# On Random Dynamical Systems Generated by White Noise Time Change of Deterministic Dynamical Systems

Authors: Mohamed Hmissi; Farida Mokchaha
Journal: Journal of Probability and Statistics
 (2022)

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## Abstract

In this paper, we apply the random time change by the real white noise to deterministic dynamical systems. We prove that the obtained random dynamical systems are solutions of some stochastic differential equations whenever the deterministic dynamical systems are solutions of ordinary differential equations.


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## Body

## 1. Introduction

Random perturbations of deterministic dynamical systems are introduced to model real phenomena, which are usually affected by external fluctuations whose resulting action would be natural to be considered as random. We refer to the monographs [1, 2] (and the references therein) for more details on the subject. A continuous time deterministic dynamical system St,x is solution of a differential equation generated by a vector field f. In general, a random perturbation of S is made either by perturbation of f by a real noise or by adding a white noise term to f (cf. Paragraph 1 below). Following [3, 4], in both cases the resulting process is a random dynamical system and many important deterministic properties are extended to analogous random properties.Moreover, the idea to consider random time changes for stochastic processes is introduced in [5] and it is extensively studied in many directions (cf. [6–13] for example). However, random time changes for dynamical systems were introduced recently in [14] as new random perturbation of dynamical systems. For a given random time τt,ω, one can consider the random process Ut,ω,x:=Sτt,ω,x, where S is a deterministic dynamical system. Following [14], the aim is to study the properties of Ut,ω,x depending on those of the initial dynamical system St,x.In this paper, we consider the random time change by the real white noise of dynamical systems and as application, we investigate the case when these systems are generated by ordinary differential equations.LetΩ=ω:ℝ⟶ℝ:ωis continuous andω0=0 equipped with the compact open topology, let W:ℝ×Ω⟶ℝ be the real white noise defined by Wtω=ωt, and let θ:ℝ×Ω⟶Ω,t,ω↦θt,ω defined by θt,ωs:=ωs+t−ωt. Then, θ generates a metric dynamical system on Ω (Definitions 1and 2).LetE be a locally compact space endowed with its Borel σ-algebra B and let S:ℝ×E⟶E such that E,B,S is a continuous dynamical system on E. Let φ:ℝ×Ω×E⟶E defined by the time change W of S, that is, φt,ω,x=SWt,ω,x. We prove first (Theorem 1) that θ,φ is a continuous random dynamical system (Definition 2) on E.Next, we suppose thatE is an open subset of ℝd and S=S1,…,Sd is the solution of the system of ordinary differential equations generated by a C1-function f=:f1,…,fd, that is,(1)yk′t=fky1t,…,ydt,yk0=xk,1≤k≤d.We prove (Theorem2) that the associated random dynamical system defined in Theorem 1 is the solution X=:X1,…,Xd of the system of stochastic differential equations.(2)dXkt=12<∇fk,f>Xtdt+fkXtdWt;Xk0=xk,1≤k≤d.Although we considered a particular (but important) random time change, we have established explicit solutions, in terms of the initial solutions, for a class of stochastic differential equations. Notice that in the previous work [14], only expected values of the solutions are investigated, in implicit form, also in terms of the initial solutions.


