Abstract
Periodic phenomena such as oscillation have been studied for many years. In this paper, we verify the stochastic version of Levinson’s conjecture, which confirmed the existence of stochastic periodic solutions for second order Newtonian systems with dissipativeness. First, we provide a stochastic Duffing’s equation to display our result. Then, we apply Wong–Zakai approximation method and Lyapunov’s method to stochastic second order Newtonian systems driven by Brownian motions. With the help of Horn’s fixed point theorem, we prove that this kind of systems is stochastic dissipative and admits periodic solutions in distribution.
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Acknowledgements
This work was supported by National Basic Research Program of China (Grant No. 2013CB834100), National Natural Science Foundation of China (Grant No. 11901231), National Natural Science Foundation of China (Grant No. 11901080), National Natural Science Foundation of China (Grant No. 11571065) and National Natural Science Foundation of China (Grant No. 11171132). We thank the anonymous referees for their valuable suggestions and comments.
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Appendices
Appendix A: Gronwall’s Inequalities and Convergence of Variables and Stochastic Processes
To obtain our results, Gronwall-type inequalities are necessary. We list two of them below which are cited from [10].
Lemma A.1
Let u, \(\Psi \) and \(\chi \) be real continuous functions in [a, b] and \(\chi \) is nonnegative. Suppose that in [a, b] we have
Then following results hold.
-
1.
([10, Theorem 1]). In [a, b],
$$\begin{aligned} u(t)\le \Psi (t)+\int ^t_a\chi (s)\Psi (s)\exp \left[ \int ^t_s\chi (r)dr\right] ds. \end{aligned}$$ -
2.
([10, Corollary 2]). Moreover if \(\Psi \) is differentiable, then
$$\begin{aligned} u(t)\le \Psi (a)\left( \int ^t_a\chi (r)dr\right) +\int ^t_a\exp \left( \int ^t_s\chi (r)dr\right) \Psi '(s)ds. \end{aligned}$$
We will state some important results about the convergence of random variables in our proofs, which are listed below. The first one is Skorokhod’s representation theorem.
Lemma A.2
([3, Theorem 6.7]) Suppose that \(\{p_n:p_n\in \mathcal {P}(\mathbb {R}^l)\}_{n=1}^\infty \) converge to \(p_0\in \mathcal {P}(\mathbb {R}^l)\) point-wise. Then there exist random variables \(\{x_n\}_{n=1}^\infty \) and \(x_0\) defined in another common probability space \((\hat{\Omega },\hat{\mathcal {F}},\hat{\mathbb {P}})\) such that
and
as \(n\rightarrow \infty \).
Although \(L^2(\mathbb {P},\mathbb {R}^{m})\) is lack of compactness, closed balls in \(L^2(\mathbb {P},\mathbb {R}^{m})\) admit weak convergent sequences due to Prohorov’s theorem (see [3, Theorems 5.1 and 5.2]).
Lemma A.3
Suppose that there is a positive constant \(R>0\) and random variables \(\{x_n\}_{n=1}^\infty \) in \(L^2(\mathbb {P},\mathbb {R}^{m})\) satisfying
Then this sequence admits a subsequence \(\{x_{n_k}\}_{k=1}^\infty \) and a random variable \(x_0\in L^2(\mathbb {P},\mathbb {R}^{m})\) such that
as \(k\rightarrow \infty \).
Proof
This is a direct corollary of [24, Theorem 3.1]. \(\square \)
Appendix B: Some Results on Conditional Expectations and Martingales
We also need some results about conditional expectations and discrete martingales. Suppose that \(\{\mathcal {V}_k\}_{k=1}^\infty \) is a class of increasing \(\sigma \)-algebras. Suppose the mentioned conditional expectations exist. Here, we denote the conditional expectation of random variable x with respect to \(\sigma \)-algebra \(\mathcal {H}\) as \(\mathbb {E}[x|\mathcal {H}]\) (which is still a random variable). For more details for conditional expectations, we recommend [30].
Lemma B.1
For any random variable \(x\in L^2(\mathbb {P},\mathbb {R}^{m})\), conditional expectations admit following properties.
-
i.
