Proof. Applying Itô's formula to (2.6), we obtain for all $t\geq 0$ and $0<\varepsilon < 1$,

(3.3)\begin{align} d(\|u^{\varepsilon}(t)\|^2) \leq& -2 \lambda \|u^{\varepsilon}(t)\|^2 \,{\rm d}t + 2 {\rm Re\ } \left( u^{\varepsilon}(t), f(u^{\varepsilon}(t-\rho)) \right) \,{\rm d}t \nonumber\\ & + 2 {\rm Re\ } \left( u^{\varepsilon}(t), g \right) \,{\rm d}t + \varepsilon \sum_{k \in \mathbb{N}} \left\| h_k + \sigma_k( u^{\varepsilon}(t-\rho) ) \right\|^2 \,{\rm d}t \nonumber\\ & + 2 \sqrt{\varepsilon}{\rm Re} \sum_{k\in \mathbb{N}} \left( u^{\varepsilon}(t),h_k + \sigma_{k}( u^{\varepsilon}(t-\rho) ) \right) {\rm d}W_k(t). \end{align}

Let $\gamma >0$ be the positive constant satisfying (3.1). Then we get from (3.3) that

(3.4)\begin{align} e^{\gamma t}\mathbb{E} [ \|u^{\varepsilon}(t)\|^2 ] & \leq \mathbb{E}[\|u^{0}\|^2] + (\gamma-2\lambda) \int_{0}^{t} e^{\gamma s} \mathbb{E}[\|u^{\varepsilon}(s)\|^2] \,{\rm d}s \nonumber\\ & \quad + 2{\rm Re\ } \int_{0}^{t} e^{\gamma s} \mathbb{E}\left[\left( u^{\varepsilon}(s), f(u^{\varepsilon}(s-\rho)) \right)\right]\,{\rm d}s\nonumber\\ & \quad + 2{\rm Re\ }\int_{0}^{t} e^{\gamma s} \mathbb{E}\left[\left( u^{\varepsilon}(s), g \right)\right] \,{\rm d}s \nonumber\\ & \quad + \varepsilon \sum_{k \in \mathbb{N}} \int_{0}^{t}e^{\gamma s} \mathbb{E}\left[\left\| h_k + \sigma_k( u^{\varepsilon}(s-\rho) )\right\|^2 \right] \,{\rm d}s. \end{align}

We now deal with the right-hand side of (3.4). For the third term on the right-hand side of (3.4), by Young's inequality and (2.2) we have

(3.5)\begin{align} & 2{\rm Re} \int_{0}^{t} e^{\gamma s} \mathbb{E} \left[ \left( u^{\varepsilon}(s), f(u^{\varepsilon}(s-\rho)) \right) \right] \,{\rm d}s \nonumber\\ & \quad \leq 2 \int_{0}^{t} e^{\gamma s} \mathbb{E} \left[ \| u^{\varepsilon}(s) \| \|f(u^{\varepsilon}(s-\rho)) \| \right] \,{\rm d}s \nonumber\\ & \quad \leq \sqrt{2}\beta_0 \int_{0}^{t} e^{\gamma s} \mathbb{E} \left[ \|u^{\varepsilon}(s)\|^2 \right] \,{\rm d}s +\frac{1}{\sqrt{2} \beta_0} \int_{0}^{t} e^{\gamma s} \mathbb{E} \left[ \| f(u^{\varepsilon}(s-\rho)) \|^2 \right] \,{\rm d}s\nonumber\\ & \quad \leq \sqrt{2}\beta_0 \int_{0}^{t} e^{\gamma s} \mathbb{E} \left[ \| u^{\varepsilon}(s) \|^2 \right] {\rm d}s + \frac{\sqrt{2}\|\alpha\|^2}{\gamma\beta_0} (e^{\gamma t}-1) \nonumber\\ & \quad\quad + \sqrt{2} \beta_0 e^{\gamma \rho} \int_{-\rho}^{t-\rho} e^{\gamma s} \mathbb{E} \left[ \| u^{\varepsilon}(s) \|^2 \right] {\rm d}s \nonumber\\ & \quad \leq \sqrt{2} \beta_0 (1+e^{\gamma \rho}) \int_{0}^{t} e^{\gamma s} \mathbb{E} \left[ \| u^{\varepsilon}(s) \|^2 \right] {\rm d}s \nonumber\\ & \quad \quad + \sqrt{2}\beta_0e^{\gamma \rho} \int_{-\rho}^{0} \mathbb{E} \left[ \|\xi(s)\|^2 \right] {\rm d}s + \frac{\sqrt{2}\|\alpha\|^2}{\gamma\beta_0} e^{\gamma t} . \end{align}

