1 Introduction
Periodic solutions are a very active research topic of the qualitative theory of ordinary differential equations. Given a dynamical system, it is important to investigate the existence, number, and positions of periodic solutions as well as the behavior of their nearby trajectories. For example, for polynomial vector fields in the plane, it is essential to derive an upper bound of the number of limit cycles and discuss their relative positions. This is the second part of Hilbert’s 16th problem. Another example is the famous Poincaré–Bendixson theorem. It plays a fundamental role in the qualitative theory of differential equations in the plane because it provides a useful method to check the existence of periodic solutions and to find their positions (cf. [Reference Bendixson3, Reference Hale10, Reference Nemytskii and Stepanov18, Reference Poincaré19]). In the past 50 years, a lot of progress has been made for periodic solutions and the global structure of dynamical systems (cf. [Reference Frances9, Reference Ji, Qi, Shen and Yi11, Reference Li, Liu and Wang13, Reference Liu and Wang15, Reference Wang and Yu21–Reference Yu23, Reference Zheng, Xia and Zheng26]).
When we study random dynamical systems, it is natural to consider the counterparts of fixed points and periodic solutions. In the literature, the counterparts are called stationary solutions and random periodic solutions, respectively. Stationary solutions have attracted lots of attention, and a series of results have been obtained (cf. [Reference Arnold2, Reference Liu and Zhao14, Reference Mohammed, Zhang and Zhao17, Reference Zhang and Zhao24]). In [Reference Zhao and Zheng25], Zhao and Zheng introduced for the first time the concept of random periodic solutions and gave a sufficient condition for their existence. In [Reference Feng and Zhao7], Feng and Zhao introduced random periodic measures and discussed the close relationship between random periodic solutions and random periodic measures. We call the reader’s attention to [Reference Dong, Zhang and Zheng5, Reference Dong, Zhang and Zheng6, Reference Feng, Zhao and Zhou8] for some other recent works on random periodic solutions.
Note that the period T of the random periodic solutions introduced in [Reference Zhao and Zheng25] is deterministic and uniform for all random paths
$\omega $
. However, for many random dynamical systems induced by random or stochastic differential equations, the solutions exhibit some periodic behaviors, while the periods depend on
$\omega $
. To deal with these phenomena, we introduce in this paper the novel concept of weak random periodic solutions. It is easy to see that any random periodic solution is a weak random periodic solution. But, in general, a weak random periodic solution might not be a random periodic solution.
The remainder of this paper is organized as follows. In Section 2, we introduce the concept of weak random periodic solutions and present a useful criterion for their existence. In Section 3, we give the definition of weak random periodic measures and show that the existence of weak random periodic solutions implies the existence of weak random periodic measures. Further, we establish the existence of invariant measures for random dynamical systems by virtue of their weak random periodic solutions. In Section 4, we use concrete examples to illustrate the weak random periodic phenomena of dynamical systems induced by random and stochastic differential equations.
2 Definition and existence of weak random periodic solutions
First, let us recall the concepts of fixed point and periodic solution. Let E be a Polish space with Borel
$\sigma $
-algebra
${\mathcal {B}}(E)$
. For a deterministic dynamical system
$\Psi :\mathbb {R}\times E\rightarrow E$
, a fixed point is a point
$x\in E$
such that
$$ \begin{align*}\Psi(t)x=x,\ \ \ \ \forall t\in\mathbb{R}. \end{align*} $$
A periodic solution with period
$T>0$
is a
${\mathcal {B}}(\mathbb {R})$
-measurable function
$Y: \mathbb {R}\rightarrow E$
such that
$$ \begin{align*}\Psi(t)Y(s)=Y(t+s),\ \ Y(t+T)=Y(t),\ \ \ \ \forall s,t\in\mathbb{R}. \end{align*} $$
Suppose that
$\Psi :\mathbb {R}\times \Omega \times E\rightarrow E$
is a measurable random dynamical system on
$(E,{\mathcal {B}}(E))$
over a metric dynamical system
$(\Omega ,{\mathcal {F}},P,(\theta _t)_{t\in \mathbb {R}})$
. Then, for
$\omega \in \Omega $
,
$$ \begin{align} \Psi(0,\omega)=\mathrm{id}_E,\ \ \Psi(t+s,\omega)=\Psi(t,\theta_s\omega)\Psi(s,\omega),\ \ \ \ \forall s,t\in\mathbb{R}. \end{align} $$
A stationary solution (cf. [Reference Arnold2]) of
$\Psi $
is a random variable
$Y: {\Omega }\rightarrow E$
such that for almost all
$\omega \in \Omega $
,
$$ \begin{align*}\Psi(t,\omega)Y(\omega)=Y(\theta_t\omega),\ \ \ \ \forall t\in\mathbb{R}. \end{align*} $$
A random periodic solution with period
$T>0$
(see [Reference Feng and Zhao7, Reference Zhao and Zheng25]) is a
${\mathcal {B}}(\mathbb {R})\times {\mathcal F}$
-measurable function
$Y:\mathbb {R}\times \Omega \rightarrow E$
such that for almost all
$\omega \in \Omega $
,
$$ \begin{align*}\Psi(t,\theta_s\omega)Y(s,\omega)=Y(t+s,\omega),\ \ Y(s+T,\omega)=Y(s,\theta_T\omega),\ \ \ \ \forall s,t\in\mathbb{R}. \end{align*} $$
The period T in the above definition is nonrandom. For many applications, this is not satisfactory. Here is a simple example. Suppose that
$X(\varpi )$
is a positive random variable. Consider the following random differential equation (RDE):
$$ \begin{align} \frac{d^2x(t)}{dt^2}=\sin(X(\varpi)(t+s)), \end{align} $$
where
$s\in \mathbb {R}$
. The periodic solution of (2.2) is given by
$$ \begin{align*}x(t)=\frac{-\sin(X(\varpi)( t+s))}{X(\varpi)^2}+r,\ \ \ \ r\in\mathbb{R}, \end{align*} $$
whose period
$T\varpi =\frac {2\pi }{X(\varpi )}$
is random.
To deal with the phenomenon of random periods, we now introduce the concept of weak random periodic solution of a random dynamical system.
Definition 2.1 A weak random periodic solution of
$\Psi $
is a pair of measurable maps
$Y: \mathbb {R}\times \Omega \rightarrow E$
and
$T:\Omega \rightarrow (0,\infty )$
such that for almost all
$\omega \in \Omega $
,
$$ \begin{align} \Psi(t,\theta_s\omega)Y(s,\omega)=Y(t+s,\omega),\ \ Y(s+T\omega,\theta_{-T\omega}\omega)=Y(s,\omega),\ \ \ \ \forall s,t\in\mathbb{R}. \end{align} $$
Obviously, if T is a constant map, then the weak random periodic solution is reduced to the random periodic solution. For the existence of weak random periodic solutions, we have the following useful criterion.
