1 Introduction

Consider the following slow-fast system of abstract stochastic evolution equations with additive noise

$$\begin{aligned} {\left\{ \begin{array}{ll} dU_t^{\epsilon }=A_1U_t^{\epsilon } dt+F(U_t^{\epsilon },V_t^{\epsilon })dt+dW^{Q_1}_t, \\ U_0^{\epsilon }=u \in H, \\ dV_t^{\epsilon }=\epsilon ^{-1}A_2V_t^{\epsilon }dt+\epsilon ^{-1} G(V_t^{\epsilon })dt+\epsilon ^{-1/2}dW^{Q_2}_t,\\ V_0^{\epsilon }=v \in K, \end{array}\right. } \end{aligned}$$
(1)

where \(\epsilon >0\) is a small parameter representing the ratio of time scales beween the slow component of the system \(U^\epsilon \) and the fast one \(V^\epsilon \). Here H, K are Hilbert spaces, \(A_1,A_2\) are unbounded linear operators on H, K respectively and \(W^{Q_1}, W^{Q_2}\) are Wiener processes on H, K respectively.

Slow-fast systems are very used in applications since it is very natural for real-world systems to present very different time-scales. We refer the reader for example to [23] for applications to physics, [32] to chemistry, [34] to neurophysiology, [1, 13, 21, 22] to mathematical finance (see also [12] for a slightly different financial model) and the references therein.

A natural idea is then to study the behaviour of the system when \(\epsilon \rightarrow 0\). In particular under certain hypotheses it is known that the slow component \(U^\epsilon \) converges to the solution U of the so called averaged equation

$$\begin{aligned} {\left\{ \begin{array}{ll} dU_t=A_1U_t dt+\overline{F}(U_t)dt+dW^{Q_1}_t, \\ U_0=u, \end{array}\right. } \end{aligned}$$

where

$$\begin{aligned} \overline{F}(u)= \int _K F(u,v) \mu (dv), \end{aligned}$$

and \(\mu \) is the invariant measure related to the fast motion, i.e.

$$\begin{aligned} {\left\{ \begin{array}{ll} dv_t^{v}=A_2v_t^{v} dt+ G(v_t^{v})dt+d w^{Q_2}_t,\\ v_0^{v}=v \in K. \end{array}\right. } \end{aligned}$$

Note that the equation for U is uncoupled from \(V^\epsilon \). This fact is known as averaging principle and it is fundamental in applications since U captures the effective dynamic of \(U^\epsilon \) (which is usually the most interesting variable in applications) and it is then a rigorous dimensionality reduction of the original system.

The first general result for the averaging principle for finite-dimensional stochastic differential equations can be found in [27]. For generalizations and improvements see [13, 17, 18, 23, 31, 40, 41] and the references therein. It is important to mention that the drift of the fast equation is allowed to depend also on the slow component (i.e. fully coupled system) and the stochastic perturbations of the slow and fast equations are allowed to be multiplicative, i.e. the diffusion coefficient of the slow equation can depend on both the slow and fast variables. Moreover when the diffusion coefficient of the slow equation is independent of the fast variable then a strong convergence in probability is obtained. Otherwise only a weak convergence can be proved.

The averaging principle for infinite dimensional systems follows more delicated arguments: for this we refer to [5, 6, 8, 9, 14, 15, 24, 39] and the references therein. Also for infinite dimensional systems the previous comment about the dependence of the diffusion coefficients holds. See also [19, 20, 38] for optimal control problems of slow-fast systems in infinite dimension.

However for numerical applications it is very important to know the speed of convergence for which \(U^\epsilon \rightarrow U\), e.g. see [3, 29]. For the study of the order of convergence for finite dimensional systems we refer to [16, 25, 28, 30, 35] and the references therein. It is important to mention that the order of convergence can be studied in two ways: in the strong sense and in the weak sense. Moreover the optimal order for the strong and weak convergence are known to be 1/2 and 1 respectively.

Recently the problem of estimating the order of convergence in the averaging principle for infinite dimensional systems systems is being addressed by researchers:

in [4] the author (generalizing his previous work [2]) considers a slow-fast stochastic reaction diffusion system with additive noise. Both the weak and strong orders of convergence are obtained: in particular under strong regularity of the noise (it is for example assumed that the covariance operator is trace class but for the precise statement see [4]) it is proved that the strong order of convergence is 1/2 and the weak order is 1 with both orders being optimal. Instead under more general assumptions on the noise only weaker orders of convergence are obtained for both the strong and weak convergence.

In [34] a 1-dimensional fully coupled reaction-diffusion system is considered and the strong order of convergence is proved to be 1/2 under very strong assumptions on the covariance operators of the noises, i.e. \(Tr (\Delta ^{1/2} Q_i)< \infty \), where \(\Delta \) is the Laplacian.

In [36] the strong order of convergence for a fully coupled slow-fast stochastic system is studied. Here it is assumed that the the covariance operators of the noises are trace-class and moreover that \(Tr (-A_1 Q_1) < \infty \).

See also [26] where the weak order of convergence for a stochastic wave equation with fast oscillation given by a fast reaction-diffusion stochastic system is proved to be 1. Also here it assumed \(Tr ( Q_i)< \infty \).

Indeed in all these papers the case \(Tr ( Q_i)= \infty \), which is very important for applications as it happens very naturally for example when the stochastic perturbation is a white noise i.e. \(Q_i=I\), can’t be treated.

In this manuscript we are then interested in studying the strong order of convergence for the slow-fast infinite-dimensional system of stochastic evolution equations (1) where \(W^{Q_1}, W^{Q_2}\) are general Wiener processes on H, K respectively with covariance operators \(Q_1,Q_2\) with \(Tr(Q_1)=+ \infty \), \(Tr(Q_2) = +\infty \) possibly. Under some hypotheses, see Hypotheses 1, 2, 3, 4, 5 below, we prove that the strong order of convergence is 1/2 which is known to be optimal. In particular we show in Theorem 1 that

$$\begin{aligned} \mathbb {E}\left[ \sup _{t\in [0,T]} \left| U^\epsilon _t-U_t\right| ^2 \right] \le C \epsilon \end{aligned}$$

where \(U_t\) is the solution of the averaged equation. Notice that this result is much stronger than [4, 36] where \(\sup _{t \in [0,T]}\) is outside the expectation.

The key tool in the proof of Theorem 1 is Proposition 3. The proof of this proposition is based on a technical result, i.e. Lemma 6.3, which is inspired by [7], and it is a consequence of the mixing properties of the fast motion, i.e. Lemmas 4.5, 4.6, 4.7. We recall that [7] studies the normal deviations, i.e. the weak convergence of \(Z^\epsilon :=(U^\epsilon -U)/ \sqrt{\epsilon }\), when the equation for the slow component has no stochastic perturbation (\(Q_1=0\)).

Finally we discuss an application of our theory to a 1-dimensional slow-fast stochastic reaction diffusion system where the stochastic perturbation is given by a white noise both in time and space which to the best of our knowledge, as said before, can not be treated by the existing literature.

The paper is organized as follows: in Sect. 2 we introduce the problem in a formal way and we state the assumptions that we will use. In Sect. 3 we prove some a-priori estimates. In Sect. 4 we prove some results related to the fast motion. In Sect. 5 we study the well posedness of the averaged equation. In Sect. 6 we prove some preliminary results. In Sect. 7 we prove that the order of convergence is 1/2 and we give an application of our theory.

2 Setup and Assumptions

In this section we define the notation and the assumptions for the rest of the paper.

H, K will be Hilbert spaces with scalar products \(\langle \cdot ,\cdot \rangle _H, \langle \cdot ,\cdot \rangle _K\) and \(|\cdot |_H, |\cdot |_K\) the induced norms.

\(B_B(H)\) will denote the space of bounded functions \(\phi :H \rightarrow \mathbb {R}\) with the sup norm \(|\cdot |_{H,\infty }\).

Lip(H) will denote the set of Lipschitz functions \(\phi :H \rightarrow \mathbb {R}\) and set

$$\begin{aligned}{}[\phi ]_{\text{ H,Lip } }=\sup _{x \ne y} \frac{|\phi (x)-\phi (y)|}{|x-y|_H}. \end{aligned}$$

\(\mathcal {L}(H)\) will denote the space of linear bounded operators from H to H, endowed with the operator norm

$$\begin{aligned} \left\| L\right\| _H=\sup _{|x|_H=1}|Lx|_H. \end{aligned}$$

Next denote by \(\mathcal {L}_2(H)\) the space of Hilbert-Schmidt operators endowed with the norm

$$\begin{aligned} \left\| L\right\| _{\mathcal {L}_2(H)}=(Tr(L^*L))^{1/2}. \end{aligned}$$

The analogous spaces \(B_B(K),Lip(K),\mathcal {L}(K),\mathcal {L}_2(K)\) are defined for the Hilbert space K with the corresponding norms \(|\cdot |_{K,\infty },[\cdot ]_{\text{ K,Lip } },\left\| \cdot \right\| _K,\left\| \cdot \right\| _{\mathcal {L}_2(K)}\).

In order to simplify the notation we will omit the subscripts K and H in the various norms when no confusion is possible.

\(\mathcal {B}(H)\) and \(\mathcal {B}(K)\) will denote the Borel sigma-algebra in H and K respectively.

Consider now the following infinite dimensional system for \(0 \le t \le T < \infty \)

$$\begin{aligned} {\left\{ \begin{array}{ll} dU_t^{\epsilon }=A_1U_t^{\epsilon } dt +F(U_t^{\epsilon },V_t^{\epsilon })dt+dW^{Q_1}_t,\\ U_0^{\epsilon }=u \in H, \\ dV_t^{\epsilon }=\epsilon ^{-1}A_2V_t^{\epsilon } dt+\epsilon ^{-1} G(V_t^{\epsilon })dt+\epsilon ^{-1/2}dW^{Q_2}_t, \\ V_0^{\epsilon }=v \in K, \end{array}\right. } \end{aligned}$$
(2)

where

  • \(\epsilon >0\) is a parameter,

  • \(A_1 :D(A_1) \subset H \rightarrow H\), \(A_2 :D(A_2) \subset K \rightarrow K\) are linear operators,

  • \(F :H\times K\rightarrow H\), \(G :K\rightarrow K\),

  • \(W^{Q_1}, W^{Q_2}\) are independent cylindrical Wiener processes on H, K respectively with covariance operator \(Q_1,Q_2\) respectively and they are defined on some probability space \((\Omega ,\mathcal {F},\mathbb {P})\) with a normal filtration \(\mathcal {F}_t\), \(t \ge 0 \).

We now state the assumptions that we will use throughout the work:

Hypothesis 1

\(A_1 :D(A_1) \subset H \rightarrow H\) is a linear operator generator of an analytical semigroup \(e^{A_1 t}\) on H, \(t \ge 0 \).

Moreover there exist an orthonormal basis \(\{e_k\}_{k \in \mathbb {N}}\) of H and \(\{\alpha _k\}_{k \in \mathbb {N}} \subset (0, +\infty )\) such that

$$\begin{aligned} A_1 e_k = -\alpha _k e_k. \end{aligned}$$

Moreover we assume that there exist \(\zeta >0,n \ge 2\) integer and \( 1/(2n)< \beta < 1/3\) such that

$$\begin{aligned} \sum _{k=1}^{\infty } \alpha _{k}^{-\zeta } < +\infty \end{aligned}$$

and

$$\begin{aligned} \sum _{k=1}^{\infty } \alpha _{k}^{n(\zeta +2\beta -1)-\zeta }< +\infty . \end{aligned}$$

Remark 2.1

Hypothesis 1 is necessary in the proof of Proposition 3. It holds for example when \(A_1= \Delta \) is the Laplacian on [0, L]. Indeed in this case it is well known that \(\alpha _k = C k^2\) and then the two series converge for example by choosing \(\zeta =\frac{3}{5}\), \(n=3\), \(\beta =\frac{1}{5}\).

From Hypothesis 1 we have the following spectral representation:

$$\begin{aligned} e^{At}x=\sum _{k=1}^{\infty }e^{-\alpha _k t}\langle x,e_k \rangle e_k,\quad \forall t >0. \end{aligned}$$
(3)

Moreover we can define the fractional powers of \(-A_1\) denoted by \((-A_1)^{\theta }\) for \(\theta \ge 0\) with domain \(D((-A_1)^\theta )\). We will denote by \(|\cdot |_{\theta }\) the norm \(|\cdot |_{D((-A_1)^\theta )}=|(-A_1)^\theta \cdot |\).

The following standard results holds, e.g. see [33, Chapter 2, Theorem 6.13],

$$\begin{aligned} \Vert (-A_1)^\theta e^{At} \Vert \le C_\theta t^{-\theta } e^{-\nu t}, \quad \forall \theta \ge 0, { t > 0}, \end{aligned}$$
(4)

for some \(\nu >0\) and

$$\begin{aligned} |e^{A_1t}x-x| \le C_\theta t^{\theta }|x|_{\theta }, \quad \forall x \in D\left( (-A_1)^\theta \right) , 0< \theta \le 1, t \ge 0. \end{aligned}$$
(5)

Hypothesis 2

\(A_2 :D(A_2) \subset K \rightarrow K\) is a linear operator generator of a \(\mathcal {C}_0\)-semigroup \(e^{A_2 t}\) on K, \(t \ge 0 \).

Moreover there exists \(\lambda >0\) such that

$$\begin{aligned} \left\| e^{A_2 t }\right\| \le e^{-\lambda t}, \end{aligned}$$
(6)

for every \(t\ge 0\).

