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

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.

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 is solution of a differential equation generated by a vector field . In general, a random perturbation of is made either by perturbation of by a real noise or by adding a white noise term to (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 , one can consider the random process , where is a deterministic dynamical system. Following [14], the aim is to study the properties of depending on those of the initial dynamical system .

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.

Let equipped with the compact open topology, let be the real white noise defined by , and let defined by . Then, generates a metric dynamical system on (Definitions 1and 2).

Let be a locally compact space endowed with its Borel -algebra and let such that is a continuous dynamical system on . Let defined by the time change of , that is, . We prove first (Theorem 1) that is a continuous random dynamical system (Definition 2) on .

Next, we suppose that is an open subset of and is the solution of the system of ordinary differential equations generated by a -function , that is,

We prove (Theorem 2) that the associated random dynamical system defined in Theorem 1 is the solution of the system of stochastic differential equations.

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, denotes the real line endowed with its Borel -algebra . Moreover, denotes the usual inner product in and if is a -function.

Definition 1. A dynamical system (DS) is a triplet where is a measurable space, such thatIf is measurable, then the DS is said to be measurable.
If is a topological space endowed with its Borel -algebra and if is continuous, then the DS is said to be continuous.
In this paper, we consider essentially two different types of DS:(1)The infinite dimensional case: , where is an infinite dimensional space. A first example is equipped with the compact open topology and the associated Borel -algebra. We define the translation shift by the following equation: Then, is a measurable DS, called DS of translations on . Under additional assumptions, we may define a probability on such that the DS becomes a metric DS called usually real noise. Another important example of infinite dimensional DS, is the Wiener DS on . It will be considered in the second paragraph.(2)The finite dimensional case: where is a locally compact space endowed with its Borel -algebra and is continuous. In this case, the DS is said to be deterministic. The global solutions of ordinary differential equations on an open subset of are the most important examples and they will be treated in the third paragraph.

Definition 2. Let be a locally compact space endowed with its Borel -algebra .
A measurable random dynamical system (RDS) defined on consists of two ingredients:(1)A metric DS, i.e., a measurable DS endowed with a probability measure which is -invariant, that is,(2)A cocycle over , i.e., a mapping which is measurable and satisfying And the cocycle equation:Such a RDS is denoted by or simply by if there is no confusion.
      is said to be continuous, if is continuous for -almost every .
Let be a measurable RDS on . We may associate the skew product defined by the following equation:Then, is a measurable DS on the product space endowed with the tensor product -algebra .
A first standard class of RDS are solutions of random differential equations (RDE): Let be a metric DS (for example the translation shift defined by (4), let be an open subset of , and be measurable such that, for each , is continuous and is locally-Lipschitz. Then, the random differential equation,admits a unique solution and is a continuous RDS on .
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]).

Let equipped with the compact open topology, let be the associated Borel -algebra, and let , which is defined by the following equation:

There exists, by a classical result (Kolmogorov extension theorem), a unique probability measure on such that the process has stationary and independent increments and has normal distribution with mean 0 and variance .

Let , which is defined by the following equation:

Then, is a metric DS, called the Wiener or Brownian DS. Moreover is called Wiener shift on and , 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 process as random time change. Combining equations (7) and (8), we get the following equation:According to [20], is in fact an extension to of the white noise process on . Therefore, by equations (7) and (9), 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. Let be a locally compact space endowed with its Borel -algebra and let such that is a continuous DS on . Let be the metric Wiener DS and let the associated real white noise. We define by the following equation:Then, is a continuous RDS on .

Proof. defined by equation (13) is measurable as composition of measurable applications. Indeed, is measurable by definition of . Let , then is trivially measurable. Also, is measurable by definition of and therefore is measurable. Similarly, is continuous for for almost every as composition of continuous applications.
Moreover, using equation (13) and the first part of equation (3), we obtain the following equation:Since by the well definition of .
It remains to prove that satisfies the cocycle equation (7). Let . By applying equation (13) to and , we obtain the following equation:By using equation (12), equation (15) becomes the following equation:Finally, by applying the second part of equation (3) in equation (16), we get the following equation: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 when 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 DS is the appropriate sample space in order to interpret stochastic differential equations (SDE) as RDS. Indeed, the associated white noise allows to define Itô stochastic integral by putting (cf. [21] Chapter 3).

Let be an open subset of endowed with its Borel -algebra . A SDE on is of the formwhere are locally-Lipschitz functions.

According to [3] Chapter 6, equation (13) admits a unique solution and is a continuous RDS on .

The following useful result is a particular case of the so-called Itô Formula.

Lemma 1. Let be a -function and let , then

Proof. We refer to [21], Theorem 4.1.2 (where is denoted by ): by taking and , formula (4.1.7) of [21] becomes the following equation:since by formula (4.1.7) of [21].
Next, let be an open subset of . For a given -function , we consider the associated autonomous first order ordinary differential equation (ODE)The autonomous ODE (21) is said to be generated by .
We noticed that for , the considered ODE is in fact a system. Indeed, if , and , then equation (21) is equivalent toSince is a -function, then equation (21) admits a unique solution for each initial value . 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 for each . In fact, we have defined a system satisfying the following equation:It is well known that is a continuous DS on (cf. [15, 16, 23]). It is said to be generated by ODE (21).
By applying Theorem 1, we come to the second result of this paper.

Theorem 2. Let be the real white noise and let be the solution of the system of ODE’s generated by a -function , that is,where . Then, the RDS defined by equation (13) is the solution of the system of SDE’s

Proof. Let . We apply Lemma 1 to the function for . First,By derivation of equation (26), we get the following equation by using equations (24) and (26).Hence,Let . Notice first that from equation (24)Moreover, we have the following equation:By using equations (20)–(22), the Itô formula (19) gives the following equation:This completes the proof.

Remark 3. In Theorem 2, we have associated a SDE to a given ODE by the random time change . Our approach is completely different from the classical idea, mentioned in the introduction, which consists of adding a stochastic term (say ) to an ODE (of the form ) in order to obtain the SDE .

Example 1. Recall that is the Wiener DS and is the associated white noise. For simplicity, we suppose that .
For the one-dimensional case, Theorem 2 reads as follows: Let be the DS solution of the ODE generated by a -function . Then, is the solution of the SDEFor example, if , then . Hence, the SDEAdmits a unique solution given by .

Example 2. For the two-dimensional case, consider the system of ODE’swhere is a -function and let the DS solution of this system. Then, defined by is the solution of the system of SDE’sWith the initial condition .

Example 3. We illustrate the three-dimensional case by considering an example on . We consider the system of ODEsLet be the DS solution of this system. Then, defined by is the solution of the system of SDE’sWith the initial condition .

Remark 4. Theorem 2 can be read as follows: we consider the SDEwhere are two -functions. We consider the ODEAnd let be the solution of the ODE (39). Then, is the solution of the SDE (38) if and only ifwhere and .
Hence, we have reduced the resolution of a class of stochastic differential equations to the resolution of the associated ordinary differential equations.

Data Availability

No data were used to support the findings of this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

We want to thank the anonymous referees for their valuable comments which improved the quality of the paper.