([30, Proposition 1 (ii)]) For any \(k_1\le k_2\),
$$\begin{aligned} \mathbb {E}[\mathbb {E}[x|\mathcal {V}_{k_1}]|\mathcal {V}_{k_2}]=\mathbb {E}[\mathbb {E}[x|\mathcal {V}_{k_2}]|\mathcal {V}_{k_1}]=\mathbb {E}[x|\mathcal {V}_{k_1}]~a.s. \end{aligned}$$and in particular,
$$\begin{aligned} \mathbb {E}[\mathbb {E}[x|\mathcal {V}_{k_1}]]=\mathbb {E}[x]. \end{aligned}$$ -
ii.
([30, Theorem 4]) For any \(k\in \mathbb {N}\) and \(q\in [1,\infty ]\),
$$\begin{aligned} \varpi \left( \mathbb {E}[x|\mathcal {V}_k]\right) \le \mathbb {E}\left[ \varpi (x)|\mathcal {V}_k\right] \end{aligned}$$if \(\varpi :\mathbb {R}\rightarrow \mathbb {R}^+\) is a continuous convex function and \(\mathbb {E}[\cdot |\mathcal {V}_k]:L^q(\mathbb {P},\mathbb {R}^{m})\rightarrow L^q(\mathbb {P},\mathbb {R}^{m})\) is a positive linear contraction on all \(L^q(\mathbb {P},\mathbb {R}^{m})\). Namely,
$$\begin{aligned} \Vert \mathbb {E}[x|\mathcal {V}_k]\Vert _q\le \Vert x\Vert _q. \end{aligned}$$
It is clear that \(\{\mathbb {E}[x|\mathcal {V}_k]\}_{k\in \mathbb {N}}\) is a discrete martingale with respect to \(\{\mathcal {V}_k\}_{k\in \mathbb {N}}\). For the convergence of discrete martingales, Doob’s martingale convergence theorem and Lévy’s zero one law are helpful tools. We state a version in [33] as follows.
Lemma B.2
([33, Corollary 5.2.4]) For any \(x\in L^q(\mathbb {P},\mathbb {R}^{m})\) with \(p\in [1,\infty )\),
In particular if x is \(\bigvee \limits ^\infty _{k=0}\mathcal {V}_k\)-measurable,
Moreover,
uniformly with respect to \(x\in \bar{S}_R\) for any fixed \(r>0\).
Proof
Equations (B.22)–(B.24) are proved in [33, Corollary 5.2.4]. While the uniform convergence in (B.24) comes from two facts. First fact is stated in Lemma B.1 ii. Second fact is that for any \(x\in S_R\) and \(k_1\ge k_2\ge 0\),
That is, \(\{\mathbb {E}[x|\mathcal {V}_k]\}\) is a submartingale. Following a similar argument in Corollary 5.2.4 of [33], this sequence of random variables converges to x uniformly on \(S_R\) in mean square. \(\square \)
Appendix C: Proof of Lemma 3.1
We verify the dissipativity of (10) via Lyapunov’s method. First, let
One can verify that there exist \(\mu ,\nu >0\) such that
One can verify that conditions (H1)–(H3) hold for f and g. For any smooth enough function \(\mathcal {U}:\mathbb {R}^+\times \mathbb {R}^l\times \mathbb {R}^l\rightarrow \mathbb {R}\), define operator \(\mathcal {L}\) as
where \({{\,\mathrm{\mathbf{tr}}\,}}\) denotes trace of matrix and \({{\,\mathrm{\mathbf{Hess}}\,}}\) denotes the Hessian matrix.