Hypothesis 3

There exist \(L_F, L_G>0\) such that

$$\begin{aligned}&|F(x_2,y_2)-F(x_1,y_1)|\le L_F (|x_2-x_1|+|y_2-y_1| ),\\&|G(y_2)-G(y_1)|\le L_G |y_2-y_1| , \end{aligned}$$

for every \(x_1,x_2 \in H, y_1,y_2 \in K\).

Moreover we assume that

$$\begin{aligned} L_G < \lambda . \end{aligned}$$

We remark that this implies that \(A_2+G(\cdot )\) is strongly dissipative, i.e. set

$$\begin{aligned} \delta =\frac{\lambda -L_G}{2}>0, \end{aligned}$$
(7)

then it holds:

$$\begin{aligned} \langle A_2 (y_2-y_1) + G(y_2) -G(y_1), y_2-y_1\rangle \le -2\delta |y_2-y_1|^2, \end{aligned}$$
(8)

for every \(y_1,y_2 \in D(A_2)\).

Hypothesis 4

There exist \(C>0\), \(\gamma \in (0,1/2)\) such that

$$\begin{aligned}&\left\| e^{A_1t}Q_1^{1/2} \right\| _{\mathcal {L}_2}\le C (t \wedge 1)^{-\gamma }, \\&\left\| e^{A_2t}Q_2^{1/2} \right\| _{\mathcal {L}_2}\le C (t \wedge 1)^{-\gamma } , \end{aligned}$$

for every \(t>0\).

Hypothesis 5

Assume that \(Q_2\) is invertible (with inverse \(Q_2^{-1} \in {\mathcal {L}}(K)\)).

Remark 2.2

Hypotheses 4, 5 hold for example choosing \(A_1= A_2=A\) to be the Laplacian on [0, L] and \(Q_1=Q_2=Q=I\).

Indeed Hypothesis 5 is immediately satisfied. Moreover by setting \(H=K=L^2([0,L])\), we have that \(A e_k =-C k^2 e_k\) for some orthonormal basis \(\{ e_k\}\) of eigenvectors of A. Thus, by the spectral representation (3) with \(\alpha _k=C k^2\), we have:

$$\begin{aligned} \left\| e^{A t}Q^{1/2} \right\| _{\mathcal {L}_2}^2&=\left\| e^{A t} \right\| _{\mathcal {L}_2}^2 =\sum _{h=1}^\infty |e^{At}e_h|^2 =\sum _{h=1}^{\infty } e^{-2Ch^2t} \le \int _{0}^{\infty } e^{-2Ch^2t} {dh}\\&=\frac{1}{\sqrt{2Ct}}\int _{0}^{\infty } e^{-y^2} dy = C t ^{-1/2}, \end{aligned}$$

where the inequality follows since \(\forall t > 0\) the function \(h \mapsto e^{-Ch^2 t}\) is non-increasing. This shows that Hypothesis 4 holds with \(\gamma =1/4.\)

Proposition 1

Let \(u \in H, v \in K\), under Hypotheses 1, 2, 3, 4, 5 there exists a unique mild solution of (2) given by

$$\begin{aligned} {\left\{ \begin{array}{ll} U_t^{\epsilon }=e^{A_1t}u +\int _0^te^{A_1(t-s)} F(U_s^{\epsilon },V_s^{\epsilon })ds +\int _0^te^{A_1(t-s)} dW^{Q_1}_s, \\ V_t^{\epsilon }=e^{A_1t}v +\int _0^te^{A_2(t-s)} G(V_s^{\epsilon })ds +\int _0^te^{A_2(t-s)} dW^{Q_2}_s, \end{array}\right. } \end{aligned}$$
(9)

for every \(t \in [0,T]\).

Proof

See e.g. [10]. \(\square \)

In the sequel we will always assume that Hypotheses  1, 2, 3, 4, 5 hold. Moreover \(C>0\) will denote a generic constant independent of \(\epsilon \) which may change from line to line.

3 A Priori Estimates

In this section we prove some classical a-priori estimates for the slow and fast components.

In the following for every \(t \ge 0\) denote by

$$\begin{aligned} \Gamma ^{1}_t=\int _0^t e^{A_1 (t-s)} dW_s^{Q_1} \end{aligned}$$

and

$$\begin{aligned} \Gamma ^{2 \epsilon }_t= \frac{1}{\sqrt{\epsilon }}\int _0^t e^{A_2 (t-s)/ \epsilon } dW_s^{Q_2} \end{aligned}$$

the stochastic convolutions.

First we prove some estimates related to \(\Gamma ^{1}_t\) and \(\Gamma ^{2 \epsilon }_t\).

Lemma 3.1

$$\begin{aligned} \mathbb {E}\left[ \sup _{t\in [0,T]} \left| \Gamma ^{1}_t\right| _{\theta }^p \right] < +\infty , \end{aligned}$$

for every \(0 \le \theta < 1/2-\gamma \), \(p \ge 1\).

Proof

Fix \(0<\eta < 1/2\), by the factorization method, e.g. see [10, Chapter 5, Section 3], we have:

$$\begin{aligned} \Gamma ^{1}_t=C \int _0^t e^{A_1(t - \rho )}(t-\rho )^{\eta -1}Y_\rho d\rho \end{aligned}$$

where

$$\begin{aligned} Y_\rho =\int _0^\rho e^{A_1 (\rho -s)} (\rho -s)^{-\eta }dW_s^{Q_1}. \end{aligned}$$

Now fix \(0 \le \theta < 1/2-\gamma \), then by Holder’s inequality:

$$\begin{aligned} \mathbb {E} \left[ \sup _{t \le T} \left| \Gamma ^{1}_t \right| ^p_\theta \right] \le C \int _0^T \mathbb {E} |Y_\rho |^p_\theta d\rho , \end{aligned}$$
(10)

for every \(p> 1/\eta >2\).

Now we estimate \(\mathbb {E} |Y_\rho |^p_\theta \); since \(Y_\rho \) is a Gaussian random variable, by Ito’s isometry, (4) and Hypothesis 4 we have:

$$\begin{aligned} \mathbb {E} |Y_\rho |^p_\theta&{ \le C \left( {\mathbb {E}} |Y_\rho |_\theta ^2 \right) ^{p/2} = } C \left( \int _0^\rho \left\| (-A_1)^\theta e^{A_1 (\rho -s)} (\rho -s)^{-\eta } Q_1^{1/2}\right\| ^2_{\mathcal {L}_2} ds \right) ^{p/2} \\&\le C \left( \int _0^\rho (\rho -s)^{-2\eta } \left\| (-A_1)^\theta e^{A_1 (\rho -s)/2}\right\| ^2 \left\| e^{A_1 (\rho -s)/2} Q_1^{1/2}\right\| ^2_{\mathcal {L}_2} ds \right) ^{p/2} \\&\le C \left( \int _0^\rho (\rho -s)^{-2(\eta +\theta +\gamma )} ds \right) ^{p/2} < C, \end{aligned}$$

for every \(\rho \le T\) and \(\theta , \eta \) such that \(0 \le \theta + \eta < 1/2 -\gamma \).

Next inserting the last inequality in (10) and recalling that \(p > 1/\eta \), which yields \( p > (1/2 - \gamma - \theta )^{-1}\) we obtain the thesis of the Lemma for \(0 \le \theta < 1/2-\gamma \), \(p> (1/2 - \gamma - \theta )^{-1}>2\).

Finally by Holder’s inequality we have the thesis of the Lemma. \(\square \)

Lemma 3.2

Let \(p\ge 2\), then there exists \(C=C(p)>0\) such that

$$\begin{aligned} \sup _{t>0} \mathbb {E}\left[ \left| \Gamma ^{2 \epsilon }_t\right| ^p \right] \le C, \end{aligned}$$

for every \(\epsilon >0\).

Proof

For \(t>0\), by [10, Theorem 4.36] and by our hypotheses we have

$$\begin{aligned} \mathbb {E}\left[ \left| \Gamma ^{2 \epsilon }_t\right| ^p \right]&\le C \left( \int _0^t \frac{1}{\epsilon } \left\| e^{A_2(t-s)/\epsilon } Q_2^{1/2}\right\| _{\mathcal {L}_2}^2ds \right) ^{p/2}\\&= C \left( \int _0^{t/ \epsilon } \left\| e^{A_2 \rho } Q_2^{1/2}\right\| _{\mathcal {L}_2}^2 d \rho \right) ^{p/2}\\&\le C \left( \int _0^{t/ \epsilon } \left\| e^{A_2 \rho /2} \right\| ^2 \left\| e^{A_2 \rho /2} Q_2^{1/2}\right\| _{\mathcal {L}_2}^2 d \rho \right) ^{p/2} \\&\le C \left( \int _0^{t/ \epsilon } e^{-\lambda \rho } \left( \frac{\rho }{2} \wedge 1 \right) ^{-2 \gamma } d \rho \right) ^{p/2} \\&\le C \left( \int _0^{\infty } e^{-\lambda \rho } \left( \frac{\rho }{2} \wedge 1 \right) ^{-2\gamma } d \rho \right) ^{p/2} \le C , \end{aligned}$$

so that the thesis is proved. \(\square \)

Lemma 3.3

Let \(p \ge 2\) then there exists \(C=C(T,p)>0\) such that

$$\begin{aligned} \mathbb {E}\left[ \sup _{t\in [0,T]} \left| U^\epsilon _t\right| ^p \right] \le C (1+|u|^p+|v|^p) \end{aligned}$$
(11)

and

$$\begin{aligned} \sup _{t\in [0,T]} \mathbb {E}\left[ \left| V^\epsilon _t\right| ^p \right] \le C (1+|u|^p+|v|^p), \end{aligned}$$
(12)

for every \(\epsilon >0\).

Proof

Define

$$\begin{aligned} \Lambda ^{1\epsilon }_t=U_t^\epsilon -\Gamma ^{1}_t, \end{aligned}$$

so that

$$\begin{aligned} d \Lambda ^{1\epsilon }_t= A_1 \Lambda ^{1\epsilon }_t dt + F(\Lambda ^{1\epsilon }_t+\Gamma ^{1}_t, V^\epsilon _t)dt. \end{aligned}$$

By Young’s inequality and Hypotheses 1,  3 we have:

$$\begin{aligned} \frac{1}{p} \frac{d}{d t}\left| \Lambda ^{1\epsilon }_t \right| ^{p}&= \langle A_{1} \Lambda ^{1\epsilon }_t, \Lambda ^{1 \epsilon }_t \rangle \left| \Lambda ^{1\epsilon }_t\right| ^{p-2}+ \langle F(\Lambda ^{1\epsilon }_t+\Gamma ^{1}_t, V^{\epsilon }_t ) , \Lambda ^{1 \epsilon }_t \rangle \left| \Lambda ^{1\epsilon }_t\right| ^{p-2}\\&\le C\left| \Lambda ^{1\epsilon }_t\right| ^{p}+C\left| F(\Lambda ^{1\epsilon }_t+\Gamma ^{1}_t, V^{\epsilon }_t ) \right| ^{p} \\&\le C\left| \Lambda ^{1\epsilon }_t\right| ^{p}+C\left( 1+\left| \Gamma ^{1 }_t\right| ^{p}+\left| V^{\epsilon }_t\right| ^{p}\right) , \end{aligned}$$

for every \(t \le T\).

Then by the comparison Theorem we have

$$\begin{aligned} \left| \Lambda ^{1\epsilon }_t\right| ^p \le C |u|^p + C \int _0^t \left( 1+\left| \Gamma ^{1}_s\right| ^p+ \left| V^{\varepsilon }_s\right| ^p \right) ds, \end{aligned}$$
(13)

for every \(t \le T\).

Then by the definition of \(\Lambda ^{1\epsilon }\) and this last inequality it follows

$$\begin{aligned} \begin{aligned} \left| U_t^\epsilon \right| ^{p} \le&C \left( 1+|u|^{p}+\int _{0}^{t} \left( \left| \Gamma _s^1\right| ^{p} +\left| V^{\epsilon }_s\right| ^{p} \right) d s +\left| \Gamma _t^1\right| ^{p} \right) , \end{aligned} \end{aligned}$$
(14)

for every \(t \le T\).

Now by Lemma 3.1 we have:

$$\begin{aligned} \mathbb {E} \left[ \sup _{t \le \tau }\left| U_t^\epsilon \right| ^{p} \right] \le C \left( 1+|u|^{p}+\int _{0}^{\tau } \mathbb {E}\left| V^{\epsilon }_r\right| ^{p} dr \right) , \end{aligned}$$
(15)

for every \(\tau \le T\).

