# Introducing Randomness into First-Order and Second-Order Deterministic Differential Equations

Authors: John F. Moxnes; Kjell Hausken
Journal: Advances in Mathematical Physics
 (2010)

---

## Abstract

We incorporate randomness into deterministic theories and compare analytically and numerically some well-known stochastic theories: the Liouville process, the Ornstein-Uhlenbeck process, and a process that is Gaussian and exponentially time correlated (Ornstein-Uhlenbeck noise). Different methods of achieving the marginal densities for correlated and uncorrelated noise are discussed. Analytical results are presented for a deterministic linear friction force and a stochastic force that is uncorrelated or exponentially correlated.


---

## Body

## 1. Introduction

Stochastic theories model systems which develop in time and space in accordance with probabilistic laws.1 Essential in stochastic theories is how randomness is accounted for. For Markov [1] processes2, which are an important class of stochastic processes, the state value of Xt+Δt at time t+Δt is given by the state value at time t, plus a state value of a “random variable” at time t. 3 A random “disturbance” in a Markov process may possibly influence all subsequent values of the realization. The influence may decrease rapidly as the time point moves into the future. Five methods to account for randomness are (1) to assume deterministic equations to determine the stochastic process for say a particle and apply Monte Carlo simulations or analytical methods [2, 3] to find probability distributions, (2) to use the Liouville equation with or without added terms for the probability density per se, for example, for a particle, (3) to use ordinary differential equations for the statistical moments of the probability distribution, (4) to use the “hydrodynamic approach,” which is to specify constitutive relations in an equation set akin to what is used in hydrodynamic formulations of gas flow  [4], and (5) in quantum physics one can use traditional quantization rules, Nelson’s [5] stochastic version of quantum mechanics, or path integration methods. This article mainly confines attention to method 1, addressed in all sections of the article. Method 2 is briefly addressed in the last part of Section 4. Methods 3, 4, and 5 are briefly outlined in Appendix F. We first outline methods 1 and 2 and thereafter specify the contribution of this article towards the end of the introduction.For method 1 allowing randomness in the initial values for a particle and for a deterministic noise commonly implies more realistic models of physical situations than, for example, the Liouville process where a realization of the stochastic process is constructed deterministically allowing randomness only in the initial conditions of the particle. In the Stratonovich [3] method noise is incorporated in a deterministic equation by adding a deterministic noise term that can be integrated in the normal Riemann sense. The Stratonovich integral occurs as the limit of time-correlated noise (so-called colored noise) when the correlation time of the noise term approaches zero. Monte Carlo simulations or analytical methods can be used to “count up” the probability density. The increment in the Einstein-Smoluchowski theory of Brownian motion during a time interval Δt is proportional with the so-called drift, plus a nondeterministic noise as a time-uncorrelated Gaussian random term (called Gaussian white noise), with mean zero and variance proportional to Δt. This mathematical model was first treated rigorously by Ito [2] (See also [6, 7]). The work by Bachelier [8], Einstein [9, 10], Smoluchowski [11, 12], Wigner [13], Ornstein-Uhlenbeck (1930), and Ito [2] constitute foundations. A system with time-uncorrelated noise is usually just a coarse-grained version of a more fundamental microscopic model with correlation. Thus depending on the problem in consideration, the Stratonovich (deterministic noise) or the Ito model (stochastic noise) could be appropriate approximations. For additive noise the Ito model gives the same answer as the Stratonovich model. But for multiplicative noise the results are different. (For a recent treatment of different interpretations of stochastic differential equations see, e.g., Lau and Lubensky [14].) Roughly, the reason for this difference is that the Stratonovich integral, which is a Riemann integral, applies on functions with bounded variations. The Ito integral applies on functions without bounded variations (i.e., white noise).In physics or engineering applications second-order ordinary differential equations are often used as models. Unfortunately, second-order processes are more difficult to address than first-order processes. The second-order differential equation is first written as the mathematical equivalent set of two first-order equations, and then randomness is incorporated into the first-order equations either by Ito or Stratonovich interpretations by defining two stochastic differential equations for the two random variables{Xt} and {Yt} [6, 7]. In the well-known Langevin models used in physics, or in any second-order process driven by noise force, a noise term is added to the velocity (say {Yt}), but not to the position (say {Xt}). The position follows as the integral of the velocity. The Ornstein-Uhlenbeck (1930) theory of Brownian motion is expressed in terms of a Markov process in the bi-dimensional phase space. Masoliver [15] studied second-order processes driven by a dichotomous exponentially time-correlated noise force, while Heinrichs [16] studied second-order processes driven by Gaussian exponential time-correlated noise force. See also Bag [17] and Bag et al. [18] for studies on stochastic processes and entropy production. Langevin models are very fruitful as a starting point for quantum noise phenomena [19].In method 2, randomness is implemented without using differential equations and Monte Carlo simulations, by studying the Liouville equation as such with our without added terms, for example, for a particle. In the Liouville process the probability density is a solution of the so-called Liouville equation4. But a traditional picture of applying realistic randomness is through collision integrals that fulfill local conservation laws of mass, momentum, and energy [20, 21]. It is usually considered in direct reference to the Boltzmann kinetic theory. Microscopic collision rules are established based on time symmetry, but then a rapid decay of correlations amounts to assuming friction [22, 23]. For a second-order process driven by dichotomous noise with exponential correlation Masoliver [15] found a third-order partial equation for the joint density distribution. For a second-order process driven by Gaussian noise with exponential correlation Hienrichs [16] found a Fokker-Planck equation with time variable coefficients for the joint distribution. In the phase-space formulation of quantum physics the Wigner quasijoint distribution is commonly used [24]. The equation for the joint distribution includes additional terms compared to the bidimensional Liouville equation. (See the review articles by Hillery et al. [25] and Lee [26] for a review of quantum phase-space distributions.) No corresponding second-order stochastic differential equation is constructed.A system with time-uncorrelated noise is usually just a coarse-grained version of a more fundamental microscopic model with time correlation. It is therefore of interest to study models with correlation. Bag et al. [18] introduced correlation by increasing the number of differential equations and applying uncorrelated noise throughout. This approach obviously increases the system complexity. This article shows that time-correlated noise can be mimicked by time-uncorrelated noise and time-dependent noise without increasing the number of equations in the equation set. To provide a benchmark we start in Section 2 by considering a one-dimensional system based on recurrence relations without correlation. We first show how the recurrence relation, which is usually applied for Gaussian noise only ( that means corresponding to the Ito integral) can be used to develop equations for a more general noise that is multifractal. We develop the equation for variances and covariances. Gaussian processes give that higher-order moments follow from second-order moments.  Our first order system has not been analyzed in the manner proposed in this paper, which is needed to develop alternative accounts of introducing correlation. We compare different methods of achieving the main equations and study when the time-dependent uncorrelated noise can mimic exponential correlated noise. Section 3 studies first order systems with correlation, which are compared with the systems without correlation in Section 2. Such comparison has not been made in the earlier literature. Section 4 considers second-order stochastic processes driven by time-uncorrelated or correlated noise. Proceeding to the second-order allows capturing a larger fraction of real life processes which makes the approach more realistic. We show that a second-order system driven by an exponential time-correlated noise force can be mimicked by adding time-uncorrelated noise both to the position and to the velocity5. Instead of expanding the equation set, noise is added separately to each of the two dimensions (exemplified with position and velocity) of the two-dimensional system. Section 5 concludes.