## 2. Dynamical and Random Dynamical Systems

For the following classical concepts, we refer essentially to Parts I, II, and IV of [4] (cf. also [15–19]).Throughout the paper,ℝ denotes the real line endowed with its Borel σ-algebra R. Moreover, ⋅,⋅ denotes the usual inner product in ℝd,d∈ℕ and ∇g=∂g/∂x1,…,∂g/∂xd if g:ℝd⟶ℝ is a C1-function.Definition 1.
A dynamical system (DS) is a tripletΓ,G,ψ where Γ,G is a measurable space, ψ:ℝ×Γ⟶Γ,t,y↦ψty:=ψt,y such that(3)ψ0,y=y;ψs+t,y=ψs,ψt,y,s,t∈ℝ,y∈Γ.
Ift,y⟶ψt,y is ℛ⊗G,G measurable, then the DS Γ,G,ψ is said to be measurable.
IfΓ is a topological space endowed with its Borel σ-algebra G and if t,y⟶ψt,y is continuous, then the DS Γ,G,ψ is said to be continuous.
In this paper, we consider essentially two different types of DS:(1)
The infinite dimensional case:Γ,G,ψ=Ω,ℱ,θ, where Ω is an infinite dimensional space. A first example is Ω:=ω:ℝ⟶ℝ:ωis continuous equipped with the compact open topology and ℱ:=ℬΩ the associated Borel σ-algebra. We define the translation shift θ:ℝ×Ω⟶Ω by the following equation:(4)θt,ωs≔=ωs+t,s,t∈ℝ,ω∈Ω.Then,Ω,ℱ,θ is a measurable DS, called DS of translations on ℝ. Under additional assumptions, we may define a probability P on Ω,ℱ such that the DS Ω,ℱ,θ becomes a metric DS called usually real noise.Another important example of infinite dimensional DS, is the Wiener DS onℝ. It will be considered in the second paragraph.(2)
The finite dimensional case:Γ,G,ψ=E,ℬ,S where E is a locally compact space endowed with its Borel σ-algebra ℬ and S is continuous. In this case, the DS E,ℬ,S is said to be deterministic. The global solutions S of ordinary differential equations on an open subset of ℝd,d≥1 are the most important examples and they will be treated in the third paragraph.Definition 2.
LetE be a locally compact space endowed with its Borel σ-algebra ℬ.
A measurable random dynamical system (RDS) defined onE consists of two ingredients:(1)
A metric DS, i.e., a measurable DSΩ,ℱ,θ endowed with a probability measure P which is θ-invariant, that is,(5)Pθt−1F=PF,F∈ℱ,t∈ℝ.(2)
A cocycle overθ, i.e., a mapping φ:ℝ×Ω×E⟶E,t,ω,x↦φt,ω,x which is ℛ⊗ℱ⊗ℬ,ℬ measurable and satisfying (6)φ0,ω,x=x,ω∈Ω,x∈E.
And the cocycle equation:(7)φs+t,ω,x=φs,θtω,φt,ω,x,s,t∈ℝ,ω∈Ω,x∈E.
Such a RDS is denoted byΩ,ℱ,θ,E,φ or simply by θ,φ if there is no confusion.
θ,φ is said to be continuous, if t,x⟶φt,ω,x is continuous for P-almost every ω∈Ω.
Letθ,φ be a measurable RDS on E. We may associate the skew product Φ:ℝ×Ω×E⟶Ω×E defined by the following equation:(8)Φtω,x≔=θt,ω,φt,ω,x;t∈ℝ,x∈E,ω∈Ω.
Then,Ω×E,ℱ⊗ℬ,Φ is a measurable DS on the product space Ω×E endowed with the tensor product σ-algebra ℱ⊗ℬ.
A first standard class of RDS are solutions of random differential equations (RDE): LetΩ,ℱ,θ be a metric DS (for example the translation shift defined by (4), let E be an open subset of ℝd, and h:Ω×E⟶E be measurable such that, for each ω∈Ω,t,x⟶hθtω,x, t,x⟶hθtω,x is continuous and x⟶hθtω,x is locally-Lipschitz. Then, the random differential equation,(9)dXtω=hθtω,Xtωdt;X0ω=x∈E.
admits a unique solutionφt,ω,x=Xtxω and θ,φ is a continuous RDS on E.
A second important class of RDS are solutions of stochastic differential equations (SDE). They will be investigated in the third paragraph.