Now we proceed in a similar way to [19, proof of Lemma 3.10], i.e. set

$$\begin{aligned} \Lambda ^{2\epsilon }_t=e^{\frac{2 \delta }{\epsilon } t}\left( V_t^\epsilon -\Gamma ^{2\epsilon }_t \right) , \end{aligned}$$

so that

$$\begin{aligned} d\Lambda ^{2\epsilon }_t = \frac{2 \delta }{\epsilon } \Lambda ^{2\epsilon }_t dt+ \frac{1}{\epsilon } A_2 \Lambda ^{2\epsilon }_t dt + \frac{1}{\epsilon } e^{\frac{2 \delta }{\epsilon } t} G(e^{-\frac{2 \delta }{\epsilon }}\Lambda ^{2\epsilon }_t + \Gamma ^{2\epsilon }_t) dt. \end{aligned}$$

Now by (8) it follows

$$\begin{aligned} d |\Lambda ^{2\epsilon }_t|^2&= \frac{4 \delta }{\epsilon } |\Lambda ^{2\epsilon }_t|^2 + \frac{2}{\epsilon } \langle A_2\Lambda ^{2\epsilon }_t, \Lambda ^{2\epsilon }_t \rangle dt + \frac{2}{\epsilon } e^{2 \delta t / \epsilon } \langle G(e^{-2 \delta t/ \epsilon } \Lambda ^{2\epsilon }_t + \Gamma ^{2\epsilon }_t), \Lambda ^{2\epsilon }_t\rangle dt \\&\quad - \frac{2}{\epsilon } e^{2 \delta t / \epsilon } \langle G(\Gamma ^{2\epsilon }_t), \Lambda ^{2\epsilon }_t\rangle dt + \frac{2}{\epsilon } e^{2 \delta t / \epsilon } \langle G(\Gamma ^{2\epsilon }_t), \Lambda ^{2\epsilon }_t\rangle dt \\&\le \frac{2}{\epsilon } e^{2 \delta t / \epsilon } \langle G( \Gamma ^{2\epsilon }_t), \Lambda ^{2\epsilon }_t\rangle dt . \end{aligned}$$

Now similarly to [19, proof of Lemma 3.10] fix \(\theta >0\), differentiate \(f(x)= \sqrt{x+ \theta }\) and use the previous inequality, then:

$$\begin{aligned} |\Lambda ^{2\epsilon }_t |&\le \sqrt{|\Lambda ^{2\epsilon }_t |^2+ \theta } \\&\le \sqrt{|v|^2+\theta }+\int _0^t \frac{1}{\epsilon } \frac{1}{\sqrt{|\Lambda ^{2\epsilon }_s|^2+\theta }} e^{\frac{2 \delta }{\epsilon } s} |G(\Gamma ^{2\epsilon }_s)||\Lambda ^{2\epsilon }_s| ds . \end{aligned}$$

Now by dominated convergence for \(\theta \rightarrow 0\) we have:

$$\begin{aligned} |\Lambda ^{2\epsilon }_t | \le |v| + \int _0^t \frac{1}{\epsilon } e^{\frac{2 \delta }{\epsilon } s} |G( \Gamma ^{2\epsilon }_s)| ds, \end{aligned}$$

so that by recalling the definition of \(\Lambda ^{2\epsilon }_t\) we have:

$$\begin{aligned} |V_t^\epsilon |&\le e^{-\frac{2 \delta }{\epsilon } t} |v| + \int _0^t \frac{1}{\epsilon } e^{-\frac{2 \delta }{\epsilon } (t-s)} |G( \Gamma ^{2\epsilon }_s)| ds +| \Gamma ^{2\epsilon }_t |\\&= e^{-\frac{2 \delta }{\epsilon } t} |v| + \int _0^{t/\epsilon } e^{-2 \delta (t / \epsilon - s)} |G( \Gamma ^{2\epsilon }_{\epsilon s})| ds +| \Gamma ^{2\epsilon }_t |. \end{aligned}$$

Then, by Holder’s inequality and Lemma 3.2, we have:

$$\begin{aligned} \mathbb {E}|V_t^\epsilon |^p&\le C |v|^p + C \mathbb {E} \left| \int _0^{t / \epsilon } e^{-2 \delta (t / \epsilon - s)} |G( \Gamma ^{2\epsilon }_{\epsilon s})| ds \right| ^p + C\mathbb {E} \left| \Gamma ^{2\epsilon }_t \right| ^p \\&\le C |v|^p + C \left( \int _0^{t / \epsilon } e^{-\frac{ 2\delta p}{ (p-1)} (t / \epsilon - s)} ds \right) ^{p-1} \int _0^{t / \epsilon } e^{-2\delta p (t / \epsilon - s)} \mathbb {E}|G(\Gamma ^{2\epsilon }_{\epsilon s})| ^p ds\\&\quad + C \mathbb {E} \left| \Gamma ^{2\epsilon }_t \right| ^p \\&= C |v|^p + C \left( \int _0^{t / \epsilon } e^{-\frac{ 2\delta p}{ (p-1)} \sigma } d\sigma \right) ^{p-1} \int _0^{t } \frac{1}{\epsilon } e^{-2\delta p (t - s)/ \epsilon } \mathbb {E}|G(\Gamma ^{2\epsilon }_{ s})| ^p ds\\&\quad +C \mathbb {E}\left| \Gamma ^{2\epsilon }_t \right| ^p \\&\le C |v|^p + C \left( \int _0^{t / \epsilon } e^{-\frac{ 2\delta p}{ (p-1)} \sigma } d\sigma \right) ^{p-1} \int _0^{t } \frac{1}{\epsilon } e^{-2\delta p (t - s)/ \epsilon } (1+ \mathbb {E}|\Gamma ^{2\epsilon }_s|^p ) ds\\&\quad +C \mathbb {E}\left| \Gamma ^{2\epsilon }_t \right| ^p \\&\le C |v|^p + C \left( \int _0^{t / \epsilon } e^{-\frac{ 2\delta p}{ (p-1)} \sigma } d\sigma \right) ^{p-1} \int _0^{t } \frac{1}{\epsilon } e^{-2\delta p (t - s)/ \epsilon } ds +C \\&\le C(1+ |v|^p). \end{aligned}$$

This proves (12).

Finally inserting (12) into (15) we have (11). \(\square \)

Lemma 3.4

Let \(0< \alpha < 1/2 - \gamma \), \(u \in D((-A_1)^\alpha )\), \(v \in K\), then there exists \(C=C(T,\alpha )>0\) such that

$$\begin{aligned} \mathbb {E}\left[ \sup _{t\in [0,T]} \left| U^\epsilon _t\right| _{\alpha }^2 \right] \le C (1+|u|_\alpha ^2+|v|^2), \end{aligned}$$

for every \(\epsilon >0\).

Proof

Consider for \(t \le T\)

$$\begin{aligned} U_t^{\epsilon }=e^{A_1t}u +\int _0^te^{A_1(t-s)} F(U_s^{\epsilon },V_s^{\epsilon })ds +\int _0^te^{A_1(t-s)} dW^{Q_1}_s. \end{aligned}$$
(16)

First as \(u \in D((-A_1)^\alpha )\) we have:

$$\begin{aligned} \sup _{t \le T}|e^{A_1 t}u|_\alpha ^2=\sup _{t \le T}|e^{A_1 t}(-A_1)^\alpha u|^2 \le C |u|_\alpha ^2. \end{aligned}$$

Moreover by (4) and by Lemma 3.3 we have:

$$\begin{aligned}&\mathbb {E}\left[ \sup _{t\in [0,T]} \left| \int _0^t e^{A_1(t-s)} F(U_s^{\epsilon },V_s^{\epsilon }) ds \right| _\alpha ^2 \right] \\&\quad \le \mathbb {E}\left[ \sup _{t\in [0,T]} \left| \int _0^t (t-s)^{-\alpha } \left| F(U_s^{\epsilon },V_s^{\epsilon }) \right| ds \right| ^2 \right] \\&\quad \le C \mathbb {E}\left[ \sup _{t\in [0,T]} \int _0^t (t-s)^{-2\alpha } ds \int _0^t \left| F(U_s^{\epsilon },V_s^{\epsilon }) \right| ^2 ds \right] \\&\quad \le C \int _0^T \left( 1+ \mathbb {E}| U_s^{\epsilon }|^2 +\mathbb {E} |V_s^{\epsilon } |^2 \right) ds \\&\quad \le C (1+|u|^2+|v|^2). \end{aligned}$$

Finally by Lemma 3.1 we have:

$$\begin{aligned} \mathbb {E} \left[ \sup _{t\in [0,T]} \left| \Gamma ^1_t\right| ^2_\alpha \right] < +\infty . \end{aligned}$$

By considering (16), calculating \( \mathbb {E}\left[ \sup _{t\in [0,T]} \left| U^\epsilon _t\right| _{\alpha }^2 \right] \) and using the last three inequalities we have the thesis. \(\square \)

Lemma 3.5

Let \(0< \alpha < 1/2 - \gamma \), \(u \in D((-A_1)^\alpha )\), \(v \in K\), then there exists \(C=C(T,\alpha )>0\) such that

$$\begin{aligned} \mathbb {E}\left| U_{t+h}^\epsilon - U_{t}^\epsilon \right| ^2 \le C |h|^{2 \alpha } (1+|u|_\alpha ^2+|v|^2), \end{aligned}$$

for every \(\epsilon >0\), \(0 \le t \le T\), \(h\ge 0\) such that \(t+h \le T\).

Proof

For \(0 \le t \le T\), \(h\ge 0\) such that \(t+h \le T\) we have

$$\begin{aligned} U^\epsilon _{t+h}-U^\epsilon _{t}= & {} (e^{A_1 h}-I) U^\epsilon _{t} + \int _t^{t+h} e^{A_1(t+h-s)}F(U^\epsilon _{s},V^\epsilon _{s}) ds\nonumber \\{} & {} + \int _t^{t+h} e^{A_1(t+h-s)} dW^{Q_1}_s. \end{aligned}$$
(17)

Consider the first term on the right-hand-side, as \(u \in D((-A_1)^\alpha )\) by (5) and by Lemma 3.4 we have:

$$\begin{aligned} \mathbb {E} \left| (e^{A_1 h}-I) U^\epsilon _{t} \right| ^2&\le C |h|^{2 \alpha } \mathbb {E} \left| U^\epsilon _{t} \right| ^2_\alpha \\&\le C |h|^{2 \alpha } (1+|u|_\alpha ^2+|v|^2). \end{aligned}$$

Consider now the second term on the right-hand-side, then by Lemma 3.3 we have:

$$\begin{aligned} \mathbb {E} \left| \int _t^{t+h} e^{A_1(t+h-s)}F(U^\epsilon _{s},V^\epsilon _{s}) ds \right| ^2&\le C |h| \int _t^{t+h} \left( 1+\mathbb {E} |U^\epsilon _{s}|^2+\mathbb {E} |V^\epsilon _{s}|^2 \right) ds \\&\le C |h|^2 (1+|u|^2+|v|^2). \end{aligned}$$

Finally for the third term on the right-hand-side by Ito’s isometry and Hypothesis 4 we have:

$$\begin{aligned} \mathbb {E} \left| \int _t^{t+h} e^{A_1(t+h-s)} dW_s^{Q_1} \right| ^2&=\int _t^{t+h} \left\| e^{A_1(t+h-s)} Q_1^{1/2}\right\| _{\mathcal {L}_2}^2 ds\\&\le C \int _t^{t+h} |t+h-s|^{-2\gamma } ds \\&= C |h|^{1-2 \gamma }. \end{aligned}$$

As by assumption \(2 \alpha \wedge 2 \wedge (1-2 \gamma )= 2\alpha \) then we have the thesis. \(\square \)

4 Fast Motion

In this section we study some classical properties of the fast motion. Consider

$$\begin{aligned} {\left\{ \begin{array}{ll} dv^v_s=A_2v^v_sds+ G(v^v_s)ds+d w^{Q_2}_s, \\ v^v_0=v \in K, \end{array}\right. } \end{aligned}$$
(18)

for every \(s \ge 0\) and for some \(Q_2\)-Wiener process \(w_s^{Q_2}\).

First define the semigroup related to the fast motion by

$$\begin{aligned} P_s \phi (v) =\mathbb {E} \left[ \phi (v_s^{v})\right] , \end{aligned}$$
(19)

for every \(\phi \in B_B(K)\), \(s \ge 0\).

Next recall that \(\delta \) was defined by (7), then we have:

Lemma 4.1

$$\begin{aligned} \mathbb {E}\left[ \left| v_s^{v_1}-v_s^{v_2} \right| ^2\right] \le e^{-4\delta s} |v_1-v_2|^2, \end{aligned}$$

for every \(s \ge 0\), \(v_1,v_2 \in K\).

Proof

Define \(\rho _s=v_s^{v_1}-v_s^{v_2}\), then by (8) we have:

$$\begin{aligned} \frac{d}{ds} |\rho _s|^2&= 2 \langle A_2 \rho _s,\rho _s \rangle +2\langle G(v_s^{v_1})-G(v_s^{v_2}),\rho _s \rangle \\&\le - 4 \delta |\rho _s|^2 , \end{aligned}$$

for every \(s \ge 0\), \(v_1,v_2 \in K\).

Then by taking the expectation and applying the comparison Theorem we have the thesis. \(\square \)

Next we can show:

Lemma 4.2

Let \(p\ge 1\), then there exists \(C=C(p)>0\) such that:

$$\begin{aligned} \mathbb {E}\left[ \left| v_s^{v} \right| ^p\right] \le C (1+e^{-\delta p s} |v|^p), \end{aligned}$$

for every \(s \ge 0\), \(v \in K\).

Proof

Define

$$\begin{aligned} {\tilde{\Gamma }}_s^{Q_2}=\int _0^s e^{A_2(s-r)}d w_r^{Q_2}. \end{aligned}$$

First by Burkholder’s inequality and Hypotheses 2 and 4 similarly to what is done for Lemma 3.2 we have:

$$\begin{aligned} \mathbb {E}\left[ \left| {\tilde{\Gamma }}^{Q_2}_s\right| ^p \right]&\le C \left( \int _0^{s} \left\| e^{A_2 (s-r)} Q_2^{1/2}\right\| _{\mathcal {L}_2}^2 dr \right) ^{p/2} \nonumber \\&\le C \left( \int _0^{s} e^{-\lambda (s-r)} \left( \frac{s-r}{2} \wedge 1 \right) ^{-2 \gamma } d r \right) ^{p/2} \le C , \end{aligned}$$
(20)

for every \(s \ge 0\).