Proposition 2.2 If there exist measurable maps
$Y_0:\Omega \rightarrow E$
and
$T:\Omega \rightarrow (0,\infty )$
such that for almost all
$\omega \in \Omega $
,
$$ \begin{align} Y_0(\omega)=\Psi(T\omega,\theta_{-T\omega}\omega)Y_0(\theta_{-T\omega}\omega), \end{align} $$
then the random dynamical system
$\Psi $
has a weak random periodic solution.
Proof For
$\omega \in \Omega $
, define
$Y(0,\omega )=Y_0(\omega )$
and
$$ \begin{align} Y(t,\omega):=\Psi(t,\omega)Y(0,\omega), \ \ t\in \mathbb{R}. \end{align} $$
Then, by (2.1), (2.4), and (2.5), we obtain that for almost all
$\omega \in \Omega $
,
$$ \begin{align*} \Psi(t,\theta_s\omega)Y(s,\omega)&=\Psi(t,\theta_s\omega)\Psi(s,\omega)Y(0,\omega)\\ &=\Psi(t+s,\omega)Y(0,\omega)\\ &=Y(t+s,\omega),\ \ \ \ \forall s,t\in\mathbb{R}, \end{align*} $$
and
$$ \begin{align*} Y(s,\omega)&=\Psi(s,\omega)Y(0,\omega)\\ &=\Psi(s,\omega)\Psi(T\omega,\theta_{-T\omega}\omega)Y(0,\theta_{-T\omega}\omega)\\ &=\Psi(s+T\omega,\theta_{-T\omega}\omega)Y(0,\theta_{-T\omega}\omega)\\ &=Y(s+T\omega,\theta_{-T\omega}\omega),\ \ \ \ \forall s\in \mathbb{R}. \end{align*} $$
Therefore,
$(Y,T)$
is a weak random periodic solution of
$\Psi $
.
We next consider the weak random periodic solution of a stochastic semiflow. Denote
$$ \begin{align*}\Delta=\{(t,s)\in \mathbb{R}^2:s\le t\}. \end{align*} $$
Let
$\varphi :\Delta \times \Omega \times E \rightarrow E$
be a stochastic semiflow. Then, for
$\omega \in \Omega $
,
$$ \begin{align} \varphi(t,s,\omega)=\varphi(t,u,\omega)\circ\varphi(u,s,\omega),\ \ \ \ \forall s\le u\le t, \end{align} $$
and
$$ \begin{align*}\varphi(s,s,\omega)=\mathrm{id}_E,\ \ \ \ \forall s\in \mathbb{R}. \end{align*} $$
Definition 2.3 A weak random periodic solution of
$\varphi $
is a pair of measurable maps
$Y: \mathbb {R}\times \Omega \rightarrow E$
and
$T:\Omega \rightarrow (0,\infty )$
such that for almost all
$\omega \in \Omega $
,
$$ \begin{align} \varphi(t,s,\omega)Y(s,\omega)=Y(t,\omega),\ \ Y(s+T\omega,\theta_{-T\omega}\omega)=Y(s,\omega),\ \ \ \ \forall s\le t. \end{align} $$
3 Weak random periodic measures and invariant measures
Let
$\Psi $
be a measurable random dynamical system. Define
$$ \begin{align*}\Upsilon_t(\omega,x)=(\theta_t\omega,\Psi(t,\omega)x),\ \ \ \ \omega\in\Omega,\,x\in E,\, t\in\mathbb{R}. \end{align*} $$
Denote by
${\mathcal {P}}(\Omega \times E)$
the set of all probability measures on
$(\Omega \times E, \mathcal {F}\otimes \mathcal {B}(E))$
.
Definition 3.1 A weak random periodic probability measure of
$\Psi $
is a pair of measurable maps
$\mu : \mathbb {R}\times \Omega \rightarrow {\mathcal {P}}(\Omega \times E)$
and
$T:\Omega \rightarrow (0,\infty )$
such that for almost all
$\omega \in \Omega $
,
$$ \begin{align*}{\Upsilon}_t\mu(s,\omega)=\mu({t+s},\omega), \ \ \mu(s+T\omega,\theta_{-T\omega}\omega)=\mu(s,\omega), \ \ \ \ \forall s,t\in\mathbb{R}. \end{align*} $$
Theorem 3.2 If a random dynamical system
$\Psi :\mathbb {R}\times \Omega \times E\rightarrow E$
has a weak random periodic solution
$Y:\mathbb {R}\times \Omega \rightarrow E$
and
$T:\Omega \rightarrow (0,\infty )$
, then it has a weak random periodic probability measure. Additionally, if for almost all
$\omega \in \Omega $
,
$$ \begin{align} T\omega=T(\theta_s\omega),\ \ \ \ \forall s\in \mathbb{R}, \end{align} $$
then
$\Psi $
has an invariant probability measure whose random factorization is supported by
$$ \begin{align*}L^{\omega}:=\{Y(s,\theta_{-s}\omega):s\in[0,T\omega)\}. \end{align*} $$
Proof For
$s\in \mathbb {R}$
and
$\omega \in \Omega $
, define
$$ \begin{align*} \mu(s,\omega)(A)=\delta_{Y(s,\omega)}(A_{\theta_s\omega}),\ \ \ \ A\in \mathcal{F}\otimes\mathcal{B}(E), \end{align*} $$
where
$A_{\omega }$
is the
$\omega $
-section of A. Then,
$\mu (s,\omega )\in {\mathcal {P}}(\Omega \times E)$
.
We have
$$ \begin{align*} (\Upsilon^{-1}_t(A))_{\omega}=\{x:(\theta_t\omega,\Psi(t,\omega)x)\in A\}=\{x:\Psi(t,\omega)x\in A_{\theta_t\omega}\}=\Psi^{-1}(t,\omega)A_{\theta_t\omega}. \end{align*} $$
Then, by (2.3), we obtain that for almost all
$\omega \in \Omega $
,
$$ \begin{align*} {\Upsilon}_t\mu(s,\omega)(A) &=\mu(s,\omega)({\Upsilon}^{-1}_t(A))\nonumber\\ &=\delta_{Y(s,\omega)}(({\Upsilon}^{-1}_t(A))_{\theta_s\omega})\nonumber\\ &=\delta_{\Psi(t,\theta_s\omega)Y(s,\omega)}(A_{\theta_{t+s}\omega})\nonumber\\ &=\delta_{Y(t+s,\omega)}(A_{\theta_{t+s}\omega})\nonumber\\ &=\mu({t+s},\omega)(A) \end{align*} $$
and
$$ \begin{align*} \mu({s+T\omega},\theta_{-T\omega}\omega)(A) &=\delta_{Y({s+T\omega},\theta_{-T\omega}\omega)}(A_{\theta_{s}\omega})\nonumber\\ &=\delta_{Y(s,\omega)}(A_{\theta_{s}\omega})\nonumber\\ &=\mu({s},\omega)(A). \end{align*} $$
Thus,
$\mu $
is a weak random periodic probability measure of
$\Psi $
.
