Lyapunov Stability, Parallelizablity, and Symmetry of Random Dynamical Systems

Abstract

This article aims to study parallelizable random dynamical systems by examining them through the terms of dissipation and stochastic Lyapunov functions. It is demonstrated that any random variable that is not a random fixed point admits a tube, and every non-wandering point is within one. The Lyapunov function is employed to characterize the asymptotic stability of compact and closed random sets. The section of a random dynamical system is used to define the parallelizable random dynamical system, and it is proven that a random dynamical system is parallelizable if and only if it admits a section. Furthermore, the principle of Lyapunov used this characterization to study the parallelizability of random dynamical systems. The concept of symmetry is defined, and then its impact on the behavior of stochastic dynamic systems, particularly the Lorenz system, is discussed. In addition, by using an appropriate stochastic Lyapunov function, we have shown that the random Lorenz system is parallelizable.

1. Introduction

The parallelizability in dynamical systems refers to the ability to solve a system of equations in parallel by dividing the problem into smaller and independent tasks that can be treated concurrently. Therefore, the concept of parallelism plays a fundamental role in various fields, such as complex systems analysis, numerical simulation, and high-performance computing. Many dynamical systems, particularly complicated ones, need extensive computations and enormous datasets. For instance, it can take a lot of processing resources to solve nonlinear differential equations in a high-dimensional phase space. By dividing the problem among several processors or machines, parallelizing the calculations makes it possible to model and solve these systems more effectively.

At last, parallelizability improves the analysis and use of dynamical systems, especially for challenging and computationally intensive issues. Large-scale dataset management, real-time solutions, and faster simulations are made possible by effectively dividing the task.

This article focuses on studying the parallelism in random dynamical systems and analyzing its effects on the system behavior.

The parallelism is an important concepts in the study of dynamical systems due to its close connection with the concepts of dispersibility and stability. This concept has been studied extensively for deterministic dynamical systems by many researchers. J. Dugundji and H. Antosiewicz ([1], Theorem 3) proved that, on a locally compact separable metric space, a deterministic dynamical system is parallelizable if and only if it is dispersive.

In ref. [2], T. Jenkins and W. Johnson provided the necessary and sufficient condition for parallelizability in a locally compact separable space.

In ref. [3,4], T. Ura defined numerous kinds of isomorphisms for local dynamical systems, addressing the concept of parallelizability in the sense mentioned in ref. [1] and using the condition “the final limit set of the prolongational limit set is an empty set” to show that the parallelizability corresponding to type-n isomorphisms (n = 1, 2) are equivalent.

In ref. [5], O. H’ajek generalized the Dugundji–Antosiewicz theorem by removing the separation condition and interchanging the local compactness condition with the Lindelöfness condition.

In ref. [6], J. Egawa completed the discussion of the results presented in refs. [1,2,3], and [5]. He studied the global parallelizability of local dynamical systems.

In ref. [7], T. Ura and J. Egawa re-examined the parallelizability of local dynamical systems from the definite point of view and categorized parallel drifts on homeomorphic phase spaces by isomorphism.

In refs. [8,9,10], B. Garay proved that dispersive dynamical systems may not be parallelizable. Also, he investigated the connection between parallelizability and asymptotic stability. Moreover, he constructed a parallelizable dynamical system with uniformly bounded trajectories.

In refs. [11,12], Z. Le’sniak studied the parallelizability of flows of free mappings; he proved that the boundary of every equivalence class of a certain equivalence relation is a union of orbits, and that at most two of the boundary orbits of a class can be contained in this class.

In ref. [13], J. Souza, T. Pacífico, and H. Tozatti presented a note on the parallelizability of local dynamical systems. They provided an improvement to the classical theory presented in ref. [5] by eliminating all topological assumptions.

Here, we mention some works involving different applications for stochastic Lyapunov function. This principle is used to describe many dynamic properties.

In ref. [14], X. Mao reviews the uses of Lyapunov’s second method in the study of numerous properties of stochastic differential equations. His article emphasizes the original thoughts and methods industrialized lately.

In ref. [15], L. Arnold and B. Schmalfuss used the stochastic Lyapunov function to study the stability of random sets in random dynamics.

In ref. [16] (see also the references therein), the stochastic Lyapunov function was used to study the attractor–repeller pair decomposition and Morse decomposition for compact metric space in the random setting.

In ref. [17], A. Hmissi, F. Hmissi, and M. Hmissi used the stochastic Lyapunov function to describe the gradient RDSs.

In ref. [18], F. Visentin introduced the conditions that were sufficient to apply Lyapunov theory for differential equations to stochastic stability in probability.

In ref. [19], X. Ju, Ailing Qi, and Jintao Wang studied the Strong Morse–Lyapunov functions for Morse decompositions of attractors of random dynamical systems.

In ref. [20], E. Üçer studied the Lyapunov function-based criteria for ship rolling in random beam seas.

In ref. [21], I.J. Kadhim and M. Imran used the stochastic Lyapunov function to study the asymptotic stability, uniformly stable, compact asymptotically stable of random sets in random dynamics.

We also refer to some references through which Lyapunov stability was studied. For example, see refs. [22,23,24].

The main aim of this article is to study the parallelizability for random dynamical systems in a new and different way from the methods used in the aforementioned sources, by using stochastic Lyapunov functions and asymptotic stability on one hand, and the principle of dispersion and prolongational limit sets on the other hand.

This work is organized into six main sections. In Section 2, some prerequisites and facts about RDSs are stated. In Section 3, the notion “section” of random dynamical systems is defined and studied. It is shown that a section of RDSs is closed random, and it cannot be written as a union of disjoint random closed sets if the phase space is connected and vice versa. Also, we prove that any random variable that is not a random fixed point admits a tube. In Section 4, the stochastic Lyapunov function is used to study the stability of random sets. Two important results are proven in this section. The first one, an invariant random compact set is asymptotically stable if and only if there exists a Lyapunov function defined on this random set. The second result, an invariant random closed set is asymptotically stable if and only if there exists a Lyapunov function defined on this random set which satisfies the conditions (i) and (ii) of Theorem 9. In Section 5, parallelizable random dynamical systems are studied by using the dissipation and the stochastic Lyapunov function. Several main results are proven in this section. A random dynamical system is parallelizable if and only if it admits a section. The parallelizability and dispersivity are equivalent when the phase space is a locally compact, separable Banach space. However, in general, parallelizability is not implied by dispersivity. Finally, parallelizability is preserved by an isomorphism. In Section 6, the concept of symmetry in random dynamic systems is studied, and the applications of symmetry are also discussed.

In Section 7, the random Lorenz system is introduced as an application of the parallelizable RDS; this fact is proved by using the stochastic Lyapunov function.

2. Basic Facts About Random Dynamical Systems

In this section, some essential facts about random dynamical systems that are related to our work are given. Let the triple $(\mathcal{\Omega },\mathcal{A},\mathbb{P})$ represent the probability space where $\Omega$ is a non-empty, $\mathcal{A}$ be a $\sigma \_$ algebra on $\Omega$, $\mathbb{P}$ is the probability measure on $\mathcal{A}$, and $(X,‖·‖)$ represent to a Banach space. We will denote the open (closed) ball centered at the set $M$ with radius $\epsilon$ by the symbol $S(M,\epsilon )$ (respectively, $S [M,\epsilon ]$). The set of random variables from $\Omega$ to $X$ is denoted by $X^{\mathrm{\Omega }}$.

Definition 1

([24,25]). The 5-tuple $(\mathbb{R},\mathcal{\Omega },\mathcal{A},\mathbb{P},\theta )$ (or just $\theta$) is said to be a metric dynamical system (MDS) if $\theta$ is measurable action of $\mathbb{R}$ on $\Omega$ and $\mathbb{P}({\theta }_{t}A)=\mathbb{P}(A)$, for every $A\in \mathcal{A}$, $t\in \mathbb{R}.$

Definition 2

([24,25]). A random dynamical system (RDS) over an MDS $\theta$ is a mapping $\phi :\mathbb{R}\times \mathrm{\Omega }\times X⟶X$, which is measurable, and the mappings $\phi \left(t,\omega \right)≔\phi \left(t,\omega ,\cdot \right):X⟶X$ form a cocycle over $\theta$., i.e., they satisfy $\phi \left(0,\omega \right)={id}_X$ for all $\omega \in \mathrm{\Omega }$, and $\phi \left(t+s,\omega \right)=\phi \left(t,{\theta }_s\omega \right)\circ \phi \left(s,\omega \right)$ for all $s,t\in \mathbb{R}$, $\omega \in \mathrm{\Omega }$. An RDS is denoted by $(\theta ,\phi ).$

Definition 3

([25,26,27]).

(a) 

The random set is a set-valued function $M:\mathrm{\Omega }⟶X$ such that the function $\Psi :\mathrm{\Omega }⟶\mathbb{R}$ defined by $\Psi \left(\omega \right)≔d(x,M\left(\omega \right))$ is measurable for every $x\in X.$

(b) 

For some  $y\in X$  and some random variable  $r:\mathrm{\Omega }⟶{\mathbb{R}}^{+}$, consider the set

$\mathcal{A}≔\left\{x:{dis}_X\left(x, y\right)\le r\left(\omega \right), for all \omega \in \Omega \right\}$. A random set $S\left(\omega \right)$ is called tempered if

$$S\left(\omega \right)\subset \mathcal{A}\mathit{for}\mathit{every}\omega \in \Omega ,$$

 and$r:\mathrm{\Omega }⟶{\mathbb{R}}^{+}$  is called tempered random variable (TRV); this means

$${\sup }_{\mathrm{t}\in \mathbb{T}}\left\{{\mathrm{e}}^{-\mathrm{\gimel }\left|\mathrm{t}\right|}\left|\mathrm{r}({\mathrm{\theta }}_{\mathrm{t}}\mathrm{\omega })\right|\right\}<\mathrm{\infty }, for every \mathrm{\gimel }>0 and \mathrm{\omega }\in \mathrm{\Omega }.$$

Definition 4

([24,28,29]). The random set ${\gamma }_M^{{\mathbb{R}}^{+}}\left(\omega \right)≔\bigcup _{\tau \in {\mathbb{R}}^{+}}\phi (\tau ,{\theta }_{-\tau }\omega )M({\theta }_{-\tau }\omega )$ is called the ${\mathbb{R}}^{+}\_$trajectory of a random set $M:\mathrm{\Omega }⟼\mathcal{B}\left(X\right)$.

Definition 5

([24,25]). A random fixed point of $(\theta ,\phi )$ is the measurable function $x:\mathrm{\Omega }⟶X$ such that, for all $t\in \mathbb{R}$, we have $\mathbb{P}\left\{\omega :\phi \left(t,\omega \right)x\left(\omega \right)=x\left({\theta }_t\omega \right)\right\}=1$. The set of all random fixed points will be denoted by ${Fix}_{\phi }(X)$.

Definition 6

([28]). The random set $M(\omega )$ is called ${\mathbb{R}}^{+}\_$ invariant if

$$\mathbb{P}\left\{\omega \in \tilde{\mathrm{\Omega }}:\phi \left(\tau ,\omega \right)M\left(\omega \right)\subseteq M\left({\theta }_{\tau }\omega \right), for every \tau \in {\mathbb{R}}^{+}\right\}=1.$$

Definition 7

([27,28,29,30]). The ${\mathbb{R}}^{+}\_$ omega limit set of a random set $M(\omega )$ is defined as follows:

$${\Gamma }_M^{{\mathbb{R}}^{+}}\left(\omega \right)=\left\{y\in X:\exists \mathrm{n}\mathrm{e}\mathrm{t} \left\{t_n\right\}\in {\mathbb{R}}^{+},t_n⟶\infty , \left\{x_n\right\}\in M\left({\theta }_{-t_n}\omega \right)with \phi (t_n,{\theta }_{-t_n}\omega )x_n⟶y\right\}.$$

For more details about the Omega limit set (in random case), refer to refs. [27,31].

The definition that follows is in line with what was stated in ref. [20]. Here, the phase space will be regarded as a Banach space.

Definition 8

([32]). Let$M(\omega )$ be a random set in RDS $(\theta ,\phi )$.