Now set \(\rho _s=v^{v}_s-{\tilde{\Gamma }}_s^{Q_2}\). Then for \(p\ge 2\) we have:

$$\begin{aligned} \frac{1}{p}\frac{d}{ds} |\rho _s|^p&= \langle A_2 \rho _s,\rho _s \rangle |\rho _s|^{p-2} +\langle G(\rho _s+{\tilde{\Gamma }}_s^{Q_2})-G({\tilde{\Gamma }}_s^{Q_2}),\rho _s \rangle |\rho _s|^{p-2}\\&\quad + \langle G({\tilde{\Gamma }}_s^{Q_2}),\rho _s \rangle |\rho _s|^{p-2} \\&\le - 2 \delta |\rho _s|^p+ C (1+|{\tilde{\Gamma }}_s^{Q_2}|)|\rho _s|^{p-1} \\&\le - \delta |\rho _s|^p+ C (1+|{\tilde{\Gamma }}_s^{Q_2}|^p) . \end{aligned}$$

Then by the comparison Theorem and (20) we have:

$$\begin{aligned} \mathbb {E}\left| v_s^{u,v} \right| ^p&\le C \mathbb {E} |\rho _s|^p + C \mathbb {E} |{\tilde{\Gamma }}_s^{Q_2}|^p \\&\le C \left[ e^{-\delta p s}|v|^p + \int _0^s e^{-\delta p (s-r)} \left( 1+\mathbb {E} |{\tilde{\Gamma }}_r^{Q_2}|^p \right) dr + \mathbb {E} |{\tilde{\Gamma }}_s^{Q_2}|^p \right] \\&\le C \left( e^{-\delta p s}|v|^p +1\right) . \end{aligned}$$

\(\square \)

Now by [11, Theorem 6.3.3] there exists a unique invariant measure \(\mu \) for the semigroup \(P_t\). Moreover we have:

Lemma 4.3

We have:

$$\begin{aligned} \int _K |z| \mu (dz) < \infty . \end{aligned}$$
(21)

Proof

Fix \(N>0\), then by definition of invariant measure and Lemma 4.2 we have for every \(s>0\)

$$\begin{aligned} \int _K (|z|\wedge N) \mu (dz)&= \int _K \mathbb {E} \left[ |v_s^{z}| \wedge N \right] \mu (dz) { \le \int _K \left\{ \left[ C \left( 1 + e^{-\delta s} |z|\right) \right] \wedge N \right\} \mu (dz)}\\&\le \int _K C \left( 1\wedge N + e^{-\delta s} |z|\wedge N \right) \mu (dz)\\&\le C \left( 1 + e^{-\delta s} \int _K (|z|\wedge N) \mu (dz) \right) , \end{aligned}$$

where we have used the fact that \((a+b)\wedge c \le a \wedge c + b \wedge c\) for every \(a,b,c \ge 0.\) By choosing \(s>0\) large enough we have

$$\begin{aligned} \int _K (|z|\wedge N) \mu (dz) \le C, \end{aligned}$$

for some \(C>0\) independent of N. Letting \(N \rightarrow \infty \) by the monotone convergence Theorem we have the result. \(\square \)

Next we study the convergence to equlibrium of the semigroup of the fast motion, i.e. we prove:

Lemma 4.4

There exists \(C>0\) such that

$$\begin{aligned} \left| P_s \phi (v)-\int _K \phi (z) \mu (dz) \right| \le C \left[ \phi \right] _{Lip}e^{-2\delta s} (1+|v|), \end{aligned}$$

for every \(s \ge 0\), \(v \in K\), \(\phi \in Lip(K)\).

Moreover there exists \(C>0\) such that

$$\begin{aligned} \left| P_s \phi (v)-\int _K \phi (z) \mu (dz) \right| \le C |\phi |_{\infty } e^{-\delta s} (s \wedge 1)^{-1/2} (1+|v|) , \end{aligned}$$

for every \(s > 0\), \(\phi \in B_B(K)\), \(v \in K\).

Proof

First for every \(\phi \in Lip(K)\) by Lemma 4.1 we have:

$$\begin{aligned} \begin{aligned} \left| P_{s} \phi (v_2)-P_{s} \phi (v_1)\right|&\le [\phi ]_{\text {Lip }} \mathbb {E}\left[ \left| v^{v_2}_s-v^{ v_1}_s\right| \right] \\&\le [\phi ]_{\text{ Lip } } e^{-2\delta s}|v_2-v_1|, \end{aligned} \end{aligned}$$
(22)

for every \(s\ge 0\), \(v_1,v_2 \in K\).

Now let \(s>0\), by definition of invariant measure, (22) and Lemma  4.3 we have:

$$\begin{aligned} \left| P_s \phi (v)-\int _K \phi (z) \mu (dz) \right|&= \left| \int _K \left( P_{s} \phi (v)-P_{s} \phi (z) \right) \mu (dz)\right| \nonumber \\&\le \left[ \phi \right] _{Lip} e^{-2\delta s} \int _{K} |v-z| \mu (dz) \nonumber \\&\le C \left[ \phi \right] _{Lip} e^{-2\delta s} (1+|v|) , \end{aligned}$$
(23)

for every \(v \in K\), \(\phi \in Lip(K)\) so that we have the first inequality.

Now thanks to Hypothesis 5 we can apply [10, Theorem 9.32] to have the Bismut-Elworthy formula:

$$\begin{aligned} \sup _{u \in H} |D P_s \phi |_{\infty } \le C (s \wedge 1)^{-1/2} |\phi |_{\infty }, \end{aligned}$$
(24)

for every \(s>0\), \(\phi \in B_B(K)\).

Now by the semigroup property, the regularizing property of the semigroup (24) and by (22) we have:

$$\begin{aligned} \left| P_{s} \phi (v_2)-P_{s} \phi (v_1)\right|&= \left| P_{s/2} \left( P_{s/2} \phi \right) (v_2) -P_{s/2} \left( P_{s/2} \phi \right) (v_1) \right| \nonumber \\&\le \left[ P_{s / 2} \phi \right] _{\text {Lip}} e^{-2\delta \frac{s}{2}}|v_2-v_1| \nonumber \\&\le C |\phi |_{\infty }(s \wedge 1)^{-\frac{1}{2}} e^{-\delta s}|v_2-v_1|, \end{aligned}$$
(25)

for every \(s>0\), \(v_1,v_2 \in K\).

Finally similarly to before by (25) for \(s>0\) we have:

$$\begin{aligned} \left| P_s \phi (v)-\int _K \phi (z) \mu (dz) \right|&= \left| \int _K \left( P_{s} \phi (v)-P_{s} \phi (z) \right) \mu (dz)\right| \\&\le C |\phi |_{\infty }(s \wedge 1)^{-\frac{1}{2}} e^{-\delta s} \int _{K} |v-z| \mu (dz) \\&\le C|\phi |_{\infty }(s \wedge 1)^{-\frac{1}{2}} e^{-\delta s} (1+|v|) , \end{aligned}$$

for every \(v \in K\), \(\phi \in B_B(K)\). \(\square \)

Now we study the mixing properties of the semigroup of the fast motion. To this purpose define for \(0 \le s \le t \le \infty \), \(v \in K\)

$$\begin{aligned} \mathcal {H}_s^{t}(v)= \sigma (v_r^{v}, 0 \le s \le r \le t ). \end{aligned}$$

Then a classical consequence of Lemmas 4.2, 4.4 is the following mixing result whose proof is the same of [7, Lemma 3.2] and is reported in the appendix for completeness.

Lemma 4.5

There exists \(C>0\) such that

$$\begin{aligned}{} & {} \sup \left\{ \left| \mathbb {P} (B_1 \cap B_2)-\mathbb {P} (B_1) \mathbb {P} (B_2) \right| : B_1 \in \mathcal {H}_0^{t}(v), B_2 \in \mathcal {H}_{t+s}^{\infty }(v) \right\} \\{} & {} \quad \le C e^{-\delta s} s^{-1/2} (1+|v|), \end{aligned}$$

for every \(0 \le s \le t\), \(v \in K\).

Now Lemma 4.5 implies the following classical result, e.g. see [37] (see also [7, Proposition 3.3]). The proof can be found in the appendix for completeness.

Lemma 4.6

There exists \(C>0\) such that for every \(0\le s_1 \le t_1 < s_2 \le t_2\) and \(\xi _i\) \(\mathcal {H}_{s_i}^{t_i}(v)\)-measurable \(i=1,2\) and \(|\xi _i|\le 1\) a.s \(i=1,2\)

$$\begin{aligned} \left| \mathbb {E}\left[ \xi _{1} \xi _{2}\right] - \mathbb {E} \xi _{1} \mathbb {E} \xi _{2}\right| \le C \frac{e^{-\delta \left( s_{2}-t_{1}\right) }}{\sqrt{s_{2}-t_{1}}}\left( 1+|v|\right) . \end{aligned}$$

Since in our case \(|\xi _i|\) will not be bounded by 1 we need the following result which is similar to [7, Proposition 3.3]. Also in this case we postpone the proof in the appendix.

Lemma 4.7

Let \(\rho \in (0,1)\). Then there exists \(C=C(\rho )>0\) such that for every \(0\le s_1 \le t_1 < s_2 \le t_2\) and \(\xi _i\) \(\mathcal {H}_{s_i}^{t_i}(v)\)-measurable, \(i=1,2\) satisfying for some \(K_i=K_i(\rho )>0\)

$$\begin{aligned} \left( \mathbb {E}\left| \xi _{i}\right| ^{\frac{2}{1-\rho }} \right) ^{\frac{1-\rho }{2}}= K_i < \infty \end{aligned}$$
(26)

then:

$$\begin{aligned} \left| \mathbb {E}\left[ \xi _{1} \xi _{2}\right] - \mathbb {E} \xi _{1} \mathbb {E} \xi _{2}\right| \le C K_1^{\frac{2}{2+\rho }} K_2^{\frac{2}{2+\rho }} (K_1+K_2)^{\frac{2 \rho }{2+\rho }} \left( \frac{e^{-\delta ( s_{2}-t_{1})}}{\sqrt{s_{2}-t_{1}}} \left( 1+|v|\right) \right) ^{\frac{\rho }{2+\rho }}. \end{aligned}$$

5 Averaged Equation

In this section we introduce the averaged equation and we prove its well-posedness.

$$\begin{aligned} \overline{F}(u)=\int _K F(u,v)\mu (dv), \end{aligned}$$
(27)

for every \(u\in H\) and consider the so called averaged equation:

$$\begin{aligned} {\left\{ \begin{array}{ll} dU_t=A_1U_t dt+\overline{F}(U_t)dt+dW^{Q_1}_t,\\ U_0=u, \end{array}\right. } \end{aligned}$$
(28)

for every \(t\le T\).

Now we prove the well-posedness of the averaged equation:

Proposition 2

Equation (28) admits a unique mild solution given by:

$$\begin{aligned} U_t=e^{A_1t}u +\int _0^te^{A_1(t-s)} \overline{F} (U_s)ds +\int _0^te^{A_1(t-s)} dW^{Q_1}_s, \end{aligned}$$

for every \(t \in [0,T]\).

Moreover for every \(p>0\) there exists \(C=C(T,p)>0\) such that

$$\begin{aligned} \mathbb {E}\left[ \sup _{t\in [0,T]} \left| U_t\right| ^p \right] \le C (1+|u|^p). \end{aligned}$$

Proof

In order to prove the first part of the Proposition it is sufficient to prove that \({\overline{F}}\) is Lipschitz (e.g. see [10]). But this follows from the Lipschitz continuity of F, indeed:

$$\begin{aligned} |{\overline{F}} (u_1) - {\overline{F}} (u_2)| \le \int _K | F(u_1,v)-F(u_2,v) |\mu (dv) \le L_F |u_1-u_2|, \quad \forall u_1,u_2 \in H. \end{aligned}$$

Hence we obtain the Lipschitzianity of \({\overline{F}}\) and the first claim of the Proposition.

The proof of the second claim of the Proposition follows by a standard application of Gronwall’s Lemma. \(\square \)

6 Preliminary Results

In this section we prove a technical result, i.e. Lemma 6.3, which is inspired by [7, Lemma 4.2] and follows by the mixing properties of the fast motion studied in Sect. 4, in particular Lemma 4.7. In order to prove it we proceed with similar techniques to the ones of the proof of [7, Lemma 4.2].

Fix \({\overline{\xi }} \in {\mathcal {C}}([0,T];H)\), \(v \in K, h\in H\) with \(|h|=1\), and define

$$\begin{aligned} \Phi _h(r) {:}{=}\langle F( {\overline{\xi }}_{r},V_r^\epsilon )-\overline{F}({\overline{\xi }}_r),h\rangle \quad \forall r \le T \end{aligned}$$

and

$$\begin{aligned} \Psi _h(r) {:}{=}\Phi _h(\epsilon r)&= \langle F({\overline{\xi }}_{\epsilon r},V_{\epsilon r}^\epsilon )-\overline{F}({\overline{\xi }}_{\epsilon r}),h\rangle . \end{aligned}$$

Moreover let \(n \in \mathbb {N}\) and set:

$$\begin{aligned} \begin{aligned} J_{ j}\left( r_{1}, \ldots , r_{2 n}\right) {:}{=}\left| \mathbb {E} \prod _{i=1}^{2 n} \Psi _h\left( r_{i}\right) -\mathbb {E} \prod _{i=1}^{2 j} \Psi _h\left( r_{i}\right) \mathbb {E} \prod _{i=2 j+1}^{2 n} \Psi _h\left( r_{i}\right) \right| , \end{aligned} \end{aligned}$$
(29)

for every \(1\le j\le n\), \(0 \le r_1 \le \ldots \le r_{2n} \le T/\epsilon \).