Let $\alpha _1>0$ be a constant satisfying (3.1). By Young's inequality we get

(3.6)\begin{equation} 2{\rm Re }\int_{0}^{t} e^{\gamma s} \mathbb{E} \left[ \left( u^{\varepsilon}(s), g \right) \right] \,{\rm d}s \leq \alpha_1\int_{0}^{t} e^{\gamma s} \mathbb{E} \left[ \| u^{\varepsilon}(s) \|^2 \right] \,{\rm d}s + \frac{\|g\|^2}{\alpha_1\gamma} e^{\gamma t}. \end{equation}

For the last term on the right-hand side of (3.4), by (2.4) we obtain

(3.7)\begin{align} & \varepsilon\sum_{k\in \mathbb{N}} \int_{0}^{t} e^{\gamma s}\mathbb{E}[\| h_k+\sigma_k(u^{\varepsilon}(s-\rho))\|^2] \,{\rm d}s\nonumber\\ & \quad \leq 2\sum_{k\in \mathbb{N}} \int_{0}^{t}e^{\gamma s}\mathbb{E}[\| h_k\|^2]\,{\rm d}s + 2\sum_{k\in \mathbb{N}} \int_{0}^{t}e^{\gamma s}\mathbb{E}[\|\sigma_k(u^{\varepsilon}(s-\rho)) \|^2] \,{\rm d}s\nonumber\\ & \quad \leq \frac{2\|h\|^2}{\gamma}e^{\gamma t}+\frac{4\|\delta\|^2}{\gamma}e^{\gamma t} + 4\|\beta\|^2 \int_{0}^{t} e^{\gamma s}\mathbb{E}\big[\|u^{\varepsilon}(s-\rho)\|^2\big]\,{\rm d}s\nonumber\\ & \quad \leq \frac{2\|h\|^2}{\gamma}e^{\gamma t} + \frac{4\|\delta\|^2}{\gamma}e^{\gamma t} + 4\|\beta\|^2 e^{\gamma \rho}\int_{0}^{t} e^{\gamma s}\mathbb{E}\big[\|u^{\varepsilon}(s)\|^2\big]\,{\rm d}s\nonumber\\ & \quad \quad + 4\|\beta\|^2 e^{\gamma \rho}\int_{-\rho}^{0} \mathbb{E}\big[\|\xi(s)\|^2\big]\,{\rm d}s. \end{align}

It follows from (3.4)–(3.7) that for all $t\geq 0$,

\begin{align*} & e^{\gamma t}\mathbb{E} [ \|u^{\varepsilon}(t)\|^2] \\ \nonumber & \quad \leq \mathbb{E} [\|u^{0}\|^2] +\left[ \gamma-2\lambda+\sqrt{2}\beta_0(1+e^{\gamma \rho}) +\alpha_1 +4\|\beta\|^2e^{\gamma \rho} \right] \int_{0}^{t} e^{\gamma s} \mathbb{E}[\|u^{\varepsilon}(s)\|^2]{\rm d}s\\ & \quad \quad +(\sqrt{2}\beta_0+4\|\beta\|^2)e^{\gamma \rho}\int_{-\rho}^{0} \mathbb{E}\left[\|\xi(s)\|^2\right]\,{\rm d}s +\frac{\sqrt{2}\|\alpha\|^2}{\gamma\beta_0}e^{\gamma t}\\ & \quad \quad +\frac{\|g\|^2}{\alpha_1\gamma}e^{\gamma t} +\frac{2\|h\|^2}{\gamma}e^{\gamma t} +\frac{4\|\delta\|^2}{\gamma}e^{\gamma t}, \end{align*}

which along with (3.1) indicates that for all $t\geq 0$,

\begin{align*} \mathbb{E} [ \|u^{\varepsilon}(t)\|^2] & \leq e^{- \gamma t}\mathbb{E} [\|u^{0}\|^2] +(\sqrt{2}\beta_0+4\|\beta\|^2)e^{- \gamma (t-\rho)} \int_{-\rho}^{0} \mathbb{E}\left[\|\xi(s)\|^2\right]\,{\rm d}s \nonumber\\ & \quad +\frac{\sqrt{2}\|\alpha\|^2}{\gamma\beta_0}+\frac{\|g\|^2}{\alpha_1\gamma} +\frac{2\|h\|^2}{\gamma} +\frac{4\|\delta\|^2}{\gamma}. \end{align*}

This implies (3.2), and thus completes the proof.

