ω− 1

βsin(2ωt) − ω

β2(1 − e−2βt)+

2

βe−βt sin(ωt) − 2ω

β2(e−βt cos(ωt) − 1)

]

23

9 Appendix D

We compare the P.D.F. ρ(x, t) of the fractional Wiener process D∗α x = ξ(t)

with the solution ρ(x, t) of the time fractional diffusion equation

∂ρ

∂t=

t2α−2

2 Γ2(α)

∂2ρ

∂x2D∗

α ρ =1

2

∂2ρ

∂x2(D1)

Choosing t0 > 0 as initial time allows to express the variances for any α > 0according to

σ2(t) =1

Γ2(α)

t2α−1 − t2α−10

2α − 1 α 6= 12

log tt0

α = 12

σ2(t) =(t − t0)

α

Γ(α + 1)(D2)

More generally the even moments are given by the recurrences, we write forα > 1/2 choosing t0 = 0

µ2n(t) =

(

t2α−1

Γ2(α) (2α − 1)

)n

cn cn = (2n − 1)cn−1

µ2n(t) = tn α cn cn = (2n − 1) nΓ((n − 1)α + 1)

Γ(nα + 1)cn−1 (D4)

where the second one is obtained from D∗α µ2n = 2n(2n−1)µ2n−1, which follows

from (D1). The characteristic functions defined as the Fourier transforms withrespect to x are given by

G(t, k) = e−1

2k2σ2(t) G(t, k) = Eα

(

−1

2k2 tα

)

≡∞∑

n=0

(

−12k2 tα

)n

Γ(nα + 1)(D5)

We recall that a distribution is said to be stable of order α if the distributionof X = (x1 + . . . + xn)/n1/α is the same for any n. This implies that thecharacteristic function G(k) has to fulfill the condition Gn(k/n1/α) = G(k)and this implies G(k) = A e−c |k|α. Taking the Laplace transform H(s, x) =Ls(ρ(t, x) ) with respect to t and recalling that L(D∗

αf) = sα Ls(f)−sα−1 f(0)we find

sα H(s, x) =∂2H

∂x2H(s, x) = A(s) e−kα/2 x (D6)

which proves that ρ is a stable distribution of order α/2 with respect to twhereas ρ(x, t) is stable of order 2 with respect to x. The very different nature

24

of the above processes is shown by the behaviour of the variances which isσ2(t) ∝ t1−2(1−α) and σ2(t) ∝ t1−(1−α)

25

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