For
$A\in \mathcal {F}\otimes \mathcal {B}(E)$
, define
$$ \begin{align*}\tilde{\mu}(A):=\int_{\Omega}\frac{1}{T\omega}\int_0^{T\omega}\mu(s,\omega)(A)dsP(d\omega). \end{align*} $$
By (2.3), (3.1), and the measure preserving property of
$\{\theta _t\}$
, we get
$$ \begin{align*} \tilde{\mu}(A)=&\int_{\mathbb{R}}\int_{\Omega}\frac{\delta_s{([0,T\omega])}\cdot\delta_{Y(s,\omega)}(A_{\theta_{s}\omega})}{T\omega}P(d\omega)ds\\ =&\int_{\mathbb{R}}\int_{\Omega}\frac{\delta_s{([0,T\omega])}\cdot\delta_{Y(s+T\omega,\theta_{-T\omega}\omega)}(A_{\theta_{s}\omega})}{T\omega}P(d\omega)ds\\ =&\int_{\mathbb{R}}\int_{\Omega}\frac{\delta_s{([0,T\omega])}\cdot\delta_{Y(s+T\omega,\theta_{-s-T\omega}\omega)}(A_{\omega})}{T\omega}P(d\omega)ds\\ =&\int_{\Omega}\int_{\mathbb{R}}\frac{\delta_s{([0,T\omega])}\cdot\delta_{Y(s+T\omega,\theta_{-s-T\omega}\omega)}(A_{\omega})}{T\omega}dsP(d\omega)\\ =&\int_{\Omega}\int_{\mathbb{R}}\frac{\delta_s{([T\omega,2T\omega])}\cdot\delta_{Y(s,\theta_{-s}\omega)}(A_{\omega})}{T\omega}dsP(d\omega)\\ =&\int_{\mathbb{R}}\int_{\Omega}\frac{\delta_s{([T\omega,2T\omega])}\cdot\delta_{Y(s,\theta_{-s}\omega)}(A_{\omega})}{T\omega}P(d\omega)ds\\ =&\int_{\mathbb{R}}\int_{\Omega}\frac{\delta_s{([T\omega,2T\omega])}\cdot\delta_{Y(s,\omega)}(A_{\theta_{s}\omega})}{T\omega}P(d\omega)ds\\ =&\int_{\Omega}\frac{1}{T\omega}\int_{T\omega}^{2T\omega}\mu(s,\omega)(A)dsP(d\omega). \end{align*} $$
Repeating this argument, we can show that
$$ \begin{align*} \tilde{\mu}(A)=\int_{\Omega}\frac{1}{T\omega}\int_{k(T\omega)}^{(k+1)(T\omega)}\mu(s,\omega)(A)dsP(d\omega),\ \ \ \ \forall k\in\mathbb{N}, \end{align*} $$
which implies that
$$ \begin{align} \tilde{\mu}(A)=&\int_{\Omega}\lim_{N\rightarrow\infty}\frac{1}{N}\int_0^{N}\delta_{Y(s,\omega)}(A_{\theta_{s}\omega})dsP(d\omega)\nonumber\\ =&\lim_{N\rightarrow\infty}\frac{1}{N}\int_0^{N}\int_{\Omega}\delta_{Y(s,\omega)}(A_{\theta_{s}\omega})P(d\omega)ds. \end{align} $$
By (2.3), (3.2), and the measure preserving property of
$\{\theta _t\}$
, we obtain that
$$ \begin{align*} \Upsilon_t\tilde{\mu}(A)&=\tilde{\mu}(\Upsilon^{-1}_t(A))\\ &=\lim_{N\rightarrow\infty}\frac{1}{N}\int_0^{N}\int_{\Omega}\delta_{Y(s,\omega)}((\Upsilon^{-1}_t(A))_{\theta_{s}\omega})P(d\omega)ds\\ &=\lim_{N\rightarrow\infty}\frac{1}{N}\int_0^{N}\int_{\Omega}\delta_{\Psi(t,\theta_s\omega)Y(s,\omega)}(A_{\theta_{t+s}\omega})P(d\omega)ds\\ &=\lim_{N\rightarrow\infty}\frac{1}{N}\int_0^{N}\int_{\Omega}\delta_{Y(t+s,\omega)}(A_{\theta_{t+s}\omega})P(d\omega)ds\\ &=\lim_{N\rightarrow\infty}\frac{1}{N}\int_0^{N}\int_{\Omega}\delta_{Y(s,\omega)}(A_{\theta_{s}\omega})P(d\omega)ds\\ &=\tilde{\mu}(A). \end{align*} $$
Let
$\pi _{\Omega }:\Omega \times E\rightarrow \Omega $
,
$\pi _{\Omega }(\omega ,x)=\omega $
, be the projection onto
$\Omega $
. By (3.2) and the measure preserving property of
$\{\theta _t\}$
, we get
$\tilde {\mu }\circ \pi ^{-1}_{\Omega }=P$
. Hence,
$\tilde {\mu }$
is an invariant probability measure of
$\Psi $
(cf. [Reference Arnold2, Definition 1.4.1]).
By (3.1) and the measure preserving property of
$\{\theta _t\}$
, we get
$$ \begin{align*} \tilde{\mu}(A)=&\int_{\mathbb{R}}\int_{\Omega}\frac{\delta_s{([0,T\omega])}\cdot\delta_{Y(s,\omega)}(A_{\theta_{s}\omega})}{T\omega}P(d\omega)ds\\ =&\int_{\mathbb{R}}\int_{\Omega}\frac{\delta_s{([0,T\omega])}\cdot\delta_{Y(s,\theta_{-s}\omega)}(A_{\omega})}{T\omega}P(d\omega)ds\\ =&\int_{\Omega}\frac{1}{T\omega}\int_0^{T\omega}\delta_{Y(s,\theta_{-s}\omega)}(A_{\omega})dsP(d\omega). \end{align*} $$
Then, the random factorization of
$\tilde {\mu }$
is given by
$$ \begin{align*}(\tilde{\mu})_{\omega}=\frac{1}{T\omega}\int_0^{T\omega}\delta_{{Y}(s,\theta_{-s}\omega)}ds, \end{align*} $$
which is supported by
$L^{\omega }$
. Therefore, the proof is complete.
