2. Preliminaries

Let <sup>Ω</sup>; <sup>F</sup>; f g Ft <sup>t</sup> <sup>≥</sup> <sup>0</sup>; <sup>P</sup> be the complete probability space with a filtration f g Ft <sup>t</sup> <sup>≥</sup> <sup>0</sup> satisfying the usual conditions

• f g Ft <sup>t</sup> <sup>≥</sup> <sup>0</sup> is an increasing family of sub algebras containing negligible sets of F and is continuous at right.

$$F\_{\curvearrowright} = \sigma\{\uplus\_{t\geq 0} F\_t\}\dots$$

Let a Brownian motion W tð Þ, adapted to Ft f g ; t ≥ 0 , i.e., Wð Þ¼ 0 0, ∀t ≥ 0,W tð Þ is Ft�measurable. We consider the SDE

$$\begin{cases} d\mathbf{x} = a(t, \mathbf{x})dt + b(t, \mathbf{x})dW\_t \\ \mathbf{x}(t\_0) = \mathbf{z}. \end{cases} \tag{1}$$

sup t ≥ 0

k kx <sup>B</sup> ¼ sup t ≥ 0

The following result proves that the solution of the SDE (1) is a Markov process.

a u; <sup>X</sup>ð Þ <sup>t</sup>;<sup>x</sup> ð Þ <sup>u</sup> � �du <sup>þ</sup>

Then the process Xt, solution of SDE (1), is a Markovian process with a transition function

p tð Þ¼ ; <sup>x</sup>;s; <sup>A</sup> P Xð Þ <sup>t</sup>;<sup>x</sup> ð Þ<sup>s</sup> <sup>∈</sup> <sup>A</sup>

Let p sð Þ ; x; t; A be a transition function; we construct a Markov process with an initial arbitrary distribution. In a particular case, for t > s, we associate with the function p sð Þ ; x; t; A a family <sup>X</sup>ð Þ <sup>s</sup>;<sup>z</sup> ð Þ <sup>t</sup>; <sup>ω</sup> of a Markov process such that the processes <sup>X</sup>ð Þ <sup>s</sup>;<sup>z</sup> ð Þ <sup>t</sup>; <sup>ω</sup> exist with initial point

P Xð Þ <sup>s</sup>;<sup>z</sup> ð Þ¼ <sup>t</sup>; <sup>ω</sup> <sup>z</sup>

Définition 1. A stochastic process X tð Þ ; ω is said to be periodic with period T Tð Þ > 0 if its finite dimensional distributions are periodic with periodic T, i.e., for all m ≥ 0, and t1, t2, …tm ∈ R<sup>þ</sup> the joint distributions of the stochastic processes Xt1þkTð Þ ω , Xt2þkTð Þ ω , …XtmþkTð Þ ω are independent of k

Remark 1. If X tð Þ ; ω is T�periodic, then m tðÞ¼ EX tð Þ,v tðÞ¼ VarX tð Þ are T�periodic, in this case,

Définition 2. The function p sð Þ¼ ; x; t; A P Xð Þ <sup>t</sup> ∈ A=Xs for 0 ≤ s ≤ t, is said to be periodic if

ðs t

Theorem 1. ([5], Th. 2, p. 466) Assume that a tð Þ ; x and b tð Þ ; x satisfy the hypothesis of the theorem ([5], Th. 1, p. 461) and that Xð Þ <sup>t</sup>;<sup>x</sup> ð Þ<sup>s</sup> is a process such that for s<sup>∈</sup> t, <sup>∞</sup><sup>Þ</sup> for all t <sup>&</sup>gt; <sup>t</sup><sup>0</sup> is a solution of SDE

we consider in B the norm

� � is the Banach space.