## 2. A First-Order Time-Uncorrelated Process with Additive Noise

This section provides a first order process which accounts for randomness. Assume that the differential equationẊ(t)=f(X(t)) has been used to describe a physical phenomenon. Assume that this equation is found to no longer hold due to results from a more developed experimental set-up. The question is then as follows: how should this equation be (a) modified or (b) reinterpreted, to be more realistic? Assume that we use (b), assuming that the original equation is reinterpreted to mean ∂E(X(t))/∂t=E(f(X(t))), where E means expectation of a stochastic process. (Another possibility is E(Ẋ(t))=E(f(X(t))). These two interpretations are in general different since this one demands a derivative path.) The next question is then how to construct a stochastic theory such that expectation, variance, and higher-order moments can be calculated.Stochastic or nonstochastic integrals can rarely be solved in analytic form, making numerical algorithms an important tool. Assume thatdXt is the change in this quantity during a small time interval from t to t+h (we will let the time spacing approach zero). Assume that this change is proportional with the time interval dt=h and with  f(Xt), which gives the recurrence relation:(2.1)dXt=defXt+h-Xt=modf(Xt)h,⇒Xt+h=Xt+f(Xt)h,
where “mod” means that this is a model assumption and “def” means definition. When constructing a more developed theory accounting for noise, we can, for instance, make the initial values stochastic as in the Liouville approach or we can change the recurrence relation in (2.1). In the study of nonlinear recurrence relations Glass and Mackey [27] showed that it is possible to construct an infinite number of deterministic relations which are chaotic, but which describe a given density distribution. Thus a given density distribution has no unique recurrence relation. It appears that the broad class of Markovian theories incorporating the Gaussian white noise input provides a satisfactory approximation for a large variety of phenomena. We consider the more general stochastic equation, which we consider as a stochastic differential equation with additive noise of the Ito form: (2.2a)dXt=defXt+h-Xt=modf(Xt)h+d*ς(t,h,Xt),d*ς(t,h,Xt)=modRandomX[Distribution[0,dσ2(t,h,Xt)]],(2.2b)E(d*ς(t,h,Xt))=mod0,(2.2c)E(H(Xt,t)d*ς(t,h,Xt))=mod0,where d*ς(t,h,Xt) is not a differential in the Riemann sense and is therefore denoted by the “*”. σ2(t,h,Xt) is a function and dσ2(t,h,Xt) is the differential change of σ2(t,h,Xt) with respect to h during the time step h. We model d*ς(t,h,Xt) as a stochastic variable where Distribution is a distribution of expectation zero and variance E(d*ς2)= dσ2. E() means expectation. H(x,t) is any arbitrary function. The expectation of d*ς(t,h,Xt) in (2.2b) is zero. We assume no correlation. Equation (2.2c) is assumed in order to develop (2.5). Equation (2.2c) is valid when using Ito calculus (no bounded variation) but is not valid when using Stratonovich calculus. When d*ς(t,h,Xt)=0, we define, as determined by (2.2a)–(2.2c), what we call a Liouville recurrence relation, which defines the Liouville process, where only the initial values are stochastic.Conceptually, we can easily generate realizations by applying (2.2a)–(2.2c) on a computer. Assume that we perform M (which approaches infinity) different runs up to time tk, with a constant time spacing h (which approaches zero). Let H(x,t) be the arbitrary function that we apply to each number. Thus we achieve the set of numbers, to read (2.3)H(Xt01,t0),H(Xt11,t1),H(Xt21,t2),…H(Xtk1,tk):Run1H(Xt02,t0),H(Xt12,t1),H(Xt22,t2),…H(Xtk2,tk):Run2⋮H(Xt0M,t0),H(Xt1M,t1),H(Xt2M,t2),…H(XtkM,tk):RunM
Applying Taylor expansion, the expectation E() is achieved when M goes to infinity, that is,(2.4)E(H[Xt+h,t+h])=E(H[Xt+f(Xt)h+d*ς,t+h])=E{(12!)H[Xt,t]+Ḣ[Xt,t]h+(DH[Xt,t])(f(Xt)h+d*ς)+(12!)(D2H[Xt,t])[h2f(Xt)2+2f(Xt)hd*ς+d*ς2]+⋯},
where the “dot” above a variable means the time derivative ∂/∂t, that is, Ḣ=∂H/∂t, and “D” means the space derivative D=∂/∂Xt, that is, DH[Xt,t]=∂H[Xt,t]/∂Xt. We can conceptually collect all tracks that pass through a given Xt. We next assume that (a) the variance is E(d*ς2)=dσ2=modg2(t,Xt)dt=g2(t,Xt)h for small time steps, where gi(t,Xt),i=2,3,4,…, is some well-behaved function to be specified exogenously, and (b) that higher-order moments also have the same powers of h, akin to multifractal phenomena [28]. The terms E[DH[Xt,t])dς] and E[(1/2!)(D2H[Xt,t])2f(Xt)hdς] equal zero due to (2.2c). Thus for n>1,dσn=E(d*ςn)=modgn(t,Xt)h,⇒σ̇n=gn(t,X(t)). This gives(2.5)Limh→>0ddtE(H[Xt,t])=E(Ḣ[Xt,t]+f(Xt)D(H[Xt,t])+(12!)g2(t,Xt)(D2H[Xt,t])+(13!)g3(t,Xt)(D3H[Xt,t])+⋯),Af()=def∂∂t()+f(x)D()+(12!)g2(t,x)D2()+(13!)g3(t,x)D3()+⋯.Af is the forward infinitesimal operator of the process. By setting H(X,t)=X as a special case, we easily find that ∂E(Xt)/∂t=E(f(Xt)).We introduce the probability density ρ(t,x) of Xt. We choose H() time independent and can then write (2.5) by definition as(2.6)∫ρ̇(t,x)H(x)dx=∫ρ(t,x)f(x)DH(x)dx+(12!)∫ρ(t,x)g2(t,x)D2H(x)dx+⋯.
(Only natural boundary conditions are used in this article and the space integration limits are suppressed.) Integrating (2.6) by parts and assuming natural boundary conditions gives
(2.7)∫H(x)ρ̇(t,x)dx=-∫H(x)D(ρ(t,x)f(x))dx+12!∫H(x)D2(ρ(t,x)g2(t,x))dx+⋯.
Equation (2.7) is valid for all H(x), and thus (2.8)ρ̇(t,x)=-D(ρ(t,x)f(x))+(12!)D2(ρ(t,x)g2(t,x))+(13!)D3(ρ(t,x)g3(t,x))+⋯=-D(ρ(t,x)f(x))+(12!)D2(ρ(t,x)σ̇2(t,x))+(13!)D3(ρ(t,x)σ̇3(t,x))+⋯.
In fact, if our uncorrelated random term is Gaussian, all odd moments are zero, and even moments of higher order than two are of higher order than h (see Appendix A). This gives σ̇3(t,x)=g3(t,x)=σ̇4(t,x)=g4(t,x)=σ̇5(t,x)=g5(t,x)=⋯=0, implying the partial differential equation called the forward Fokker-Planck equation or forward Kolmogorov equation [29]. We can from (2.8) easily calculate the derivative of the variance of Xt which for a time-dependent (and uncorrelated) random term with σ̇2=modg2(t) becomes (2.9)∂∂tVar(Xt)=∂∂tE(Xt2)-2E(Xt)∂∂tE(Xt)=∂∂t(∫ρ(t,x)x2)dx-2E(Xt)E(f(Xt))=∫x2(-D(ρ(t,x)f(x))+(12!)D2(ρ(t,x)σ̇2)+(13!)D3(ρ(t,x)σ̇3)+⋯)dx-2E(Xt)E(f(Xt))=∫2(ρ(t,x)f(x)x)dx-2E(Xt)E(f(Xt))+σ̇2(t)=σ̇2(t)+2Cov(f(Xt),Xt).