## 3. Time Change by Real White Noise

For the following standard notions, we refer essentially to [20], Part IV of [4], and Chapter 2 of [21] (cf. also [17]).LetΩ=ω:ℝ⟶ℝ:ωbe continuous andω0=0 equipped with the compact open topology, let ℱ:=ℬΩ be the associated Borel σ-algebra, and let W:ℝ×Ω⟶ℝ, which is defined by the following equation:(10)Wt,ω≔=ωt,t∈ℝ,ω∈Ω.There exists, by a classical result (Kolmogorov extension theorem), a unique probability measureP on Ω,ℱ such that the process W has stationary and independent increments and Wt,.−Ws,. has normal distribution with mean 0 and variance t−s.Letθ:ℝ×Ω⟶Ω,t,ω↦θt,ω, which is defined by the following equation:(11)θt,ωs≔=ωs+t−ωt,s,t∈ℝ,ω∈Ω.Then,Ω,ℱ,P,θ is a metric DS, called the Wiener or Brownian DS. Moreover θ is called Wiener shift on ℝ and W, which is called real white noise.The Wiener DS is the appropriate sample space in order to interpret stochastic differential equations (SDE) as RDS.Remark 1.
The main idea of this paper is to investigate the real white noise processW:ℝ×Ω⟶ℝ as random time change. Combining equations (7) and (8), we get the following equation:(12)Ws+t,ω=Wt,ω+Ws,θt,ω,s,t∈ℝ,ω∈Ω.
According to [20], W is in fact an extension to ℝ of the white noise process on 0,∞. Therefore, by equations (7) and (9), W is (the extension of) a random time in the sense of Definition 2.1 of [14].
Now we come to our first result.Theorem 1.
LetE be a locally compact space endowed with its Borel σ-algebra ℬ and let S:ℝ×E⟶E such that E,ℬ,S is a continuous DS on E. Let Ω,ℱ,P,θ be the metric Wiener DS and let W the associated real white noise. We define φ:ℝ×Ω×E⟶E by the following equation:(13)φt,ω,x=S0t,ω,x,t∈ℝ,x∈E,ω∈Ω.
Then,θ,φ is a continuous RDS on E.Proof.
φ defined by equation (13) is ℛ⊗ℱ⊗ℬ,ℬ measurable as composition of measurable applications. Indeed, W:ℝ×Ω⟶ℝ is ℛ⊗ℱ,ℛ measurable by definition of W. Let Ix:=x,x∈E, then W,I:ℝ×Ω×E⟶ℝ×E;t,ω,x↦Wt,ω,x is trivially ℛ⊗ℱ⊗ℬ,ℛ⊗ℬ measurable. Also, S:ℝ×E⟶E is ℛ⊗ℬ,ℬ measurable by definition of S and therefore φ=SoW,I is ℛ⊗ℱ⊗ℬ,ℬ measurable. Similarly, t,x⟶φt,ω,x is continuous for P for almost every ω∈Ω as composition of continuous applications.
Moreover, using equation (13) and the first part of equation (3), we obtain the following equation:(14)φ0,ω,x=SW0,ω,x=S0,x=x,ω∈Ω,x∈E.
SinceW0,ω=ω0=0 by the well definition of Ω.
It remains to prove thatφ satisfies the cocycle equation (7). Let s,t∈ℝ,x∈E,ω∈Ω. By applying equation (13) to ω¯:=θt,ω and x¯:=φt,ω,x=SWt,ω,x, we obtain the following equation:(15)φs,θt,ω,φt,ω,x=SWs,θt,ω,SWt,ω,x.
By using equation (12), equation (15) becomes the following equation:(16)φs,θt,ω,φt,ω,x=SWs+t,ω−Wt,ω,SWt,ω,x.
Finally, by applying the second part of equation (3) in equation (16), we get the following equation:(17)φs,θt,ω,φt,ω,x=SWs+t,ω−Wt,ω+Wt,ω,x=SWs+t,ω,x=φs+t,ω,x,
in view of Formula (13).Remark 2.
(1)
Equation (12) is called helix equation. We refer to [22] for a detailed study of equation (12).(2)
The particular RDS obtained in Theorem 1 is constructed by real white noise time change of a deterministic DS. It seems to be worthwhile to investigate this RDS according to the general theory of RDS as presented in the monograph [4].(3)
In the next paragraph, we deepen the study of this RDS in the particular case whenS is generated by an ordinary differential equation.