<sup>X</sup>ð Þ <sup>t</sup>;<sup>x</sup> ð Þ¼ <sup>s</sup> <sup>x</sup> <sup>þ</sup>

2.2. Notions of periodicity and boundedness

this process is said to be T�periodic in the wide sense.

p sð Þ ; x; t þ s; A is periodic in s:

2.1. Markov property

B; : k k<sup>B</sup>

z in s, i.e.,

ð Þ k ∈Z :

Ex t j j ð Þ <sup>2</sup> ,

> E xj j<sup>2</sup> � �<sup>1</sup> 2

An Extension of Massera's Theorem for *N*-Dimensional Stochastic Differential Equations

ðs t

� �:

b u; <sup>X</sup>ð Þ <sup>t</sup>;<sup>x</sup> ð Þ <sup>u</sup>

� � <sup>¼</sup> <sup>1</sup> (3)

� �dWu (2)

http://dx.doi.org/10.5772/intechopen.73183

59

in <sup>Ω</sup>; <sup>F</sup>; f g Ft <sup>t</sup> <sup>≥</sup> <sup>0</sup>; <sup>P</sup> :

The functions a tð Þ ; <sup>x</sup> : <sup>R</sup><sup>þ</sup> � <sup>R</sup><sup>n</sup> ! <sup>R</sup><sup>n</sup> and b tð Þ ; <sup>x</sup> : <sup>R</sup><sup>þ</sup> � <sup>R</sup><sup>n</sup> ! <sup>R</sup><sup>n</sup>�<sup>m</sup> are measurable. We suppose that Ft is the completion of σ Wr f g ; t<sup>0</sup> ≤ r ≤ t for all t ≥ t0, and the initial condition z is independent of Wt, for <sup>t</sup> <sup>≥</sup> <sup>t</sup><sup>0</sup> and E zj j<sup>p</sup> <sup>&</sup>lt; <sup>∞</sup>.

Suppose that the functions a tð Þ ; x and b tð Þ ; x satisfy the global Lipschitz and the linear growth conditions

$$\exists k > 0, \forall t \in \mathbb{R}\_+, \forall \mathbf{x}, \mathbf{y} \in \mathbb{R}^n : \left\| \boldsymbol{a}(t, \mathbf{x}) - \boldsymbol{a}(t, \mathbf{y}) \right\| + \left\| \boldsymbol{b}(t, \mathbf{x}) - \boldsymbol{b}(t, \mathbf{y}) \right\| \le k \| \mathbf{x} - \mathbf{y} \|$$

and

$$||a(t, \mathfrak{x})||^p + ||b(t, \mathfrak{x})||^p \le k^p (1 + ||\mathfrak{x}||^p)$$

We know that if a and b satisfy these conditions, then the system (1) admits a single global solution.

We note by B the space of random Ft�measurable functions x tð Þ for all t, satisfying the relation

$$\sup\_{t \ge 0} E|\mathfrak{x}(t)|^2 \,\_{\prime}$$

we consider in B the norm

The existence of periodic solutions for differential equations has received a particular interest. We quote the famous results of Massera [9]. In its approach, Massera was the first to establish a relation between the existence of bounded solutions and that of a periodic solution for a

dx ¼ a tð Þ ; x dt þ b tð Þ ; x dWt

Let <sup>Ω</sup>; <sup>F</sup>; f g Ft <sup>t</sup> <sup>≥</sup> <sup>0</sup>; <sup>P</sup> be the complete probability space with a filtration f g Ft <sup>t</sup> <sup>≥</sup> <sup>0</sup> satisfying the

• f g Ft <sup>t</sup> <sup>≥</sup> <sup>0</sup> is an increasing family of sub algebras containing negligible sets of F and is

F<sup>∞</sup> ¼ σf g ∪<sup>t</sup> <sup>≥</sup> <sup>0</sup>Ft :

Let a Brownian motion W tð Þ, adapted to Ft f g ; t ≥ 0 , i.e., Wð Þ¼ 0 0, ∀t ≥ 0,W tð Þ is Ft�measurable.

dx ¼ a tð Þ ; x dt þ b tð Þ ; x dWt

The functions a tð Þ ; <sup>x</sup> : <sup>R</sup><sup>þ</sup> � <sup>R</sup><sup>n</sup> ! <sup>R</sup><sup>n</sup> and b tð Þ ; <sup>x</sup> : <sup>R</sup><sup>þ</sup> � <sup>R</sup><sup>n</sup> ! <sup>R</sup><sup>n</sup>�<sup>m</sup> are measurable. We suppose that Ft is the completion of σ Wr f g ; t<sup>0</sup> ≤ r ≤ t for all t ≥ t0, and the initial condition z is

Suppose that the functions a tð Þ ; x and b tð Þ ; x satisfy the global Lipschitz and the linear growth