We now consider weak random periodic measures and invariant measures of a semiflow
$\varphi $
. Define
$\overline {E}:=\mathbb {R}\times E$
and
$$ \begin{align*}\overline{\Psi}(t,\omega)(s,x)=(t+s,\varphi(t+s,s,\theta_{-s}\omega)x),\ \ \ \ \omega\in\Omega,\,s\in\mathbb{R},\, x\in E,\, t\ge0. \end{align*} $$
Then,
$\overline {\Psi }:[0,\infty )\times \Omega \times \overline {E}\rightarrow \overline {E}$
is a measurable random dynamical system on
$(\overline {E},{\mathcal {E}}(\overline {E}))$
over the metric dynamical system
$(\Omega ,{\mathcal {F}},P,(\theta _t)_{t\in \mathbb {R}})$
. Assume that
$\varphi $
has a weak random periodic solution
$(Y,T)$
. Define
$$ \begin{align*}\overline{Y}(s,\omega)=(s,Y(s,\omega)),\ \ \ \ \omega\in\Omega,\,s\in\mathbb{R}, \end{align*} $$
and
$$ \begin{align*}\eta_t(s,x)=(t+s,x),\ \ \ \ \omega\in \Omega,\, s,t\in\mathbb{R},\, x\in E. \end{align*} $$
Then, by (2.7), we obtain that for almost all
$\omega \in \Omega $
,
$$ \begin{align} \overline{\Psi}(t,\theta_s\omega)\overline{Y}(s,\omega)=\overline{Y}(t+s,\omega),\ \ \overline{Y}(s+T\omega,\theta_{-T\omega}\omega)=\eta_{T\omega}\circ\overline{Y}(s,\omega),\ \ s\in\mathbb{R},\, t\ge0.\ \ \ \ \end{align} $$
Define
$$ \begin{align*}\overline{\Upsilon}_t(\omega,s,x)=(\theta_t\omega,\overline{\Psi}(t,\omega)(s,x)),\ \ \ \ \omega\in\Omega,\,s\in\mathbb{R},\, x\in E,\, t\ge0, \end{align*} $$
and
$$ \begin{align*}\tilde{\eta}_t(\omega,s,x)=(\omega,t+s,x),\ \ \ \ \omega\in\Omega,\,s,t\in\mathbb{R},\, x\in E. \end{align*} $$
Denote by
${\mathcal {P}}(\Omega \times \overline {E})$
the set of all probability measures on
$(\Omega \times \overline {E}, \mathcal {F}\otimes \mathcal {B}(\overline {E}))$
. Let
$\pi _{\Omega }:\Omega \times \overline {E}\rightarrow \Omega $
,
$\pi _{\Omega }(\omega ,(s,x))=\omega $
, be the projection onto
$\Omega $
.
Definition 3.3 A weak random periodic probability measure of
$\varphi $
is a pair of measurable maps
$\mu : \mathbb {R}\times \Omega \rightarrow {\mathcal {P}}(\Omega \times \overline {E})$
and
$T:\Omega \rightarrow (0,\infty )$
such that for almost all
$\omega \in \Omega $
,
$$ \begin{align*}\overline{\Upsilon}_t\mu(s,\omega)=\mu({t+s},\omega), \ \ \mu({s+T\omega},\theta_{-T\omega}\omega)\circ \tilde{\eta}_{T\omega}=\mu(s,\omega), \ \ \ \ \forall s\in\mathbb{R},\, t\ge0. \end{align*} $$
Theorem 3.4 If a stochastic semiflow
$\varphi :\Delta \times \Omega \times E \rightarrow E$
has a weak random periodic solution
$Y:\mathbb {R}\times \Omega \rightarrow E$
and
$T:\Omega \rightarrow (0,\infty )$
, then it has a weak random periodic probability measure. If in addition (3.1) holds for almost all
$\omega \in \Omega $
, then there exists a weak-invariant probability measure
$\tilde {\mu }$
on
$\mathcal {F}\otimes \mathcal {B}(\overline {E})$
satisfying
$\tilde {\mu }\circ \pi ^{-1}_{\Omega }=P$
,
$$ \begin{align*}\overline{\Upsilon}_t\tilde{\mu}(A)=\tilde{\mu}(A),\ \ \ \ \forall A\in \mathcal{F}\otimes\{\emptyset,\mathbb{R}\}\times \mathcal{B}({E}),\,t\ge 0, \end{align*} $$
and its random factorization is supported by
$$ \begin{align*}L^{\omega}:=\{\overline{Y}(s,\theta_{-s}\omega):s\in[0,T\omega)\}. \end{align*} $$
Proof For
$s\in \mathbb {R}$
and
$\omega \in \Omega $
, define
$$ \begin{align*} \mu(s,\omega)(A)=\delta_{\overline{Y}(s,\omega)}(A_{\theta_{s}\omega}),\ \ \ \ A\in \mathcal{F}\otimes\mathcal{B}(\overline{E}), \end{align*} $$
where
$A_{\omega }$
is the
$\omega $
-section of A. Then,
$\mu (s,\omega )\in {\mathcal {P}}(\Omega \times \overline {E})$
.
We have
$$ \begin{align*} (\overline{\Upsilon}^{-1}_t(A))_{\omega}&=\{(s,x):(\theta_t\omega,\overline{\Psi}(t,\omega)(s,x))\in A\}\\ &=\{(s,x):\overline{\Psi}(t,\omega)(s,x)\in A_{\theta_t\omega}\}\\ &=\overline{\Psi}^{-1}(t,\omega)A_{\theta_t\omega}. \end{align*} $$
Then, by (3.3), we obtain that for almost all
$\omega \in \Omega $
,
$$ \begin{align*} {\overline{\Upsilon}}_t\mu(s,\omega)(A) &=\mu(s,\omega)({\overline{\Upsilon}}^{-1}_t(A))\nonumber\\ &=\delta_{\overline{Y}(s,\omega)}(({\Upsilon}^{-1}_t(A))_{\theta_{s}\omega})\nonumber\\ &=\delta_{\overline{\Psi}(t,\theta_{s}\omega)\overline{Y}(s,\omega)}(A_{\theta_{t+s}\omega})\nonumber\\ &=\delta_{\overline{Y}(t+s,\omega)}(A_{\theta_{t+s}\omega})\nonumber\\ &=\mu({t+s},\omega)(A) \end{align*} $$
and
$$ \begin{align*} \mu({s+T\omega},\theta_{-T\omega}\omega)(\tilde{\eta}_{T\omega} A)&=\delta_{\overline{Y}({s+T\omega},\theta_{-T\omega}\omega)}((\tilde{\eta}_{T\omega}A)_{\theta_{s}\omega})\nonumber\\ &=\delta_{\eta_{T\omega}\circ \overline{Y}(s,\omega)}(\eta_{T\omega}A_{\theta_{s}\omega})\\ &=\delta_{\overline{Y}(s,\omega)}(A_{\theta_{s}\omega})\nonumber\\ &=\mu({s},\omega)(A). \end{align*} $$
Thus,
$\mu $
is a weak random periodic probability measure of
$\varphi $
.
For
$A\in \mathcal {F}\otimes \mathcal {B}(\overline {E})$
, define
$$ \begin{align*}\tilde{\mu}(A):=\int_{\Omega}\frac{1}{T\omega}\int_0^{T\omega}\mu(s,\omega)(A)dsP(d\omega). \end{align*} $$
Then, by using (3.3) and following the same argument of the proof of Theorem 3.2, we can complete the proof.