According to Itô’s formula, we have
Now take a nonnegative function \(h:\mathbb {R}\rightarrow \mathbb {R}^+\) such that
-
(a)
\(h(t)\le \frac{1}{\lambda ^2}\;\forall \;t\ge 0\);
-
(b)
there exist a constant \(\sigma >0\) and a sequence \(\{t_k\}^{+\infty }_{k=1}\) satisfying \(\lim \limits _{k\rightarrow +\infty }t_k=+\infty \) such that
$$\begin{aligned} \int ^{t_{k+1}}_{t_{k}}h(s)ds\ge \sigma \;\forall \;k\in \mathbb {N}_+; \end{aligned}$$ -
(c)
there exists \(M>0\) such that
$$\begin{aligned} \int ^t_0e^{-\int ^t_sh(r)dr}ds\le M. \end{aligned}$$
Therefore by Itô’s formula, we have
Since e is continuous and \(\mathcal {T}\)-periodic with respect to t, there exists a constant \(\kappa >0\) such that
Note that
where K is a positive constant. When \(\lambda \) is large enough, there exist \(K_1,K_2>0\) such that
Hence,
Therefore by (C.25) and Condition (b), we have
Without loss of generality, let \(t_1=0\) in \(\{t_k\}^{+\infty }_{k=1}\). Condition (b) leads to that
for all \(t\ge t_k\). Therefore, we have
Let
When \(\Vert (x_0,y_0)\Vert _2<B_1\) for any \(B_1>0\), there exists \(k_0\in \mathbb {N}_+\) such that for all \(t\ge t_{k_0}\),
which means that system (10) is \(B_0\)-dissipative. Moreover, since (x(t), y(t)) is continuous, there exists a constant \(M_0\ge B_0\) such that for all \(t\in \mathbb {R}^+\),
Appendix D: Proof of Lemma 3.2
Since system (11) is \(B_0\)-dissipative, there exists a \(T_{B_1}\) for any \(B_1>0\) and \(x_0\in L(\mathbb {P},\mathbb {R}^{m})\) satisfying \(\Vert x_0\Vert _2\le B_1\) such that
for all \(t\ge T_{B_1}\).
On the other hand, according to Theorem 2.1, for every fixed \(N\in \mathbb {N}\) satisfying \(N\mathcal {T}\ge T_{B_1}\), \(\{X_\epsilon \}\) converge to X uniformly on \([0,N\mathcal {T}]\) as \(\epsilon \rightarrow 0\). Thus for all \(\nu >0\), there exists an \(\epsilon _{\nu ,N}\) such that
for all \(0<\epsilon <\epsilon _{\nu ,N}\), \(t\in [0,N\mathcal {T}]\) and \(\Vert x_0\Vert _2<\infty \).
Therefore when \(\Vert x_0\Vert _2\le B_1\) and \(t\in [T_{B_1},N\mathcal {T}]\), we have
This ends the proof.\(\square \)
In fact, we can estimate \(X_\epsilon \) by Theorem 2.1 and Gronwall’s inequality.
Lemma D.1
Suppose that conditions (H1)–(H2) hold. Then for all \(t\in [0,T]\) for any fixed \(T\in \mathbb {R}^+\), \(x_0\) satisfying \(\Vert x_0\Vert _2\le B_0\) and \(\nu >0\) there exist \(K>0\) and \(\varepsilon >0\) such that for \(0<\epsilon <\varepsilon \),
Proof
Rewrite (11) in integral Itô’s form and we have
for \(i=1,\ldots ,m\). Here \(F(t,x)=(F^1(t,x),\ldots ,F^{m})\) also satisfies conditions (H1)–(H2). Thus we have,
Therefore by Gronwall’s inequality, we have
Then similar to Lemma 3.2, we have for any \(x_0\) satisfying \(\Vert x_0\Vert _2\le B_0\), \(t\in [0,T]\) and \(\nu >0\), there exist \(K(B_0,T)>0\) and \(\varepsilon >0\) such that for \(0<\epsilon <\varepsilon \),
\(\square \)
Appendix E: Proof of Lemma 3.3
When proving the existence of periodicity for stochastic differential equations, we will apply Horn’s fixed point theory.
Lemma E.1
([14, Theorem 6]) Suppose that \(S_0\subset S_1\subset S_2\) are convex subsets of some Banach space \(\mathcal {H}\). Moreover, \(S_0\), \(S_2\) are compact and \(S_1\) is open relative to \(S_2\). Suppose \(\Gamma :S_2\rightarrow \mathcal {H}\) is continuous mapping and there is a integer \(M_0\in \mathbb {N}\) such that
Then \(\Gamma \) admits a fixed point in \(S_0\).
Now we can prove Lemma 3.3.