<sup>∃</sup><sup>k</sup> <sup>&</sup>gt; <sup>0</sup>, <sup>∀</sup>t<sup>∈</sup> <sup>R</sup>þ, <sup>∀</sup>x, y<sup>∈</sup> <sup>R</sup><sup>n</sup> : k k a tð Þ� ; <sup>x</sup> a tð Þ ; <sup>y</sup> <sup>þ</sup> k k b tð Þ� ; <sup>x</sup> b tð Þ ; <sup>y</sup> <sup>≤</sup> k xk k � <sup>y</sup>

k k a tð Þ ; <sup>x</sup> <sup>p</sup> <sup>þ</sup> k k b tð Þ ; <sup>x</sup> <sup>p</sup> <sup>≤</sup> <sup>k</sup><sup>p</sup> <sup>1</sup> <sup>þ</sup> k k<sup>x</sup> <sup>p</sup> ð Þ

We know that if a and b satisfy these conditions, then the system (1) admits a single global

We note by B the space of random Ft�measurable functions x tð Þ for all t, satisfying the relation

(1)

x tð Þ¼ <sup>0</sup> z:

In this work, we will prove an extension of Massera's theorem for the following:

nonlinear ODE.

2. Preliminaries

usual conditions

We consider the SDE

in <sup>Ω</sup>; <sup>F</sup>; f g Ft <sup>t</sup> <sup>≥</sup> <sup>0</sup>; <sup>P</sup> :

conditions

and

solution.

independent of Wt, for <sup>t</sup> <sup>≥</sup> <sup>t</sup><sup>0</sup> and E zj j<sup>p</sup> <sup>&</sup>lt; <sup>∞</sup>.

continuous at right.

nonlinear SDE in dimension n ≥ 2

58 Differential Equations - Theory and Current Research

$$\|\|\mathbf{x}\|\|\_{B} = \sup\_{t \ge 0} \left( E|\mathbf{x}|^2 \right)^{\frac{1}{2}}$$

B; : k k<sup>B</sup> � � is the Banach space.

#### 2.1. Markov property

The following result proves that the solution of the SDE (1) is a Markov process.

Theorem 1. ([5], Th. 2, p. 466) Assume that a tð Þ ; x and b tð Þ ; x satisfy the hypothesis of the theorem ([5], Th. 1, p. 461) and that Xð Þ <sup>t</sup>;<sup>x</sup> ð Þ<sup>s</sup> is a process such that for s<sup>∈</sup> t, <sup>∞</sup><sup>Þ</sup> for all t <sup>&</sup>gt; <sup>t</sup><sup>0</sup> is a solution of SDE

$$X^{(t,x)}(\mathbf{s}) = \mathbf{x} + \int\_{t}^{s} a\left(u, X^{(t,x)}(u)\right) du + \int\_{t}^{s} b\left(u, X^{(t,x)}(u)\right) dW\_{u} \tag{2}$$

Then the process Xt, solution of SDE (1), is a Markovian process with a transition function

$$p(t, x; s, A) = P\left(X^{(t, x)}(s) \in A\right).$$

Let p sð Þ ; x; t; A be a transition function; we construct a Markov process with an initial arbitrary distribution. In a particular case, for t > s, we associate with the function p sð Þ ; x; t; A a family <sup>X</sup>ð Þ <sup>s</sup>;<sup>z</sup> ð Þ <sup>t</sup>; <sup>ω</sup> of a Markov process such that the processes <sup>X</sup>ð Þ <sup>s</sup>;<sup>z</sup> ð Þ <sup>t</sup>; <sup>ω</sup> exist with initial point z in s, i.e.,

$$P\left(X^{(s,z)}(t,\omega)=z\right)=1\tag{3}$$

### 2.2. Notions of periodicity and boundedness

Définition 1. A stochastic process X tð Þ ; ω is said to be periodic with period T Tð Þ > 0 if its finite dimensional distributions are periodic with periodic T, i.e., for all m ≥ 0, and t1, t2, …tm ∈ R<sup>þ</sup> the joint distributions of the stochastic processes Xt1þkTð Þ ω , Xt2þkTð Þ ω , …XtmþkTð Þ ω are independent of k ð Þ k ∈Z :

Remark 1. If X tð Þ ; ω is T�periodic, then m tðÞ¼ EX tð Þ,v tðÞ¼ VarX tð Þ are T�periodic, in this case, this process is said to be T�periodic in the wide sense.