4 Examples
In this section, we use examples to illustrate the weak random periodic phenomena of dynamical systems induced by random and stochastic differential equations.
First, we investigate the periodic behavior of RDEs of type (2.2) by virtue of weak random periodic solutions.
Example 4.1 Let
$X(\varpi )$
be a positive random variable and
$a_k,b_k\in \mathbb {R}$
,
$1\le k\le N$
, for some
$N\in \mathbb {N}$
. Consider the following RDE:
$$ \begin{align} \frac{d^2x(t)}{dt^2}=\sum_{k=1}^N[a_k\sin(kX(\varpi)(t+s))+b_k\cos(kX(\varpi)(t+s))], \end{align} $$
where
$s\in \mathbb {R}$
. Note that equation (4.1) is equivalent to
$$ \begin{align} \left\{ \begin{array}{@{}l} dx_1(t)=x_2(t)dt,\\ dx_2(t)=\left\{\sum_{k=1}^N[a_k\sin(kX(\varpi)(t+s))+b_k\cos(kX(\varpi)(t+s))]\right\}dt. \end{array} \right. \end{align} $$
Denote by
$\nu $
the distribution of
$X(\varpi )$
. Define
$$ \begin{align*}V=\left\{(x,y):x\in(0,\infty),\,y\in\left[0,\frac{2\pi}{x}\right)\right\}. \end{align*} $$
We equip
$(V,{\mathcal {B}}(V))$
with the probability measure
$P_V$
:
$$ \begin{align*}P_V(A)=\int_0^{\infty}\int_0^{\infty}\frac{x1_A(x,y)}{2\pi}dy\nu(dx),\ \ \ \ A\in {\mathcal{B}}(V). \end{align*} $$
Define
$$ \begin{align*}g_{x,y}(t)=\sum_{k=1}^N[a_k\sin(kx(t+y))+b_k\cos(kx(t+y))],\ \ \ \ t\in \mathbb{R},\,(x,y)\in V, \end{align*} $$
and
$$ \begin{align*}\Omega=\{g_{x,y}:(x,y)\in V\}. \end{align*} $$
Set
$J:V\mapsto \Omega , J(x,y)=g_{x,y}$
. Define
$$ \begin{align*}{\mathcal{F}}=J({\mathcal{B}}(V)),\ \ P=P_V\circ J^{-1}, \end{align*} $$
and
$$ \begin{align*}\theta_t\omega(s)=\omega(t+s),\ \ \ \ \omega\in\Omega,\,s,t\in\mathbb{R}. \end{align*} $$
Then,
$(\Omega ,{\mathcal {F}},P,(\theta _t)_{t\in \mathbb {R}})$
is a metric dynamical system and equation (4.2) is equivalent to the following RDE:
$$ \begin{align} \left\{ \begin{array}{@{}l} dx_1(t)=x_2(t)dt,\\ dx_2(t)=\omega(t)dt. \end{array} \right. \end{align} $$
The random dynamical system
$\Psi :\mathbb {R}\times \Omega \times \mathbb {R}^2\rightarrow \mathbb {R}^2$
induced by equation (4.3) is given by
$$ \begin{align*} &\Psi(t,g_{x,y})(x_1,x_2)\\=&\left( x_1+x_2t-\sum_{k=1}^{N}\frac{a_k[\sin(kx (t+y) )-kxt\cos(kxy)-\sin(kxy)]}{k^2x^2}\right.\\&\ \ \ \ \ \ -\sum_{k=1}^{N}\frac{b_k[\cos(kx (t+y) )+kxt\sin(kxy)-\cos(kxy)]}{k^2x^2},\\&\left.\ \ \ x_2+\sum_{k=1}^{N}\frac{-a_k[\cos(kx (t+y) )-\cos(kxy)]+b_k[\sin(kx (t+y) )-\sin(kxy)]}{kx}\right), \end{align*} $$
where
$t\in \mathbb {R}$
,
$(x,y)\in V$
, and
$(x_1,x_2)\in \mathbb {R}^2$
. Fix a
${\mathcal {B}}(\mathbb {R})$
-measurable function
$h:\mathbb {R}\rightarrow \mathbb {R}$
. For
$t\in \mathbb {R}$
and
$(x,y)\in V$
, define
$$ \begin{align} &Y(t,g_{x,y})\nonumber\\=&\left(h(x) -\sum_{k=1}^{N}\frac{a_k\sin(kx (t+y) )+b_k\cos(kx (t+y) )}{k^2x^2},\right.\nonumber\\&\left.\ \ \ \sum_{k=1}^{N}\frac{-a_k\cos(kx (t+y) )+b_k\sin(kx (t+y) )}{kx}\right)\end{align} $$
and
$$ \begin{align*}Tg_{x,y}=\frac{2\pi}{x}. \end{align*} $$
Then,
$(Y,T)$
is a weak random periodic solution of
$\Psi $
. Further, by Theorem 3.2, we know that
$\Psi $
has an invariant probability measure.
Remark 4.2 Theorem 3.2 shows that, if a random dynamical system has a weak random periodic solution, then it has an invariant probability measure induced by this solution. On the other hand, Example 4.1 shows that different weak random periodic solutions can be obtained for the same random dynamical system through choosing different functions h in (4.4). It is interesting to consider how weak random periodic solutions affect the ergodicity of random dynamical systems.
Next, we consider a system of RDEs driven by periodic multiplicative noises.
Example 4.3 Suppose that
$d\ge 1$
. Denote by
$C(\mathbb {R};\mathbb {R}^d)$
and
$C^1(\mathbb {R};\mathbb {R}^d)$
the spaces of all continuous and continuously differentiable
$\mathbb {R}^d$
-valued functions on
$\mathbb {R}$
, respectively. We equip
$C(\mathbb {R};\mathbb {R}^d)$
with topology of locally uniform convergence. Define
$\Omega =C^1(\mathbb {R};\mathbb {R}^d)$
and
$$ \begin{align*}{\mathcal{F}}=\{A\cap \Omega:A\in{\mathcal{B}}(C(\mathbb{R};\mathbb{R}^d))\}. \end{align*} $$
For
$\omega =(\omega _1,\dots ,\omega _d)\in \Omega $
and
$s,t\in \mathbb {R}$
, set
$(\theta _t\omega )(s)= \omega (t + s)-\omega (t)$
.