The general Liouville solution of (2.8) is easily found, to read(2.10)vL(x)=def-ax,ρL(t,x)=e-atA(xe-at),∫A(u)du=1.A() is an arbitrary function such that the integral is equal to 1. Say that the force is a linear friction force f(x)=mod-ax and that σ̇2=modξ02(1-Exp(-t/τ)), where  ξ0 and τ are parameters. The time derivative of the variance becomes according to (2.9)
(2.11)V̇ar(Xt)=-2aV̇ar(Xt)+ξ02(1-Exp(-t/τ)),t0=0⇒Var(Xt)=Exp(-2at)Var(Xt0)+ξ02Exp(-2at)[Exp(2at)-12a+τExp(-t(1/τ-2a))-11-2aτ].
Say that we will formulate a continuous equation in time that corresponds to (2.2a)–(2.2c), to read Ẋ(t)=f(X(t))+ξ(t)=f(X(t))+ς̇(t),ς̇(t)=dς/dt=defξ(t), where ξ(t) is an arbitrary noise function. We let ξ(t) be deterministic (the randomness is then only in the initial values of  ξ(t); see Appendix E as an example). The deterministic approach generally gives that ξ(t) can be integrated in the traditional Riemann sense. The Stratonovich integral occurs as the limit of colored noise if the correlation time of the noise term approaches zero. A quite common and different integral is the Ito integral for the uncorrelated situation and Gaussian noise.  This integral of ξ(t) cannot be integrated in the Riemann sense due to lack of bounded variation. However, for additive noise the Stratonovich and Ito models give the same answer. To match the noise in (2.2a)–(2.2c) we set that expectation is zero and that the noise is uncorrelated, to read in the Stratonovich sense (indeed, also in the Ito sense since we will achieve the same result for additive noise): (2.12)Ẋ(t)=f(X(t))+ξ(t)⇒X(t)=X0+∫0t(f(X(t′))+ξ(t′))dt′,(a),E(X(t))=E(X(0))+∫0tE(f(X(t′))dt',E(Ẋt)=E(f(X(t)),(b),E(ξ(t))=mod0,(c),E(ξ(t)ξ(t′))=modσ̇2(t)δ(t-t′),(e),
where δ() is the Dirac delta function that accounts for the lack of correlation. This gives that
(2.13)∂∂tVar(X(t))=∂∂tE(X(t)2)-2E(X(t))∂∂tE(X(t))=2E[X(t)(f(X(t))+ξ(t))]-2E(X(t))E(f(X(t)))=2E[∫0tξ(u)ξ(t)du+∫0tf(X(u))ξ(t)du]+2Cov(X(t),f(X(t)))=2∫0tσ̇2(t)δ(u-t)du+2E[∫0tf(X(u))duξ(t)]+2Cov(X(t),f(X(t)))=σ̇2+2Cov(X(t),f(X(t)))+2E[∫0tf(X(u))duξ(t)].
Equation (2.13) is equal to (2.9) if 2E[∫0tf(X(u))duξ(t)]=0, which we will find is correct if f is a linear force. As another example, set that the continuous process in space of Xt is in fact an approximation to a discrete process in space. Assume as an example that Xt≈Nt is the number of cells that die randomly. We find that the drift term (first order term) of the Fokker-Planck equation (2.8) is related to the diffusion term (second-order term) by g2=f2 (see Appendix B). Apply the example of linear friction, that is, f(x)=mod-ax. We then have for the time continuous case that (2.14)Ẋ(t)=-aX(t)+ξ(t).
Equation (2.14) can be solved as
(2.15)Ẋ(t)=-aX(t)+ξ(t)⇒Ẋ(t)Exp(at)+aX(t)Exp(at)=Exp(at)ξ(t)⇒d(X(t)Exp(at))=dt(Exp(at)ξ(t))⇒X(t)Exp(at)=X(0)+∫0tExp(au)ξ(u)du⇒X(t)=X(0)Exp(-at)+Exp(-at)∫0tExp(au)ξ(u)du,t0=0.
This gives the covariance as(2.16)Cov[X(t),X(t′)]=E[(X(t)-E(X(t)))(X(t′)-E(X(t′)))]=E[(X(0)Exp(-at)-E[X(0)]Exp(-at)+Exp(-at)∫0tExp(au)ξ(u)du)×(X(0)Exp(-at′)-E[X(0)]Exp(-at′)+Exp(-at′)∫0t′Exp(av)ξ(v)dv)]=Exp(-at-at′)Cov[X(0),X(0)]+Exp(-at-at′)E[X(0)∫0t'Exp(av)ξ(v)dv+X(0)∫0tExp(au)ξ(u)du]+Exp(-at-at′)E[∫0t∫0t′Exp(au+av)ξ(u)ξ(v)dudv]=Exp(-at-at′)Var[X(0)]+Exp(-at-at′)∫0t∫0t′Exp(au+av)E(ξ(u)ξ(v))dudv.
We have used that the covariance of the initial value X(0) and the noise is zero, to read E(X(0)ξ(u))=E(X(0))E(ξ(u))=0. The integral in (2.16), which is quite general, can be calculated explicitly for a time-dependent uncorrelated noise where E(ξ(t)ξ(t′))=σ̇2(t)δ(t-t′),σ̇2(t)=ξ02λ(t), to read(2.17)Cov[X(t),X(t′)]-Exp(-at-at′)Var[X(0)]=ξ02Exp(-at-at′)∫0t′∫0tExp(au+av)λ(u)δ(u-v)dudv=ξ02Exp(-at-at')∫0t'∫-v-v+tExp(2av)λ(z+v)Exp(az)δ(z)dzdv=ξ02Exp(-at-at′)∫0t'Exp(2av)λ(v)dv,t>t′.
We use our example λ(t)=mod1-Exp(-t/τ) (we will compare this process with a correlated process in the next section). This gives from (2.17) when t>t′(2.18)Cov[X(t),X(t′)]=Exp(-at-at′)Var[X(0)]+ξ02Exp(-at-at′)∫0t′Exp(2av)(1-Exp(-v/τ))dv=Exp(-at-at′)Var[X(0)]+ξ02Exp(-at-at′)[Exp(2at′)-12a-Exp((2a-1/τ)t′)-12a-1/τ],lima→0Cov[X(t),X(t′)]=Var[X(0)]+ξ02[t′+τ(Exp(-t′/τ)-1)],limτ→0Cov[X(t),X(t′)]=Exp(-at-at′)Var[X(0)]+ξ02Exp(-at-at′)Exp(2at′)-12a,lima→0τ→0Cov[X(t),X(t′)]=Var[X(0)]+ξ02t′,t>t′.
The variance follows for t=t′ (actually t=t′ demands a more careful analysis, but it turns out that setting t=t′ in (2.18) is correct), to read(2.19)Var[X(t)]=Exp(-2at)Var[X(0)]+ξ02Exp(-2at)[Exp(2at)-12a+τExp(-t(1/τ-2a))-11-2aτ],lima→0Var[X(t)]=Var[X(0)]+ξ02(t+τ(Exp(-t/τ))-1)),limτ→0Var[X(t)]=Exp(-2at)Var[X(0)]+ξ022a[1-Exp(-2at)],lima→0τ→0Var[X(t)]=Var[X(0)]+ξ02t,lim2aτ→1Var[X(t)]=Exp(-2at)Var[X(0)]+ξ022a[1-Exp(-2at)].
The time derivative of the variance becomes (2.20)∂∂tVar[X(t)]=-2aExp(-2at)Var[X(0)]+ξ02[Exp(-2at)+τ-Exp(-t/τ)/τ+2aExp(-2at)1-2aτ]=-2aVar[X(t)]+ξ02[Exp(-2at)+-Exp(-t/τ)+2aτExp(-2at)1-2aτ]+ξ02[1-Exp(-2at)+2aτExp(-t/τ)-Exp(-2at)1-2aτ]=-2aVar[X(t)]+ξ02[1-Exp(-tτ)]=-2aVar[X(t)]+σ̇2.
Thus (2.19) gives the same solution as in (2.20) since 2Cov(f(Xt),Xt)=-2aVar(Xt) for our linear friction force. The expectation is given by E(Xt)=E(X0)e-at. For Gaussian processes (correlated or uncorrelated) the variance is important since higher-order moments follow from second-order moments (variance). By comparing with (2.13) and (2.20) we see that 2E[∫0tf(X(u))duξ(t)] = 0 for a linear deterministic force f. However, this can be found more easily for a linear force by using (2.14) since (2.21)2E[∫0tf(X(u))duξ(t)]=-2aE[∫0tX(u)duξ(t)]=2E[∫0t[X(0)Exp(-au)ξ(t)+Exp(-at)∫0uExp(av)ξ(v)ξ(t)dv]du]=2∫0tExp(-au)∫0uExp(av)E[ξ(v)ξ(t)]dvdu=2∫0tExp(-au)∫0uExp(av)σ̇2(v)δ(v-t)dvdu=2∫0tExp(-au)∫-tu-tExp(av)σ̇2(t+z)δ(z)dzdu=0.
We further find from (2.18) for the special case that λ=1 that
(2.22)Cov[X(t),X(t′)]=ξ022aExp(-a(t-t')),t≫1/a,t′≫1/a.
Thus in this special case X(t) is for large times exponentially correlated.We can write (2.8) as (2.23)ρ̇(t,x)=-D(ρ(t,x)v(t,x)),v(t,x)=deff(x)-12!D(g2(t,x)ρ(t,x))ρ(t,x)-13!D2(g3(t,x)ρ(t,x))ρ(t,x)-⋯,
where v(t,x) is a current velocity, which can be arbitrary in a general theory. Generally, we can let the increment depend on the probability density at time t, to read(2.24)Xt+h=Xt+f(Xt)h+d*ς(t,h,Xt,ρ(t,Xt)),d*ς(t,h,Xt,ρ(t,Xt))=RandomX[Distribution[0,dσ2(t,h,Xt,ρ(t,Xt))]].
Realizations are easily generated by a computer. All realizations have to be calculated in parallel such that the density can be “counted up” at each time t before a new time step is calculated. This process is not Markovian.