Proof. For any $t\geq \rho$ and $m \in \mathbb {N}$, let

\[ \tau_{m}^{t}= \inf\{s\geq t:\|u^{\varepsilon}(s)\|> m\},\]

and we set $\tau _{m}^{t}=\infty$ if $\{s\geq t:\|u^{\varepsilon }(s)\|> m\}=\emptyset$.

For any $T> 0$, applying Itô's formula to (2.6), we have

(3.8)\begin{align} & \mathbb{E}[\|u^{\varepsilon}((t+T)\wedge \tau_{m}^{t}) \|^2] \nonumber\\ & \quad \leq \mathbb{E}[\|u^{\varepsilon}(t)\|^2] - 2 \lambda \int_{t}^{ (t+T) \wedge \tau_{m}^{t} } \mathbb{E} [ \|u^{\varepsilon}(s)\|^2 ] \,{\rm d}s \nonumber\\ & \quad \quad + 2 {\rm Re}\int_{t}^{(t+T)\wedge \tau_{m}^{t}} \mathbb{E} \left[ \left( u^{\varepsilon}(s), f(u^{\varepsilon}(s-\rho)) \right) \right] {\rm d}s \nonumber\\ & \quad \quad + 2 {\rm Re}\int_{t}^{ (t+T)\wedge \tau_{m}^{t} } \mathbb{E} \left[ \left( u^{\varepsilon}(s), g \right) \right] {\rm d}s \nonumber\\ & \quad \quad + \varepsilon \sum_{k\in \mathbb{N}} \int_{t}^{(t+T)\wedge \tau_{m}^{t}} \mathbb{E} [ \| h_k + \sigma_k(u^{\varepsilon}(s-\rho)) \|^2 ] \,{\rm d}s. \end{align}

For the third term on the right-hand side of (3.8), by (2.2) we obtain

(3.9)\begin{align} & 2{\rm Re}\int_{t}^{(t+T)\wedge \tau_{m}^{t}} \mathbb{E}\left[\left(u^{\varepsilon}(s),f(u^{\varepsilon}(s-\rho))\right)\right]\,{\rm d}s \nonumber\\ & \quad \leq 2\beta_0^2 \int_{t}^{(t+T)\wedge \tau_{m}^{t}}\mathbb{E}[\|u^{\varepsilon}(s-\rho)\|^2]\,{\rm d}s +2\|\alpha\|^2 T +\int_{t}^{(t+T)\wedge \tau_{m}^{t}}\mathbb{E}[\|u^{\varepsilon}(s)\|^2]\,{\rm d}s. \end{align}

For the fourth term on the right-hand side of (3.8), by Young's inequality we have

(3.10)\begin{equation} 2{\rm Re}\int_{t}^{(t+T)\wedge \tau_{m}^{t}}\mathbb{E}\big[\big(u^{\varepsilon}(s),g\big)\big]\,{\rm d}s \leq \int_{t}^{(t+T)\wedge \tau_{m}^{t}}\mathbb{E}[\|u^{\varepsilon}(s)\|^2]\,{\rm d}s +\|g\|^2\,T. \end{equation}

For the last term on the right-hand side of (3.8), by (2.4) we get

(3.11)\begin{align} & \varepsilon\sum_{k \in \mathbb{N}} \int_{t}^{ (t+T) \wedge \tau_{m}^{t} } \mathbb{E} [ \| h_k + \sigma_k(u^{\varepsilon}(s-\rho)) \|^2 ] \,{\rm d}s \nonumber\\ & \quad \leq 2 \|h\|^2\,T + 4 \|\delta\|^2\,T + 4 \|\beta\|^2 \int_{t}^{ (t+T)\wedge \tau_{m}^{t} } \mathbb{E} [ \|u^{\varepsilon}(s-\rho) \|^2 ] \,{\rm d}s. \end{align}

From (3.8)–(3.11) and lemma 3.1, it follows that there exists a positive constant $C_{T,R}$ depending only on $R$ and $T$ such that for all $t\geq \rho$,