Proof of Theorem 3.1
Throughout this proof, we adopt a following notation,
Step 1 approximation of initial state
Let \(w:\Omega \rightarrow \mathbb {R}\) be a one-dimensional random variable according with normal distribution N(0, 1) independent of \(x_0\) and W with respect to \((\Omega ,\mathbb {P},\mathcal {F})\). The density function and distribution function of w are
Since \(F_w:\mathbb {R}\cup \{\pm \infty \}\rightarrow [0,1]\) is monotone increasing, \(F_w^{-1}: [0,1] \rightarrow \mathbb {R}\cup \{\pm \infty \}\) is well-defined. For any \(k\in \mathbb {N}_+\), let
Thus, for any \(j\in \{1,2,\ldots ,2^k\}\),
For any fixed \(x_0\in S_{B_1}\) denoted as \(x_0=(x_0^1,\ldots ,x_0^{m})\), let
It is easy to see that \(\Omega _{k,0}\cup \Omega _{k,1}=\Omega \) and \(\Omega _{k,0}\cap \Omega _{k,1}=\emptyset \). By Chebyshev’s inequality,
Let \(Z_k\) be the set of l-dimensional integer vectors \(\varvec{\lambda }=(\lambda ^1,\ldots ,\lambda ^{m})\) in the block \([-k^32^k,k^32^k)^{m}\). Take a set-valued map \(\Lambda _k:Z_k\rightarrow \sigma \left( \mathbb {R}^{m}\right) \) as
for all \({\varvec{\lambda }}\in Z_k\). Let
Let
Then for any \(j_1\ne j_2\) and \(\varvec{\lambda }_1\ne \varvec{\lambda }_2\),
Moreover,
and
Let
Then \(\{\chi _{k,0}\}\cup \{\chi _{k,j,\varvec{\lambda }}\}_{1\le j\le 2^k}^{\varvec{\lambda }\in Z_k}\) spans a subspace of \(L^2(\mathbb {P},\mathbb {R}^{m})\) with finite dimension. Let
It is clear that \(x_k\in L^2(\mathbb {P},\mathbb {R}^{m})\) for all \(k\in \mathbb {N}_+\). Then for any \(k\in \mathbb {N}_+\) and almost every \(\omega \in \Omega _{k,1}\),
Meanwhile,
Thus,
Moreover,
Then for any \(k_1<k_2\),
Thus, \(\{x_k\}\) is a Cauchy sequence in \(L^2(\mathbb {P},\mathbb {R}^{m})\). Therefore,
In the left of this proof, some notations are adopted. For \(k\in \mathbb {N}\),
Then for any fixed \(k\in \mathbb {N}\), \(L^2_k(\mathbb {P},\mathbb {R}^{m})\) is a subspace of \(L^2(\mathbb {P},\mathbb {R}^{m})\) with finite dimensions which can be viewed as \(\mathbb {R}^{m_k}\) where
\(\{\sigma _k\}_{k\in \mathbb {N}}\) is an increasing class of sub-algebras of \(\mathcal {F}\). Moreover,
Step 2 estimate of \({\varvec{X_\epsilon (t,\omega ,x_k)}}\)
Since (14) is \(B_0\)-dissipative,
Since that \(\lim \limits _{k\rightarrow \infty }\Vert x_k-x_0\Vert _2=0\), for any \(\varrho >0\), there exists a \(k_{\varrho }\in \mathbb {N}_+\) such that for any \(k>k_{\varrho }\)
Thus, we have
By (H1)–(H2),
Then by Gronwall’s inequality,
Thus,
Moreover,
Then for any \(\nu >0\), let \(\varrho ,\epsilon _{B_1,\varrho ,N}>0\) small enough such that
and
Then by Lemma B.1 ii, for \(k>k_\varrho \), \(0<\epsilon <\min \{\epsilon _{\nu ,N},\epsilon _{B_1,\varrho ,N}\}\), \(n\in \mathbb {N}\) and \(t\in [T_{B_1},N\mathcal {T}]\),
Step 3 construction of Poincaré map
Without loss of generality, assume that for any \(k\in \mathbb {N}\), \(1\le j\le 2^k\) and \(\varvec{\lambda }\in Z_k\), \(\Omega _{k,0}\) and \(\Omega _{k,j,\varvec{\lambda }}\) are not zero measure sets. By Lemma 2.1 (1) and (3), we know that for all \(t\in \mathbb {R}_+\) and \(x_0\in L^2(\mathbb {P},\mathbb {R}^{m})\), \(X_\epsilon (t,\omega ,x_0)\) is in \(L^2(\mathbb {P},\mathbb {R}^{m})\) and is \(\mathcal {F}^{t+\epsilon }_0\)-measurable. This leads to that \(\mathbb {E}[X_\epsilon (t,\omega ,x_{k_1})|\sigma _{k_2}]\) exists for any \(k_1\le k_2\). Let
for \(k\le n\). Note that \(\sigma _k\) is generated by finite subsets of \(\Omega \), \(X_{\epsilon ,n}\) can be represented as