## 3. First-Order Stochastic Processes with Exponential Correlation

In Section2 we analyzed first order uncorrelated processes with additive noise. The Fokker-Plank equation was applicable for the uncorrelated processes. It turns out that in some cases uncorrelated processes with additive noise can in fact mimic correlated processes. Assume as an example that the random term is exponentially correlated, with correlation E(ξ(t)ξ(t′))=mod(ξ02/(2τ))Exp(-|t-t′|/τ), where τ is a correlation time parameter. In the limit when τ approaches zero we achieve uncorrelated noise, to read Limτ→0ξ02Exp(-|t-t′|/τ)/(2τ)=ξ02δ(t-t′). Notice that an equation for ξ(t) that in fact will generate this exponentially correlated noise for large times (see (2.16)) follows simply by a scaling of the parameters in (2.14), to read
(3.1)ξ̇(t)=-ξ(t)τ+ψ(t),
where ψ(t) is white noise with E(ψ(t)ψ(t′))=τ2ξ02δ(t-t′).The last line in (2.16) is general. For exponentially correlated noise with linear friction we achieve
(3.2)Cov[X(t),X(t′)]-Exp(-a(t+t′))Var(X0)=Exp(-a(t+t′))∫0t′∫0tExp(a(u+v))E(ξ(u)ξ(v))dudv=Exp(-a(t+t′))ξ022τ∫0t′∫0tExp(a(u+v))E(-|u-v|τ)dudv=Exp(-a(t+t′))ξ022τ∫0t′∫-v-v+tExp(2av)Exp(-|z|τ+az)dzdv=Exp(-a(t+t′))ξ022τ[∫0t′Exp(2av)(∫-v0Exp(z(1τ+a))dz+∫0-v+tExp(-z(1τ-a))dz)dv]=Exp(-a(t+t′))ξ022τ[∫0t′Exp(2av)(1-Exp(-v(1/τ+a))1/τ+a-Exp((v-t)(1/τ-a))-11/τ-a)dv]=Exp(-a(t+t′))ξ022τ(Exp(2at′)-12a(1/τ+a)+Exp(2at′)-12a(1/τ-a)+Exp(-t′(1/τ-a))(1/τ+a)(1/τ-a)+-1-Exp(t′(1/τ+a)-t(1/τ-a))+Exp(-t(1/τ-a))(1/τ+a)(1/τ-a)),t>t′=Exp(-a(t+t′))ξ02(Exp(2at′)-12a(1+aτ)(1-aτ)+Exp(-t′(1/τ-a))-12(1/τ+a)(1-τa)+-Exp(t′(1/τ+a)-t(1/τ-a))+Exp(-t(1/τ-a))2(1/τ+a)(1-τa)),t>t′.
Equation (3.2) then implies (3.3)lima→0Cov[X(t),X(t′)]=Var[X(0)]+ξ02(t′+τ2(Exp(-t′τ)-1-Exp(t′-tτ)+Exp(-tτ))),limτ→0Cov[X(t),X(t')]=Exp(-a(t+t′))Var[X(0)]+ξ022a(Exp(-a(t-t′))-Exp(-a(t+t′))),lima→0τ→0Cov[X(t),X(t′)]=Var[X(0)]+ξ02t′.
The variance follows when t=t′, to read(3.4a)Var[X(t)]=Exp(-2at)Var(X0)+Exp(-2at)ξ02(Exp(2at)-12a(1+aτ)(1-aτ)+τ2Exp(-t(1/τ-a))-1-Exp(2at)2(1+aτ)(1-aτ)),(3.4b)lima→0Var[X(t)]=Var(X0)+ξ02(t+τ(Exp(-tτ)-1)),(3.4c)limτ→0Var[X(t)]=Exp(-2at)Var(X0)+ξ022a(1-Exp(-2at)),(3.4d)lima→0τ→0Var[X(t)]=Var(X0)+ξ02t,(3.4e)limaτ→1Var[X(t)]=Exp(-2at)Var(X0)+ξ022a(1+aτ)(1-Exp(-2at)),(3.4f)lima→0t/τ≪1Var[X(t)]=Var(X0)+ξ02(t+τ(1-tτ+(t/τ)22-1))=Var(X0)+ξ02t22τ+⋯.Equations (3.4d) and (3.4e) show that when t≫τ, we achieve the uncorrelated noise with Hurst exponent one half, while when t≪τ, we achieve a fractional noise with Hurst exponent one. (For the concept of fractional Brownian motion see Peitgen et al. [30, section 9.5, page 491]. See also M. Rypdal and K. Rypdal [31, 32] for fractional Brownian motion models descriptive for avalanching systems.) We can compare the two different stochastic processes, the one with linear friction and uncorrelated time-dependent noise where E(ξ(t)ξ(t′))=ξ02λ(t)δ(t-t′)=σ̇2(t)δ(t-t′), and the one with linear friction and exponentially correlated noise of the type E(ξ(t)ξ(t′))=ξ02/(2τ)Exp(-|t-t′|/τ). The covariances and variances are given by the solution (2.18)-(2.19) and (3.2)–(3.4a), (3.4b) (3.4c), (3.4d), (3.4e), and (3.4f), respectively. In the limit when a approaches zero, the variances become equal, but the covariances remain unequal, to read(3.5)Var[X(t)]=Var[X(0)]+ξ02(t+τ(Exp(-tτ)-1)),a=0:uncorrelatedandcorrelated,Cov[X(t),X(t′)]=Var[X(0)]+ξ02[t′+τ(Exp(-t′τ)-1)],a=0:uncorrelated,Cov[X(t),X(t′)]=Var[X(0)]+ξ02(t′+τ2(Exp(-t′τ)-1-Exp(t′-tτ)+Exp(-tτ))),a=0:correlated.
Indeed, the variances can be calculated easily when a=0, to read without correlation(3.6)X(t)=X0+∫0tξ(u)du,Var[X(t)]=Exp[(X(0)+∫0tξ(u)du-E[X(0)])(X(0)+∫0tξ(v)dv-E[X(0)])]=Var[X(0)]+E(∬0tξ(u)ξ(u)dudv)=Var[X(0)]+ξ02∬0t(1-Exp(-uτ))δ(u-v)dudv=Var[X(0)]+∫0t∫-vt-v(1-Exp(-v+zτ))δ(z)dzdv=Var[X(0)]+ξ02∫0t(1-Exp(-vτ))dv=Var[X(0)]+ξ02(t+τ(Exp(-tτ)-1)),∂∂tVar[X(t)]=ξ02(1-Exp(-tτ))=σ̇2(t).
With exponential correlation we achieve that (3.7)Var[X(t)]-Var[X(0)]=E(∬0tξ(u)ξ(u)dudv)=ξ022τ∬0tExp(-|u-v|τ)dudv=ξ022τ∫0t∫-vt-vExp(-|z|τ)dzdv=ξ022τ∫0t[∫-v0Exp(zτ)dz+∫0t-vExp(-zτ)dz+]dv=ξ022∫0t[2-Exp(-vτ)-Exp(-t-vτ)]dv=ξ022[2t+τ(Exp(-tτ)-1)-τ(1-Exp(-tτ))]=ξ02(t+τ(Exp(-tτ)-1)).