Définition 2. The function p sð Þ¼ ; x; t; A P Xð Þ <sup>t</sup> ∈ A=Xs for 0 ≤ s ≤ t, is said to be periodic if p sð Þ ; x; t þ s; A is periodic in s:

Définition 3. The Markov families Xð Þ <sup>t</sup>0;<sup>z</sup> ð Þ <sup>ω</sup> are said to be p�uniformly bounded pð Þ <sup>&</sup>gt; <sup>2</sup> , if ∀α > 0, ∃θ αð Þ > 0, ∀t ≥ t0:

$$\|\|z\|\|\_{\mathcal{B},p} \le \alpha \Rightarrow \left\|\|X^{(t\_0,z)}(\omega)\|\right\|\_{\mathcal{B},p} \le \theta(\alpha)$$

We denote <sup>X</sup>ð Þ <sup>t</sup>0;<sup>z</sup> ð Þ <sup>ω</sup> as the family of all Markov process for <sup>t</sup><sup>0</sup> <sup>∈</sup> <sup>ℝ</sup>þand <sup>z</sup> in <sup>L</sup><sup>p</sup> :

Remark 2. It is easy to see that all Lp�borné Markov processes Xt, i:e∃<sup>M</sup> <sup>&</sup>gt; <sup>0</sup>; <sup>∀</sup><sup>t</sup> <sup>≥</sup> <sup>t</sup><sup>0</sup> : k k Xt <sup>p</sup> B,p ≤ M; � � is p�uniformly bounded.

Lemme 1. ([6], Theorem 3.2 and Remark 3.1, pp. 66–67) A necessary and sufficient condition for the existence of a Markov T�periodic Xð Þ <sup>t</sup>0;<sup>z</sup> ð Þ <sup>ω</sup> with a given T�periodic transition function p sð Þ ; <sup>x</sup>; <sup>t</sup>; <sup>A</sup> , is that for some t0, z, Xð Þ <sup>t</sup>0;<sup>z</sup> ð Þ <sup>ω</sup> are uniformly stochastically continuous and

$$\lim\_{R \to \infty} \lim\_{L \to \infty} \inf \frac{1}{L} \int\_{t\_0}^{t\_0 + L} p(t\_0, z; t, \overline{\mathcal{U}}\_{\mathbb{R}, p}) dt = 0 \tag{4}$$

k k<sup>z</sup> B, <sup>p</sup> <sup>≤</sup> <sup>α</sup> ) <sup>X</sup>ð Þ <sup>t</sup>0;<sup>z</sup> ð Þ <sup>ω</sup> �

p t0; z; t; UR, <sup>p</sup> � � ≤

p t0; z; t; UR,p

that is, Eq. (4). From Lemma 1, we have a T�periodic Markov process.

(

H1) the functions a tð Þ ; x and b tð Þ ; x are T�periodic in t. H2) the functions a tð Þ ; x and b tð Þ ; x satisfy the condition

Lemme 3. ([13], Lemme 3.4) Assume that a tð Þ ; x and b tð Þ ; x verify

We prove the Massera's theorem for the SDE in dimension n ≥ 2:

where ϕ is a concave non-decreasing function.

We assume that this SDE satisfies the conditions as in Section 2 after Eq. (1).

� �dt ≤ lim

we get

Then

0 ≤ lim <sup>R</sup>!<sup>∞</sup> lim L!∞

> θp ð Þ α RP zð Þ <sup>¼</sup> <sup>0</sup>,

¼ lim R!∞

3. Main result

Let the SDE

Suppose that

inf <sup>1</sup> L ð<sup>t</sup>0þ<sup>L</sup> t0

� �

� B, p ≤ θ αð Þ

1 RP zð Þ <sup>θ</sup><sup>p</sup>

R!∞

dx ¼ a tð Þ ; x dt þ b tð Þ ; x dWt

E at k k ð Þ ; <sup>x</sup> <sup>p</sup> ð Þþ E bt k k ð Þ ; <sup>x</sup> <sup>p</sup> ð Þ <sup>≤</sup> <sup>η</sup>, p <sup>&</sup>gt; <sup>2</sup>

then, the solutions of periodic SDE (6) are uniformly stochastically continuous.