(a) 

The first ${\mathbb{R}}^{+}\_$ prolongational limit set of $M(\omega )$ is the random set $J_M^{{\mathbb{R}}^{+}}\left(\omega \right)$ with the property that $y\in J_M^{{\mathbb{R}}^{+}}\left(\omega \right)$ if and only if there exist sequences $\left\{x_n\right\}$ in $X$ and $\left\{t_n\right\}$ in ${\mathbb{R}}^{+}$ with $t_n⟶+\mathrm{\infty }$ such that

$$\underset{n⟶+\infty }{\lim }{\inf }_{z\in M{(\theta }_{-t_n}\omega )}‖x_n-z‖=0 \mathit{and} \underset{n⟶+\infty }{\lim }‖\phi \left(t_n,{\theta }_{-t_n}\omega \right)x_n-y‖=0$$

(b) 

The first ${\mathbb{R}}^{+}\_$prolongation of a random set $M(\omega )$ is the random set $D_M^{{\mathbb{R}}^{+}}\left(\omega \right)$ with the property that $y\in D_M^{{\mathbb{R}}^{+}}\left(\omega \right)$ if and only if there exist sequences $\left\{x_n\right\}$ in $X$ and $\left\{t_n\right\}$ in ${\mathbb{R}}^{+}$ such that

$$\underset{n⟶+\infty }{\lim }{\inf }_{z\in M{(\theta }_{-t_n}\omega )}‖x_n-z‖=0 and \underset{n⟶+\infty }{\lim }‖\phi \left(t_n,{\theta }_{-t_n}\omega \right)x_n-y‖=0$$

Definition 9

([32]). A random variable $x:\mathrm{\Omega }⟶X$ is said to be ${\mathbb{R}}^{+}\_$ non-wandering if $x\in J_x^{{\mathbb{R}}^{+}}\left(\omega \right)$.

Definition 10

([32]). Consider an RDS $(\theta ,\phi )$. A random variable $x:\mathrm{\Omega }⟶X$ is said to be ${\mathbb{R}}^{+}\_$ wandering if $\phi \left(t,{\theta }_{-t}\omega \right)B\left({\theta }_{-t}\omega \right)\cap B\left(\omega \right)=\varnothing$, for $t\in {\mathbb{R}}^{+}$ for some random ball $B$ centered at $x$.

Definition 11

([32]). Consider the RDS $(\theta ,\phi )$. If for every random variables $x,y:\mathrm{\Omega }⟶X$ there exist random balls $U,V$ centered at $x,y$, respectively, such that $\phi \left(t,{\theta }_{-t}\omega \right)U\left({\theta }_{-t}\omega \right)\cap V\left(\omega \right)=\varnothing$, for $t\in {\mathbb{R}}^{+}$, then $(\theta ,\phi )$ will said to be ${\mathbb{R}}^{+}\_$ dispersive.

Theorem 1

([32]). The RDS $(\theta ,\phi )$ is ${\mathbb{R}}^{+}\_$ dispersive if and only if, for each $x \in X$, $J_x^{{\mathbb{R}}^{+}}\left(\omega \right)$ is empty.

Theorem 2

([32]). The RDS $(\theta ,\phi )$ is ${\mathbb{R}}^{+}\_$ dispersive if and only if, for each $x\in X$,$D_x^{{\mathbb{R}}^{+}}\left(\omega \right)={\gamma }_x^{{\mathbb{R}}^{+}}\left(\omega \right)$and there are no random periodic trajectories or random fixed points.

Theorem 3

([32]). Let $M(\omega )$ be a closed random set in the RDS $(\theta ,\phi )$. If $M(\omega )$ is invariant and asymptotically stable, then

(a) 

$J_x^{{\mathbb{R}}^{+}}(\omega )\subset M(\omega )$ for each $x\in A_M\left(\omega \right),$

(b) 

$J_x^{{\mathbb{R}}^{+}}\left(\omega \right)\cap A_M(\omega )=\varnothing$ for each $x\in A_M(\omega )-M(\omega )$.

3. The Section and the Tube for Random Dynamical Systems

For the study of parallelizable dynamical systems, the two concepts, section and tube, are essential resources. Consequently, we study them for RDSs in this section. The wandering points are used to prove the sufficient condition for the existence of the section. Additionally, we demonstrate that any random variable does not represent a random fixed point for RDSs which is called a tube.

Definition 12.

A section of the RDS $(\theta ,\phi )$ is a random set $\mathbb{S}(\mathit{\omega })$ with the property that, for every $\mathit{x}\in \mathit{X}$, there exists a mapping $\tau :\mathrm{\Omega }\times X⟶\mathbb{R}$, such that

$$\mathbb{P}\left\{\omega :x\in \phi (-\tau \left(\omega ,x\right),{\theta }_{\tau \left(\omega ,x\right)}\omega )\mathbb{S}({\theta }_{{\tau }_{\omega }\left(x\right)}\omega \right\}=1.$$

Lemma 1.

If $\mathbb{S}$ is a section of $(\theta ,\phi )$ such that $\tau :\mathrm{\Omega }\times X⟶\mathbb{R}$ is continuous for every $\omega \in \mathrm{\Omega }$, then

(i)  

$\mathbb{S}(\omega )$ is a closed random set in $X,$

(ii) 

$\mathbb{S}(\omega )$ cannot be written as a union of disjoint random closed sets if and only if $X$ is connected.

Proof of (i).

Fix $\omega \in \mathrm{\Omega }$. If $\left\{x_n\right\}$ is a sequence in $\mathbb{S}(\omega )$ and $x_n ⟶x\in X$. Since ${\tau }_{\omega }:X⟶\mathbb{R}$ is continuous for every $\omega \in \mathrm{\Omega }$, then $\underset{n⟶\infty }{\lim }\left|{\tau }_{\omega }\left(x_n \right)-{\tau }_{\omega }(x)\right|=0$ for every $\omega \in \mathrm{\Omega }$. Since $\phi \left(t,\omega \right):X⟶X$ is continuous for every $t\in \mathbb{R}$ and every $\omega \in \mathrm{\Omega },$ then $\underset{n⟶\infty }{\lim }‖\phi \left(t,\omega \right)x_n-\phi \left(t,\omega \right)x‖=0$ for every $t\in \mathbb{R}$ and every $\omega \in \mathrm{\Omega }$. In particular,$\underset{n⟶\infty }{\lim }‖\phi \left({\tau }_{\omega }(x_n),\omega \right)x_n ⟶\phi \left({\tau }_{\omega }(x),\omega \right)x‖=0$ for every $\omega \in \mathrm{\Omega }$. Since ${\tau }_{\omega }(x_n)=0$ for each $n$, we get ${\tau }_{\omega }(x)=0$. Thus, by the definition of $\tau$ and $\mathbb{S}$, we have $\phi \left({\tau }_{\omega }(x),\omega \right)x=\phi \left(0,\omega \right)x=x\in \mathbb{S}({\theta }_{{\tau }_{\omega }\left(x\right)}\omega )$. So, $S$ is a closed random set in $X$. □

Proof of (ii).

Suppose that $X$ is connected. Assume the contrary, that $\mathbb{S}(\omega )$ is disconnected. So, $\mathbb{S}(\omega )$ can be written as a union of two random closed sets, say, ${\mathbb{S}}_1(\omega ),{\mathbb{S}}_2(\omega )$, such that ${\mathbb{S}}_1(\omega )\cap {\mathbb{S}}_2(\omega )=\varnothing$. As $\mathbb{P}\left\{\omega :{\gamma }_{\mathbb{S}}^{\mathbb{R}}\left(\omega \right)=X\right\}=1$, then $\mathbb{P}\left\{\omega :{\gamma }_{{\mathbb{S}}_1\cup {\mathbb{S}}_2}^{\mathbb{R}}\left(\omega \right)=X\right\}=1$. Note that ${\gamma }_{{\mathbb{S}}_1}^{\mathbb{R}}\left(\omega \right)\cap {\gamma }_{{\mathbb{S}}_2}^{\mathbb{R}}\left(\omega \right)=\varnothing$. To show that ${\gamma }_{{\mathbb{S}}_1}^{\mathbb{R}}(\omega )$ and ${\gamma }_{{\mathbb{S}}_2}^{\mathbb{R}}\left(\omega \right)$ are closed. If $x\in \overline{{\gamma }_{{\mathbb{S}}_1}^{\mathbb{R}}\left(\omega \right)}$, then there exists a net $\left\{x_n\right\}$ in ${\gamma }_{{\mathbb{S}}_1}^{\mathbb{R}}\left(\omega \right)$, such that $x_n⟶x$. Since $\tau$ is continuous, then $\underset{n⟶\infty }{\lim }\left|{\tau }_{\omega }\left(x_n\right)-{\tau }_{\omega }(x)\right|=0$. Since $\phi \left(t,{\theta }_{-t}\omega \right):X⟶X$ and $\phi \left(\omega ,x\right):\mathbb{R}⟶X$ are continuous, then we have $\underset{n⟶\infty }{\lim }‖\phi \left({\tau }_{\omega }(x_n),\omega \right)x_n-\phi \left({\tau }_{\omega }(x),\omega \right)x‖=$. Since $\left\{\phi \left({\tau }_{\omega }(x_n),\omega \right)x_n\right\}$ is a net in ${\mathbb{S}}_1$ and ${\mathbb{S}}_1$ is closed, then $\phi \left({\tau }_{\omega }(x),\omega \right)x\in {\mathbb{S}}_1(\theta \left({\tau }_{\omega }\left(x\right)\omega \right))$. Then $x\in \phi \left(-{\tau }_{\omega }(x),{\theta }_{{\tau }_{\omega }(x)}\omega \right){\mathbb{S}}_1({\theta }_{{\tau }_{\omega }\left(x\right)}\omega )\subset {\gamma }_{{\mathbb{S}}_1}^{\mathbb{R}}\left(\omega \right)$.

Therefore, ${\gamma }_{{\mathbb{S}}_1}^{\mathbb{R}}\left(\omega \right)$ is closed. In a similar way we can show that ${\gamma }_{{\mathbb{S}}_2}^{\mathbb{R}}\left(\omega \right)$. Thus $X$ is not connected, and this implies a contradiction. The converse can be proven in a similar way. □

Definition 13.

A random open set $U(\omega )$ in $X$ will be called a tube if there exists an open neighborhood $\aleph$ of $e$ and a random subset $\mathbb{S}(\omega )\subset U(\omega )$, such that

(i) 

${\gamma }_{\mathbb{S}}^{\aleph }\left(\omega \right)≔\left\{\phi \left(t,{\theta }_{-t}\omega \right)\mathbb{S}({\theta }_{-t}\omega ):t\in \aleph \right\}\subset U(\omega )$, and

(ii) 

there is a function ${\tau }_{\omega }:U(\omega )⟶\mathbb{R}$, with ${\tau }_{\omega }(x)\in \aleph$, with $\phi \left({\tau }_{\omega }(x),{\theta }_{-{\tau }_{\omega }(x)}\omega \right)x\in \mathbb{S}({\theta }_{-{\tau }_{\omega }(x)}\omega )$.

Remark 1.

Since $p\in (\omega )$ implies $p\in {\gamma }_{\mathbb{S}}^{\aleph }\left(\omega \right)$, then $U(\omega )={\gamma }_{\mathbb{S}}^{\aleph }\left(\omega \right)$. Therefore $U(\omega )$ is an $\aleph$-tube with section $\mathbb{S}(\omega )$ and write $U\left(\omega \right)≔T_{\aleph }(\mathbb{S}(\omega ))$, and $\mathbb{S}(\omega )$ called an $(\aleph -U)$-section of the tube $U(\omega )$.

Lemma 2.

Let $U(\omega )$ be a $\aleph$-tube with section $\mathbb{S}(\omega )$. If $K(\omega )\subset \mathbb{S}(\omega )$ is a compact random set, then for every $\omega \in \mathrm{\Omega }$, the function ${\tau }_{\omega }$ is continuous on $\left\{\phi \left(t,{\theta }_{-t}\omega \right)K\left({\theta }_{-t}\omega \right):t\in int C\right\}$ for any compact subset $C$ of $\aleph$.

Proof. 