First we show the following result:

Lemma 6.1

Let \(0< \rho < 1\) then there exists \(C=C(T,\rho ,n)>0\) and \(\eta =\eta (\rho ,j_2-j_1)>0\) such that

$$\begin{aligned} \left| \mathbb {E} \prod _{i=j_1}^{j_2} \Psi _h (r_i) \right| \le C \left( \frac{ e^{-\delta (r_{j_2}-r_{j_2-1})}}{\sqrt{r_{j_2}-r_{j_2-1}}}\right) ^{\frac{\rho }{2+\rho }} \left( 1+\sup _{r \le T}|{\overline{\xi }}_r|^{\eta }+|v|^{ \eta }\right) , \end{aligned}$$
(30)

for every \(u \in H,v \in K\), \(1\le j_1 \le j_2 \le n\), \(0 \le r_1 \le \ldots \le r_{2n} \le T/\epsilon \).

Moreover there exists \(C=C(T,\rho ,n)>0\) and \(\eta = \eta (\rho ,n) >0\) such that

$$\begin{aligned} \begin{aligned} J_{ j}\left( r_{1}, \ldots , r_{2 n}\right) \le C \left( \frac{e^{-\delta \hat{r}_{j}}}{\sqrt{\hat{r}_{j}}}\right) ^{\frac{\rho }{2+\rho }} \left( 1+ \sup _{r \le T}|{\overline{\xi }}_r| ^{\eta }+|v|^{\eta }\right) , \end{aligned} \end{aligned}$$
(31)

where

$$\begin{aligned} \hat{r}_{j}=\max \left( r_{2 n}-r_{2 n-1}, r_{2 j+1}-r_{2 j}\right) , \end{aligned}$$

for every \(1\le j\le n\), \(0 \le r_1 \le \ldots \le r_{2n} \le T/\epsilon \).

Remark 6.2

Note that the dependence of the exponents with respect to \(j_2-j_1\) and n is not restrictive: once n has been fixed (together with \(\rho \) and T) we are allowed to take any \(0 \le j_1 \le j_2 \le n\). In this sense \(\eta =\eta (\rho , j_2-j_1)\). Of course since \(j_2-j_1 \le n\) we could choose \(\eta '=\eta '(\rho ,n) \ge \eta \) and replace \(\eta \) with \(\eta '\) in the estimates of the Lemma. However the estimate with \(\eta \) is more precise.

Proof

By the sublinearity of \(\Psi _h\) and Lemma 4.2 for every \(p\ge 1\) we have:

$$\begin{aligned} \begin{aligned} \mathbb {E}\left| \prod _{i=j_{1}}^{j_{2}} \Psi _h\left( r_{i}\right) \right| ^{p}&\le C \left( 1+\sum _{i=j_{1}}^{j_{2}} \left| {\overline{\xi }}_{\epsilon r_i} \right| ^{\left( j_{2}-j_{1}+1\right) p}+\sum _{i=j_{1}}^{j_{2}} \mathbb {E}\left| V^\epsilon _{\epsilon r_{i}}\right| ^{\left( j_{2}-j_{1}+1\right) p}\right) \\&\le C\left( 1+\sup _{r \le T}|{\overline{\xi }}_r|^{\left( j_{2}-j_{1}+1\right) p}+|v|^{\left( j_{2}-j_{1}+1\right) p}\right) , \end{aligned} \end{aligned}$$
(32)

for every \(1 \le j_1 \le j_2 \le 2n\).

Notice that \(V^{\epsilon }_{\epsilon r}\) and \(v^v_r\), defined by (18) for \(w_r^{Q_2}= W^{Q_2}_{\epsilon r}/ \sqrt{\epsilon }\), are indistinguishable for \(r \in [0,T/\epsilon ]\). Then by setting \(p=2/(1-\rho )\) and applying Lemma  4.7 to \(\xi _1=\prod _{i=1}^{j} \Psi _h \left( r_{i}\right) \), \(\xi _2=\prod _{i=j+1}^{2 n} \Psi _h \left( r_{i}\right) \) with \(K_1=C(1+\sup _{r \le T}|{\overline{\xi }}_r|^j+|v|^j)\), \(K_2=C(1+\sup _{r \le T}|{\overline{\xi }}_r|^{2n-j}+|v|^{2n-j})\) for every \(0 \le j_1< j < j_2 \le 2n\) we have:

$$\begin{aligned} \begin{aligned}&\left| \mathbb {E} \prod _{i=j_1}^{j_2} \Psi _h\left( r_{i}\right) -\mathbb {E} \prod _{i=j_1}^{j} \Psi _h \left( r_{i}\right) \mathbb {E} \prod _{i=j+1}^{j_2} \Psi _h \left( r_{i}\right) \right| \\&\qquad \le C \left( 1+\sup _{r \le T}|{\overline{\xi }}_r|^{\eta }+|v|^{\eta }\right) \left( \frac{e^{-\delta \left( r_{j+1}-r_{j}\right) }}{\sqrt{r_{j+1}-r_{j}}}\right) ^{\frac{\rho }{2+\rho }}, \end{aligned} \end{aligned}$$
(33)

where \(\eta =j_2-j_1 + \frac{\rho ({j_2}-j_1-1)}{(\rho +2)}\).

By definitions of \(\Psi _h\), the indistinguishability of \(V^{\epsilon }_{\epsilon r}\), \(v^v_r\) and by Lemma 4.4 we have:

$$\begin{aligned} \left| \mathbb {E} \Psi _h\left( r_{j_{2}}\right) \right| \le C e^{-2 \delta r_{j_{2}}}\left( 1+\sup _{r \le T}|{\overline{\xi }}_r|+|v|\right) . \end{aligned}$$
(34)

Now by the last three inequalities we have:

$$\begin{aligned} \begin{aligned} \left| \mathbb {E} \prod _{i=j_{1}}^{j_{2}} \Psi _h \left( r_{i}\right) \right|&\le \left| \mathbb {E} \prod _{i=j_{1}}^{j_{2}} \Psi _h\left( r_{i}\right) -\mathbb {E} \prod _{i=j_{1}}^{j_{2}-1} \Psi _h \left( r_{i}\right) \mathbb {E} \Psi _h \left( r_{j_{2}}\right) \right| + \mathbb {E} \Bigg | \prod _{i=j_{1}}^{j_{2}-1} \Psi _h \left( r_{i}\right) \Bigg | \big | \mathbb {E} \Psi _h \left( r_{j_{2}}\right) \big | \\&\le C \left( 1+\sup _{r \le T}|{\overline{\xi }}_r|^{ \eta }+|v|^{ \eta }\right) \left( \frac{e^{-\delta \left( r_{j_{2}}-r_{j_{2}-1}\right) }}{\sqrt{r_{j_{2}}-r_{j_{2}-1}}}\right) ^{\frac{\rho }{2+\rho }}\\&\quad + C (1+\sup _{r \le T}|{\overline{\xi }}_r|^{j_{2}-j_{1}}+|v|^{j_{2}-j_{1}}) \left( 1+\sup _{r \le T}|{\overline{\xi }}_r|+|v|\right) e^{-2 \delta r_{j_{2}}} \\&\le C\left( 1+\sup _{r \le T}|{\overline{\xi }}_r|^{ \eta }+|v|^{ \eta }\right) \left( \frac{e^{-\delta \left( r_{j_{2}}-r_{j_{2}-1}\right) }}{\sqrt{r_{j_{2}}-r_{j_{2}-1}}}\right) ^{\frac{\rho }{2+\rho }}, \end{aligned} \end{aligned}$$

where \( \eta =(j_2-j_1 +1)+ \frac{\rho (j_2-j_1)}{(\rho +2)}\). This implies (30).

Now by (30) we have:

$$\begin{aligned} \left| \mathbb {E} \prod _{i=j_1}^{2 n} \Psi _h \left( r_{i}\right) \right| \le C \left( 1+\sup _{r \le T}|{\overline{\xi }}_r|^{\eta }+|v|^{\eta }\right) \left( \frac{e^{-\delta \left( r_{2 n}-r_{2 n-1}\right) }}{\sqrt{r_{2 n}-r_{2 n-1}}}\right) ^{\frac{\rho }{2+\rho }}. \end{aligned}$$
(35)

for every \(j_1 <2n.\)

Now fix any \(j < 2n\). Then by the last inequality and (32) we have:

$$\begin{aligned} \begin{aligned}&\left| \mathbb {E} \prod _{i=1}^{2 n} \Psi _h \left( r_{i}\right) -\mathbb {E} \prod _{i=1}^{j} \Psi _h \left( r_{i}\right) \mathbb {E} \prod _{i=j+1}^{2 n} \Psi _h \left( r_{i}\right) \right| \\&\quad \le C\left( 1+\sup _{r \le T}|{\overline{\xi }}_r|^{\eta }+|v|^{\eta }\right) \left( \frac{e^{-\delta \left( r_{2 n}-r_{2 n-1}\right) }}{\sqrt{r_{2 n}-r_{2 n-1}}}\right) ^{\frac{\rho }{2+\rho }}. \end{aligned} \end{aligned}$$
(36)

Finally consider (33) with \(j_1=1,j_2=2n\) and (36). Since the function \(f(s)=e^{-\delta s} s^{-1/2}\) is decreasing we have

$$\begin{aligned} \begin{aligned}&\left| \mathbb {E} \prod _{i=1}^{2 n} \Psi _h \left( r_{i}\right) -\mathbb {E} \prod _{i=1}^{j} \Psi _h \left( r_{i}\right) \mathbb {E} \prod _{i=j+1}^{2 n} \Psi _h \left( r_{i}\right) \right| \\&\quad \le C \left( \frac{e^{-\delta \max \left( r_{2 n}-r_{2 n-1}, r_{2 j+1}-r_{2 j}\right) }}{\sqrt{\max \left( r_{2 n}-r_{2 n-1}, r_{2 j+1}-r_{2 j}\right) }}\right) ^{\frac{\rho }{2+\rho }} \left( 1+\sup _{r \le T}|{\overline{\xi }}_r|^{\eta }+|v|^{\eta }\right) \\&\quad =C \left( \frac{e^{-\delta \hat{r}_{j}}}{\sqrt{\hat{r}_{j}}}\right) ^{\frac{\rho }{2+\rho }} \left( 1+ \sup _{r \le T}|{\overline{\xi }}_r| ^{\eta }+|v|^{\eta }\right) . \end{aligned} \end{aligned}$$

This implies (31). \(\square \)

Let \(0<\alpha \), \(0 \le \beta < 1/3\) and set

$$\begin{aligned} \theta _{\alpha ,\beta }(r):=e^{-r \alpha }r^{-\beta } \quad \forall r>0. \end{aligned}$$

Now we can state and prove the main result of this section.

Lemma 6.3

Let \(n \in \mathbb {N}\) and \(0 \le \beta < 1/3\). Then there exists a constant \(C=C(T,n,\beta )>0\) and \(\eta =\eta (n)>0\) such that for every \(\epsilon >0\), \(\alpha >0\), \(u \in H,v \in K\), \(h\in H\) with \(|h|=1\) we have:

$$\begin{aligned}{} & {} \int _{[s,t]^{2n}} { \left| \mathbb {E} \prod _{i=1}^{2n} \theta _{\alpha ,\beta }(t-r_i) \langle F({\overline{\xi }}_{r_i},V_{r_i}^\epsilon )-\overline{F}({\overline{\xi }}_{ r_i}),h\rangle \right| } dr_1\ldots dr_{2n} \\{} & {} \quad \le C \left( 1+\sup _{r \le T}|{\overline{\xi }}_r|^{\eta }+|v|^{\eta }\right) \epsilon ^{n} \left( \frac{1}{\alpha } \right) ^{(1-2\beta )n}. \end{aligned}$$

Proof

Recall the definition of \(\Psi _h(r)\), then by a change of variable we have:

$$\begin{aligned} \begin{aligned} \int _{[s,t]^{2n}} { \left| \mathbb {E} \prod _{i=1}^{2n} \theta _{\alpha ,\beta }(t-r_i) \langle F({\overline{\xi }}_{r_i},V_{r_i}^\epsilon )-\overline{F}({\overline{\xi }}_{ r_i}),h\rangle \right| } dr_1\ldots dr_{2n} =\epsilon ^{2 n} { H_{\epsilon }(s, t) }, \end{aligned}\nonumber \\ \end{aligned}$$
(37)

where we have defined:

$$\begin{aligned} { H_{\epsilon }(s, t)}:=\int _{\left[ \frac{s}{\epsilon }, \frac{t}{\epsilon }\right] ^{2 n}} { \left| \mathbb {E} \prod _{i=1}^{2 n} \left( \theta _{\alpha , \beta }\left( t-\epsilon r_{i}\right) \Psi _{ h}\left( r_{i}\right) \right) \right| } d r_{1} \cdots d r_{2 n}. \end{aligned}$$