We can calculate higher-order moments for the two stochastic processes. For the special case that the time-dependent uncorrelated process is Gaussian, a Fokker-Plank equation follows as shown in Section 2 with σ̇2=ξ02(1-Exp(-t/τ)). Heinrichs [16] has proved that a Fokker-Planck equation also follows for the probability density when assuming Gaussian exponentially correlated noise (called Ornstein-Uhlenbeck noise, see Appendix A). Thus for a=0, the probability density of the time-dependent Gaussian uncorrelated noise is equal to probability density of the exponentially correlated Gaussian noise. Gaussian processes give that higher-order moments follow from the second-order moment. Thus we should expect equality of two Gaussian processes, even a correlated and uncorrelated one, if the variances are equal.With friction we can also compare the variances, to read(3.8a)Var[X(t)]=Exp(-2at)Var[X(0)]+ξ02[1-Exp(-2at)2a+τExp(-t/τ)-Exp(-2at)1-2aτ](uncorrelated),=Exp(-2at)Var[X(0)]+ξ02[1-2aτ-Exp(-2at)(1-2aτExp(-t/τ(1-2aτ)))2a(1-2aτ)](uncorrelated),(3.8b)limt→∞Var[X(t)]=ξ022a(uncorrelated),(3.8c)Var[X(t)]=Exp(-2at)Var[X(0)]+ξ02(1-Exp(-2at)2a(1+aτ)(1-aτ)+τ2Exp(-t/τ(1+aτ))-1-Exp(-2at)2(1+aτ)(1-aτ))(correlated),(3.8d)limt→∞Var[X(t)]=ξ022a(1+aτ)(correlated).We observe that the variances are different. Thus the two stochastic processes couple differently to the linear friction term. Assume that aτ≪1. The variance then becomes (3.9)Var[X(t)]=Exp(-2at)Var[X(0)]+ξ02[1-Exp(-2at)+2aτExp(-t/τ)2a],aτ≪1:uncorrelated,Var[X(t)]=Exp(-2at)Var[X(0)]+ξ02(1-Exp(-2at)+2aτExp(-t/τ)2a),aτ≪1:correlated.
In this limit the variances are not different. Thus pure exponentially time-correlated noise can be mimicked by uncorrelated noise. However, the noise couples differently to a linear deterministic friction force.Figure1 shows the Liouville variance (E(ξ(t)ξ(t′))=0), the variance assuming E(ξ(t)ξ(t′))=ξ02δ(t-t′), the variance assuming E(ξ(t)ξ(t′))=ξ02(1-Exp(-t/τ))δ(t-t′), and the variance assuming correlation E(ξ(t)ξ(t′))=(ξ02/(2τ))Exp(-|t-t′|/τ).Figure 1
The variances as a function of time assuming linear friction.Var(X(0))=2,a=1,ξ0=1.5,τ=0.8,t0=0.We have so far only analyzed first order systems with additive noise. Consider now the continuous time approach with nonlinear noise, to read(3.10)Ẋ(t)=-aX(t)+X(t)ξ(t)⇒∫0tdX(t)X(t)=-at+∫0tξ(u)du+const.,E(ξ(t)ξ(t′))=σ̇2δ(t-t′).
Using the Stratonovich integral ordinary calculus applies, the solution becomes
(3.11)X(t)=X(0)Exp(-at+∫0tξ(u)du):Stratonovich.
Assuming Gaussian noise, the expectation becomes (3.12)E[X(t)]=E[X(0)]Exp(-at)E[Exp[∫0tξ(u)du]]=E[X(0)]Exp(-at)E[1+∫0tξ(u)du+12!∬0tξ(u)ξ(v)dudv+13!∭0tξ(u)ξ(v)ξ(w)dudvdw+⋯]=E[X(0)]Exp(-at)[1+12∬0tE[ξ(u)ξ(v)]dudv+16∭0tE[ξ(u)ξ(v)ξ(w)]dudvdw+⋯]=E[X(0)]Exp(-at+σ22)σn=E(∫∭0tξ(t′)ξ(t′′)ξ(t′′′)ξ(t′′′′)⋯dt′dt′′dt′′′dt′′′′⋯︸ntimes),σ2n=Gaussian(2n)!2nn!(σ2)n.
Here we have used the algebra in (A.5)–(A.7). However, it has in the literature been considered nice to have a process that fulfills the relation (∂/∂t)E(Xt)=E(f(Xt,t)), even for multiplicative noise. It turns out that this can be archived if the time derivative path is abandoned (see Appendix C for the Ito calculus).We can mimic Gaussian-correlated noise by Gaussian-uncorrelated noise when using the Stratonovich integral for multiplicative noise also. To achieve this we must according to (3.11) have that
(3.13)∬0tE[ξ(u)ξ(v)]dudv=∬0tσ̇2(u)δ(u-v)dudv=12τ∬0tExp(-|u-v|τ)dudv
However, we have proven this already in (3.6) and (3.7). More generally assume that the correlated noise is chosen to be E[ξ(u)ξ(v)]=Cuc(u,v,τ). A solution is σ̇2(t,τ)=2∫0tCuc(u,t,τ)du, which is fulfilled for the two processes we examine in this article. When using the Ito integral in (3.10), the solution is different. Appendix C shows that the solution is E[X(t)]=E[X(0)]Exp(-at). This solution for the uncorrelated case cannot mimic the correlated case.