(3.12)\begin{align} & \mathbb{E}[\|u^{\varepsilon}((t+T)\wedge \tau_{m}^{t}) \|^2] \nonumber\\ & \quad \leq \mathbb{E}[\|u^{\varepsilon}(t)\|^2] + (2-2\lambda)\int_{t}^{(t+T)\wedge \tau_{m}^{t}}\mathbb{E}[\|u^{\varepsilon}(s)\|^2]\,{\rm d}s \nonumber\\ & \quad \quad +(\|g\|^2+4\|\delta\|^2+2\|\alpha\|^2+2\|h\|^2 )T +(2\beta_0^2+4\|\beta\|^2)\int_{t-\rho}^{t+T-\rho} \mathbb{E}[\|u^{\varepsilon}(s)\|^2]\,{\rm d}s \nonumber\\ & \quad \leq \left[1+(2-2\lambda+2\beta_0^2+4\|\beta\|^2)T\right] M_1 \left( 1+\mathbb{E}[\|u^{0}\|^2] +\int_{-\rho}^{0}\mathbb{E}[\|\xi(s)\|^2]\,{\rm d}s \right) \nonumber\\ & \quad \quad +(\|g\|^2+4\|\delta\|^2+2\|\alpha\|^2+2\|h\|^2 )T \nonumber\\ & \quad \leq C_{T,R}. \end{align}

Recalling the definition of $\tau _{m}^{t}$, we obtain by (3.12) that for all $t\geq \rho$,

\begin{align*} m^2\mathbb{P}(\{ \tau_{m}^{t}< t+T \}) & \leq \mathbb{E}\left[\|u^{\varepsilon}(\tau_{m}^{t})\|^2I_{\{\tau_{m}^{t}< t+T \}} \right]\\ & \leq \mathbb{E}[\|u^{\varepsilon}((t+T)\wedge \tau_{m}^{t})\|^2 ]\leq C_{T,R}. \end{align*}

Then, we have

(3.13)\begin{equation} \mathbb{P}(\{ \tau_{m}^{t}< t+T \}) \leq \frac{C_{T,R}}{m^2}. \end{equation}

By (3.13) we find that for every $\varepsilon '>0$, $T>0$ and $R>0$, there exists $m_1=m_1(\varepsilon ',\,T,\,R)>0$ such that for $m\geq m_1$,

\[ \mathbb{P}(\{ \tau_{m}^{t}< t+T \}) \leq \frac{\varepsilon '}{2}, \quad \forall\ t \geq \rho,\]

which implies that

(3.14)\begin{equation} \mathbb{P}\left(\left\{ \sup\nolimits_{s \in[t,t+T]}\|u^{\varepsilon}(s)\|> m \right\}\right) \leq \frac{\varepsilon '}{2}, \quad \forall\ t \geq \rho,\ m\geq m_1. \end{equation}

On the other hand, making use of (2.7) and the Chebyshev inequality, we obtain that there exists $m_2=m_2(\varepsilon ',\,R)>0$ such that

\[ \mathbb{P}\left(\left\{ \sup\nolimits_{0\leq s \leq \rho}\|u^{\varepsilon}(s)\|> m \right\}\right) \leq \frac{\mathbb{E}\left[\sup\nolimits_{0\leq s \leq \rho}\|u^{\varepsilon}(s)\|^2\right]}{m^2} \leq \frac{\varepsilon '}{2}, \quad \forall\ m\geq m_2,\]

which along with (3.14) implies that there exists $M_2=M_2(\varepsilon ',\,T,\,R)>0$ such that

\[ \mathbb{P}\left(\left\{ \sup\nolimits_{s \in [t,t+T]}\|u^{\varepsilon}(s)\|> m \right\}\right) \leq {\varepsilon '} , \quad \forall\ t \geq 0, \ m\geq M_2,\]

as desired.