By simmetry we have:

$$\begin{aligned} { H_{\epsilon }(s, t)}=C \int _{\frac{s}{\epsilon }}^{\frac{t}{\epsilon }} \int _{\frac{s}{\epsilon }}^{r_{2 n}} \ldots \int _{\frac{s}{\epsilon }}^{r_{2 }} { \left| \mathbb {E} \prod _{i=1}^{2n} \left( \theta _{\alpha , \beta }\left( t-\epsilon r_{i}\right) \Psi _{h}\left( r_{i}\right) \right) \right| } d r_{1} \cdots d r_{2 n}. \end{aligned}$$
(38)

We proceed by induction on n and to this purpose fix some \(\rho \in (0,1)\). Consider \(n=1\) then by the definition of \(\theta _{\alpha ,\beta }=e^{-r \alpha }r^{-\beta }\), (30) and some changes of variables we have

$$\begin{aligned} \epsilon ^2 { H_{\epsilon }(s, t)}= & {} 2 \epsilon ^2 \int _{\frac{s}{\epsilon }}^{\frac{t}{\epsilon }} \int _{\frac{s}{\epsilon }}^{r_{2}} \theta _{\alpha , \beta }\left( t-\epsilon r_{1}\right) \theta _{\alpha , \beta }\left( t-\epsilon r_{2}\right) { \left| \mathbb {E} \left[ \Psi _{h}\left( r_{1}\right) \Psi _{h}\left( r_{2}\right) \right] \right| } d r_{1} d r_{2} \\\le & {} C \epsilon ^2 \int _{\frac{s}{\epsilon }}^{\frac{t}{\epsilon }} \int _{\frac{s}{\epsilon }}^{r_{2}} \theta _{\alpha , \beta }\left( t-\epsilon r_{1}\right) \theta _{\alpha , \beta }\left( t-\epsilon r_{2}\right) \left( \frac{e^{-\delta \left( r_{2}-r_{1}\right) }}{\sqrt{r_{2}-r_{1}}}\right) ^{\frac{\rho }{2+\rho }} d r_{1} d r_{2} \left( 1+\sup _{r \le T}|{\overline{\xi }}_r|^{\eta }+|v|^{\eta }\right) \\= & {} C \epsilon ^{2-2 \beta } \int _{0}^{\frac{t-s}{\epsilon }} y_{2}^{-\beta } e^{-\epsilon y_{2} \alpha } \int _{y_{2}}^{\frac{t-s}{\epsilon }} y_{1}^{-\beta } e^{-\epsilon y_{1} \alpha }\left( \frac{e^{-\delta \left( y_{1}-y_{2}\right) }}{\sqrt{y_{1}-y_{2}}}\right) ^{\frac{\rho }{2+\rho }} d y_{1} d y_{2} \left( 1+\sup _{r \le T}|{\overline{\xi }}_r|^{\eta }+|v|^{\eta }\right) \\\le & {} C \epsilon ^{2-2 \beta } \int _{0}^{\frac{t-s}{\epsilon }} y_{2}^{-2 \beta } e^{-2 \epsilon y_{2} \alpha } \int _{0}^{+\infty }\left( \frac{e^{-\delta y_{1}}}{\sqrt{y_{1}}}\right) ^{\frac{\rho }{2+\rho }} d y_{1} d y_{2} \left( 1+\sup _{r \le T}|{\overline{\xi }}_r|^{\eta }+|v|^{\eta }\right) \\\le & {} C \epsilon ^{2-2 \beta } \int _{0}^{\frac{t-s}{\epsilon }} y_2^{-2 \beta } e^{-2 \epsilon y_2 \alpha } d y_2 \left( 1+\sup _{r \le T}|{\overline{\xi }}_r|^{\eta }+|v|^{\eta }\right) \\= & {} C \epsilon ^{2-2 \beta }(\epsilon \alpha )^{-(1-2 \beta )} \int _{0}^{\alpha (t-s)} r^{-2 \beta } e^{-2 r} d r \left( 1+\sup _{r \le T}|{\overline{\xi }}_r|^{ \eta }+|v|^{\eta }\right) \\\le & {} C \epsilon ^{2-2 \beta }(\epsilon \alpha )^{-(1-2 \beta )}\left( 1+\sup _{r \le T}|{\overline{\xi }}_r|^{\eta }+|v|^{\eta }\right) \\= & {} C \epsilon \alpha ^{-(1-2 \beta )}\left( 1+\sup _{r \le T}|{\overline{\xi }}_r|^{\eta }+|v|^{\eta }\right) , \end{aligned}$$

so that by (37) we have the thesis for \(n=1\).

Now assume that

$$\begin{aligned}{} & {} \int _{[s,t]^{j}} { \left| \mathbb {E} \prod _{i=1}^{j} \theta _{\alpha ,\beta }(t-\epsilon r_i) \Psi _h (r_i)\right| } dr_1\ldots dr_j\\{} & {} \quad \le C \left( 1+\sup _{r \le T}|{\overline{\xi }}_r|^{\eta _j}+|v|^{\eta _j}\right) \epsilon ^{j/2} \left( \frac{1}{\alpha } \right) ^{(1-2\beta )j/2}, \end{aligned}$$

for every even \(j<2n\) where \(\eta _j=j + \frac{\rho (j-1)}{(\rho +2)}\).

We prove that then it holds for \(j=2n\).

Set for \(r=(r_1,\ldots r_{2n}) \in (s,t)^{2n}\) with \(s \le r_1 \le \ldots \le r_{2n} \le t\) the integer j(r) such that

$$\begin{aligned} \max _{j=1, \ldots , n-1}\left( r_{2 j+1}-r_{2 j}\right) =r_{2 j(r)+1}-r_{2 j(r)}. \end{aligned}$$

and consider \(H_\epsilon (s,t)\) given by (38). Recalling the definition of \(J_j(r_1,\ldots r_{2n})\) given by (29) we have:

$$\begin{aligned} \begin{aligned} \epsilon ^{2n} {H_{\epsilon }(s, t) }&=C \epsilon ^{2n} \int _{\frac{s}{\epsilon }}^{\frac{t}{\epsilon }} \int _{\frac{s}{\epsilon }}^{r_{2 n}} \ldots \int _{\frac{s}{\epsilon }}^{r_{2 }} { \prod _{i=1}^{2n}\theta _{\alpha , \beta }\left( t-\epsilon r_{i}\right) \left| \mathbb {E} \prod _{i=1}^{2n} \Psi _{h}\left( r_{i}\right) \right| } d r_{1} \cdots d r_{2 n} \\&{ \le C \epsilon ^{2n} \int _{\frac{s}{\epsilon }}^{\frac{t}{\epsilon }} \int _{\frac{s}{\epsilon }}^{r_{2 n}} \ldots \int _{\frac{s}{\epsilon }}^{r_{2}} \prod _{i=1}^{2n} \theta _{\alpha , \beta } \left( t-\epsilon r_{i}\right) J_{ j(r)}\left( r_{1}, \ldots , r_{2 n}\right) d r_{1} \cdots d r_{2 n}}\\&\quad { +C \epsilon ^{2n} \int _{\frac{s}{\epsilon }}^{\frac{t}{\epsilon }} \int _{\frac{s}{\epsilon }}^{r_{2 n}} \ldots \int _{\frac{s}{\epsilon }}^{r_{2}} \prod _{i=1}^{2n} \theta _{\alpha , \beta } \left( t-\epsilon r_{i}\right) \left| \mathbb {E} \prod _{i=1}^{2 j(r)} \Psi _{ h}\left( r_{i}\right) \mathbb {E} \prod _{i=2j(r)+1}^{2n} \Psi _{ h}\left( r_{i}\right) \right| d r_{1} \cdots d r_{2 n}.}\end{aligned} \end{aligned}$$

Note that by definition of j(r), for every \(s \le r_1 \le \ldots \le r_{2n} \le t\), we have

$$\begin{aligned} \left| \mathbb {E} \prod _{i=1}^{2 j(r)} \Psi _{ h}\left( r_{i}\right) \mathbb {E} \prod _{i=2j(r)+1}^{2n} \Psi _{ h}\left( r_{i}\right) \right| \le \sum _{j=1}^{n-1} \left| \mathbb {E} \prod _{i=1}^{2 j} \Psi _{ h}\left( r_{i}\right) \mathbb {E} \prod _{i=2j+1}^{2n} \Psi _{ h}\left( r_{i}\right) \right| . \end{aligned}$$

It follows:

$$\begin{aligned} \begin{aligned}&{ \epsilon ^{2n} H_{\epsilon }(s, t) } \\&{ \le C \epsilon ^{2n} \int _{\frac{s}{\epsilon }}^{\frac{t}{\epsilon }} \int _{\frac{s}{\epsilon }}^{r_{2 n}} \ldots \int _{\frac{s}{\epsilon }}^{r_{2}} \prod _{i=1}^{2n} \theta _{\alpha , \beta } \left( t-\epsilon r_{i}\right) J_{ j(r)}\left( r_{1}, \ldots , r_{2 n}\right) d r_{1} \cdots d r_{2 n} } \\&\quad { +C \epsilon ^{2n} \sum _{j=1}^{n-1} \int _{\frac{s}{\epsilon }}^{\frac{t}{\epsilon }} \int _{\frac{s}{\epsilon }}^{r_{2 n}} \ldots \int _{\frac{s}{\epsilon }}^{r_{2}} \prod _{i=1}^{2n} \theta _{\alpha , \beta } \left( t-\epsilon r_{i}\right) \left| \mathbb {E} \prod _{i=1}^{2 j} \Psi _{ h}\left( r_{i}\right) \mathbb {E} \prod _{i=2j+1}^{2n} \Psi _{ h}\left( r_{i}\right) \right| d r_{1} \cdots d r_{2 n} } \\&= C \epsilon ^{2n} \int _{\frac{s}{\epsilon }}^{\frac{t}{\epsilon }} \int _{\frac{s}{\epsilon }}^{r_{2 n}} \ldots \int _{\frac{s}{\epsilon }}^{r_{2}} \prod _{i=1}^{2n} \theta _{\alpha , \beta } \left( t-\epsilon r_{i}\right) J_{ j(r)}\left( r_{1}, \ldots , r_{2 n}\right) d r_{1} \cdots d r_{2 n}\\&\quad +C \epsilon ^{2n} \sum _{j=1}^{n-1} \int _{\frac{s}{\epsilon }}^{\frac{t}{\epsilon }} \int _{\frac{s}{\epsilon }}^{r_{2 j}} \ldots \int _{\frac{s}{\epsilon }}^{r_{2}} \prod _{i=1}^{2 j} \theta _{\alpha , \beta }\left( t-\epsilon r_{i}\right) \left| \mathbb {E} \prod _{i=1}^{2 j} \Psi _{ h}\left( r_{i}\right) \right| d r_{1} \cdots r_{2 j}\\&\quad \quad \times \int _{\frac{s}{\epsilon }}^{\frac{t}{\epsilon }} \int _{\frac{s}{\epsilon }}^{r_{2(n-j)}} \cdots \int _{\frac{s}{\epsilon }}^{r_{2}} \prod _{i=1}^{2(n-j)} \theta _{\alpha , \beta }\left( t-\epsilon r_{i}\right) \left| \mathbb {E} \prod _{i=1}^{2(n-j)} \Psi _{ h}\left( r_{i}\right) \right| d r_{1} \cdots r_{2(n-j)}\\&=: \epsilon ^{2n} I_{1, \epsilon } (s,t)+ \epsilon ^{2n}I_{2, \epsilon } (s,t). \end{aligned} \end{aligned}$$
(39)

Now by (31) and the definition of j(r) we have:

$$\begin{aligned} J_{j(r)}\left( r_{1}, \ldots , r_{2 n}\right)&\le C \left( 1+\sup _{r \le T}|{\overline{\xi }}_r|^{\eta }+|v|^{\eta }\right) \left( \frac{e^{-\delta \hat{r}_{j(r)}}}{\sqrt{\hat{r}_{j(r)}}}\right) ^{\frac{\rho }{2+\rho }}\\&=C \left( 1+\sup _{r \le T}|{\overline{\xi }}_r|^{\eta }+|v|^{\eta }\right) \\&\quad \left( \frac{e^{-\delta \max \left( r_{2 n}-r_{2 n-1}, r_{2n-1}-r_{2n-2},\cdots ,r_{3}-r_{2}\right) }}{\sqrt{\max \left( r_{2 n}-r_{2 n-1}, r_{2n-1}-r_{2n-2},\cdots ,r_{3}-r_{2}\right) }}\right) ^{\frac{\rho }{2+\rho }}\\&=C \left( 1+\sup _{r \le T}|{\overline{\xi }}_r|^{\eta }+|v|^{\eta }\right) \\&\quad \frac{e^{-\frac{\delta \rho }{2+\rho } \max \left( r_{2 n}-r_{2 n-1}, r_{2n-1}-r_{2n-2},\cdots ,r_{3}-r_{2}\right) }}{\left[ \max \left( r_{2 n}-r_{2 n-1}, r_{2n-1}-r_{2n-2},\cdots ,r_{3}-r_{2}\right) \right] ^{{\bar{\rho }}}}, \end{aligned}$$

where \(\bar{\rho }=\frac{\rho }{2(2+\rho )}\).