## 4. An Application to Ordinary Differential Equations

For the following notions, we refer essentially to Chapters 3, 4, 5, and 7 of [21] (cf. also [17]).The Wiener DSΩ,ℱ,P,θ is the appropriate sample space in order to interpret stochastic differential equations (SDE) as RDS. Indeed, the associated white noise W allows to define Itô stochastic integral ∫0tfs,ωdWsω by putting Wt:=Wt,. (cf. [21] Chapter 3).LetE be an open subset of ℝd endowed with its Borel σ-algebra B. A SDE on E is of the form(18)dXt=gXtdt+fXtdWt;X0=x;t∈ℝ,x∈E,where f,g:E⟶E are locally-Lipschitz functions.According to [3] Chapter 6, equation (13) admits a unique solution φ:ℝ×Ω×E⟶E and φ,θ is a continuous RDS on E.The following useful result is a particular case of the so-called Itô Formula.Lemma 1.
Letℝ⟶ℝ;r↦ur be a C2-function and let Yt:=uWt, then(19)dYt=12d2udr2Wtdt+dudrWtdWt.Proof.
We refer to [21], Theorem 4.1.2 (where W is denoted by B): by taking Xt=Wt and gt,r=ur, formula (4.1.7) of [21] becomes the following equation:(20)dYt=0+dudrWtdWt+12d2udr2WtdWt2=dudrWtdWt+12d2udr2Wtdt,
sincedWt2=dt by formula (4.1.7) of [21].
Next, letE be an open subset of ℝd,d≥1. For a given C1-function f:E⟶E, we consider the associated autonomous first order ordinary differential equation (ODE)(21)dyt=fytdt;y0=x∈E.
The autonomous ODE (21) is said to be generated by f.
We noticed that ford≥2, the considered ODE is in fact a system. Indeed, if f=f1,…,fd,y=y1,…,yd, and x=x1,…,xd, then equation (21) is equivalent to(22)dykt=fky1t,…,ydtdt,yk0=xk,1≤k≤d.
Sincef is a C1-function, then equation (21) admits a unique solution for each initial value x∈E. The proof of this classical result can be found in chapter 2 of [23]. We suppose that the unique solution of equation (21) is global (cf. [23], chapter 3 for more details). This means that, equation (21) admits a unique solution S.,x:ℝ⟶E for each x∈E. In fact, we have defined a system S:ℝ×E⟶E,t,x↦St,x satisfying the following equation:(23)∂∂tSt,x=fSt,xS0,x=x∈E.
It is well known thatE,Ε,S is a continuous DS on E (cf. [15, 16, 23]). It is said to be generated by ODE (21).
By applying Theorem1, we come to the second result of this paper.Theorem 2.
LetW be the real white noise and let S=S1,…,Sd:ℝ×E⟶E be the solution of the system of ODE’s generated by a C1-function f=:f1,…,fd, that is,(24)∂∂tSkt,x=fkSt,x;Sk0,x=xk,1≤k≤d,where x=x1,…,xd∈E. Then, the RDS φ defined by equation (13) is the solution X=:X1,…,Xd of the system of SDE’s(25)dXkt=12<∇fk,f>Xtdt+fkXtdWt;Xk0=xk,1≤k≤d.Proof.
Let1≤k≤d. We apply Lemma 1 to the function ukr=Skr,x for 1≤k≤d,x∈E. First,(26)dukdrr=∂∂rSkr,x=fkSr,x.
By derivation of equation (26), we get the following equation by using equations (24) and (26).(27)d2ukdr2r=∂∂rfkSr,x=∂∂rfkS1r,x,…,Sdr,x=∑i=1i=d∂∂xifkSr,x∂∂rSir,x=∑i=1i=d∂∂xifkSr,xfiSr,x.