We choose
$\omega ^1,\omega ^2,\dots \in \Omega $
with periods
$T_1<T_2<\cdots $
, respectively, and
$a_1,a_2,\dots \in (0,\infty )$
satisfying
$\sum _{n=1}^{\infty }a_n=1$
. Define
$$ \begin{align*}\Omega_n:=\{\theta_t\omega^n:0\le t<T_n\},\ \ \ \ n\in\mathbb{N}. \end{align*} $$
Denote by
${\mathcal {L}}$
the Lebesgue measure on
$\mathbb {R}$
. We define a probability measure P on
$(\Omega ,{\mathcal {F}})$
by
$$ \begin{align*}P\left(\Omega\setminus \bigcup_{n=1}^{\infty}\Omega_n\right)=0, \end{align*} $$
and
$$ \begin{align*}P(\{\theta_t\omega_n:t\in A\})=\frac{a_n{\mathcal{L}}(A)}{T_n},\ \ \ \ \forall A\in {\mathcal{B}}([0,T_n)),\,n\in\mathbb{N}. \end{align*} $$
Set
$$ \begin{align*}T\omega=T_n,\ \ \ \ \forall \omega\in \Omega_n,\,n\in\mathbb{N}, \end{align*} $$
and
$$ \begin{align*}T\omega=1,\ \ \ \ \forall\omega\notin\bigcup_{n=1}^{\infty}\Omega_n. \end{align*} $$
Then,
$\{\theta _t\}$
are P-measure preserving and
$T:\Omega \rightarrow (0,\infty )$
is a measurable map such that for almost all
$\omega \in \Omega $
,
$$ \begin{align*}\omega(s+T\omega)=\omega(s),\ \ \ \ \forall s\in\mathbb{R}. \end{align*} $$
Let A be a
$d\times d$
hyperbolic matrix and
$\sigma =(\sigma _{ij})_{1\le i,j\le d}$
with
$\sigma _{ij}:\mathbb {R}^d\rightarrow \mathbb {R}$
being Lipschitz-continuous and satisfying
$\sigma _{ij}(x)=o(|x|)$
as
$|x|\rightarrow \infty $
. Consider the RDE
$$ \begin{align} dx(t)=Ax(t)dt+\sigma(x(t))d\omega(t). \end{align} $$
By [Reference Amann1, Theorem 22.1], we know that (4.5) has a
$T_n$
-periodic solution for each
$\omega ^n$
, which is denoted by
$x(t,\omega ^n)$
. For
$s,t\in \mathbb {R}$
, define
$$ \begin{align*}Y(t,\theta_s\omega^n)=x(t+s,\omega^n). \end{align*} $$
Then,
$(Y,T)$
is a weak random periodic solution of the random dynamical system induced by equation (4.5). Further, by Theorem 3.2, we know that this random dynamical system has an invariant probability measure.
The third example is concerned with a random dynamical system induced by stochastic differential equations (SDEs), which is an extension of the example given by Zhao and Zheng (see [Reference Zhao and Zheng25, Section 2]).
Example 4.4 Let
$\Omega :=C(\mathbb {R};\mathbb {R})$
and
$\{\omega (t)\}_{t\in \mathbb {R}}$
be a one-dimensional two-sided Brownian motion on the path space
$(\Omega , {\mathcal {B}}(\Omega ),P)$
with
$\theta $
being the shift operator
$(\theta _t\omega )(s)= \omega (t + s)-\omega (t)$
for
$s,t\in \mathbb {R}$
. We define an equivalence relation
$\sim $
on
$\Omega $
by
$\omega \sim \omega '$
if and only if there exists
$t\in \mathbb {R}$
such that
$\omega '=\theta _t\omega $
. For
$\omega \in \Omega $
, denote by
$[\omega ]$
the equivalence class of
$\omega $
. Let
$\Omega ':=\Omega /\sim $
be the quotient space of
$\Omega $
,
${\mathcal {B}}(\Omega '):=\{\cup _{\omega \in F}\{[\omega ]\}:F\in {\mathcal {B}}(\Omega )\}$
, and
$P'$
be the induced image measure of P on
$(\Omega ',{\mathcal {B}}(\Omega '))$
.
Before stating the example, we present a proposition. To the best of our knowledge, this is a novel result in the literature, which is of independent interest.
Proposition 4.5
$(\Omega ',{\mathcal {B}}(\Omega '),P')$
is isomorphic (mod 0) to
$[0,1]$
with the Lebesgue measure.
Proof We equip
$\Omega $
with the locally uniform metric:
$$ \begin{align*}d(\omega_1,\omega_2):=\sum_{n=1}^{\infty}\frac{1}{2^n}\max_{-n\le t\le n}(|\omega_1(t)-\omega_2(t)|\wedge 1),\ \ \ \ \omega_1,\omega_2\in\Omega. \end{align*} $$
Then,
$\Omega $
is a Polish space. By [Reference Rohlin20, pp. 14 and 24], we know that
$\Omega $
is a Lebesgue space (also called standard probability space or Lebesgue–Rohlin probability space). Note that
$(\Omega ,{\mathcal {B}}(\Omega ))$
is countably generated (cf. [Reference Mackey16, Sections 1 and 2] for the definition of countably generated). Let
$\{O_n,n\in \mathbb {N}\}$
be a sequence of open subsets of
$\Omega $
which generates
${\mathcal {B}}(\Omega )$
and separates
$\Omega $
. Define
$O^{\prime }_n:=\cup _{\omega \in O_n}\{[\omega ]\}$
,
$n\in \mathbb {N}$
. Then,
$\{O^{\prime }_n,n\in \mathbb {N}\}$
generates
${\mathcal {B}}(\Omega ')$
and separates
$\Omega '$
. Hence,
$(\Omega ',{\mathcal {B}}(\Omega '))$
is also countably generated. Further, by [Reference de la Rue4, Theorem 3.2], we conclude that
$(\Omega ',{\mathcal {B}}(\Omega '),P')$
is a Lebesgue space.
It is known that any Lebesgue space is isomorphic (mod 0) to an interval with Lebesgue measure, a finite or countable set of atoms, or a combination (disjoint union) of both (see [Reference Rohlin20, p. 20]). Then, to show that
$(\Omega ',{\mathcal {B}}(\Omega '),P')$
is isomorphic (mod 0) to
$[0,1]$
with the Lebesgue measure, it suffices to prove that the probability space
$(\Omega ',{\mathcal {B}}(\Omega '),P')$
has no atom, equivalently,
$P([w])=0$
for almost all
$\omega \in \Omega $
. Note that
$$ \begin{align*}[w]=\{\theta_t\omega:t\in[0,\infty)\}\cup\{\theta_t\omega:t\in(-\infty,0]\}. \end{align*} $$
By symmetry, it suffices to show that for almost all
$\omega \in \Omega $
,
$$ \begin{align*}P^*(\{\theta_t\omega|_{[0,\infty)}:t\in[0,\infty)\})=0, \end{align*} $$
where
$P^*$
is the restriction of P on
$C[0,\infty )$
.