Recall that \(r_{i+1} \ge r_{i}\) for every i and note that \(\max \left( r_{2 n}-r_{2 n-1}, r_{2n-1}-r_{2n-2}\right) \ge 1/2 (r_{2 n}-r_{2 n-2})\) and \(\max \left( r_{2 n}-r_{2 n-1}, r_{2n-1}-r_{2n-2},\cdots ,r_{3}-r_{2}\right) \ge \frac{1}{n} \left( r_{2 n}-r_{2 n-1}+\sum _{i=1}^{n-1} r_{2 i+1}-r_{2 i}\right) .\) Hence, since \(g(t)=1/t^{{\bar{\rho }}}\), \(f(t)=e^{-\alpha t}\) for \(\alpha >0\) are decreasing, we have (recall that C may change from line to line):

$$\begin{aligned} J_{j(r)}\left( r_{1}, \ldots , r_{2 n}\right)&\le C \left( 1+\sup _{r \le T}|{\overline{\xi }}_r|^{\eta }+|v|^{\eta }\right) \frac{e^{-\frac{\delta \rho }{2+\rho } \max \left( r_{2 n}-r_{2 n-1}, r_{2n-1}-r_{2n-2},\cdots ,r_{3}-r_{2}\right) }}{\left[ \max \left( r_{2 n}-r_{2 n-1}, r_{2n-1}-r_{2n-2}\right) \right] ^{{\bar{\rho }}}} \\&\le C \left( 1+\sup _{r \le T}|{\overline{\xi }}_r|^{\eta }+|v|^{\eta }\right) \frac{e^{-\frac{\delta \rho }{2+\rho } \max \left( r_{2 n}-r_{2 n-1}, r_{2n-1}-r_{2n-2},\cdots ,r_{3}-r_{2}\right) }}{\left( r_{2 n}-r_{2 n-2}\right) ^{\bar{\rho }}} \\&\le C \left( 1+\sup _{r \le T}|{\overline{\xi }}_r|^{\eta }+|v|^{\eta }\right) \frac{e^{-\frac{\delta \rho }{2+\rho } \frac{1}{n} \left( r_{2 n}-r_{2 n-1}+\sum _{i=1}^{n-1} r_{2 i+1}-r_{2 i}\right) }}{\left( r_{2 n}-r_{2 n-2}\right) ^{\bar{\rho }}}\\&= C \left( 1+\sup _{r \le T}|{\overline{\xi }}_r|^{\eta }+|v|^{\eta }\right) \frac{e^{-\delta _{n}\left( r_{2 n}-r_{2 n-2}+\sum _{i=1}^{n-2} r_{2 i+1}-r_{2 i}\right) }}{\left( r_{2 n}-r_{2 n-2}\right) ^{\bar{\rho }}}, \end{aligned}$$

where in the last equality we have defined \(\delta _{n}=\frac{\delta \rho }{n(2+\rho )}\) and we have used the following identity: \(\sum _{i=1}^{n-1} r_{2 i+1}-r_{2 i}=r_{2n-1}-r_{2n-2}+ \sum _{i=1}^{n-2} r_{2 i+1}-r_{2 i}\).

We can apply this last inequality in order to estimate \(I_{1,\epsilon }(s,t)\), i.e.

$$\begin{aligned} I_{1, \epsilon }(s,t)&\le C \left( 1+\sup _{r \le T}|\xi _r|^{\eta }+|v|^{\eta }\right) \int _{\frac{s}{\epsilon }}^{\frac{t}{\epsilon }} \int _{\frac{s}{\epsilon }}^{r_{2 n}} \cdots \int _{\frac{s}{\epsilon }}^{r_{2}} \frac{e^{-\delta _{n}\left( r_{2 n}-r_{2 n-2}\right) }}{\left( r_{2 n}-r_{2 n-2}\right) ^{\bar{\rho }}}\\&\quad \times \prod _{i=1}^{2 n} \vartheta _{\alpha , \beta }\left( t-\epsilon r_{i}\right) \prod _{i=1}^{n-2} e^{-\delta _{n}\left( r_{2 i+1}-r_{2 i}\right) } d r_{1} \cdots d r_{2 n}\\&=C \epsilon ^{-2 n \beta } \int _{0}^{\frac{t-s}{\epsilon }} e^{-\left( \epsilon \alpha -\delta _{n}\right) y_{2 n}} y_{2 n}^{-\beta } \int _{y_{2 n}}^{\frac{t-s}{\epsilon }} e^{-\epsilon \alpha y_{2 n-1}} y_{2 n-1}^{-\beta } \int _{y_{2 n-1}}^{\frac{t-s}{\epsilon }} \frac{e^{-\left( \epsilon \alpha +\delta _{n}\right) y_{2 n-2}}}{\left( y_{2 n-2}-y_{2 n}\right) ^{\bar{\rho }}} y_{2 n-2}^{-\beta }\\&\quad \times \int _{y_{2 n-2}}^{\frac{t-s}{\epsilon }} \cdots \int _{y_{3}}^{\frac{t-s}{\epsilon }} \prod _{i=2}^{2 n-2} e^{-\left( \epsilon \alpha +(-1)^{i} \delta _{n}\right) y_{i}} r_{i}^{-\beta } \\&\quad \int _{y_{2}}^{\frac{t-s}{\epsilon }} e^{-\epsilon \alpha y_{1}} y_{1}^{-\beta } d y_{1} \cdots d y_{2 n} \left( 1+\sup _{r \le T}|\xi _r|^{\eta }+|v|^{\eta }\right) . \end{aligned}$$

Now for \(k=1,2,3\), \(i=1,3, \ldots 2n-1\) we obtain:

$$\begin{aligned} \int _{y_{i+1}}^{\frac{t-s}{\epsilon }} e^{-k \epsilon \alpha y_{i}} y_{i}^{-k \beta } d y_{i}&=(\epsilon \alpha )^{k \beta -1} \int _{\epsilon \alpha y_{i+1}}^{(t-s) \alpha } e^{-k y_{i}} y_{i}^{-k \beta } d y_{i} \nonumber \\&\le C (\epsilon \alpha )^{k \beta -1}. \end{aligned}$$
(40)

Now consider \(i=2,4,\ldots 2n\), we have:

$$\begin{aligned} \int _{y_{i+1}}^{\frac{t-s}{\epsilon }} e^{-\left( \epsilon \alpha +\delta _{n}\right) y_{i}} y_{i}^{-\beta } d y_{i}&\le y_{i+1}^{-\beta } e^{-\left( \epsilon \alpha +\delta _{n}\right) y_{i+1}}\left( \epsilon \alpha +\delta _{n}\right) ^{-1} \nonumber \\&\le C y_{i+1}^{-\beta } e^{-\left( \epsilon \alpha +\delta _{n}\right) y_{i+1}}. \end{aligned}$$
(41)

Now by (40) and (41) we have:

$$\begin{aligned} \begin{aligned} \int _{y_{2 n-2}}^{\frac{t-s}{\epsilon }} \cdots \int _{y_{3}}^{\frac{t-s}{\epsilon }}&\prod _{i=2}^{2 n-2} e^{-\left( \epsilon \alpha +(-1)^{i} \delta _{n}\right) y_{i}} y_{i}^{-\beta } \int _{y_{2}}^{\frac{t-s}{\epsilon }} e^{-\epsilon \alpha y_{1}} y_{1}^{-\beta } d y_{1} \cdots d y_{2 n-3} \\ { }&\le (\epsilon \alpha )^{\beta -1} (\epsilon \alpha )^{(2 \beta -1)(n-2)} \\&=(\epsilon \alpha )^{n(2 \beta -1)-(3 \beta -1)}. \end{aligned} \end{aligned}$$
(42)

Now consider the remaining term in the inequality for \(I_{1,\epsilon }(s,t)\): by (40) and (41) we have

$$\begin{aligned} \begin{aligned} \int _{0}^{\frac{t-s}{\epsilon }} e^{-\left( \epsilon \alpha -\delta _{n}\right) y_{2 n}} y_{2 n}^{-\beta }&\int _{y_{2 n}}^{\frac{t-s}{\epsilon }} e^{-\epsilon \alpha y_{2 n-1}} y_{2 n-1}^{-\beta } \int _{y_{2 n-1}}^{\frac{t-s}{\epsilon }} \frac{e^{-\left( \epsilon \alpha +\delta _{n}\right) y_{2 n-2}}}{\left( y_{2 n-2}-y_{2 n}\right) ^{\bar{\rho }}} y_{2 n-2}^{-\beta } d y_{2 n-2} d y_{2 n-1} d y_{2 n} \\&\le C \int _{0}^{\frac{t-s}{\epsilon }} e^{-3 \epsilon \alpha y_{2 n}} y_{2 n}^{-3 \beta } \int _{y_{2 n}}^{\frac{t-s}{\epsilon }} \frac{e^{-\left( 2 \epsilon \alpha +\delta _{n}\right) \left( y_{2 n-1}-y_{2 n}\right) }}{\left( y_{2 n-1}-y_{2 n}\right) ^{\bar{\rho }}} d y_{2 n-1} d y_{2 n} \\&\le C \int _{0}^{\frac{t-s}{\epsilon }} e^{-3 \epsilon \alpha y_{2 n}} y_{2 n}^{-3 \beta } \int _{0}^{\infty } \frac{e^{-\delta _{n} y_{2 n-1}}}{y_{2 n-1}^{\bar{\rho }}} d y_{2 n-1} d y_{2 n} \\&\le C(\epsilon \alpha )^{3 \beta -1}. \end{aligned} \end{aligned}$$
(43)

Applying now (42) and (43) to the inequality for \(I_{1,\epsilon }(s,t)\) we have:

$$\begin{aligned} \begin{aligned} I_{1, \epsilon }&\le C \epsilon ^{-2 n \beta } (\epsilon \alpha )^{n(2 \beta -1)-(3 \beta -1)} (\epsilon \alpha )^{3 \beta -1} \left( 1+\sup _{r \le T}|{\overline{\xi }}_r|^{\eta }+|v|^{\eta }\right) \\&=C \epsilon ^{-n} \alpha ^{-(1-2 \beta )n} \left( 1+\sup _{r \le T}|{\overline{\xi }}_r|^{\eta }+|v|^{\eta }\right) . \end{aligned} \end{aligned}$$
(44)

In addition by the inductive hypothesis we have:

$$\begin{aligned} I_{2, \epsilon } \le C \epsilon ^{-n} \alpha ^{-(1-2 \beta )n} \left( 1+\sup _{r \le T}|{\overline{\xi }}_r|^{\eta }+|v|^{\eta }\right) . \end{aligned}$$
(45)

Finally applying the last two inequalities to (39) and then going back to (37) we have the thesis. \(\square \)

7 Order of Convergence

In this section we finally investigate the order of convergence. We first prove the following proposition which will be crucial in the derivation of the order of convergence.

Proposition 3

There exist \(C=C(T)>0\), \(\iota >0\) such that

$$\begin{aligned} \mathbb {E}\left[ \sup _{0 \le t \le \tau } \left| \int _{0}^t e^{A_1 (t-s)} \left( F(U_{s},V_s^\epsilon )-\overline{F}(U_s)\right) ds \right| ^2 \right] \le C \epsilon (1+|u|^{\iota }+|v|^{\iota }), \end{aligned}$$

for every \(0 \le \tau \le T\), \(\epsilon >0\), \(u \in H,v \in K\).

Proof

We proceed with similar techniques to the ones in the proof of [7, Theorem 4.1].

Define

$$\begin{aligned} Z_t^\epsilon := \int _{0}^t e^{A_1 (t-s)} \left( F(U_{s},V_s^\epsilon )-\overline{F}(U_s)\right) ds. \end{aligned}$$

We proceed using the factorization method, e.g. see [10, Chapter 5, Section 3]: fix \(\zeta >0,n \ge 2\) integer and \( 1/(2n)< \beta < 1/3\) as in Hypothesis 1, then:

$$\begin{aligned} Z_t^\epsilon = \frac{\sin {\beta \pi }}{\pi } \int _{0}^{t} (t -s)^{\beta -1} e^{A_1 ( t -s) } Y_s^{\epsilon } ds, \end{aligned}$$

where

$$\begin{aligned} Y_s^{\epsilon }=\int _{0}^{ s} (s-r)^{- \beta } e^{A_1(s-r) } \left( F(U_{r},V_r^\epsilon )-\overline{F}(U_r)\right) dr. \end{aligned}$$

By Holder’s inequality we have:

$$\begin{aligned} {\mathbb {E}} \left[ \sup _{t \in [0,\tau ]} |Z_t^\epsilon |^{2n} \right]&\le C \int _0^\tau {\mathbb {E}} |Y_s^{\epsilon }|^{2n} ds. \end{aligned}$$
(46)

We now claim that there exist \(C=C(T)>0\) and \(\eta >0\) such that

$$\begin{aligned} \mathbb {E}\left| Y^{\epsilon }_s\right| ^{2 n}\le C \epsilon ^{n}\left( 1+|u|^\eta +|v|^\eta \right) , \end{aligned}$$
(47)

for every \(0 \le t_0 \le s \le T\), \(\epsilon >0\), \(u \in H,v \in K\).