Thus, \(X_{\epsilon ,n}(t,\omega ,x_k)\in L^2_k(\mathbb {P},\mathbb {R}^{m})\). Since \(L^2_k(\mathbb {P},\mathbb {R}^{m})\subset L^2_n(\mathbb {P},\mathbb {R}^{m})\) for \(k\le n\), define Poincaré maps \(P_n:S_R\cap L^2_n(\mathbb {P},\mathbb {R}^{m})\rightarrow P_n(S_R\cap L^2_n(\mathbb {P},\mathbb {R}^{m}))\) as
Now we need to verify two assertions below:
-
(a)
\(P_n(x_k)\xrightarrow {~2~}X_\epsilon (\mathcal {T},\omega ,x_k)\) as \(n\rightarrow \infty \);
-
(b)
\(\lim \limits _{n\rightarrow \infty }d_{BL}\Big (p_{P_n^\zeta (x_k)},p_{X_{\epsilon ,n}(\zeta \mathcal {T},\omega ,x_k)}\Big )=0\) for any fixed \(k\in \mathbb {N}\) and \(\zeta =2,3,\ldots .\)
We first deal with Assertion (a) Since \(\{\sigma _k\}\) is an increasing class of \(\sigma \)-algebras, by Lemma B.1.i we have
for all \(n_1\le n_2\). Thus, \(\{P_n(x_k)\}_{n=1}^\infty \) is a discrete martingale for all \(k\in \mathbb {N}\) with respect to \(\{\sigma _n\}_{n=1}^\infty \). Then by Lemma B.2, \(\{P_n(x_k)\}\) converges both a.s. and in \(L^2(\mathbb {P},\mathbb {R}^{m})\). Moreover,
a.s. and in \(L^2(\mathbb {P},\mathbb {R}^{m})\).
For Assertion (b), we adopt induction. Consider the case that \(\zeta =2\). Note that,
Since \(X_\epsilon (\cdot ,\cdot ,\cdot )\) generates a continuous cocycle*** and \(\mathbb {P}\) is invariant under \(\theta _{\cdot }\), we have
Thus, \(\Delta _3\,\mathop {=\!=\!=\!=}\limits ^{d}\, 0\). Following the discuss of Assertion (a), we have \(\Delta _4\xrightarrow {~2~} 0\) as \(n\rightarrow \infty \). For \(\Delta _2\), we have
a.s. and in \(L^2(\mathbb {P},\mathbb {R}^{m})\) like \(\Delta _4\). Therefore, by a standard argument similar to the uniqueness of random differential equations and ordinary differential equations satisfying Lipschitz conditions, we have
a.s. and in \(L^2(\mathbb {P},\mathbb {R}^{m})\). Thus, \(\Delta _2\xrightarrow {~2~} 0\) as \(n\rightarrow \infty \). For \(\Delta _1\), note that
Following the discussion about \(\Delta _2\) and \(\Delta _4\), we have
as \(n\rightarrow \infty \). Moreover by Lemma B.1. ii, we have \(\Vert \Delta _{11}\Vert _2\rightarrow 0\) as \(n\rightarrow \infty \). Therefore, \(\Delta _1\xrightarrow {~2~} 0\) as \(n\rightarrow \infty \). Hence,
for any fixed \(k\in \mathbb {N}\). This means that Assertion (b) holds for \(\zeta =2\).
Assume that Assertion (b) holds for all \(2<\tilde{\zeta }\le \zeta \) where \(\zeta >2\). We only need is to verify that
Note that
Thus,
For \(\Delta ^\zeta _1\), observe that
By the induction hypothesis,
Moreover,
Thus,
This leads to
Then by Skorokhod’s representation theorem Lemma A.2, there exists another common probability space \((\tilde{\Omega },\tilde{\mathcal {F}},\tilde{\mathbb {P}})\), a random variable \(X^\zeta _k\) and \(\{P^\zeta _{k,n}\}\) such that
and
as \(n\rightarrow \infty \). Furthermore, by (E.34) we have
Therefore, for any \(\varphi >0\) there exists \(n_0>0\), when \(n\ge n_0\),
Together with (E.35), there is a subsequence of \(\{P^\zeta _{k,n}\}\) (assume itself without loss of generality) such that
as \(n\rightarrow \infty \). Moreover, by a similar way with \(\Delta _{11}\),
as \(n\rightarrow \infty \). Thus,
By uniqueness of solutions for (14), we have \(\Vert \Delta ^\zeta _2\Vert _2\rightarrow 0\) as \(n\rightarrow \infty \). For \(\Delta _3^\zeta \), note that
By a similar argument with \(\Delta _3\), we get that \(\Delta ^\zeta _3\xrightarrow {~d~}0\) as \(n\rightarrow \infty \). Therefore,
Hence by induction, Assertion (b) holds for all \(k\in \mathbb {N}\) and \(\zeta =2,3,\ldots \).