If $\left\{x_n\right\}$ is a sequence in $\left\{\phi \left(t,{\theta }_{-t}\omega \right)K\left({\theta }_{-t}\omega \right):t\in \mathrm{i}\mathrm{n}\mathrm{t}C\right\}$ and $x_n⟶x\in \left\{\phi \left(t,{\theta }_{-t}\omega \right)K\left({\theta }_{-t}\omega \right):t\in \mathrm{i}\mathrm{n}\mathrm{t}C\right\}$, then $\underset{n⟶\mathrm{\infty }}{\lim }\left|{\tau }_{\omega }\left(x_n\right)-{\tau }_{\omega }(x)\right|=0$. Note that, for every $\omega \in \mathrm{\Omega }$ sequence, $\left\{\phi \left(-\tau (x_n),\omega \right)x_n\right\}$ is in $K({\theta }_{-\tau }\omega )$. Since $K({\theta }_{-{\tau }_{\omega }}\omega )$ is compact, then this sequence can be assumed to be convergent. Moreover, if $\{{\tau }_{\omega }(x_n)\}$ is in $\mathrm{i}\mathrm{n}\mathrm{t}C$, then it is bounded; hence, $\{{\tau }_{\omega }(x_n)\}$ is convergent. Thus, suppose that $\phi \left({\tau }_{\omega }(x_n),\omega \right)x_n⟶x^{\mathrm{*}}\in K$, and ${\tau }_{\omega }(x_n)⟶{\tau }^{\mathrm{*}}\in \overline{\mathrm{i}\mathrm{n}\mathrm{t}C}$. Since $x_n⟶x$, thus $({\tau }_{\omega }\left(x_n\right),x_n)⟶({\tau }^{*},x)$. Since $\phi \left(·,\omega ,·\right):\mathbb{R}\times X⟶X$ is continous for every $\omega$, then $\phi ({\tau }_{\omega }\left(x_n\right),\omega ,x_n)⟶\phi ({\tau }^{*},\omega ,x)$. Therefore, $\phi \left({\tau }^{*},\omega ,x\right)=x^{*}$. This means that $\phi \left({\tau }_{\omega }\left(x_n\right),\omega ,x_n\right)⟶\phi \left({\tau }^{*},\omega ,x\right)$. Since ${\tau }^{*}\in \mathrm{i}\mathrm{n}\mathrm{t}C\subset \aleph$, we have ${\tau }^{*}={\tau }_{\omega }(x)$ by uniqueness. □

Lemma 3.

If $X$ is locally compact and $x:\mathrm{\Omega }⟶X$ is a wandering point of an RDS $(\theta ,\phi )$, then for some random set $\mathbb{S}(\omega )$ containing $x$, we have $U(\omega )={\gamma }_{\mathbb{S}}^{\mathbb{R}}(\omega )$ is open and $\mathbb{S}(\omega )$ is $(\mathbb{R}-U)$-section with ${\tau }_{\omega ,\mathbb{S}}$ is continuous on $U(\omega )$.

Proof. 

Let $V\subset X$ and $N$ be the neighborhoods according to which $x:\mathrm{\Omega }⟶X$ is wandering, and let $T_{\aleph }(W(\omega ))$ be a tubular neighborhood of $x$ with ${\tau }_{\omega ,W}$ continuous on $T_{\aleph }(W(\omega ))$. Thus, $x\in W(\omega )$, and if $\phi \left(t,{\theta }_{-t}\omega \right)y\in W({\theta }_{-t}\omega )$ for some $y\in W(\omega )$ and $t\in \mathbb{R}$, then $t\notin \aleph$ or $t=0$. We may assume $W(\omega )\subset V(\omega )$; otherwise, we need only replace $T_{\aleph }(W(\omega ))$ by $T_{\aleph }(V(\omega )\cap W(\omega ))$, which is also open in $X$. If no $\mathbb{S}(\omega )$ exists, then for every $n\ge 1$, there is an $x_n\in B(x,1/n)\cap W(\omega )$ and a $t_n\notin \aleph$, such that $\phi \left(t_n,{\theta }_{-t_n}\omega \right)x_n\in B(x,1/n)\cap W(\omega )$. Since $x$ is wandering, for each $t_n\in N$ hereafter, we may assume that $t_n⟶t_0\in \overline{N}$. This implies that $\phi \left(t_0,{\theta }_{-t_0}\omega \right)x=x$, and so $t_0=0$ because ${\gamma }_x^{\mathbb{R}}(\omega )$ is not closed. Thus, $t_n\in \aleph$ for a sufficiently large $n$, which is absurd. Therefore, there exists a set $\mathbb{S}=B(x,\epsilon )\cap W(\omega )$ such that if $y\in \mathbb{S}(\omega )$, the ${\gamma }_y^{\mathbb{R}}(\omega )$ intersects $\mathbb{S}(\omega )$ solely at $y$. Since $\mathbb{S}(\omega )$ is open in $W(\omega )$, $T_{\aleph }(\mathbb{S}(\omega ))$ is open in $X$. Thus, $T_{\mathbb{R}}\left(W(\omega )\right)=\bigcup _{t\in \mathbb{T}}\phi \left(t,{\theta }_{-t}\omega \right)T_{\aleph }(S({\theta }_{-t}\omega ))$ is open. To show that, ${\tau }_{\omega }$ is continuous on $T_{\mathbb{R}}\left(\mathbb{S}(\omega )\right)$, let $y_0\in T_{\mathbb{R}}\left(\mathbb{S}(\omega )\right)$ and let $V^{\prime }(\omega )\subset T_{\aleph }(\mathbb{S}({\theta }_{-t}\omega ))$ be a neighborhood of $\phi \left(-{\tau }_{\omega ,S}(y_0),{\theta }_{{\tau }_{\omega ,S}(y_0)}\omega \right)y_0\in \mathbb{S}(\omega )$. Then, $V(\omega )=\phi \left({\tau }_{\omega ,S}(y_0),{\theta }_{-{\tau }_{\omega ,S}(y_0)}\omega \right)V^{\prime }(\omega )$ is a neighborhood of $y_0$, such that to every $y\in V(\omega )$ there corresponds a unique $x^{\prime }\in V^{\prime }(\omega )$ for which ${\tau }_{\omega ,s}\left(y\right)={\tau }_{\omega ,S}\left(y_0\right)+{\tau }_{\omega ,W}(x^{\prime })$. Since ${\tau }_{\omega ,W}$ is continuous on $V^{\prime }(\omega )$, continuity of ${\tau }_{\omega ,S}$ on $V(\omega )$ follows. □

Theorem 4.

Any random variable that is not a random fixed point admits a tube.

Proof. 

Suppose that $x\notin {\mathrm{F}\mathrm{i}\mathrm{x}}_{\phi }(X)$, then $‖x\left(\omega \right)-\phi (T_0,\omega )x(\omega )‖>0$, for some $T_0>0$. Consider the function $\psi \left(t,\omega \right)y(\omega )≔{\int }_t^{t+T_0}‖x\left(\omega \right)-\phi (T_0,\omega )x(\omega )‖d\tau$. It follows that

$$\psi \left(t_1+t_2,\omega \right)y(\omega )≔{\int }_{t_2}^{t_2+T_0}‖x\left(\omega \right)-\phi (T_0,\omega )x(\omega )‖d\tau =\psi \left(t_2,{\theta }_{t_1}\omega \right)\phi \left(t_1,\omega \right)y(\omega )$$

Further, the function $\psi \left(t,\omega ,y(\omega )\right)$ is continuous in $\left(t,y\right)$, with

$${\displaystyle \frac{\partial }{\partial t}}\psi \left(t,\omega ,y(\omega )\right)=‖x\left(\omega \right)-\phi (t+T_0,\omega )y(\omega )‖-‖x\left(\omega \right)-\phi (t,\omega )y(\omega )‖$$

is continuous. Since

$$\begin{array}{c}{\displaystyle \frac{\partial }{\partial t}}\psi \left(0,x\left(\omega \right)\right)=‖x\left(\omega \right)-\phi \left(T_0,\omega \right)x\left(\omega \right)‖-‖x\left(\omega \right)-\phi \left(0,\omega \right)x\left(\omega \right)‖\\ >0,\end{array}$$

there is a TRV $\epsilon$ such that ${\displaystyle \frac{\partial }{\partial t}}\psi \left(0,y(\omega )\right)>0$ for $‖y\left(\omega \right)-x(\omega )‖<\epsilon (\omega )$. Define ${\tau }_0>0$ such that

$$‖\phi (s,\omega )x(\omega )-x(\omega )‖<\epsilon (\omega ), \mathrm{whenever} \left|s\right|\le 3{\tau }_0.$$

Then, in particular, $\psi \left({\tau }_0,\omega \right)x(\omega )>\psi \left(0,\omega \right)x(\omega )>\psi \left({-\tau }_0,\omega \right)x(\omega )$. Now choose TRV $\xi$ such that

$$(S\left[\phi \left({\tau }_0,\omega \right)x\left(\omega \right),\xi \left(\omega \right)\right]\cup S\left[\phi \left(-{\tau }_0,\omega \right)x\left(\omega \right),\xi \left(\omega \right)\right])\subset S(x(\omega ),\epsilon (\omega ))$$

and such that for $‖y\left(\omega \right)-\phi \left({\tau }_0,\omega \right)x(\omega )‖<\xi (\omega )$ we have $\psi \left(0,\omega \right)y(\omega )>\psi (0,\omega )x(\omega )$, and for $‖y\left(\omega \right)-\phi \left(-{\tau }_0,\omega \right)x(\omega )‖<\xi (\omega )$, we have $\psi \left(0,\omega \right)y(\omega )<\psi (0,\omega )x(\omega )$. Finally, determine TRV $\delta$ such that if $‖\phi \left({\tau }_0,\omega \right)y-\phi \left({\tau }_0,\omega \right)x(\omega )‖<\xi (\omega )$, then $‖\phi \left(-{\tau }_0,\omega \right)y-\phi \left(-{\tau }_0,\omega \right)x(\omega )‖<\xi (\omega )$, and $‖\phi \left(s,\omega \right)x\left(\omega \right)-x(\omega )‖<\epsilon (\omega )$ whenever $\left|s\right|\le 3{\tau }_0$. We show that, if $‖y(\omega )-x(\omega )‖\le \delta (\omega )$, then there is exactly one ${\tau }_{\omega }(y(\omega ))$, $\left|{\tau }_{\omega }(y(\omega ))\right|<{\tau }_0$, such that $\psi \left({\tau }_{\omega }\left(y\left(\omega \right)\right),\omega \right)y(\omega )=\psi (0,\omega )x(\omega )$. This follows from the fact that $\psi \left(t,\omega \right)y(\omega )=\psi \left(0,\phi \left(t,\omega \right)y(\omega )\right)$ is a strictly increasing function of $t$, and $\psi \left({\tau }_0,\omega \right)y(\omega )>\psi \left(0,\omega \right)x(\omega )>\psi \left({-\tau }_0,\omega \right)y(\omega )$. Consider now the open set $U(\omega )=\left\{\phi (s,\omega )S\left(x(\omega ),\delta (\omega )\right):s\in I_{{\tau }_0}\right\}$, and set

$$\mathbb{S}(\omega )=\left\{y\left(\omega \right)\in U\left(\omega \right):\mathbb{P}\left\{\omega :\psi \left(0,\omega \right)y\left(\omega \right)=\psi \left(0,\omega \right)x\left(\omega \right)\right\}=1\right\}$$

We claim that $\mathbb{S}(\omega )$ is $(2{\tau }_0-U)$-section. For this, we need prove that if $y(\omega )\in U(\omega )$, then there is a unique $\tau (y(\omega ))$, $\left|\tau (y(\omega ))\right|<2{\tau }_0$, such that $‖\phi \left({\tau }_{\omega }\left(y\left(\omega \right)\right),\omega \right)y\left(\omega \right)-\theta ({\tau }_{\omega }\left(y(\omega )\right)‖<\delta (\omega )$. Indeed, for any $y(\omega )\in U(\omega )$, we have $‖y^{\prime }\left(\omega \right)-x(\omega )‖<\delta (\omega )$, where $y^{\prime }(\omega )=\phi (t^{\prime },\omega )y(\omega )$ for some $t^{\prime }$, $\left|t^{\prime }\right|<{\tau }_0$ and for $‖y^{\prime }-x(\omega )‖<\delta (\omega )$, we have $\phi (t^{″},\omega )y^{\prime }(\omega )\in \mathbb{S}({\theta }_{t^{″}}\omega )$, for some $t^{″}$, $\left|t^{″}\right|<{\tau }_0$. Thus, $\phi \left(t^{\prime }+t^{″},\omega \right)y(\omega )=\phi \left({\tau }_{\omega }\left(y\left(\omega \right)\right),\omega \right)y(\omega )\in \mathbb{S}(\omega )$, where ${\tau }_{\omega }\left(y\right)=t^{\prime }+t^{″}$, and $\left|{\tau }_{\omega }\left(y\right)\right|\le \left|t^{\prime }\right|+\left|t^{″}\right|<2{\tau }_0$. Now, let it be possible that there are two numbers, ${\tau }^{\prime }\left(y\right)$, ${\tau }^{″}\left(y\right)$, $\left|{\tau }^{\prime }\left(y\right)\right|<2{\tau }_0$, $\left|{\tau }^{″}\left(y\right)\right|<2{\tau }_0$, such that $\phi \left({\tau }^{\prime }\left(y\right),\omega \right)y(\omega )\in \mathbb{S}(\omega )$ and $\phi \left({\tau }^{\prime \prime }\left(y\right),\omega \right)y(\omega )\in \mathbb{S}(\omega )$, and let $‖y^{\prime }\left(\omega \right)-y(\omega )‖<\delta (\omega )$, where $y^{\prime }(\omega )=\phi \left({\tau }^{\prime }\left(y\right),\omega \right)y(\omega )$, and where $\left|t^{\prime }\right|\le {\tau }_0$. Then

$$\psi \left({\tau }^{\prime }\left(y\right)-t^{\prime },\omega \right)y^{\prime }(\omega )=\psi \left({\tau }^{\prime }\left(y\right),\omega \right)y^{\prime }(\omega )=\psi \left(0,\omega \right)\phi \left({\tau }^{\prime }\left(y\right),\omega \right)y(\omega )$$

and

$$\psi \left({\tau }^{″}\left(y\right)-t^{\prime },\omega \right)y^{\prime }(\omega )=\psi \left({\tau }^{″}\left(y\right),\omega \right)y^{\prime }(\omega )=\psi \left(0,\omega \right)\phi \left({\tau }^{″}\left(y\right),\omega \right)y(\omega )$$

so that

$$\psi \left({\tau }^{\prime }\left(y\right)-t^{\prime },\omega \right)y^{\prime }(\omega )=\psi \left({\tau }^{″}\left(y\right)-t^{\prime },\omega \right)y^{\prime }(\omega )=\psi \left(0,\omega \right)x(\omega )$$