## 4. Second-Order Stochastic Processes

Unfortunately bidimensional first order processes or second-order stochastic processes are more difficult to address than one dimensional first order processes. This is so because the position at a given time depends strongly on the velocity. Removing this dependence is tricky. We construct a stochastic interpretation of the bidimensional equation set with additive noise, to read as the Ito stochastic equation(4.1)Xt+h=Xt+fX(Xt,Yt)h+d*ςX,Yt+h=Yt+fY(Xt,Yt)h+d*ςY,d*ςX=RandomX[Distribution[0,dσ2X]],d*ςY=RandomY[Distribution[0,dσ2Y]],
where the expectations are zero and dσ2X and dσ2Y are the variances. We achieve by Taylor expansion of an arbitrary function H(x,y) that(4.2)E(H[Xt+h,Yt+h])=E(H[Xt+fXh+d*ςX,Yt+fYh+d*ςY])=E{[h2fXfY+hfXd*ςY+hfYd*ςX+d*ςXd*ςY]H[Xt,Yt]+(D1H[Xt,Yt])(fXh+d*ςX)+(D2H[Xt,Yt])(fYh+d*ςY)+12!(D12H[Xt,Yt])[h2fX2+2hfXd*ςX+d*ςX2]+12!(D22H[Xt,Yt])[h2fY2+2hfYd*ςY+d*ςY2]+(D1D2H[Xt,Yt])[h2fXfY+hfXd*ςY+hfYd*ςX+d*ςXd*ςY]+⋯}.
For time-uncorrelated Gaussian random terms with no cross correlation only the terms with d*ςY2,d*ςx2 contribute to order h. Thus after some simple algebra analogous to the algebra in Section 2 we achieve the well-known Fokker-Planck equation:(4.3)ρ̇(t,x,y)=-D1(ρ(t,x,y)fX(x,y))-D2(ρ(t,x,y)fY(x,y))+(12)D12(ρ(t,x,y)g2X(t,x,y))+(12)D22(ρ(t,x,y)g2Y(t,x,y)),σ̇2X=g2X(t,x,y),σ̇2Y=g2Y(t,x,y).
In physics or engineering applications second-order differential equations are often used as models. For physical systems where we use f=ma, we set the mass of the object equal to m=1. The second-order differential equation can be written as bidimensional first order equations, to read
(4.4)Ẍt=f(Xt,Ẋt)⇒Ẋt=Yt,Ẏt=f(Xt,Yt),Xt+h=Xt+Yth,Yt+h=Yt+f(Xt,Yt)h+d*ςY.
The Langevin model for the Ornstein-Uhlenbeck (1930) process with uncorrelated Gaussian random force is a special case of (4.1)–(4.3) when assuming a random term in the velocity (Yt) equation only (due to a stochastic force), that is, ξX=0,g2X=0. As a well-known example, assume that f(Xt,Yt)=-V'(Xt,)-εYt,g2Y(Xt,Yt)=2εkT in the Ornstein-Uhlenbeck (1930) process. Thus we have a random force, a conservative nonrandom force, and a linear nonrandom friction force. k is the Boltzmann constant and T is the temperature. The Fokker-Planck equation, corresponding to an uncorrelated Gaussian random force, becomes according to (4.4)
(4.5)ρ̇(t,x,y)=-D1(ρ(t,x,y)y)-D2(ρ(t,x,y)(-V'(x)-εy))+εkTD22(ρ(t,x,y)).
It is easily verified, and well known, that a steady-state solution is given by the Boltzmann distribution, to read
(4.6)ρ(t,x,y)=cBExp(-y22kT-V(x)kT),
where cB is a constant and T is the temperature. Thus the Boltzmann distribution is achieved as a steady-state solution when assuming linear friction. Notice that when ε=0, every solution of the type ρ(t,x,y)=ρ(H) is a steady-state solution. H is now the Hamiltonian, to read H=(1/2)y2+V(x). This shows the importance of linear friction to achieve the correct steady-state solution for the uncorrelated Gaussian process.Consider now the second-order process in (4.1) with only a random force. We can find the analogous continuous time solution equation for the position, to read according to Stratonovich
(4.7)Ẍ(t)=ξY(t)⇒X(t)=X(0)+Y(0)t+∫0t∫0vξY(u)dudv.
The question is now whether uncorrelated noise can mimic correlated noise for the position, and not only for the velocity. Assume that we first calculate the variance by only applying an uncorrelated force of our now familiar type E(ξY(t)ξY(t′))=ξ02(1-Exp(-t/τ))δ(t-t′). This gives that (4.8)Ẋ(t)=Y(t)=Y0+∫0tξY(u′)du,X(t)=X(0)+Y(0)t+∫0t∫0vξY(u)dudv,∂∂tE[X(t)2]=2E[X(t)Ẋ(t)]=2E[(X(0)+Y(0)t+∫0t∫0vξY(u)dudv)×(Y(0)+∫0tξY(u′)du′)],∂∂tE[X(t)2]-2E[(X(0)+Y(0)t)Y(0)]=2∫0t∫0v∫0tE(ξY(u)ξY(u'))du'dudv=2ξ02∫0t∫0v∫0t(1-Exp(-uτ))δ(u-u′)du′dudv=2ξ02∫0t∫0v∫u-tu(1-Exp(-uτ))δ(z)dzdudv=2ξ02∫0t∫0v(1-Exp(-uτ))dudv=ξ02∫0t(2v+2τExp(-vτ)-2τ)dv=ξ02[t2+2τ2-2τ2Exp(-tτ)-2τt].