xt<sup>0</sup> <sup>¼</sup> z, E zj j<sup>p</sup> <sup>&</sup>lt; <sup>∞</sup>

ð Þ α

An Extension of Massera's Theorem for *N*-Dimensional Stochastic Differential Equations

ð Þ α lim L!∞

k k a tð Þ ; <sup>x</sup> <sup>p</sup> <sup>þ</sup> k k b tð Þ ; <sup>x</sup> <sup>p</sup> <sup>≤</sup><sup>ϕ</sup> k k<sup>x</sup> <sup>p</sup> ð Þ, p <sup>&</sup>gt; <sup>2</sup> (7)

inf <sup>1</sup> L

ð<sup>t</sup>0þ<sup>L</sup> t0 dt � �

http://dx.doi.org/10.5772/intechopen.73183

(6)

61

1 RP zð Þ <sup>θ</sup><sup>p</sup>

if the transition function p s; Xs ð Þ ; t; A satisfies the following not very restrictive assumption

$$a(R) = \sup\_{z \in \mathcal{U}\_{\beta(\mathbb{R}), p}} 0 < t\_0, t - t\_0 < Tp\left(t\_0, z; t, \overline{\mathcal{U}}\_{\mathbb{R}, p}\right) \to\_{\mathbb{R} \to \infty} 0 \tag{5}$$

for some function βð Þ R which tends to infinity as R ! ∞:

In Eq. (4), we have R∈ R<sup>∗</sup> þ:

$$\mathcal{U}\_{\mathbb{R},p} = \{ \mathbf{x} \in \mathbb{R}^n : |\mathbf{x}|^p < \mathcal{R} \}$$

$$\overline{\mathcal{U}}\_{\mathbb{R},p} = \{ \mathbf{x} \in \mathbb{R}^n : |\mathbf{x}|^p \ge \mathcal{R} \}$$

The conditions of Lemma 1 are of little use for stochastic differential equations, since the properties of transition functions of such processes are usually not expressible in terms of the coefficients of the equation. So, in the following, we will give some new useful sufficient conditions in terms of uniform boundedness and point dissipativity of systems.

Lemme 2. If Markov families Xð Þ <sup>t</sup>0;<sup>z</sup> ð Þ <sup>ω</sup> with T�periodic transition functions are uniformly bounded uniformly stochastically continuous, then there is a T�periodic Markov process.

Proof. By using a Markov inequality [13], we have

$$\begin{aligned} p\left(t\_0, z; t, \overline{\mathcal{U}}\_{\mathbb{R}, p}\right) &= \frac{1}{\mathcal{R}P(\mathcal{X}\_{t\_0} = z)} E\left|X^{(t\_0, z)}(\omega)\right|^p \\ &\leq \frac{1}{\mathcal{R}P(z)} \left\| |X^{(t\_0, z)}(\omega)|\right\|\_{\mathcal{B}, p}^p \end{aligned}$$

Then, for α > 0, ∃θ αð Þ > 0, such that for all t ≥ t<sup>0</sup>

An Extension of Massera's Theorem for *N*-Dimensional Stochastic Differential Equations http://dx.doi.org/10.5772/intechopen.73183 61

$$\left\| \|z\|\_{B,p} \le \alpha \Rightarrow \left\| \left| X^{(t\_0,z)}(\omega) \right| \right\|\_{B,p} \le \theta(\alpha) \right\| $$

we get

Définition 3. The Markov families Xð Þ <sup>t</sup>0;<sup>z</sup> ð Þ <sup>ω</sup> are said to be p�uniformly bounded pð Þ <sup>&</sup>gt; <sup>2</sup> , if

Lemme 1. ([6], Theorem 3.2 and Remark 3.1, pp. 66–67) A necessary and sufficient condition for the existence of a Markov T�periodic Xð Þ <sup>t</sup>0;<sup>z</sup> ð Þ <sup>ω</sup> with a given T�periodic transition function

� �

� B, p ≤ θ αð Þ

p t0; z; t; UR, <sup>p</sup>

0 < t0, t � t<sup>0</sup> < Tp t0; z; t; UR, <sup>p</sup>

UR,p <sup>¼</sup> <sup>x</sup> <sup>∈</sup> <sup>R</sup><sup>n</sup> : j j <sup>x</sup> <sup>p</sup> f g <sup>&</sup>lt; <sup>R</sup>