Now, $\left|{\tau }^{\prime }\left(y\right)-t^{\prime }\right|\le 3{\tau }_0$, and $\left|{\tau }^{″}\left(y\right)-t^{\prime }\right|\le 3{\tau }_0$, and ${\displaystyle \frac{\partial }{\partial t}}\psi \left(t,\omega \right)y^{\prime }(\omega )>0$ for $\left|t\right|\le 3{\tau }_0$, i.e., $\psi \left(t,\omega \right)y^{\prime }(\omega )$ is strictly increasing for $\left|t\right|\le 3{\tau }_0$. Hence, ${\tau }^{\prime }\left(y\right)-t^{\prime }={\tau }^{″}\left(y\right)-t^{\prime }$, or ${\tau }^{\prime }\left(y\right)={\tau }^{″}\left(y\right)$. □

The following theorem can be now proven as a more general result.

Theorem 5.

Let $x\notin {Fix}_{\phi }(X)$. Let $\tau >0$ be given, restricted only by $\tau <T/4$ if the motion $\phi \left(·,\omega ,x(\omega )\right):\mathbb{R}⟶X$ is periodic with least period $T$. Then there exists a tube $U(\omega )$ containing $x$ with a $(\tau -U)$-section $\mathbb{S}(\omega )$.

Theorem 6.

If $x$ is a point in a locally compact space $X$ such that $x\notin J_x^{+}\left(\omega \right)$, then there exists a tube $U(\omega )$ containing $x$, with an $\left(\mathbb{R}-U\right)$-section $\mathbb{S}(\omega )$, and ${\tau }_{\omega }:U(\omega )⟶\mathbb{R}$ is continuous for every $\omega \in \mathrm{\Omega }$.

Proof. 

In fact, there exists a tube $W(\omega )$ of $x$ such that a $\left(\tau -W\right)$-section of a random set $\mathbb{S}(\omega )\subset \mathrm{\Omega }\times U(\omega )$, and ${\tau }_{\omega }:W(\omega )⟶\mathbb{R}$ is continuous for every $\omega \in \mathrm{\Omega }$. Since $x$ is a wandering, we claim that there is a TRV $\delta$ with $S\left(x,\delta \left(\omega \right)\right)\cap \mathbb{S}(\omega )={\mathbb{S}}^{*}(\omega )$, which is an $\left(\mathbb{R}-U\right)$-section $\mathbb{S}$ of the random open set $U\left(\omega \right)={\gamma }_{{\mathbb{S}}^{*}}^{\mathbb{R}}(\omega )$, which is an $\mathbb{R}$-tube containing $x$. As there exists a TRV $\delta$ such that every trajectory ${\gamma }_y^{\mathbb{R}}(\omega )$ with $y\in {\mathbb{S}}^{*}(\omega )$ intersect ${\mathbb{S}}^{*}(\omega )$ only at the point $y$ (i.e., if $y\in {\mathbb{S}}^{*}(\omega )$, then ${\gamma }_y^{\mathbb{R}}\left(\omega \right)\cap {\mathbb{S}}^{*}\left(\omega \right)=\left\{y\right\}$). Otherwise, there will be a sequence $\left\{y_n\right\}$ in $\mathbb{S}(\omega )$, $\underset{n⟶\infty }{\lim }‖y_n⟶x‖=0$, and a sequence $\left\{p_n\right\}$ in $\mathbb{R}$, $p_n⟶\infty$, such that $\underset{n⟶\infty }{\lim }‖\phi \left(p_n,{\theta }_{-p_n}\omega \right)y_n-x‖=0$, i.e., either $x\in J_x^{+}\left(\omega \right)$ or $x\in J_x^{-}\left(\omega \right)$. However, this is a contradiction because $x$ is wandering. Thus, for some TRV $\delta$, $S\left(x,\delta \left(\omega \right)\right)\cap \mathbb{S}(\omega )={\mathbb{S}}^{*}(\omega )$ is an $\left(\mathbb{R}-U\right)$-section $\mathbb{S}$ of the random open set $U\left(\omega \right)={\gamma }_{{\mathbb{S}}^{*}}^{\mathbb{R}}(\omega )$. Furthermore, $U\left(\omega \right)$ is open, and the continuity of ${\tau }_{\omega }:U⟶\mathbb{R}$ follows from continuity; it is continuity on $W(\omega )\cap U(\omega )$, and continuity of the $\phi \left(·,\omega ,·\right):\mathbb{R}\times X⟶X$. □

Definition 14.

Given an open $\infty$-tube $U(\omega )$ with section $\mathbb{S}(\omega )$ and a continuous function ${\tau }_{\omega }:U(\omega )⟶\mathbb{R}$, and given a random set $N(\omega )$, $K(\omega )$ satisfying, $N(\omega )\subset K(\omega )\subset \mathbb{S}(\omega )$, where $N(\omega )$ is open in $\mathbb{S}(\omega )$ and $K(\omega )$ is compact, we will call ${\gamma }_K^{\mathbb{R}}(\omega )$ the random compactly tube over $K(\omega )$. Then indeed ${\tau }_{\omega }:{\gamma }_K^{\mathbb{R}}\left(\omega \right)⟶\mathbb{R}$ is continuous forevery $\omega \in \mathrm{\Omega }$.

Theorem 7.

If $X$ is locally compact and separable, and if every $x\in X$ is a wandering point, then there exists a countable collection $\left\{{\gamma }_{K_n}^{\mathbb{R}}(\omega )\right\}$ of compactly tubes which form a covering of $X$, with ${\tau }_{n,\omega }:{\gamma }_{K_n}^{\mathbb{R}}(\omega ) ⟶\mathbb{R}$ continuous on ${\gamma }_{K_n}^{\mathbb{R}}(\omega )$ for every $\omega \in \mathrm{\Omega }$ and $n=1,2,\dots$

Proof. 

The proof follows from Theorem 5 and the separability of $X$. □

Lemma 4.

A compactly $\mathbb{R}$-tube $U(\omega )$ with section $K(\omega )$ of a dispersive dynamical system $(\theta ,\phi )$ is a closed random set in $X.$

Proof. 

Let $U(\omega )={\gamma }_K^{\mathbb{R}}(\omega )$. For any sequence $\left\{x_n\right\}$ in ${\gamma }_K^{\mathbb{R}}(\omega )$, there are sequences $\left\{{\tau }_n\right\}$ and $\left\{y_n\right\}$ in $\mathbb{R}$ and $K(\omega )$, respectively, with $x_n=\phi ({\tau }_n,{\theta }_{-{\tau }_n}\omega )y_n$. Since $K(\omega )$ is compact, we may assume that $y_n⟶y\in K(\omega )$. Now, if $x_n⟶x$, then $\left\{{\tau }_n\right\}$ is bounded, so that $x\in {\gamma }_y^{\mathbb{R}}(\omega )\subset {\gamma }_K^{\mathbb{R}}(\omega )$. For otherwise, if $\left\{{\tau }_n\right\}$ contains an unbounded subsequence $\left\{{\tau }_{n_k}\right\}$, say ${\tau }_{n_k}⟶+\mathrm{\infty }$, then clearly $x\in J_y^{+}(\omega )$, which is absurd as $J_y^{+}\left(\omega \right)=\mathrm{\varnothing }$ for each $y\in X$ by Theorem 1. □

Lemma 5.

Let $U_1(\omega )$, $U_2(\omega )$ be compactly tubes of a dispersive RDS with sections $K_1(\omega )$, $K_2(\omega )$, and continuous mappings ${\tau }_{1,\omega }(x)$, ${\tau }_{2,\omega }(x)$, respectively. If $U_1(\omega ) \cap U_2(\omega )\ne \varnothing$, then $U(\omega )=U_1(\omega )\cup U_2(\omega )$ is a compactly tube with a section $K(\omega )\supset K_1(\omega )$ and a continuous mapping ${\tau }_{\omega }:U(\omega )⟶\mathbb{R}$.Moreover, if the distance of time between $K_1$ and $K_2$ along trajectories in $U_1(\omega )\cap U_2(\omega )$ is less than $\tau$ ($>0$), the distance of time between $K(\omega )$ and $K_2(\omega )$ along trajectories in $U(\omega )$ is also less than $\tau .$

Proof. 

$U_1(\omega )$ and $U_2(\omega )$ are invariant and random closed. Therefore $U_1(\omega )\cap U_2(\omega )$ is invariant and closed. Further, $U_1(\omega )\cap K_2(\omega )$ is compact and non-empty. Set ${\mathbb{S}}_2(\omega )=U_1(\omega )\cap K_2(\omega )$ and ${\mathbb{S}}_1(\omega )=U_1(\omega )\cap K_1(\omega )$. Any trajectory in $U_1(\omega )\cap U_2(\omega )$ meets $K_2(\omega )$, and therefore ${\mathbb{S}}_2(\omega )$, in precisely one point, and therefore meets $K_1(\omega )$ and ${\mathbb{S}}_1(\omega )$ in precisely one point. Therefore, for every $x\in U_1(\omega )\cap U_2(\omega )$, we have ${\tau }_1(\omega ,x)={\tau }_2(\omega ,x)+{\tau }_1(\omega ,\phi \left({\tau }_{2,\omega }(x),\omega \right)x)$. This is because $\phi \left({\tau }_{1,\omega }(x),{\theta }_{-{\tau }_{1,\omega }(x)}\omega \right)x=\phi \left({\tau }_{2,\omega }(x),{\theta }_{-{\tau }_{2,\omega }(x)}\omega \right)x$ and

$${\tau }_1\left(\omega ,\phi \left({\tau }_{2,\omega }(x),{\theta }_{-{\tau }_{2,\omega }(x)}\omega \right)x\right)=\phi \left({\tau }_{2,\omega }(x)+{\tau }_{1,\omega }(\phi \left({\tau }_{2,\omega }(x),\omega \right)x),{\theta }_{-{\tau }_{2,\omega }(x)}\omega \right)x,$$

and there are no random periodic trajectories or random fixed points. The continuous function ${\tau }_{1,\omega }(x)$ on a compact set ${\mathbb{S}}_2(\omega )$ can be extended to a continuous function ${\tau }_{\omega }(x)$ defined on $K_2(\omega )$ in which ${\tau }_{\omega }(x)\equiv {\tau }_{1,\omega }(x)$ for $x\in {\mathbb{S}}_2(\omega )$. Further, if $\left|{\tau }_{1,\omega }(x)\right|<\tau$ for $x\in {\mathbb{S}}_2(\omega )$, we can have $\left|{\tau }_{\omega }(x)\right|<\tau$ on $K_2(\omega )$. Notice now that $\{\phi \left({\tau }_{\omega }(x),{\theta }_{-{\tau }_{\omega }(x)}\omega \right)x: x\in {\mathbb{S}}_2(\omega )\}={\mathbb{S}}_1(\omega )$, and ${\tau }_{\omega }(x)$ is continuous. Since $K_2(\omega )$ is compact, then so is $\{\phi \left({\tau }_{\omega }(x),{\theta }_{-{\tau }_{\omega }(x)}\omega \right)x: x\in K_2(\omega )\}$. Now set

$$K(\omega )=K_1(\omega )\cup \{\phi \left({\tau }_{\omega }(x),{\theta }_{-{\tau }_{\omega }(x)}\omega \right)x: x\in K_2(\omega )\}$$

and define ${\tau }^{\mathrm{*}}:\mathrm{\Omega }\times \phi \left(t,{\theta }_{-t}\omega \right)K({\theta }_{-t}\omega )⟶\mathbb{R}$, where

$$\phi \left(t,{\theta }_{-t}\omega \right)K\left({\theta }_{-t}\omega \right)=\phi \left(t,{\theta }_{-t}\omega \right)K_1\left({\theta }_{-t}\omega \right)\cup \phi \left(t,{\theta }_{-t}\omega \right)K_2\left({\theta }_{-t}\omega \right)$$

as follows

$${\tau }^{*}\left(\omega ,x\right)=\left\{\begin{array}{cc}{\tau }_{1,\omega }(x),& x\in \phi \left(t,{\theta }_{-t}\omega \right)K_1\left({\theta }_{-t}\omega \right)\\ {\tau }_{2,\omega }\left(x\right)+{\tau }_{\omega }\left(\phi \left({\tau }_{2,\omega }(x),\omega \right)x\right),& x\in \phi \left(t,{\theta }_{-t}\omega \right)K_2\left({\theta }_{-t}\omega \right)\end{array}\right.$$

${\tau }_{\omega }^{\mathrm{*}}:\phi \left(t,{\theta }_{-t}\omega \right)K({\theta }_{-t}\omega )⟶\mathbb{R}$ is continuous for every $\omega \in \mathrm{\Omega }$. We need only verify that if $x\in U_1(\omega )\cap U_2(\omega )$, then

$${\tau }_{1,\omega }(x)={\tau }_{2,\omega }(x)+{\tau }_{\omega }(\phi \left({\tau }_{2,\omega }(x),\omega \right)x). \square$$

4. Stability for the Random Sets via Stochastic Lyapunov Function

This section focuses on studying the stochastic version of Lyapunov stability theory. The principle of Lyapunov stability will be used to complete our study in the later sections. In ref. [24], L. Arnold and B. Schmalfuss studied the stability of random sets by using the stochastic Lyapunov functions. They assumed that the state space of the RDS is the Euclidean space ${\mathbb{R}}^n$. In this article, the infinite-dimensional Banach space will be considered instead of ${\mathbb{R}}^n$.