However, for the correlated process with E(ξY(t)ξY(t′))=ξ02/(2τ)Exp(-|t-t′|/τ), we achieve that(4.9)∂∂tE[X(t)2]-2E[(X(0)+Y(0)t)Y(0)]=ξ02τ∫0t∫0v∫0tExp(-|u-u′|τ)du′dudv=ξ02τ∫0t∫0v∫u-tuExp(-|z|τ)dzdudv=ξ02τ∫0t∫0v[∫u-t0Exp(zτ)dz+∫0uExp(-zτ)dz+]dzdudv=ξ02∫0t∫0v[2-Exp(u-tτ)-Exp(-uτ)]dudv=ξ02∫0t[2v-τExp(-tτ)Exp(vτ)+τExp(-tτ)+τExp(-vτ)-τ]dv=ξ02[t2+τtExp(-tτ)-τt].
Thus we find that the variances of the position are unequal since V̇ar[X(t)]=Ė(X(t)2)-2E(X(t))E(Ẋ(t))=Ė(X(t)2)-2E(X(0)+Y(0)t)E(Y(0)). We have that(4.10)V̇ar[X(t)]=∂∂tE[X(t)2]-2E(X(t))E(Ẋ(t))=2E[(X(0)+Y(0)t)Y(0)]-2E(X(0)+Y(0)t)E(Y(0))+ξ02[t2+τtExp(-tτ)-τt]=Cov((X(0)+Y(0)t),Y(0))+{ξ02t2whent≫τ,ξ022τ(t3-t43τ+⋯)whent≪τ.
Thus for t≫τ we find a Brownian motion with Hurst exponent 1.5, while for t≪τ we find the Brownian motion with Hurst exponent 2. This is in agreement with the asymptotic solution found by Heinrichs [16].The equation for the joint distribution is more complicated to derive. We achieve by Taylor expansion with only a random force that(4.11)E(H[Xt+h,Yt+h])=E(H[Xt+Yth,Yt+d*ςY])=E{12!H[Xt,Yt]+(D1H[Xt,Yt])Yth+(D2H[Xt,Yt])d*ςY+12!(D12H[Xt,Yt])h2Yt2+12!(D22H[Xt,Yt])d*ςY2+(D1D2H[Xt,Yt])hYtd*ςY+⋯}.
We further achieve from (4.11)(4.12)Ė(H[Xt,Yt])=E{H[Xt,Yt]+(D1H[Xt,Yt])Yth+12!(D22H[Xt,Yt])d*ςY2+O(h2)},E(d*ςY2)=σ̇2(t)dt=ξ02(1-Exp(-tτ))dt.(D1D2H[Xt,Yt])hYtdςY is of order h2. This gives, as it should, the Fokker-Planck equation without mixed terms, to read(4.13)ρ̇(t,x,y)=-D1(ρ(t,x,y)y)+(12)D22(ξ02λ(t)ρ(t,x,y))λ(t)=def1-Exp(-tτ).
However, Heinrichs [16] found for a deterministic noise force only, that is exponentially correlated and Gaussian, that(4.14)Ẍ(t)=ξ(t),E(ξ(t)ξ(t′))=ξ02Exp(-|t-t′|/τ)2τ,ρ̇(t,x,y)=-D1(ρ(t,x,y)y)-12D1D2(ξ02β(t)ρ(t,x,y))+(12)D22(ξ02λ(t)ρ(t,x,y)),λ(t)=def1-Exp(-tτ),β(t)=def(t+τ)Exp(-tτ)-τ,
and when the correlation time τ approaches zero, we achieve the traditional Fokker-Planck equation corresponding to the Gaussian uncorrelated random force. ξ02 = 0 gives the Liouville equation. The D1D2 term is zero for τ=0 as it should be. Say that we calculate the expectation (∂/∂t)E(XY). This gives when using (4.14) that(4.15)∂∂tE[X(t)Y(t)]=∫xyρ̇(t,x,y)dxdy=∫xy[-D1(ρ(t,x,y)y)-12D1D2(ξ02β(t)ρ(t,x,y))+(12)D22(ξ02λ(t)ρ(t,x,y))]dxdy=∫ρ(t,x,y)y2dxdy-12ξ02β(t)=E[Y(t)2]-12ξ02β(t)=E[Y(t)2]-ξ022((t+τ)Exp(-tτ)-τ).
We further have for our time-dependent uncorrelated noise that (∂/∂t)E[XtYt]= E[Y(t)2] according to (4.13), or alternatively if we use (4.7) with  E(ξY(u)ξY(t))=ξ02(1-Exp(-u/τ))δ(u-t), to read(4.16)∂∂tE[X(t)Y(t)]=E[Y(t)2]+E[Ẍ(t)X(t)]=E[Y(t)2]+E[ξY(t)(X0+Y0t+∫0t∫0vξY(u)dudv)]=E[Y(t)2]+∫0t∫0vE(ξY(u)ξY(t))dudv=E[Y(t)2]+∫0t∫0vξ02(1-Exp(-uτ))δ(u-t)dudv=E[Y(t)2]+∫0t∫-tv-tξ02(1-Exp(-z+tτ))δ(z)dzdv=E[Y(t)2].
This is in disagreement with (4.15). This shows that our time-dependent uncorrelated noise cannot mimic the correlated noise for position. To check this further we have for correlated noise that(4.17)∂∂tE[X(t)Y(t)]=E[Y(t)2]+∫0t∫0vE(ξY(u)ξY(t))dudv=E[Y(t)2]+ξ022τ∫0t∫0vExp(-|u-t|τ)dudv=E[Y(t)2]+ξ022τ∫0t∫-tv-tExp(-|z|τ)dzdv=E[Y(t)2]+ξ022τ∫0t[∫-tv-tExp(zτ)dz]dv=E[Y(t)2]+ξ022∫0tExp(-tτ)[Exp(vτ)-1]dv=E[Y(t)2]+ξ022Exp(-tτ)[τExp(tτ)-τ-t]=E[Y(t)2]-ξ022[(τ+t)Exp(-tτ)-τ].
This is in agreement with (4.15). However, it is easily observed from (4.1) that (4.14) follows if (4.18)Xt+h=Xt+Yth+dςX,Yt+h=Yt+fY(Xt,Yt)h+d*ςY,fY(Xt,Yt)=0,d*ςX=RandomX[Distribution[0,dσ2X]],d*ςY=RandomY[Distribution[0,dσ2Y]],E(d*ςX)=E(d*ςY)=0,E(d*ςX2)=0,E(d*ςY2)=ξ02λ(t),E(d*ςXd*ςY)=-ξ02β(t),λ(t)=1-Exp(-tτ),β(t)=(t+τ)Exp(-tτ)-τ.
We thus find that noise must be added both to the position and to the velocity in the Ito stochastic differential equation to mimic the correlated noise force. In addition noise in velocity and position must be cross correlated. The time continuous uncorrelated Stratonovich version (but with cross correlation in velocity and position) that will mimic the solution in (4.14) is accordingly(4.19)Ẋ(t)=Y(t)+ξX(t),Ẏ(t)=ξY(t),E(ξX(t))=E(ξY(t))=0,E(ξY(t)ξY(t′))=ξ02λ(t)δ(t-t′),E(ξX(t)ξX(t′))=0,E(ξX(t)ξY(t′))=-ξ02β(t)δ(t-t′).