Now for any \(x\in S_R\), let
Then \(\bar{x}_k\in L^2_k(\mathbb {P},\mathbb {R}^{m})\). Define Poincaré maps \(P_\epsilon :S_R\rightarrow L^2(\mathbb {P},\mathbb {R}^{m})\) as
Then
a.s. and in \(L^2(\mathbb {P},\mathbb {R}^{m})\). Thus, \(P_\epsilon \) is weakly continuous and weakly compact in weak topology. Moreover by Assertion b),
for all \(\zeta =2,3,\ldots \).
Step 4 existence of periodic solution in distribution
Let \(\nu =\frac{1}{2}\) in Step 2. Let \(B_1>B_0+1>0\) be the constants in Definition 2.2. Let
It can be verified that
Then
for all \(n\in \mathbb {N}\), \(0<\epsilon <\epsilon _{\nu ,N}\) and \(t\in [T_{B_1},N\mathcal {T}]\). Thus by (E.38), there exists \(N_1\in \mathbb {N}\) such that \(N_1\mathcal {T}\ge T_{B_1}>(N_1-1)\mathcal {T}\), \(N_1<N\) and
for all \(\xi \ge N_1\) and all \(x_0\in S_{B_1}\). Moreover, by the definition of \(P_\epsilon \), we have
a.s. and in mean square. On one hand for all \(k\in \mathbb {N}\), \(\Vert x_k\Vert _2\le \Vert x_0\Vert _2\le B_1\). Thus, \(x_k\) converges to \(x_0\) in mean square uniformly on \(S_{B_1}\). That is for any \(\vartheta >0\), there exists \(k_0\in \mathbb {N}\) such that for all \(k\ge k_0\) and \(x_0\in S_{B_1}\),
Hence, we have all \(t\in [0,T]\),
where K and \(K_\epsilon \) are independent with k and \(x_0\). Therefore by Gronwall’s inequality,
That is, \(X_\epsilon (t,\omega ,x_k)\) converges to \(X_\epsilon (t,\omega ,x_0)\) in mean square uniformly with respect to \(t\in [0,T]\) and \(x_0\in S_{B_1}\) as \(k\rightarrow \infty \). On the other hand, note that
Hence by Gronwall’s inequality,
Thus by Lemma B.2, \(X_{\epsilon ,n}(t,\omega ,x_k)\) converges to \(X_\epsilon (t,\omega ,x_k)\) in mean square uniformly with respect to \(x_0\in S_{B_1}\) and \(t\in [0,T]\). Therefore for any fixed \(\zeta \), \(P^\zeta _n(x_k)\) converges to \(P^\zeta _\epsilon (x_0)\) uniformly on \(S_{B_1}\). Hence for any fixed \(\zeta \) such that \(\zeta \mathcal {T}\ge T_{B_1}\), there exists \(\tilde{k}_0\) and \(n_0\) such that for all \(x_0\in S_{B_1}\), \(k\ge \tilde{k}_1\) and \(n\ge n_0\),
Meanwhile we also have for all \(\zeta \) such that \(\zeta \mathcal {T}\le T_{B_1}\), \(\epsilon \) small enough and all \(x_0\in S_1\),
by (D.29).