Define
$$ \begin{align*}A=\left\{\omega\in C[0,\infty): \limsup_{t\rightarrow\infty}\omega(t)=\infty\ \mathrm{and}\ \liminf_{t\rightarrow\infty}\omega(t)=-\infty\right\}.\end{align*} $$
It is well known that
$P^*(A)=1$
. Fix an
$\omega _0\in A$
. We choose
$0<t_1<t_2<\infty $
such that
$\omega _0(t_1)=\omega _0(t_2)=0$
and
$$ \begin{align*}m_1:=\max_{0\le u\le t_1}\omega_0(u)>0,\ \ m_2:=\max_{t_1\le u\le t_2}\omega_0(u)>0. \end{align*} $$
Then, there exists
$\varepsilon>0$
such that
$$ \begin{align*}m_1=\max_{\varepsilon\le u\le t_1+\varepsilon}\omega_0(u),\ \ m_2=\max_{t_1+\varepsilon\le u\le t_2+\varepsilon}\omega_0(u). \end{align*} $$
Define
$$ \begin{align*}B=\left\{\omega\in C[0,\infty): \left[\max_{t_1\le u\le t_2}\omega(u)\right]-\left[\max_{0\le u\le t_1}\omega(u)\right]=m_2-m_1\right\}. \end{align*} $$
Then,
$$ \begin{align} \{\theta_t\omega_0:t\in[0,\varepsilon)\}\subset B. \end{align} $$
For
$\omega \in C[0,\infty )$
, we have
$$ \begin{align*} &\left[\max_{t_1\le u\le t_2}\omega(u)\right]-\left[\max_{0\le u\le t_1}\omega(u)\right]\\=&\ \left[\max_{t_1\le u\le t_2}\{\omega(u)-\omega(t_1)\}\right]-\left[\max_{0\le u\le t_1}\{\omega(u)-\omega(t_1)\}\right]\\=&\ \left[\max_{0\le v\le t_2-t_1}\{\omega(t_1+v)-\omega(t_1)\}\right]-\left[\max_{0\le v\le t_1}\{\omega(t_1-v)-\omega(t_1)\}\right]\\:=&\ M_2-M_1. \end{align*} $$
It is known that
$M_2$
and
$M_1$
are two independent continuous random variables such that (cf. [Reference Karatzas and Shreve12, p. 96])
$$ \begin{align*}P^*(M_2\in dx)=\frac{2}{\sqrt{2\pi (t_2-t_1)}}e^{-\frac{x^2}{2(t_2-t_1)}}dx,\ \ P^*(M_1\in dx)=\frac{2}{\sqrt{2\pi t_1}}e^{-\frac{x^2}{2t_1}}dx;\ \ \ \ x>0. \end{align*} $$
Then,
$P^*(B)=0$
, which together with (4.6) implies that
$$ \begin{align*}P^*(\{\theta_t\omega_0:t\in[0,\varepsilon)\})=0.\end{align*} $$
Applying the similar argument, we can show that for any
$s\ge 0$
, there exists
$\varepsilon _{s}>0$
such that
$$ \begin{align*}P^*(\{\theta_t(\theta_s\omega_0):t\in[0,\varepsilon_s)\})=0,\end{align*} $$
which implies that
$$ \begin{align} P^*(\{\theta_{t}\omega_0:t\in[s,s+\varepsilon_s)\})=0,\ \ \ \ \forall s\ge0. \end{align} $$
Define
$$ \begin{align*}C=\sup\{c: P^*(\{\theta_t\omega_0: t\in[0,c)\})=0\}. \end{align*} $$
Then, by (4.7), we obtain that
$C=\infty $
. Hence,
$$ \begin{align*}P^*(\{\theta_t\omega_0: t\in[0, \infty)\})=0. \end{align*} $$
Since
$\omega _0\in A$
is arbitrary, we conclude that for almost all
$\omega \in \Omega $
,
$$ \begin{align*}P^*(\{\theta_t\omega|_{[0,\infty)}:t\in[0,\infty)\})=0. \end{align*} $$
Therefore,
$(\Omega ',{\mathcal {B}}(\Omega '),P')$
is isomorphic (mod 0) to
$[0,1]$
with the Lebesgue measure.
We consider the SDE
$$ \begin{align} \left\{\begin{array}{@{}l} dx(t)=\{x(t)-y(t)-x(t)[x^2(t)+y^2(t)]\}dt+x(t)\circ d\omega(t),\\ dy(t)=\{x(t)+y(t)-y(t)[x^2(t)+y^2(t)]\}dt+y(t)\circ d\omega(t), \end{array} \right. \end{align} $$
where
$\circ d\omega (t)$
denotes the Stratonovich stochastic integral. Using polar coordinates
$$ \begin{align*}x=\rho\cos(2\pi\alpha),\ \ y=\rho\sin(2\pi\alpha), \end{align*} $$
we can transform equation (4.8) on
$\mathbb {R}^2$
to the following equation on
$[0,\infty )\times [0,\infty )$
:
$$ \begin{align} \left\{ \begin{array}{@{}l} d\rho(t)=[\rho(t)-\rho^3(t)]dt+\rho(t)\circ d\omega(t),\\ d\alpha(t)=\frac{1}{2\pi}dt. \end{array} \right. \end{align} $$
Equation (4.9) has a unique closed form solution as follows:
$$ \begin{align*}\rho(t,\alpha_0,\rho_0,\omega)=\frac{\rho_0e^{t+\omega(t)}}{(1+2\rho_0^2\int_0^te^{2s+2\omega(s)}ds)^{\frac{1}{2}}},\ \ \alpha(t,\alpha_0,\rho_0,\omega)=\alpha_0+\frac{t}{2\pi}. \end{align*} $$
One can check that
$$ \begin{align*}\rho^*(\omega)=\left(2\int_{-\infty}^0e^{2s+2\omega(s)}ds\right)^{-\frac{1}{2}} \end{align*} $$
is the stationary solution of the first equation of (4.9), i.e.,
$$ \begin{align*}\rho(t,\alpha_0,\rho^*(\omega),\omega)=\rho^*(\theta_t\omega). \end{align*} $$
By Proposition 4.5, we can choose a surjective measurable map
$T':\Omega '\rightarrow (0,\infty )$
. Define
$T\omega :=T'[\omega ]$
for
$\omega \in \Omega $
. Then, T is a measurable map on
$\Omega $
. Define
$$ \begin{align*}\Psi^*(t,\omega)(\alpha_0,\rho_0)=\left(\left(\alpha_0+\frac{t}{T\omega}\right)\ \mathrm{mod}\ 1,\ \rho(t,\alpha_0,\rho_0,\omega)\right). \end{align*} $$
We find that
$$ \begin{align*}\Psi^*(0,\omega)(\alpha,\rho)=(\alpha,\rho),\ \Psi^*(t+s,\omega)=\Psi^*(t,\theta_s\omega)\Psi^*(s,\omega),\ \ \ \ \forall (\alpha,\rho)\in [0,1)\times [0,\infty),\, s,t\in\mathbb{R}. \end{align*} $$
Hence,
$\Psi ^*(t,\omega )=(\Psi ^*_1(t,\omega ), \Psi ^*_2(t,\omega ))$
defines a random dynamical system on the cylinder
$[0,1)\times [0,\infty )$