Agreement can also be shown by studying the variance directly, to read(4.20)Y(t)=Y0+∫0tξY(u)du,X(t)=X(0)+Y(0)t+∫0t∫0vξY(u)dudv+∫0tξX(u)du,∂∂tE[X(t)2]=2E[X(t)Ẋ(t)]=2E[(X(0)+Y(0)t+∫0t∫0vξY(u)dudv+∫0tξX(u)du)×(Y(0)+∫0tξY(u)du)],∂∂tE[X(t)2]-E((X(0)+Y(0)t)Y(0))=ξ02[t2+2τ2-2τ2Exp(-tτ)-2τt]+E(∬0tξX(u)ξY(u′)dudu′)=ξ02[t2+2τ2-2τ2Exp(-tτ)-2τt]-∬0tξ02β(u)δ(u-u′)dudu′=ξ02[t2+2τ2-2τ2Exp(-tτ)-2τt]-∫0t∫-u′t-u′ξ02β(z+u′)δ(z)dzdu′=ξ02[t2+2τ2-2τ2Exp(-tτ)-2τt]-ξ02∫0t[vExp(-vτ)+τExp(-vτ)-τ]dv=ξ02[t2+2τ2-2τ2Exp(-tτ)-2τt]-ξ02(τ2-Exp(-tτ)τ(t+τ)-τ2(Exp(-tτ)-1)-τt)=ξ02[t2-τt+τtExp(-tτ)].
This agrees with the solution in (4.9). More generally, according to method 2 discussed in the introduction, we can introduce noise simply by adding terms (more or less as hoc) to the Liouville equation. Expanding on the results of (4.18) with a random and nonrandom force, consider the equation where the force is both a noise term plus a deterministic force, to follow from (4.18) when fY(Xt,Yt)≠0(4.21)ρ̇(t,x,y)=-D1(ρ(t,x,y)y)-D2(ρ(t,x,y)fY(x,y))-12D1D2(ξ02β(t)ρ(t,x,y))+(12)D22(ξ02λ(t)ρ(t,x,y)).
Equation (4.21) gives that
(4.22)∂∂tE[X(t)Y(t)]=∫xyρ̇(t,x,y)dxdy=∫xy[(12)-D1(ρ(t,x,y)y)-D2(ρ(t,x,y)fY(x,y))-12D1D2(ξ02β(t)ρ(t,x,y))+(12)D22(ξ02λ(t)ρ(t,x,y))]dxdy=∫ρ(t,x,y)y2dxdy+∫ρ(t,x,y)xfY(x,y)dxdy-12ξ02β(t)=E[Y(t)2]+E[X(t)fY(X(t),Y(t))]-12ξ02β(t).
A question is now whether the equation set (4.19) (or (4.21)) could mimic Ẍ(t)=fY(X(t),Y(t))+ξY(t), where  E(ξ(t)ξ(t′))=(ξ02/(2τ)Exp(-|t-t′|/τ). It will not. As our example we use the linear friction force, fY=-aY. The Stratonovich solution of (4.19) is(4.23)Ẏ(t)=-aY(t)+ξY(t)⇒Y(t)=Y(0)Exp(-at)+Exp(-at)∫0tExp(au)ξ(u)du,t0=0,Ẋ(t)=Y(t)+ξX(t)⇒X(t)=X(0)-1aY(0)(Exp(-at)-1)+∫0t∫0vExp(au-av)ξY(u)dudv+∫0tξX(u)du.
The solution for the variance of Y(t) is equivalent to (3.8a) substituting X→Y, which assumes time-dependent uncorrelated noise. The solution is not equal to the correlated solution in (3.8c). Thus (4.19) (or (4.21)) and Ẍ(t)=fY(X(t),Y(t))+ξY(t) in general model different physical realities, even though the models are the same when fY is set to zero. Also both models are equal in the limit where the correlation time approaches zero.By integrating with respect to the position and velocity, respectively, we achieve the equation for the marginal density of velocity and position, to read(4.24)ρ̇Y(t,y)=-D2(ρY(t,y)vY(t,y))+(12)D22(ξ02α(t)ρY(t,y)),ρ̇X(t,x)=-D1(ρX(t,x)vX(t,x)),ρX(t,x)=def∫ρ(t,x,y)dy,ρY(t,y)=def∫ρ(t,x,y)dx,vY(t,y)=def∫ρ(t,x,y)fY(x,y)ρ(t,y),vX(t,x)=def∫ρ(t,x,y)ydyρX(t,x).
The same equations apply for the Liouville process. We can find an explicit relation for vX(t,x):(4.25)ρ̇X(t,x)=-D1(ρX(t,x)vX(t,x)),vX(t,x)=∫ρ(t,x,y)ydyρX(t,x)=σ̇2(t)D1(ρX(t,x))2ρX(t,x),∫ρ(t,x,y)ydy=σ̇2(t)D1(ρX(t,x))2,σ̇2(t)=ξ02t(t-τ(1-Exp(-tτ))).


## 5. Conclusion

This article studies the construction of stochastic theories from deterministic theories based on ordinary differential equations. We incorporate randomness into deterministic theories and compare analytically and numerically some well-known stochastic theories: the Liouville process, the Ornstein-Uhlenbeck process, and a process that is Gaussian and exponentially correlated (Ornstein-Uhlenbeck noise). Different methods of achieving the marginal densities are discussed and we find an equation for the marginal density of velocity and position for a second-order process driven by an exponentially correlated Gaussian random force. We show that in some situations a noise process with exponential correlation can be mimicked by time-dependent uncorrelated noise. We show that a second-order system driven by an exponential time-correlated noise force can be mimicked by adding time-uncorrelated noise both to the position and to the velocity. For such a situation the traditional concept of force loses its meaning.



---

*Source: 509326-2010-06-02.xml*