The following definition is mentioned in ref. [30], but in this article, the phase space is Banach space.

Definition 15.

Let $(\theta ,\phi )$ be an RDS defined over the infinite dimensional Banach space, and $A(\omega )$ be a random compact set which is invariant under $(\theta ,\phi ).$

(i) 

$A(\omega )$ is called stable under $(\theta ,\phi )$ if for any $\epsilon >0$ there exists a random compact set $C(\omega )$ which is a neighborhood of $A(\omega )$ (i.e., $C(\omega )$ is a neighborhood of $A(\omega )$ for all $\omega$), such that

• $\mathbb{P}\left\{\omega :{\mathit{sup}}_{x\in C\left(\omega \right)}{\mathit{inf}}_{y\in A\left(\omega \right)}‖x-y‖\ge \epsilon \right\}<\epsilon ,$
• $C(\omega )$ is forward invariant under $(\theta ,\phi ).$

(ii) 

$A(\omega )$ is called a (global) attractor of $(\theta ,\phi )$ if for any random variable $R$

$\mathbb{P}\{\omega : \underset{t⟶\infty }{\mathit{lim}}{\mathit{inf}}_{y\in A(\omega )}‖\phi \left(t,{\theta }_{-t}\omega \right)R({\theta }_{-t}\omega )-y‖\}=0.$

(iii) 

$A(\omega )$ is called (globally) asymptotically stable if it is stable and is an attractor.

Definition 16.

Let $(\theta ,\phi )$ be an RDS in $X$, and $A(\omega )$ be a random compact set which is invariant under $(\theta ,\phi )$. A function $\mathcal{L}:\mathrm{\Omega }\times X ⟶\mathbb{R}$ is called a Lyapunov function for $A(\omega )$ (under $(\theta ,\phi )$) if it has the following properties:

(i)

$\omega ⟼\mathcal{L}(\omega , x)$ is measurable for each $x\in X$, and $x⟼\mathcal{L}(\omega , x)$ is continuous for each $\omega \in \mathrm{\Omega }$;

(ii)

$\mathcal{L}$ is uniformly unbounded, i.e., $\underset{‖x‖⟶\mathrm{\infty }}{\lim }\mathcal{L}(\omega ,x)=\mathrm{\infty }$ for all $\omega$;

(iii)

$\mathcal{L}$ is positive-definite, i.e., $\mathcal{L}(\omega ,x)=0$  for $x\in A(\omega )$, and $\mathcal{L}(\omega ,x)>0$ for $x\notin A(\omega )$;

(iv)

$\mathcal{L}$ is strictly decreasing along orbits of $(\theta ,\phi )$, i.e.,

$$\mathcal{L}({\theta }_t\omega , \phi (t, \omega , x))<\mathcal{L}(\omega , x) \mathit{for}\mathit{all} t>0 \mathit{and} x\notin A(\omega ), (6.1)$$

Theorem 8.

Let $(\theta ,\phi )$ be an RDS in $X$, and let $A(\omega )$ be a random compact set which is invariant under $(\theta ,\phi )$. Then, $A(\omega )$ is asymptotically stable if and only if there exists a Lyapunov function for $A(\omega )$. If $A(\omega )$ is asymptotically stable, then the Lyapunov function can be chosen to satisfy

$$\mathcal{L}\left({\theta }_t\omega , \phi \left(t, \omega , x\right)\right)=e^{-t}\mathcal{L}(\omega , x) \mathit{for}\mathit{all} t\in \mathbb{R}, x\in X$$

on a $\theta$-invariant set of full measure.

Proof. 

For the proof, see ref. [13]. Corollary 2 only replaces the phase space with infinite-dimensional Banach space rather than ${\mathbb{R}}^n$. □

Theorem 9.

Let $M(\omega )$ be a closed random set. Then, $M(\omega )$ is asymptotically stable if and only if there is a function $\mathcal{L}:\mathrm{\Omega }\times X⟶\mathbb{R}$ which satisfies both the condition (i)–(iii) of Definition 16 and the following conditions

(i)

there exist strictly increasing functions $\alpha \left(\mu \right), \beta \left(\mu \right), \alpha \left(0\right)=\beta \left(0\right)=0$, defined for $\mu >0$, such that

$$\alpha ({\inf }_{y\in M(\omega )}‖x-y‖)<\mathcal{L}(\omega ,x)<\beta ({\inf }_{y\in M(\omega )}‖x-y‖)$$

(ii)

$\mathcal{L}\left({\theta }_t\omega ,\phi \left(t,\omega \right)x\right)<\mathcal{L}(\omega ,x)$ for all $x\in X$, for all $\omega \in \mathrm{\Omega }$, $t>0$, and if there is a TRV $\delta$ such that if ${\inf }_{y\in M(\omega )}‖x-y‖<\delta \left(\omega \right)$, but $x\notin M(\omega )$ then

$$\mathcal{L}\left({\theta }_t\left(\omega \right),\phi \left(t,\omega \right)x\right)<\mathcal{L}(\omega ,x)\mathrm{for} t>0,\mathrm{and} \underset{t⟶+\infty }{\lim }\mathcal{L}\left({\theta }_t\left(\omega \right),\phi \left(t,\omega \right)x\right)=0$$

Proof of i.

Sufficiency. Given $\epsilon >0$ set $m_0=\mathrm{i}\mathrm{n}\mathrm{f}\{\mathcal{L}\left(\omega ,x\right):{\inf }_{y\in M\left(\omega \right)}‖x-y‖\ge \epsilon \}$. By Definition 16 (iii), $m_0>0$. Then, for $x \in M(\omega )$, there is a tempered random variable $\delta :\Omega \to {\mathbb{R}}^{+}$ such that $\mathcal{L}(\omega ,y)<m_0$ for $y\in S(x,\delta \left(\omega \right))$. This is also possible by Definition 16 (i) and (iii). We claim that ${\gamma }_{S\left(x,\delta \left(\omega \right)\right)}^{{\mathbb{R}}^{+}}(\omega )\subset S(M\left(\omega \right),\epsilon )$; otherwise, there is $y\in S(x,\delta \left(\omega \right))$ and $t\ge 0$ such that ${\inf }_{z\in M\left(\omega \right)}‖\phi \left(t,{\theta }_{-t}\omega \right)y-z‖=\epsilon$. But then, $\mathcal{L}({\theta }_t\left(\omega \right),\phi \left(t,\omega \right)y)\le \mathcal{L}(\omega ,y)<m_0$ on one hand by Theorem 9 (iii), and also

$$\mathcal{L}\left({\theta }_t\omega ,\phi \left(t,\omega \right)y\right)=\mathrm{i}\mathrm{n}\mathrm{f}\{\mathcal{L}\left(\omega ,x\right):{\inf }_{y\in M\left(\omega \right)}‖x-y‖\ge \epsilon \}$$

as ${\inf }_{z\in M\left(\omega \right)}‖\phi \left(t,{\theta }_{-t}\omega \right)y-z‖=\epsilon$, that is $\mathcal{L}\left({\theta }_t\omega ,\phi \left(t,\omega \right)y\right)\ge m_0$. This contradiction proves the result.

Proof of ii.

Necessity. Let $M(\omega )$ be stable. Define

$$\mathcal{L}\left(\omega ,x\right)={\sup }_{t\ge 0}{\displaystyle \frac{{\inf }_{x\in M\left(\omega \right)}‖\phi \left(t,{\theta }_{-t}\omega \right)x-y‖}{1+{\inf }_{x\in M\left(\omega \right)}‖\phi \left(t,{\theta }_{-t}\omega \right)x-y‖}}$$

This $\mathcal{L}\left(\omega ,x\right)$ is defined on $X$. The theorem is proven. □

Remark 2.

Condition (ii) in the Theorem 9 is equivalent to the requirement that for every $\epsilon >0$, there is a tempered random variable $\delta :\Omega \to {\mathbb{R}}^{+}$, such that

$$\mathbb{P}\left\{\omega :\mathcal{L}\left(\omega ,x\right)\ge \delta \left(\omega \right)\right\}=1\mathit{whenever} {\inf }_{y\in M\left(\omega \right)}‖x-y‖\ge \epsilon ,$$

also, for any sequence $\left\{x_n\right\}$,

$$\mathbb{P}\left\{\omega :\mathcal{L}\left(\omega ,x_n\right)\to 0\right\}=1\mathit{whenever} x_n\to x\in M(\omega ).$$

5. ${\mathbb{R}}^{+}\_$ Dispersive and Parallelizability for Random Dynamical Systems

The parallelizable RDSs is given [16] as a special case of gradient-like, where the authors proved that the following statements are equivalents:

(1)

$(\theta ,\phi )$ is a gradient-like RDS.

(2)

$(\theta ,\phi )$ possesses a continuous random section $\mathbb{S}\subset \Omega \times E.$

(3)

$(\theta ,\phi )$ possesses a continuous and strict Lyapunov function.

(4)

The backward RDS associated to $(\theta ,\phi )$ is gradient-like.

In this section, some new properties of parallelizable RDS are given, and the parallelizable RDS is characterized using dispersive RDS, which is given in ref. [31].

Definition 17.

Let $(\theta ,\phi )$ be an RDS. If there is a random set $\mathbb{S}(\omega )$ and a mapping $h: \mathrm{\Omega }\times X⟶\mathbb{R}\times \mathrm{\Omega }\times \mathbb{S}(\omega )$ satisfy the following conditions:

(i) 

$h: \mathrm{\Omega }\times X⟶\mathbb{R}\times \mathrm{\Omega }\times \mathbb{S}(\omega )$ is measurable.

(ii) 

$h_{\omega }: X⟶\mathbb{R}\times (\omega )$ is a homeomorphism for every $\omega \in \mathrm{\Omega }$.

(iii) 

${\gamma }_{\mathbb{S}}^{\mathbb{R}}\left(\omega \right)=\left\{\phi \left(t,{\theta }_{-t}\omega \right)\mathbb{S}(\omega ):t\in \mathbb{R}\right\}=X$.

(iv) 

$h\left(\phi \left(t,{\theta }_{-t}\omega \right)x\right)=(t,\omega , x)$ for every $x\in \mathbb{S}(\omega )$ and $t\in \mathbb{R}$.

Then$(\theta ,\phi )$ is called parallelizable.

Theorem 10.

An RDS $(\theta ,\phi )$ is parallelizable if and only if it admits a section $\mathbb{S}(\omega )$ such that ${\tau }_{\omega }:X⟶\mathbb{R}$ is continuous for every $\omega \in \mathrm{\Omega }.$

Proof. 

Suppose that $(\theta ,\phi )$ is parallelizable, then by Definition 17, there is a random set $\mathbb{S}(\omega )$ and a function $h: \mathrm{\Omega }\times X⟶\mathbb{R}\times \mathrm{\Omega }\times \mathbb{S}(\omega )$, which satisfies

(i)

$h: \mathrm{\Omega }\times X⟶\mathbb{R}\times \mathrm{\Omega }\times \mathbb{S}(\omega )$ is measurable.