Hence,(28)d2ukdr2r=<∇fk,f>Sr,x.
LetXktωx:=SkWt,x. Notice first that from equation (24)(29)Xk0=SkW0,x=Sk0,x=xk.
Moreover, we have the following equation:(30)Xt≔=X1t,…,Xdt=SWt.
By using equations (20)–(22), the Itô formula (19) gives the following equation:(31)dXkt=12∇fk,fSWtdt+fkSWtdWt=12∇fk,fXtdt+fkXtdWt.
This completes the proof.Remark 3.
In Theorem2, we have associated a SDE to a given ODE by the random time change W. Our approach is completely different from the classical idea, mentioned in the introduction, which consists of adding a stochastic term (say gXtdWt) to an ODE (of the form dXt=fXtdt) in order to obtain the SDE dXt=fXtdt+gXtdWt.Example 1.
Recall thatΩ,ℱ,P,θ is the Wiener DS and W is the associated white noise. For simplicity, we suppose that E:=ℝd.
For the one-dimensional case, Theorem2 reads as follows: Let S:ℝ×ℝ⟶ℝ be the DS solution of the ODE generated by a C1-function f:ℝ⟶ℝ. Then, t,ω,x↦φt,ω,x:=SWt,ω,x is the solution X of the SDE(32)dXt=12f′fXtdt+fXtdWt;X0=x∈ℝ.
For example, iffx=x1−x,x∈ℝ, then St,x=xx+1−xe−t−1;t,x∈ℝ. Hence, the SDE(33)dXt=12Xt1−3Xt+2X2tdt+Xt1−XtdWt;X0=x∈ℝ.
Admits a unique solution given byφt,ω,x=xx+1−xe−Wt,ω−1.Example 2.
For the two-dimensional case, consider the system of ODE’s(34)dy1=f1y1,y2dtdy2=f2y1,y2dty10,y20=x1,x2=x∈ℝ2,where f:=f1,f2:ℝ2⟶ℝ2 is a C1-function and let S:ℝ×ℝ2⟶ℝ2 the DS solution of this system. Then, φ=φ1,φ2:ℝ×Ω×ℝ2⟶ℝ2 defined by φt,ω,x:=SWtω,x is the solution X:=X1,X2 of the system of SDE’s(35)dX1t=12∂f1∂x1f1+∂f1∂x2f2Xtdt+f1XtdWtdX2t=12∂f2∂x1f1+∂f2∂x2f2Xtdt+f2XtdWt.
With the initial conditionX10,X20=x1,x2=x∈ℝ2.Example 3.
We illustrate the three-dimensional case by considering an example onℝ3. We consider the system of ODEs(36)dy1=y2y3dtdy2=y1+y3dtdy3=y12dt.
LetS:ℝ×ℝ3⟶ℝ3 be the DS solution of this system. Then, φ=φ1,φ2,φ3:ℝ×Ω×ℝ3⟶ℝ3 defined by φt,ω,x:=SWtω,x is the solution X:=X1,X2,X3 of the system of SDE’s(37)dX1t=12X1tX3t+X32t+X2tX12tdt+X2tX3tdWtdX2t=12X2tX3t+X12tdt+X1t+X3tdWtdX3t=X1tX2tX3tdt+X12tdWt.
With the initial conditionX10,X20,X30=x1,x2,x3=x∈ℝ3.Remark 4.
Theorem2 can be read as follows: we consider the SDE(38)dXt=gXtdt+fXtdWt;X0=x;t∈ℝ,x∈E,where f,g:E⟶E are two C1-functions. We consider the ODE(39)dyt=fytdt;y0=x∈E.
And letSt,x;t∈ℝ,x∈E be the solution of the ODE (39). Then, φt,ω,x:=SWt,ω,x;t∈ℝ,ω∈Ω,x∈E is the solution of the SDE (38) if and only if(40)gk=<∇fk,f>;1≤k≤d,where f=:f1,…,fd and g=:g1,…,gd.
Hence, we have reduced the resolution of a class of stochastic differential equations to the resolution of the associated ordinary differential equations.



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