Indeed first recall the spectral representation (3). Then by Parseval’s identity, Holder’s inequality, Hypothesis 1 and the properties of conditional expectations we have:

$$\begin{aligned} \mathbb {E} \left| Y^{\epsilon }_s\right| ^{2 n}=&\mathbb {E} \left( \sum _{k=1}^{\infty } \alpha _k^{-\frac{(n-1)\zeta }{ n}} \alpha _k^{\frac{(n-1)\zeta }{ n}} \left| \int _{0}^{s}(s-r)^{-\beta } e^{-(s-r) \alpha _{k}}\langle F(U_{r},V_r^\epsilon )-\overline{F}(U_r), e_{k}\rangle d r\right| ^{2}\right) ^{n} \\&\le \left( \sum _{k=1}^{\infty } \alpha _{k}^{-\zeta }\right) ^{n-1} \sum _{k=1}^{\infty } \Big [ \alpha _{k}^{(n-1)\zeta } \mathbb {E} \left| \int _{0}^{s}(s-r)^{-\beta } e^{-(s-r) \alpha _{k}}\langle F(U_{r},V_r^\epsilon )-\overline{F}(U_r), e_{k}\rangle d r\right| ^{2n} \Bigg ] \\&\le C \sum _{k=1}^{\infty } \Big [ \alpha _{k}^{(n-1)\zeta } \mathbb {E} \left| \int _{0}^{s}(s-r)^{-\beta } e^{-(s-r) \alpha _{k}}\langle F(U_{r},V_r^\epsilon )-\overline{F}(U_r), e_{k}\rangle d r\right| ^{2n} \Bigg ] \\&= C\sum _{k=1}^{\infty } \Big [ \alpha _{k}^{(n-1)\zeta } \mathbb {E}\int _{[0, s]^{2 n}} \prod _{i=1}^{2 n}\left( s-r_{i}\right) ^{-\beta } e^{-\left( s-r_{i}\right) \alpha _{k}} \langle F(U_{r_i},V_{r_i}^\epsilon )-\overline{F}(U_{r_i}), e_{k} \rangle d r_{1} \cdots d r_{2 n} \Big ] \\&= C\sum _{k=1}^{\infty } \Bigg [ \alpha _{k}^{(n-1)\zeta } \mathbb {E} \Bigg [ \mathbb {E} \Bigg [ \int _{[0, s]^{2 n}} \prod _{i=1}^{2 n}\left( s-r_{i}\right) ^{-\beta } e^{-\left( s-r_{i}\right) \alpha _{k}} \langle F(U_{r_i},V_{r_i}^\epsilon )-\overline{F}(U_{r_i}), e_{k} \rangle \\&\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad d r_{1} \cdots d r_{2 n} \Bigg | \left\{ U_r, \forall r \le s \right\} \Bigg ] \Bigg ] \Bigg ]. \end{aligned}$$

Now consider the conditional expectation on the right-hand-side. Fixing \(s \ge 0\), due to the independence of the averaged component \(\{U_r, \forall r \le s \}\) (which is \(\sigma \{ W_r^{Q_1}, \forall r \le s \} -\)measurable) and the fast component \(\{V_r^\epsilon , \forall r \le s \}\) (which is \(\sigma \{ W_r^{Q_2}, \forall r \le s \}- \)measurable being independent of \(\sigma \{ W_r^{Q_1}, \forall r \le s \} \)), for every \({\overline{\xi }} \in C([0,s]; H)\) we have

$$\begin{aligned} \mathbb {E} \Bigg [ \int _{[0, s]^{2 n}}&\prod _{i=1}^{2 n}\left( s-r_{i}\right) ^{-\beta } e^{-\left( s-r_{i}\right) \alpha _{k}} \langle F(U_{r_i},V_{r_i}^\epsilon )-\overline{F}(U_{r_i}), e_{k} \rangle d r_{1} \cdots d r_{2 n} \Bigg | \{U_r, \forall r \le s \}= \{{\overline{\xi }}_r, \forall r \le s \} \Bigg ] \\&= \mathbb {E} \Bigg [ \int _{[0, s]^{2 n}} \prod _{i=1}^{2 n}\left( s-r_{i}\right) ^{-\beta } e^{-\left( s-r_{i}\right) \alpha _{k}} \langle F({\overline{\xi }}_{r_i},V_{r_i}^\epsilon )-\overline{F}({\overline{\xi }}_{r_i}), e_{k} \rangle d r_{1} \cdots d r_{2 n} \Bigg ]\\&{\le \int _{[0, s]^{2 n}} \Bigg | \mathbb {E} \Bigg [ \prod _{i=1}^{2 n}\left( s-r_{i}\right) ^{-\beta } e^{-\left( s-r_{i}\right) \alpha _{k}} \langle F({\overline{\xi }}_{r_i},V_{r_i}^\epsilon )-\overline{F}({\overline{\xi }}_{r_i}), e_{k} \rangle \Bigg ] \Bigg | d r_{1} \cdots d r_{2 n} } \\&\le C \left( 1+\sup _{r \le T}|{\overline{\xi }}_r|^{\eta }+|v|^{\eta }\right) \epsilon ^{n} \left( \frac{1}{\alpha _k} \right) ^{(1-2\beta )n}, \end{aligned}$$

where the last inequality follows by Lemma 6.3.

Hence we have:

$$\begin{aligned} \mathbb {E} \Bigg [ \int _{[0, s]^{2 n}}&\prod _{i=1}^{2 n}\left( s-r_{i}\right) ^{-\beta } e^{-\left( s-r_{i}\right) \alpha _{k}} \langle F(U_{r_i},V_{r_i}^\epsilon )-\overline{F}(U_{r_i}), e_{k} \rangle d r_{1} \cdots d r_{2 n} \Bigg | \{U_r, \forall r \le s \}\Bigg ] \\&\le C \left( 1+\sup _{r \le T}|U_r|^{\eta }+|v|^{\eta }\right) \epsilon ^{n} \left( \frac{1}{\alpha _k} \right) ^{(1-2\beta )n}. \end{aligned}$$

Inserting this inequality in the one for \(\mathbb {E} \left| Y^{\epsilon }_s\right| ^{2 n}\) we have:

$$\begin{aligned} \mathbb {E}\left| Y^{\epsilon }_s\right| ^{2 n}&\le C \epsilon ^{n}\left( 1+{\mathbb {E}} \left[ \sup _{r \le T}|U_{r}|^\eta \right] +|v|^\eta \right) \sum _{k=1}^{\infty } \alpha _{k}^{(n-1)\zeta -(1-2 \beta )n}\\&\le C \epsilon ^{n}\left( 1+|u|^\eta +|v|^\eta \right) \sum _{k=1}^{\infty } \alpha _{k}^{n(\zeta +2\beta -1)-\zeta }. \end{aligned}$$

The series on the right-hand-side is convergent by Hypothesis 1 and we have (47), so that the claim is proved.

Inserting (47) into (46) we have:

$$\begin{aligned} {\mathbb {E}} \left[ \sup _{t \in [0,\tau ]} |Z_t^\epsilon |^{2n} \right] \le C \epsilon ^{n} \left( 1+|u|^\eta +|v|^\eta \right) . \end{aligned}$$

Finally by Holder’s inequality we have the thesis:

$$\begin{aligned} \mathbb {E}\left[ \sup _{t \in [0,\tau ]} \left| Z_t^\epsilon \right| ^{2 } \right]&\le \left( \mathbb {E}\left[ \sup _{t \in [0,\tau ]} \left| Z_t^\epsilon \right| ^{2 n} \right] \right) ^{1/n} \le C \epsilon \left( 1+|u|^\iota +|v|^\iota \right) , \end{aligned}$$

for \(\iota =\eta / n\). \(\square \)

We can now state and prove the main Theorem of this work:

Theorem 1

Let \(u \in H\), \(v \in K\) and assume Hypotheses 1, 2, 3, 4, 5. Then there exists \(C=C(T,|u|,|v|)>0\) such that

$$\begin{aligned} \mathbb {E}\left[ \sup _{t\in [0,T]} \left| U^\epsilon _t-U_t\right| ^2 \right] \le C \epsilon , \end{aligned}$$

for every \(\epsilon >0\).

Proof

For \(t \in [0,T]\) we have:

$$\begin{aligned} U_t^\epsilon - U_t = \int _{0}^t e^{A_1 (t-s)} \left( F(U_{s}^\epsilon ,V_s^\epsilon )-\overline{F}(U_s)\right) ds, \end{aligned}$$

so that:

$$\begin{aligned} \left| U_t^\epsilon - U_t \right|&\le \left| \int _{0}^t e^{A_1 (t-s)} \left( F(U_{s}^\epsilon ,V_s^\epsilon )-F(U_{s},V_s^\epsilon )\right) ds \right| \\&\quad +\left| \int _{0}^t e^{A_1 (t-s)} \left( F(U_{s},V_s^\epsilon )-\overline{F}(U_s)\right) ds \right| . \end{aligned}$$

Now let \(0 \le \tau \le T\) and compute

$$\begin{aligned} \mathbb {E}\left[ \sup _{0 \le t \le \tau } \left| {\tilde{U}}_t^\epsilon - U_t \right| ^2 \right]&\le 2 \mathbb {E}\left[ \sup _{0 \le t \le \tau } \left| \int _{0}^t e^{A_1 (t-s)} \left( F(U_{s}^\epsilon ,V_s^\epsilon )-F(U_{s},V_s^\epsilon )\right) ds \right| ^2 \right] \\&\quad + 2 \mathbb {E}\left[ \sup _{0 \le t \le \tau }\left| \int _{0}^t e^{A_1 (t-s)} \left( F(U_{s},V_s^\epsilon )-\overline{F}(U_s)\right) ds \right| ^2 \right] . \end{aligned}$$

For the first term on the right-hand-side by the Lipschitz continuity of F we have:

$$\begin{aligned} \mathbb {E}\left[ \sup _{0 \le t \le \tau }\left| \int _{0}^t e^{A_1 (t-s)} \left( F(U_{s}^\epsilon ,V_s^\epsilon )-F(U_{s},V_s^\epsilon )\right) ds \right| ^2 \right]&\le C \int _{0}^\tau \mathbb {E} \left| U_s^{\epsilon }-U_s\right| ^2 ds . \end{aligned}$$

For the second term on the right-hand-side by Proposition 3 we have:

$$\begin{aligned} \mathbb {E}\left[ \sup _{0 \le t \le \tau } \left| \int _{0}^t e^{A_1 (t-s)} \left( F(U_{s},V_s^\epsilon )-\overline{F}(U_s)\right) ds \right| ^2 \right] \le C \epsilon . \end{aligned}$$

Putting everything together we have:

$$\begin{aligned} \mathbb {E}\left[ \sup _{0 \le t \le \tau } \left| U_t^\epsilon - U_t \right| ^2 \right] \le C \left( \epsilon + \int _{0}^\tau \mathbb {E} \left| U_s^{\epsilon }-U_s\right| ^2 ds \right) , \end{aligned}$$
(48)

for every \(\tau \le T\).

Then by Gronwall’s Lemma we have the thesis of the Theorem:

$$\begin{aligned} \mathbb {E}\left[ \sup _{0 \le t \le T} \left| U_t^\epsilon - U_t \right| ^2 \right] \le C \epsilon . \end{aligned}$$

\(\square \)

Finally we can provide an application to which our theory can be applied and which is not covered by the existing literature.

Example 1

Consider the following fully coupled slow-fast stochastic reaction-diffusion system:

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{\partial u_{\varepsilon }}{\partial t}(t, \xi )= \frac{\partial ^2}{\partial \xi ^2} u_{\varepsilon }(t, \xi )+f\left( \xi , u_{\varepsilon }(t, \xi ), v_{\varepsilon }(t, \xi )\right) + {\dot{w}}_1(t, \xi ),\\ \frac{\partial v_{\varepsilon }}{\partial t}(t, \xi )= \frac{1}{\varepsilon }\left[ \left( \frac{\partial ^2}{\partial \xi ^2} - \lambda \right) v_{\varepsilon }(t, \xi )+g\left( \xi , v_{\varepsilon }(t, \xi )\right) \right] +\frac{1}{\sqrt{\varepsilon }} {\dot{w}}_2(t, \xi ),\\ u_{\varepsilon }(0, \xi )=u (\xi ), \quad v_{\varepsilon }(0, \xi )=v(\xi ), \quad \xi \in [0,L],\\ u_{\varepsilon }(t, \xi )= v_{\varepsilon }(t, \xi )=0, \quad t \ge 0, \quad \xi =0,L, \end{array}\right. } \end{aligned}$$
(49)

where

  • \(t \in [0,T], \xi \in [0,L]\),

  • \(\epsilon \in (0,1]\) is a small parameter representing the ratio of time-scales between the two variables of the system \(u_\epsilon \) and \(v_\epsilon \),

  • \(u_\epsilon \) and \(v_\epsilon \) are the slow and fast components respectively,

  • \(u,v \in H= L^2[0,T]\) are the initial conditions,

  • \(\lambda >0\),

  • \(f,g :[0,L] \times \mathbb {R} \rightarrow \mathbb {R}\) are Lipschitz functions uniformly wrt \(\xi \) with Lipschitz constants \(L_f,L_G\) respectively and \(L_G < \lambda \),

  • \({\dot{w}}_1, {\dot{w}}_2\) are independent white noises both in time and space.

Then it is well known [6] that (49) can be rewritten in the abstract form (2) where \(H=K=L^2[0,T]\), \(F:H\times H\rightarrow H\), \(G:H \rightarrow H\) are the Nemytskii operators of f, g respectively, i.e.

$$\begin{aligned} F(x,y)(\xi )&=f(\xi ,x(\xi ),y(\xi )), \\ G(y)(\xi )&=g(\xi ,y(\xi )). \end{aligned}$$

In this setting the hypotheses of Theorem 1 are satisfied (recall Remarks 2.1, 2.2) so that the result can be applied.