(ii)

$h_{\omega }: X⟶\mathbb{R}\times \mathbb{S}(\omega )$ is homeomorphism for every $\omega \in \mathrm{\Omega }$.

(iii)

${\gamma }_{\mathbb{S}}^{\mathbb{R}}\left(\omega \right)=\left\{\phi \left(t,{\theta }_{-t}\omega \right)\mathbb{S}(\omega ):t\in \mathbb{R}\right\}=X$.

(iv)

$h\left(\phi \left(t,{\theta }_{-t}\omega \right)x\right)=(t,\omega ,x)$ for every $x\in \mathbb{S}(\omega )$ and $t\in \mathbb{R}$.

To show that $\mathbb{S}:\mathrm{\Omega }⟶2^X$ is a section. Let $x\in X$, then $x\in {\gamma }_{\mathbb{S}}^t\left(\omega \right)$. Thus, for some $y\in \mathbb{S}(\omega )$ and $t\in \mathbb{R}$, we have $x=\phi \left(t,{\theta }_{-t}\omega \right)y$, so that $y=\phi \left(-t,\omega \right)x\in \mathbb{S}(\omega )$. Define $\tau :\mathrm{\Omega }\times X⟶\mathbb{R}$ by ${\tau }_{\omega }(x)=-t$. Then $\phi \left({\tau }_{\omega }(x),\omega \right)x\in \mathbb{S}(\omega )$, and hence $\mathbb{S}(\omega )$ is a section of $(\theta ,\phi )$. Suppose that $(\theta ,\phi )$ has a section $\mathbb{S}(\omega )$ with ${\tau }_{\omega }:X⟶\mathbb{R}$ is continuous for every $\omega \in \mathrm{\Omega }$. Then ${\gamma }_{\mathbb{S}}^{\mathbb{R}}\left(\omega \right)=\left\{\phi \left(t,{\theta }_{-t}\omega \right)\mathbb{S}({\theta }_{-t}\omega ):t\in \mathbb{R}\right\}=X$. Now, define $h:\mathrm{\Omega }\times X⟶\mathbb{R}\times \mathrm{\Omega }\times \mathbb{S}(\omega )$ by $h\left(\omega ,x\right)=(-{\tau }_{\omega }\left(x\right),{\theta }_{-{\tau }_{\omega }\left(x\right)}\omega ,\phi \left({\tau }_{\omega }\left(x\right),\omega \right)x)$. Then, $h_{\omega }\left(x\right)$ is $1-1$ for every $\omega$. Since ${\tau }_{\omega }\left(x\right)$ and $\phi \left({\tau }_{\omega }\left(x\right),{\theta }_{-\tau \left(x\right)}\omega \right)$ are continuous, then $h_{\omega }\left(x\right)$ is continuous for every $\omega$. Define $h^{-1}:\mathbb{R}\times \mathrm{\Omega }\times \mathbb{S}(\omega )⟶\mathrm{\Omega }\times X$ by $h^{-1}\left(t,\omega ,x\right)=({\theta }_{-t}\omega ,\phi \left(t,{\theta }_{-t}\omega \right)x)$. It is easy to show that the inverse of $h$ is the function $h^{-1}$. Also, $h^{-1}(\omega ):\mathbb{R}\times \mathbb{S}(\omega )⟶X$ is continuous for every $\omega$. Therefore, $h_{\omega }$ is a homeomorphism for every $\omega$. Consequently, $(\theta ,\phi )$ is parallelizable. □

Theorem 11.

For the RDS $\left(\theta ,\phi \right),$ the parallelizable and dispersive are equivalent when the phase space $X$ is locally compact separable Banach space.

Proof. 

Only the sufficiency part needs proof. By Theorem 10, it is enough to show that $X$ admits a section $\mathbb{S}$ such that ${\tau }_{\omega }:X⟶\mathbb{R}$ is continuous for every $\omega \in \mathrm{\Omega }$. Theorem 7 implies that there exists a countable cover $\{U_n(\omega )\}$ of $X$ by compactly tubes $U_n(\omega )$, with $K_n(\omega )$ as a section and continuous functions ${\tau }_{\omega }^n:X⟶\mathbb{R}$. We construct a compactly tube cover as follows. Set $K_1(\omega )=K^1(\omega )$, and $U_1(\omega )=U^1(\omega )$. Beginning with $U^1(\omega )$ and $U_2(\omega )$, we use Lemma 5 to extend $K^1(\omega )$ to a compact set $K^2(\omega )$, hence finding the compactly tube $U^2(\omega )=U^1(\omega )\bigcup U_2(\omega )$ with ${\tau }_{\omega }^2:U^2(\omega )⟶\mathbb{R}$ is continuous for every $\omega \in \mathrm{\Omega }$. This guarantees $K^1(\omega )$ is an invariant. After we found the $U^n(\omega )$, choose it with $U_{n+1}(\omega )$, and create likewise $U^{n+1}(\omega )$ with $K^{n+1}(\omega )\supset K^n(\omega )$ and ${\tau }_{\omega }^{n+1}:U^{n+1}(\omega )⟶\mathbb{R}$ continuous for every $\omega \in \mathrm{\Omega }$. Now set $\mathbb{S}(\omega )=\bigcup K^n(\omega )$ then $X={\gamma }_{\mathbb{S}}^t(\omega )$, and the function ${\tau }_{\omega }:X⟶\mathbb{R}$ defined by ${\tau }_{\omega }(x)≔{\tau }_{\omega }^n(x)$ for $x\in U^n(\omega )$ and $\omega \in \mathrm{\Omega }$ is continuous on $X$ for every $\omega \in \mathrm{\Omega }$, with the property that $\phi ({\tau }_{\omega }\left(x\right),{\theta }_{-{\tau }_{\omega }(x)}\omega )x\in \mathbb{S}(\omega )$. Moreover, ${\tau }_{\omega }(x)$ is unique for each $x\in X$ and $\omega \in \mathrm{\Omega }$. Thus, $X$ admits a section $\mathbb{S}(\omega )$ such that ${\tau }_{\omega }:X⟶\mathbb{R}$ is continuous for every $\omega \in \mathrm{\Omega }$. The RDS $(\theta ,\phi )$ is parallelizable. □

Corollary 1.

If $(\theta ,\phi )$ is RDS with the phase space $X$ is locally compact separable Banach space, then the following holds:

(1)

The RDS $(\theta ,\phi )$ is parallelizable if and only if the prolongation limit set $J_x^{+}\left(\omega \right)=\varnothing$ for every x ∈ X.

(2)

Let $M(\omega )$ be an asymptotically stable closed invariant random set. The set $A_M(\omega )-M (\omega )$ is parallelizable if it is locally compact and covers a countable dense set.

Proof. 

The proof follows from Theorem 3 and Theorem 10. □

Example 1.

Consider the stochastic differential system

$$d{x}_1=-x_{1}dt+{\sigma }_{1}d{B}_1(t), d{x}_2=x_{2}dt+{\sigma }_{2}d{B}_2(t)$$

(1)

To put the system (1) in the framework of RDS, we model white noise as a metric dynamical system as follows: Let $W$ be the space of continuous functions $\omega :\mathbb{R}⟶\mathbb{R}$, which satisfy $\omega (0)=0$, let $\mathcal{F}$ be the Borel sigma-algebra induced by the compact-open topology of $W$, and let $\mathbb{P}$ be the Wiener measure on $(W,\mathcal{F})$, i.e., the distribution on $\mathcal{F}$ of a standard Wiener process with two-sided time. The shift ${\theta }_t$ is defined by ${\theta }_t\omega (s) :=\omega (t+s)-\omega (t)$. Then, $(W,\mathcal{F},\mathbb{P},{({\theta }_t)}_{t\in \mathbb{R}})$ is an ergodic metric dynamical system ‘‘driving’’ the stochastic differential system (1), and $W_t(\omega )=\omega (t)$. We refer to ref. [4] for details. The solution of the system (1) can be explicitly given by

$$X\left(t\right)=\left(x_1(0)e^{-t}+{\int }_0^t{\sigma }_1{e}^{\left(s-t\right)}d{B}_1\left(s\right),x_2(0)e^t+{\int }_0^t{\sigma }_2{e}^{(t-s)}d{B}_2(s).\right)$$

(2)

Then, $\phi :\mathbb{R}\times \mathrm{\Omega }\times {\mathbb{R}}^2⟶{\mathbb{R}}^2$ is defined by

$$\phi \left(t,\omega \right)X\left(t\right)≔\left(x_1{e}^{-t}+{\int }_0^t{\sigma }_1{e}^{\left(s-t\right)}d{B}_1\left(s\right),x_2{e}^t+{\int }_0^t{\sigma }_2{e}^{(t-s)}d{B}_2(s)\right)$$

which forms a cocycle over $\theta$. Thus $(\theta ,\phi )$ is the RDS generated by the system (1). Let ${\mathbb{R}}_{*}^2≔\left\{\left(x,y\right)\in {\mathbb{R}}^2:x\ne 0,y\ne 0\right\}$. Since ${\mathbb{R}}_{*}^2$ is invariant, we can restrict $\phi \left(t,\omega \right)$ on ${\mathbb{R}}_{*}^2$ to provide a new RDS $(\theta ,{\phi }_{*})$ in which ${\phi }_{*}\left(t,\omega \right)≔{\left.\phi \left(t,\omega \right)\right|}_{{\mathbb{R}}_{*}^2}$. Now, since $J_z^{+}\left(\omega \right)=\varnothing$ for every $z\in {\mathbb{R}}_{*}^2$, then by Theorem 1 (with $\mathbb{T}=\mathbb{R}$ and $P={\mathbb{R}}^{+}$), $(\theta ,{\phi }_{*})$ is dispersive RDS. Since the space ${\mathbb{R}}_{*}^2$ is a locally compact separable metric space, then by Theorem 11, $(\theta ,{\phi }_{*})$ is parallelizable.

Example 2.

Consider the system of stochastic differential equations

$$d{x}_1=f\left(x_1,x_2\right)dt+{\sigma }_{1}d{B}_1(t), d{x}_2={\sigma }_{2}d{B}_2(t).$$

where $f:{\mathbb{R}}^2⟶\mathbb{R}$ is continuous. Furthermore, when $n\in {\mathbb{Z}}^{+}$, assume that $f\left(x_1,x_2\right)=0$ if $\left(x_1,x_2\right)=(n, 1/n)$ and $f\left(x_1,x_2\right)>0$ otherwise. Let $X≔{\mathbb{R}}^2/\bigcup _{n=1}^{\infty }{I}_n$, where $I_n≔\left\{\left(x_1,x_2\right)\in {\mathbb{R}}^2:x_1\le n, x_2=1/n\right\},$ $n=1,2,3,\dots$. Note that $X$ is not locally compact. Now consider the RDS generated from the given system on $X$ over an MDS $\mathrm{\theta }=(\mathrm{\Omega },\mathcal{F},\mathbb{P},\{{\mathrm{\theta }}_{\mathrm{t}}:\mathrm{t}\in \mathbb{R}\})$. This RDS is dispersive but not parallelizable.

Example 3.

Consider Example 3 with $X={\mathbb{R}}^2$ ($X$ is locally compact); then, $J_x^{+}\left(\omega \right)=\varnothing$ for every x ∈ X. Then, by Corollary 1 (1), the RDS is parallelizable.

Theorem 12.

Let $M(\omega )$ be a closed invariant random set. If $M(\omega )$ is uniformly asymptotically stable, then $A_M(\omega )-M(\omega )$ is parallelizable.

Proof. 