UR,p <sup>¼</sup> <sup>x</sup> <sup>∈</sup> <sup>R</sup><sup>n</sup> : j j <sup>x</sup> <sup>p</sup> f g <sup>≥</sup><sup>R</sup>

The conditions of Lemma 1 are of little use for stochastic differential equations, since the properties of transition functions of such processes are usually not expressible in terms of the coefficients of the equation. So, in the following, we will give some new useful sufficient

Lemme 2. If Markov families Xð Þ <sup>t</sup>0;<sup>z</sup> ð Þ <sup>ω</sup> with T�periodic transition functions are uniformly bounded

RP Xð Þ <sup>t</sup><sup>0</sup> ¼ z

≤ 1 E Xð Þ <sup>t</sup>0;<sup>z</sup> ð Þ <sup>ω</sup> � � � � p

> � p B,p

RP zð Þ <sup>X</sup>ð Þ <sup>t</sup>0;<sup>z</sup> ð Þ <sup>ω</sup> � � �

conditions in terms of uniform boundedness and point dissipativity of systems.

uniformly stochastically continuous, then there is a T�periodic Markov process.

� � <sup>¼</sup> <sup>1</sup>

p t0; z; t; UR, <sup>p</sup>

:

� �

� �dt <sup>¼</sup> <sup>0</sup> (4)

� �!<sup>R</sup>!<sup>∞</sup><sup>0</sup> (5)

B,p ≤ M;

k k<sup>z</sup> B, <sup>p</sup> <sup>≤</sup> <sup>α</sup> ) <sup>X</sup>ð Þ <sup>t</sup>0;<sup>z</sup> ð Þ <sup>ω</sup> �

Remark 2. It is easy to see that all Lp�borné Markov processes Xt, i:e∃<sup>M</sup> <sup>&</sup>gt; <sup>0</sup>; <sup>∀</sup><sup>t</sup> <sup>≥</sup> <sup>t</sup><sup>0</sup> : k k Xt <sup>p</sup>

We denote <sup>X</sup>ð Þ <sup>t</sup>0;<sup>z</sup> ð Þ <sup>ω</sup> as the family of all Markov process for <sup>t</sup><sup>0</sup> <sup>∈</sup> <sup>ℝ</sup>þand <sup>z</sup> in <sup>L</sup><sup>p</sup>

p sð Þ ; <sup>x</sup>; <sup>t</sup>; <sup>A</sup> , is that for some t0, z, Xð Þ <sup>t</sup>0;<sup>z</sup> ð Þ <sup>ω</sup> are uniformly stochastically continuous and

if the transition function p s; Xs ð Þ ; t; A satisfies the following not very restrictive assumption

inf <sup>1</sup> L ð<sup>t</sup>0þ<sup>L</sup> t0

lim <sup>R</sup>!<sup>∞</sup> lim L!∞

z∈ Uβð Þ <sup>R</sup> ,p

αð Þ¼ R sup

for some function βð Þ R which tends to infinity as R ! ∞:

þ:

Proof. By using a Markov inequality [13], we have

Then, for α > 0, ∃θ αð Þ > 0, such that for all t ≥ t<sup>0</sup>

∀α > 0, ∃θ αð Þ > 0, ∀t ≥ t0:

60 Differential Equations - Theory and Current Research

is p�uniformly bounded.

In Eq. (4), we have R∈ R<sup>∗</sup>

$$p\left(t\_0, z; t, \overline{\mathcal{U}}\_{\mathbb{R}, p}\right) \le \frac{1}{\mathcal{R}P(z)} \theta^p(\alpha)$$

Then

$$\begin{aligned} 0 \le \lim\_{R \to \infty} \lim\_{L \to \infty} \inf \frac{1}{L} \int\_{t\_0}^{t\_0+L} p\left(t\_0, z; t, \overline{\mathcal{U}}\_{R,p}\right) dt \le \lim\_{R \to \infty} \frac{1}{RP(z)} \Theta^p(\alpha) \left(\liminf\_{L \to \infty} \frac{1}{L} \int\_{t\_0}^{t\_0+L} dt\right) \\ = \lim\_{R \to \infty} \frac{\Theta^p(\alpha)}{RP(z)} = 0, \end{aligned}$$

that is, Eq. (4). From Lemma 1, we have a T�periodic Markov process.