. Next, we transform the random dynamical system
$\Psi ^*$
back to
$\mathbb {R}^2$
. For
$(x,y)\in \mathbb {R}^2, x =\rho \cos (2\pi \alpha ), y =\rho \sin (2\pi \alpha )$
, define
$$ \begin{align*}\Psi(t,\omega)(x,y)=\left(\Psi^*_2(t,\omega)(\alpha,\rho)\cdot\cos[2\pi\Psi^*_1(t,\omega)(\alpha,\rho)],\, \Psi^*_2(t,\omega)(\alpha,\rho)\cdot\sin[2\pi\Psi^*_1(t,\omega)(\alpha,\rho)]\right). \end{align*} $$
Now, we investigate weak random periodic solutions of the random dynamical system
$\Psi $
. Fix an
$\alpha _0\in [0,1)$
and define
$$ \begin{align*}Y(t,\omega)=\left(\rho^*(\theta_t\omega)\cos\left[2\pi\alpha_0+\frac{2\pi t}{T\omega}\right],\,\rho^*(\theta_t\omega)\sin\left[2\pi\alpha_0+\frac{2\pi t}{T\omega}\right]\right). \end{align*} $$
Then, we have
$$ \begin{align*} &\Psi(T\omega,\theta_{-T\omega}\omega)Y_0(\theta_{-T\omega}\omega)\\=&\ \Psi(T\omega,\theta_{-T\omega}\omega)(\rho^*(\theta_{-T\omega}\omega)\cos(2\pi\alpha_0),\rho^*(\theta_{-T\omega}\omega)\sin(2\pi\alpha_0))\\=&\ (\Psi_2^*(T\omega,\theta_{-T\omega}\omega)\cdot\cos[2\pi\Psi^*_1(T\omega,\theta_{-T\omega}\omega)(\alpha_0,\rho^*(\theta_{-T\omega}w))],\\&\ \,\Psi_2^*(T\omega,\theta_{-T\omega}\omega)\cdot\sin[2\pi\Psi^*_1(T\omega,\theta_{-T\omega}\omega)(\alpha_0,\rho^*(\theta_{-T\omega}w))])\\=&\ \left(\rho(T\omega,\alpha_0,\rho^*(\theta_{-T\omega}w),\theta_{-T\omega}\omega)\cdot\cos\left[2\pi\left(\left(\alpha_0+\frac{T\omega}{T(\theta_{-T\omega}\omega)}\right)\ \ \mathrm{mod}\ 1\right)\right],\right.\\&\ \ \ \left.\rho(T\omega,\alpha_0,\rho^*(\theta_{-T\omega}w),\theta_{-T\omega}\omega)\cdot\sin\left[2\pi\left(\left(\alpha_0+\frac{T\omega}{T(\theta_{-T\omega}\omega)}\right)\ \ \mathrm{mod}\ 1\right)\right]\right)\\=&\ (\rho^*(\omega)\cos(2\pi\alpha_0),\,\rho^*(\omega)\sin(2\pi\alpha_0))\\=&\ Y_0(\omega),\end{align*} $$
which implies that (2.4) holds. Therefore, by Proposition 2.2, we find that
$(Y,T)$
is a weak random periodic solution of
$\Psi $
. Further, by Theorem 3.2, we conclude that
$\Psi $
has an invariant probability measure.
Remark 4.6 By Proposition 4.5, we know that there are different choices of the random period map
$T'$
. Hence, Example 4.7 implies that the weak random periodic solution of a random dynamical system is not necessarily a random periodic solution defined as in [Reference Feng and Zhao7, Reference Zhao and Zheng25].
The last example is related to SDEs in random environments.
Example 4.7 Let Z be an
$\mathbb {N}$
-valued random variable on a probability space
$(\Omega _1,{\mathcal {F}}_1,P_1)$
. Suppose that
$d\ge 1$
,
$\Omega _2:=C(\mathbb {R};\mathbb {R}^d)$
, and
$\{\omega _2(t)\}_{t\in \mathbb {R}}$
is a d-dimensional two-sided Brownian motion on the path space
$(\Omega _2, {\mathcal {B}}(\Omega _2),P_2)$
with
$\theta _2$
being the shift operator
$(\theta _{2,t}\omega _2)(s)= \omega _2(t + s)-\omega _2(t)$
for
$s,t\in \mathbb {R}$
. Let
$m\in \mathbb {N}$
,
$T_n>0$
,
$b_{n,i}: \mathbb {R}\times \mathbb {R}^m\rightarrow \mathbb {R}$
,
$\sigma _{n,ij}: \mathbb {R}\times \mathbb {R}^m\rightarrow \mathbb {R}$
,
$1\le i\le m,\, 1\le j\le d$
,
$n\in \mathbb {N}$
. Suppose
$$ \begin{align*}b_{n,i}(t+T_n,x)=b_{n,i}(t,x),\ \ \sigma_{n,ij}(t+T_n,x)=\sigma_{n,ij}(t,x),\ \ \ \ t\in \mathbb{R},\, x\in \mathbb{R}^m,\,n\in\mathbb{N}. \end{align*} $$
Assume that, for each
$n\in \mathbb {N}$
, the stochastic semiflow
$\varphi _n$
induced by the following SDE has a random periodic solution
$\{Y_n(t,\omega _2)\}$
with period
$T_n$
:
$$ \begin{align*}dx_n(t)=b_n(t,x_n(t))dt+\sigma_n(t,x_n(t))dw_2(t). \end{align*} $$
We refer the reader to [Reference Dong, Zhang and Zheng5–Reference Feng, Zhao and Zhou8] for some concrete examples of random periodic solutions of nonautonomous SDEs.
Let
$(\Omega ,{\mathcal {F}},P)=(\Omega _1,{\mathcal {F}}_1,P_1)\times (\Omega _2, {\mathcal {B}}(\Omega _2),P_2)$
and
$\theta _t(\omega _1,\omega _2)=(\omega _1,\theta _{2,t}(\omega _2))$
. We consider the following SDE in random environment:
$$ \begin{align*}dx(t)=b_Z(t,x(t))dt+\sigma_Z(t,x(t))dw_2(t), \end{align*} $$
which induces a stochastic semiflow
$\varphi $
:
$$ \begin{align*}\varphi(t,s,\omega_1,\omega_2)=\varphi_{Z(\omega_1)}(t,s,\omega_2),\ \ \ \ (\omega_1,\omega_2)\in\Omega,\,s\le t. \end{align*} $$
Define
$$ \begin{align*}T(\omega_1,\omega_2)=T_{Z(\omega_1)},\ \ Y(t,\omega_1,\omega_2)=Y_{Z(\omega_1)}(t,\omega_2),\ \ \ \ (\omega_1,\omega_2)\in\Omega,\, t\in\mathbb{R}. \end{align*} $$
Then,
$(Y,T)$
is a weak random periodic solution of
$\varphi $
, which has a weak-invariant probability measure by Theorem 3.4.
Acknowledgment
We cordially thank the anonymous referee for valuable comments and suggestions.