Let \(B_2=\max \{K(B_1,T_{B_1})+\frac{1}{2},B_1\}\). Then \(\bar{S}_{B_0+\frac{1}{2},n}\subset S_{B_1,n}\subset \bar{S}_{B_2,n}\) are convex subsets of \(L^2_n(\mathbb {P},\mathbb {R}^{m})\) with \(\bar{S}_{B_0+\frac{1}{2}_n}\) and \(S_{B_2,n}\) compact and \(S_{B_1,n}\) open relative to \(\bar{S}_{B_2,n}\). Meanwhile, \(\bar{S}_{B_0+\frac{1}{2},n}\) and \(\bar{S}_{B_2,n}\) are compact with respect to the topology of \(\mathcal {P}(\mathbb {R}^{m})\). Moreover,
Then by Horn’s fixed point theorem Lemma E.1, there exists a random variable \(x_{\epsilon ,\mathcal {T},n}\in \bar{S}_{B_0+\frac{1}{2},n}\) such that
Since \(\{x_{\epsilon ,\mathcal {T},n}\}_{n=1}^\infty \subset \bar{S}_{B_0+\frac{1}{2}}\), according to Lemma A.3 there exists a subsequence \(\{x_{\epsilon ,\mathcal {T},n_k}\}\) and a random variable \(x_{\epsilon ,\mathcal {T}}\in \bar{S}_{B_0+\frac{1}{2}}\) such that
Thus by weak uniqueness of solutions for (14)
Note that \(X_{\epsilon ,n}(t,\omega ,x)\) converges to \(X_{\epsilon }(t,\omega ,x)\) as \(n\rightarrow \infty \) in mean square uniformly with respect to \(t\in [0,T]\) and \(x\in S_{B_0+\frac{1}{2}}\), we can obtain uniform convergence of this sequence in distribution. Therefore for any \(\vartheta >0\) there exists an \(\tilde{n}_0\) and a \(\tilde{k}_1\) such that for all \(n\ge \tilde{n}_0\) and \(k\ge \tilde{k}_1\) we have
Hence, there exists \(K_0\ge \tilde{k}_1\) satisfying \(n_{K_0}\ge \tilde{n}_0\) such that for all \(k\ge K_0\),
That is,
Namely,
Since \(X_\epsilon (t,\omega ,x_{\epsilon ,\mathcal {T}})\) is a solution of (14) starting from \(x_{\epsilon ,\mathcal {T}}\), we have
Then, \(\{X_\epsilon (t,\omega ,x_{\epsilon ,\mathcal {T}})\}_{t\in \mathbb {R}_+}\) is a periodic solution of (14) in distribution. \(\square \)
Appendix F: Proof of Lemma 3.4
According to the hypotheses, there exists \(N_0\in \mathbb {N}\) such that for all \(n\ge N_0\), \(\frac{1}{n}\le \varepsilon \) and
Hence, \(\{x_{\frac{1}{n},\mathcal {T}}\}_{n=N_0}^{\infty }\) is a bounded sequence in \(L^2(\mathbb {P},\mathbb {R}^{m})\). By Lemma A.3, there exists a random variable \(x_{\mathcal {T}}\in S_{R}\) and a subsequence of \(\{x_{\frac{1}{n},\mathcal {T}}\}\) (still denote as itself) such that
as \(n\rightarrow \infty \). Then by Lemma A.2, there exists a common probability space \((\bar{\Omega },\bar{\mathcal {F}},\bar{\mathbb {P}})\), random variables \(\bar{x}_{\mathcal {T}}\) and \(\{\bar{x}_{n,\mathcal {T}}\}\) such that
and
as \(n\rightarrow \infty \). This leads to that
By Theorem 2.1, \(X_\epsilon (t,\omega ,x)\xrightarrow {~2~}X(t,\omega ,x)\) uniformly on \([0,N\mathcal {T}]\) and uniformly respect to \(x\in S_{R}\). Thus for any \(\nu >0\), there exists a constant \(\tilde{N}_1\in \mathbb {N}\) such that for all \(n_1\ge \tilde{N}_1\) and all \(n_2\in \mathbb {N}\),
On the other hand, according to the uniqueness of solutions for stochastic differential equations, there exists a constant \(\tilde{N}_2\in \mathbb {N}\) such that for all \(n_2\ge \tilde{N}_2\),
Therefore, when \(n\ge \max \{\tilde{N}_1,\tilde{N}_2\}\), we have
That is,
as \(n\rightarrow \infty \). Together with (F.39), we have
With a similar discussion with \(X_\epsilon (t,\omega ,x_{\epsilon ,\mathcal {T}})\), \(\{X(t,\omega ,x_{\mathcal {T}})\}_{t\in \mathbb {R}_+}\) is a periodic solution of (11) in distribution.
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Jiang, X., Li, Y. & Yang, X. Existence of Periodic Solutions in Distribution for Stochastic Newtonian Systems. J Stat Phys 181, 329–363 (2020). https://doi.org/10.1007/s10955-020-02583-3
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DOI: https://doi.org/10.1007/s10955-020-02583-3