Define a function $\psi :A_M(\omega )⟶\mathbb{R}$ satisfying the properties of Theorem 9 because $M(\omega )$ is uniformly asymptotically stable. Also, there is an $\alpha >0$ so that $S[M, \alpha ]\subset A_M(\omega )$, and such that for any $\epsilon >0$, there is a $T>0$ with the property that ${\gamma }_y^t(\omega )\subset S [M(\omega ), \alpha ]$ for every $x\in S [M, \alpha ]$, where $y≔\phi \left(T,{\theta }_{-T}\omega \right)x$. Now, let $m_0=\inf \{\psi \left(x\right): {\inf }_{\mathrm{z}\in M(\omega )}‖x-\mathrm{z}‖=\alpha \}$. Indeed, $m_0>0$. Consider now any set ${\mathbb{S}}_{\eta }(\omega )≔\left\{x\in X:{\inf }_{\mathrm{z}\in M\left(\omega \right)}‖x-\mathrm{z}‖\le \alpha \mathrm{a}\mathrm{n}\mathrm{d} \psi \left(x\right)=\eta \right\}$. We claim that if $\eta <m_0$, then ${\mathbb{S}}_{\eta }(\omega )$ is a section of the flow in $A_M(\omega )-M$ such that there is a unique continuous function $\tau :\mathrm{\Omega }\times X⟶\mathbb{R}$, mapping $A_M(\omega )-M(\omega )$ into $\mathbb{R}$, such that for each $x\in A_M(\omega )-M(\omega ),$ $\phi ({\tau }_{\omega }\left(x\right),{\theta }_{-{\tau }_{\omega }\left(x\right)}\omega )x$. Then, by Corollary 1, $A_M(\omega )-M(\omega )$ is parallelizable. To show that, ${\mathbb{S}}_{\eta }(\omega )$ admits the above enunciated properties, consider the set $P_{\eta }(\omega )=\{x\in X:{\inf }_{\mathrm{z}\in M\left(\omega \right)}‖x-\mathrm{z}‖\le \alpha \mathrm{a}\mathrm{n}\mathrm{d} \psi \left(x\right)\le \eta \}$. Indeed, $P_{\eta }(\omega )\subset A_M(\omega )$ and $P_{\eta }(\omega )\supset M(\omega )$. Any trajectory in $A_M(\omega )-M(\omega )$ can meet the set ${\mathbb{S}}_{\eta }(\omega )$ at a single point at most because, if a trajectory in $A_M(\omega )-M(\omega )$ admits two points $x_1$, $x_2$ on ${\mathbb{S}}_{\eta }(\omega )$, then we may assume that $x_2=\phi (t,{\theta }_{-t}\omega )x_1$ where $t>0$. However, since ${\mathbb{S}}_{\eta }(\omega )\cap M=\varnothing$, and $\psi \left(x_2\right)=\psi \left(\phi \left(t,{\theta }_{-t}\omega \right)x_1\right)<\psi \left(x_1\right)$, which is a contradiction. To show that any trajectory in $A_M(\omega )-M$ meets ${\mathbb{S}}_{\eta }(\omega )$, if $x\notin P_{\eta }(\omega )$, there exists a $t>0$ with $\phi \left(t,{\theta }_{-t}\omega \right)x\in P_{\eta }(\omega )$. But $\left\{\phi \left(s,{\theta }_{-s}\omega \right)x:s\in [0,t]\right\}\cap \partial P_{\eta }(\omega )\ne \varnothing$. However, $S_{\eta }(\omega )\equiv \partial P_{\eta }(\omega )$. If $x\in P_{\eta }(\omega )$, then we claim that there exists $t<0$ with $\phi \left(t,{\theta }_{-t}\omega \right)x\in {\mathbb{S}}_{\eta }(\omega )$; otherwise, ${\gamma }_x^{{\mathbb{R}}^{-}}\left(\omega \right)\subset Int P_{\eta }(\omega )$. Thus, we can put $\delta ={\inf }_{t\le 0}{\inf }_{\mathrm{z}\in M(\omega )}‖\phi \left(t,{\theta }_{-t}\omega \right)x-\mathrm{z}‖$, and $\delta >0$. By uniform asymptotic stability, it can be assumed that $T>0$ such that ${\inf }_{\mathrm{z}\in M(\omega )}‖y-\mathrm{z}‖\le \alpha (\omega )$ implies ${\inf }_{\mathrm{z}\in M(\omega )}‖\phi \left(T,{\theta }_{-T}\omega \right)y-\mathrm{z}‖<\delta /2$. Thus, $y=\phi \left(-T, \omega \right)x\in P_{\eta }(\omega )\subset S [M(\omega ), \alpha ]$, but ${\inf }_{\mathrm{z}\in M(\omega )}‖\phi \left(T,{\theta }_{-T}\omega \right)y-\mathrm{z}‖>\delta /2$, which is a contradiction, and any trajectory in $A_M(\omega )-M(\omega )$ meets ${\mathbb{S}}_{\eta }(\omega )$ at a single point. Define $\tau :\mathrm{\Omega }\times (A_M(\omega )-M(\omega ))⟶\mathbb{R}$ by the requirement that $\phi \left({\tau }_{\omega }(x),{\theta }_{-{\tau }_{\omega }(x)}\omega \right)y\in {\mathbb{S}}_{\eta }(\omega )$ for $x\in A_M(\omega )-M(\omega )$. We claim that ${\tau }_{\omega }:(A_M(\omega )-M(\omega ))⟶\mathbb{R}$ is uniquely defined and is continuous for every $\omega \in \mathrm{\Omega }$. For any $x \in A_M(\omega )-M(\omega )$, and $\epsilon >0$, the point $y=\phi \left({\tau }_{\omega }\left(x\right)+\epsilon ,{\theta }_{-{\tau }_{\omega }\left(x\right)-\epsilon }\omega \right)y \in Int P_{\alpha }(\omega )$.·There is, therefore, a neighborhood $N_y$ of $y$ such that $N_y\subset P_{\alpha }(\omega )$. The set $N^{+}=N_y({\tau }_{\omega }\left(x\right)+\epsilon )$ forms a neighborhood of x and ${\tau }_{\omega }\left(u\right)\le {\tau }_{\omega }\left(x\right)+\epsilon$ for each $u\in N^{+}$. Again, $z\equiv \phi \left({\tau }_{\omega }\left(x\right)+\epsilon ,{\theta }_{-{\tau }_{\omega }\left(x\right)-\epsilon }\omega \right)x \in A_M(\omega )-P_{\alpha }(\omega )$. Note that $A_M(\omega )-P_{\alpha }(\omega )$ is open. Hence, there exists a neighborhood $N_z$ of $z$ so that $N_z\subset (A_M(\omega )-P_{\alpha }(\omega ))$. Then $N^{-}=N_z(-{\tau }_{\omega }\left(x\right)+\epsilon )$ is a neighborhood of $x$, and note that, for each $u\in N^{-}$, we have ${\tau }_{\omega }\left(u\right)\ge {\tau }_{\omega }\left(x\right)-\epsilon$. Thus, if $z$ is in the neighborhood $N_{\epsilon }=N^{+}\cap N^{-}$ of $x$, we have ${\tau }_{\omega }\left(x\right)-\epsilon \le {\tau }_{\omega }\left(z\right)\le {\tau }_{\omega }\left(x\right)+\epsilon$. Thus, ${\tau }_{\omega }$ is continuous on $A_M(\omega )-M(\omega )$. We claim that for every $\omega \in \mathrm{\Omega }$, ${\tau }_{\omega }\left(x\right)⟶+\infty$ as $x⟶ M(\omega )$, $x\in A_M(\omega )-M(\omega )$. Otherwise, there is a $T>0$ and a sequence $\left\{x_n\right\}$ in $A_M(\omega )-M(\omega )$, with $x_n⟶ M(\omega )$ and $-T\le \tau \left(x_n\right)\le 0$. Since the sequence $\left\{{\tau }_{\omega }\left(x_n\right)\right\}$ is bounded, it has a convergent subsequence. Therefore, it can be assumed that ${\tau }_{\omega }\left(x_n\right)⟶\tau$ for every $\omega \in \mathrm{\Omega }$, where $-T\le \tau \le 0$. Then, by the continuity of $\phi \left(t,{\theta }_{-t}\omega \right):X⟶X$ (This function is continuous because $X$ is Banach space, and hence it is smooth manifold, see ref. [6].) we have $\phi \left({\tau }_{\omega }\left(x_n\right),{\theta }_{-\tau \left(x_n\right)}\omega \right)x_n⟶\phi \left(\tau ,{\theta }_{-\tau }\omega \right)x$. As $M(\omega )$ is invariant, $\phi \left(\tau ,{\theta }_{-\tau }\omega \right)x\in M(\omega )$; on the other hand, $\phi \left({\tau }_{\omega }\left(x_n\right),{\theta }_{-{\tau }_{\omega }\left(x_n\right)}\omega \right)x_n\in \partial P_{\alpha }(\omega )$. which is compact. Therefore, $\phi \left(\tau ,{\theta }_{-\tau }\omega \right)x\in \partial P_{\alpha }(\omega )$ but $\partial P_{\alpha }(\omega )\cap M(\omega )=\varnothing$; this is a contradiction. Define $\mathcal{L}:\mathrm{\Omega }\times A_M(\omega )⟶\mathbb{R}$ by

$$\mathcal{L}\left(\omega ,x\right)=\left\{\begin{array}{cc}0,& x\in M(\omega )\\ e^{{\tau }_{\omega }\left(x\right)}& x\in A_M(\omega )-M(\omega )\end{array}\right.$$

Then this function is continuous on $A_M(\omega )$. It is positive for $x\notin M(\omega )$ and

$$\mathcal{L}\left({\theta }_t\omega ,\phi \left(t,\omega \right)x\right)=e^{-t} \mathcal{L}\left(\omega ,x\right).$$

Finally, to show that, $\mathcal{L}\left(\omega ,x\right)$ is uniformly unbounded, since

$$A_M(\omega )=\bigcup \{\phi \left(m,{\theta }_{-m}\omega \right)P_{\alpha }(\omega ):m\in \left[-n, 0\right],n\in \mathbb{N}\}.$$

The sets $\phi \left(m,{\theta }_{-m}\omega \right)P_{\alpha }(\omega )$ are compact and positively invariant random sets for every $m\in \left[-n,0\right]$. Observe that, if $x\notin \phi \left(m,{\theta }_{-m}\omega \right)P_{\alpha }(\omega )$ for each $m\in \left[-n,0\right]$, then ${\tau }_{\omega }\left(x\right)>n$, so that $\mathcal{L}\left(\omega ,x\right)>e^n$. □

6. Symmetry of Random Dynamical Systems

This section is dedicated to discussing the concept of symmetry in random dynamical systems. The symmetry of random dynamical systems is a generalization of the symmetry of deterministic dynamical systems. We will illustrate during the discussion the types of symmetry used in the research and their significance to the posed problem. Additionally, we will discuss the applications of symmetry. This field has been the subject of extensive research recently. We make reference to several recent advancements [33,34,35,36,37].

Symmetry will be defined in a manner similar to that mentioned in ref. [26].

Definition 18.

The two RDSs, $(\theta ,{\phi }_1)$ and $(\theta ,{\phi }_2)$ (with phase spaces $X$ and $Y$, respectively), are said to be symmetric and written as $(\theta ,{\phi }_1){\cong }_{\mathcal{T}}(\theta ,{\phi }_2)$, if there is a function $\mathcal{T}:\Omega \times X⟶Y$ such that

(i) 

The function ${\mathcal{T}}_{\omega }:X⟶Y$, ${\mathcal{T}}_{\omega }\left(x\right)≔\mathcal{T}\left(\omega , x\right)$ is diffeomor-phism for every $\omega \in \Omega .$

(ii) 

The function ${\mathcal{T}}_x:\Omega$ → $X,$ ${\mathcal{T}}_x\left(\mathrm{\omega }\right)≔\mathcal{T}\left(\omega ,x\right)$ is bi-measurable for every $x\in X$.

(iii) 

$\mathbb{P}\left\{\omega :{\phi }_2 (t, \omega , \mathcal{T}(\omega , x))=\mathcal{T}({\theta }_t \omega , {\phi }_1 (t, \omega , x))\right\}=1$; for every $x\in X$.

If ${\phi }_1={\phi }_2=\phi$ , then the RDS $(\theta ,\phi )$ is called symmetry. The function $\mathcal{T}$ is called a function of symmetry. Any property that is invariant under the influence of a symmetry function is called a symmetry property.

Proposition 1.

Let $(\theta ,\phi ){\cong }_{\mathcal{T}}(\theta ,\psi )$. If $(\theta ,\phi )$ is a parallelizable RDS, then so is $(\theta ,\psi ).$

Proof. 

If $(\theta ,\phi )$ is parallelizable RDS, then there is a random set $\mathbb{S}(\omega )$ and a function $h: \mathrm{\Omega }\times X⟶\mathbb{R}\times \mathrm{\Omega }\times \mathbb{S}(\omega )$ satisfying the conditions of Definition 17. It is easy to show that ${\mathbb{S}}^{\prime }:\mathrm{\Omega }⟶ 2^Y$ and $h^{\prime }: \mathrm{\Omega }\times Y⟶\mathbb{R}\times \mathrm{\Omega }\times {\mathbb{S}}^{\prime }$, satisfying the Definition 17, where ${\mathbb{S}}^{\prime }≔{\mathcal{T}}_{\omega }\circ \mathbb{S}$ and $h^{\prime }≔({Id}_{\mathbb{R}}\times {\mathcal{T}}_{\omega })\circ h_{\omega }\circ {\mathcal{T}}_{\omega }^{-1}$. □

Proposition 2.

Let $(\theta ,\phi ){\cong }_T(\theta ,\psi )$. If $(\theta ,\phi )$ is an ${\mathbb{R}}^{+}\_$ dispersive RDS, then so is $(\theta ,\psi ).$

Proof. 

Suppose that $(\theta ,\phi )$ is a ${\mathbb{R}}^{+}\_$dispersive RDS. Then, by Theorem 1, $J_x^{{\mathbb{R}}^{+}}\left(\omega \right)=\varnothing$ for every $x\in X$. Since

$$J_{{\mathcal{T}}_{\omega }(x)}^{{\mathbb{R}}^{+}}\left(\omega \right)={\mathcal{T}}_{\omega }\left(J_x^{{\mathbb{R}}^{+}}\left(\omega \right)\right)={\mathcal{T}}_{\omega }(\varnothing )=\varnothing ,$$

then $(\theta ,\psi )$ is an ${\mathbb{R}}^{+}\_$dispersive. □

Proposition 3.

Let $(\theta ,\phi ){\cong }_{\mathcal{T}}(\theta ,\psi )$. If $A(\omega )$ is an asymptotically stable compact (closed) random set in $(\theta ,\phi )$, then $B\left(\omega \right)={\mathcal{T}}_{\omega }(A\left(\omega \right))$ is an asymptotically stable compact (closed) random set in $(\theta ,\psi ).$

Proof. 

Suppose that $A(\omega )$ is an asymptotically stable compact (closed) random set in $(\theta ,\phi )$. First note that, because ${\mathcal{T}}_{\omega }$ is diffeomorphism (and hence homeomorphism), then the set $B\left(\omega \right)={\mathcal{T}}_{\omega }(A\left(\omega \right))$ is a compact (closed) random set in $Y$. Now, since $A(\omega )$ is asymptotically stable compact (closed), then there exists a function $\mathcal{L}:\mathrm{\Omega }\times X⟶\mathbb{R}$ which satisfies the hypothesis of Theorem 8 (Theorem 9). Define ${\mathcal{L}}^{*}:\mathrm{\Omega }\times Y⟶\mathbb{R}$ by ${\mathcal{L}}^{*}\left(\omega ,y\right)≔\mathcal{L}(\omega ,{\mathcal{T}}_{\omega }^{-1}\left(y\right))$ for every $(\omega ,y)\in \mathrm{\Omega }\times Y$. It is not difficult to show that ${\mathcal{L}}^{*}$ satisfies the hypothesis of Theorem 8 (Theorem 9). Consequently, $B\left(\omega \right)={\mathcal{T}}_{\omega }(A\left(\omega \right))$ is a compact (closed) random set in $Y$. □

Corollary 2.

The parallelizability, dissipativity, and asymptotic stability are symmetry properties.

Proof. 

The proof follows from Propositions 1, 2, and 3, respectively.

7. Application

Consider the (deterministic) system of ordinary differential equations

$$\begin{array}{c}{\displaystyle \frac{dx}{dt}}=\sigma (y-x)\\ {\displaystyle \frac{dy}{dt}}=\rho x-y-xz{\displaystyle \frac{dz}{dt}}=xy-\beta z\end{array}$$

(3)

This system is referred to as “the Lorenz deterministic system” (one can see, for example, [38]), where

• $t$ represents time.
• $x$ represents the intensity of convection motion.
• $y$ represents the temperature difference between uphill and downhill currents.
• $z$ represents the misrepresentation (from linearity) of the vertical temperature profile.

The constants in (3) are defined as follows: $\sigma$ called the Brandt number, $\rho ={\displaystyle \frac{R_a}{R_c}}$, $\beta ={\displaystyle \frac{4}{1+a^2}}$, a constant associated with the given space. Here, as in ref. [38],$a$ is a number associated with the concerned region, $R_a$ is the Rayleigh number, and $R_c={\pi }^4{a}^{-2}{(1+a^2)}^3$.

Suppose that the real noises are perturbing the parameters $\sigma ,\rho ,$ and $\beta >0$, i.e.,

$$\begin{array}{c}\sigma \left(\omega \right)=\sigma +\xi (\omega ),\\ \rho \left(\omega \right)=\rho +\eta (\omega ),\\ \beta \left(\omega \right)=\beta +\zeta (\omega ).\end{array}$$

(4)

Assume that the perturbed parameters are still positive and $\mathrm{\rho }(\mathrm{\omega })<\mathrm{\sigma }(\mathrm{\omega })\le 1$ almost surely. Thus, the random ODE corresponding to deterministic ODE (3) is given by

$$\begin{array}{c}{\displaystyle \frac{dx}{dt}}=\mathrm{\sigma }({\mathrm{\theta }}_{\mathrm{t}}\mathrm{\omega })(y-x)\\ {\displaystyle \frac{dy}{dt}}=\mathrm{\rho }({\mathrm{\theta }}_{\mathrm{t}}\mathrm{\omega })x-y-xz\\ {\displaystyle \frac{dz}{dt}}=xy-\mathrm{\beta }\left({\mathrm{\theta }}_{\mathrm{t}}\mathrm{\omega }\right)z,\end{array}$$

(5)

or, in a compact form ${\displaystyle \frac{dX}{dt}}=F({\mathrm{\theta }}_{\mathrm{t}}\mathrm{\omega },\mathrm{X})$, where $F:\mathrm{\Omega }\times {\mathbb{R}}^3⟶{\mathbb{R}}^3$ defined by

$$F\left({\mathrm{\theta }}_{\mathrm{t}}\mathrm{\omega },\mathrm{X}\right)=(\mathrm{\sigma }\left({\mathrm{\theta }}_{\mathrm{t}}\mathrm{\omega }\right)\left(\mathrm{y}-\mathrm{x}\right),\mathrm{\rho }\left({\mathrm{\theta }}_{\mathrm{t}}\mathrm{\omega }\right)\mathrm{x}-\mathrm{y}-\mathrm{x}\mathrm{z},\mathrm{x}\mathrm{y}-\mathrm{\beta }\left({\mathrm{\theta }}_{\mathrm{t}}\mathrm{\omega }\right)\mathrm{z})$$

Then the perturbed random ordinary differential equations generate an RDS, and we denote it by $\mathrm{\phi }$; see ref. [5] for details. Consider the function $\mathcal{L}:\mathrm{\Omega }\times {\mathbb{R}}^3⟶\mathbb{R}$ defined by $\mathcal{L}\left(\omega ,x,y,z\right)=x^2+y^2+z^2$ and define the random set

$$D_r\left(\omega \right)≔\left\{\left(x,y,z\right):\mathcal{L}\left(\omega ,x,y,z\right)\le r\right\}.$$

$D_r\left(\omega \right)$ is thus obviously a random compact set. Furthermore, throughout the orbits of $\phi$, the derivative of $\mathcal{L}$ with respect to $t$ is

$$\begin{array}{c}{\displaystyle \frac{d\mathcal{L}}{dt}}=2x\left(\sigma \left(y-x\right)\right)+2y\left(\rho x-y-xz\right)+2z(xy-\beta z)\\ =-2\sigma x^2-2y^2-2\beta z^2+2\left(\rho +\sigma \right)xy\\ \le -\left(\sigma -\rho \right)x^2-\left(2-\rho -\sigma \right)y^2-2\beta z^2\\ <0.\end{array}$$

whenever $(\mathrm{x},\mathrm{y},\mathrm{z})\ne (0,0,0)$ by the assumption $\mathrm{\rho }\left(\mathrm{\omega }\right)<\mathrm{\sigma }(\mathrm{\omega })\le 1$, where $\mathrm{\rho }=\mathrm{\rho }({\mathrm{\theta }}_{\mathrm{t}}\mathrm{\omega })$, $\mathrm{\sigma }=\mathrm{\sigma }({\mathrm{\theta }}_{\mathrm{t}}\mathrm{\omega })$, $\mathrm{\beta }=\mathrm{\beta }({\mathrm{\theta }}_{\mathrm{t}}\mathrm{\omega })$ and $\mathrm{X}=\mathrm{X}(\mathrm{t},\mathrm{\omega })$ with $\mathrm{X}:=(\mathrm{x},\mathrm{y},\mathrm{z})$. Therefore, for every $\mathrm{X}\in {\mathrm{D}}_{\mathrm{r}}(\mathrm{\omega })$ and every $\mathrm{t}>0$ we have $\mathcal{L}({\mathrm{\theta }}_{\mathrm{t}}\mathrm{\omega },\mathrm{\phi }(\mathrm{t},\mathrm{\omega })\mathrm{X})<\mathcal{L}(\mathrm{\omega },\mathrm{X})\le \mathrm{r}$, i.e., $\mathrm{\phi }(\mathrm{t},\mathrm{\omega })\mathrm{X}\in {\mathrm{D}}_{\mathrm{r}}({\mathrm{\theta }}_{\mathrm{t}}\mathrm{\omega })$. Consequently, $\mathcal{L}$ is the Lyapunov function. Thus, $(\theta ,\phi )$ possesses a continuous and strict Lyapunov function. Hence, we conclude that $(\theta ,\phi )$ is parallelizable RDS [16].

It is worth mentioning that the Lorenz system is symmetry. This can be demonstrated by choosing $\mathcal{T}:\Omega \times {\mathbb{R}}^3⟶{\mathbb{R}}^3$ to be such that $\mathcal{T}\left(\omega ,\left(x,y,z\right)\right)≔\left(-x,-y,z\right).$

To clarify the ideas presented in this section, we will provide a numerical simulation with some graphs. Note that the parameter will be represented by the order triplet $\gamma =(\rho ,\sigma ,\beta )$ for every figure. If not specified, $\sigma$ and $\beta$ are fixed at $\sigma =10$ and $\beta =8/3$.

MatLab’s ODE45 (version 2020) is utilized for every simulation, with an error allowed of $1\times {10}^{-9}$. A black point indicates where the initial conditions are in each graph. Unless otherwise indicated, $T=\left[0,100\right]$ is the time interval displayed for every graph.

For different initial conditions and $\mathrm{\rho }:\mathrm{\Omega }⟶(0,1)$, a number of tests demonstrate global convergence. One can see the bifurcation as $\rho$ increases from $0.9$ to $1.2$. For $\rho >1$, the origin is no longer stable.

Figure 1 illustrates how a small alteration to the starting condition can result in a significant shift in the trajectory. The $x$-coordinate versus time is shown on the right side of 6.1. Up until roughly $t=40$, the trajectory is almost stable.

In Figure 2, we observe from the graphs that, when the equation includes a random element, sudden changes in long-term patterns can be noticed. The graph shows stability for a certain period, followed by a sudden change in behavior due to random fluctuations. Additionally, irregular fluctuations appear. Consequently, these fluctuations are unpredictable, which can make the movement more erratic. The graph illustrates how the paths continuously change due to randomness.

8. Conclusions

In this article, the parallelism in stochastic dynamics was studied through random Lyapunov function, dispersion, and prolongational limit set. A number of new results were reached, highlighting the impact of parallelism on the long-term behavior of stochastic dynamical systems. We review some of them here. Any random variable that is not a random fixed point admits a tube. The relationship between the concepts, a prolongational limit set and a section, is discussed, and we came to the conclusion stated in Theorem 6. Also, we prove that a closed random set is asymptotically stable if and only if there is a Lyapunov function that satisfies the conditions (i) and (ii) of Theorem 9. A characterization of the concept of parallelizable RDS is given in the sense of the concept of the RDS section, and this is explained in Theorem 10. Theorem 11 shows that for any RDS, the parallelizable and dispersive are equivalent when the phase space is a locally compact separable Banach space. However, in general, parallelizability is not implied by dispersivity. The relationship between the parallelizable property of RDS and the prolongational limit set and the prolongation set, on the one hand, and the relationship of the parallelism property to the dispersion property contained in ref. [21], on the other hand, has been shown. In addition, we find the relationship between the parallelizable and the region of attraction of the asymptotically stable closed invariant random set; this is stated in Corollary 1. One of the important results we have reached during this research is the relationship between parallelizable RDS and the region of attraction of the uniformly asymptotically stable closed invariant random set. This is stated in Theorem 12.

Also, we show that the parallelizability, dispersivity, and asymptotic stability are invariant under symmetry.

Finally, by using an appropriate stochastic Lyapunov function, we conclude that the RDS generated by the random Lorenz system is parallelizable.

The study demonstrates, through the introduction of the random Lorenz system, that there are significant applications for the notion of parallelism. Looking at the future prospects, upcoming research may contribute to the development of the theoretical foundation of the parallelism, in addition to developing effective and important practical models. This may be possible through the study of the parallelizability for RDSs defined on the plane as phase space and proving that such an RDS is either parallelizable, or it has a single random fixed point, which is a global Poincaré center, or the set of all random fixed points is equal to the plane.