Proof. For any $\varepsilon '>0$, it follows from remark 3.3 that there exists $m_3=m_3(\varepsilon ',\,R)>0$ such that for any $t \geq \rho$,

(3.15)\begin{equation} \mathbb{P}\left(\left\{\sup_{t \leq s \leq t+\rho} \|u^{\varepsilon}_s\|_{C([-\rho,0];l^2)}\leq m_3 \right\}\right) \geq 1-\frac{\varepsilon'}{2}. \end{equation}

For each $r\geq \rho$, define a stopping time $\tau _r$ by

\[ \tau_r= \inf\{s\geq r: \|u^{\varepsilon}_s\|_{C([-\rho,0];l^2)}> m_3\},\]

and we set $\tau _r=\infty$ if $\{s\geq r: \|u^{\varepsilon }_s\|_{C([-\rho,0];l^2)}> m_3\}=\emptyset$. By (3.15) we know that

(3.16)\begin{equation} \mathbb{P}\{\tau_r< r+\rho\} \leq \frac{\varepsilon'}{2},\quad \forall\ r\geq \rho. \end{equation}

By (2.6) we have for any $\rho \leq r \leq t$,

\begin{align*} \|u^{\varepsilon}(t)-u^{\varepsilon}(r)\| & \leq (\lambda+4)\int_{r}^{t}\|u^{\varepsilon}(s)\|\,{\rm d}s +\int_{r}^{t}\|u^{\varepsilon}(s)\|^3\,{\rm d}s +\int_{r}^{t}\|f(u^{\varepsilon}(s-\rho))\|\,{\rm d}s\nonumber\\ & \quad +\|g\||t-r| +\sqrt{\varepsilon}\Bigg\|\sum_{k \in \mathbb{N}}\int_{r}^{t} (h_k+\sigma_k(u^{\varepsilon}(s-\rho)))\,{\rm d}W_k(s)\Bigg\|, \end{align*}

and hence for any $r\geq \rho$, $0<\eta <\rho$ and $p>1$, we get

(3.17)\begin{align} & \sup_{t \in [r,r+\eta]}\|u^{\varepsilon}(t\wedge \tau_r)-u^{\varepsilon}(r)\|^{2p} \nonumber\\ & \quad \leq 5^{2p-1}(\lambda+4)^{2p} \sup_{t \in [r,r+\eta]}\left(\int_{r}^{t \wedge \tau_r} \|u^{\varepsilon}(s)\| \,{\rm d}s\right)^{2p} \nonumber\\ & \quad \quad +5^{2p-1} \sup_{t \in [r,r+\eta]}\left(\int_{r}^{t \wedge \tau_r}\|u^{\varepsilon}(s)\|^3\,{\rm d}s\right)^{2p} \nonumber\\ & \quad \quad +5^{2p-1}\sup_{t \in [r,r+\eta]}\left(\int_{r}^{t \wedge \tau_r}\|f(u^{\varepsilon}(s-\rho))\|\,{\rm d}s\right)^{2p} +5^{2p-1}\|g\|^{2p}\eta^{2p} \nonumber\\ & \quad \quad +5^{2p-1}\varepsilon^{p}\left[\sup_{t \in [r,r+\eta]}\Big\|\sum_{k\in\mathbb{N}}\int_{r}^{t \wedge \tau_r} (h_k+\sigma_k(u^{\varepsilon}(s-\rho)))\,{\rm d}W_k(s)\Big\|^{2p}\right]. \end{align}

By (2.2), (2.4), the Hölder inequality and the Burkholder–Davis–Gundy (BDG) inequality, it follows from (3.17) that for any $r\geq \rho$, $0<\eta < \min \{1,\, \rho \}$,

(3.18)\begin{align} & \mathbb{E}\left[\sup_{t \in [r,r+\eta]}\|u^{\varepsilon}(t\wedge \tau_r)-u^{\varepsilon}(r)\|^{2p} \right] \nonumber\\ & \quad \leq 5^{2p-1}(\lambda+4)^{2p}m_3^{2p}\eta^{2p} +5^{2p-1}m_3^{6p}\eta^{2p} +5^{2p-1}(2\|\alpha\|^2+2\beta_0^2m_3^2)^{p}\eta^{2p} \nonumber\\ & \quad \quad +5^{2p-1}\|g\|^{2p}\eta^{2p} +5^{2p-1}C_p(4\|\delta\|^2+4\|\beta\|^2m_3^2+2\|h\|^2)^{p}\eta^{p} \nonumber\\ & \quad \leq C_{0}\eta^{p}(1+\eta^{p}) \leq 2C_{0} \eta^{p}, \end{align}

where $C_p$ is the coefficient of the BDG inequality and $C_{0}$ is a positive constant independent of $\eta$, $r$ and $\varepsilon \in (0,\,1)$. From (3.16) and (3.18), we can derive that for any $\delta _1>0$ and $t \geq 2\rho$,

(3.19)\begin{align} & \mathbb{P}\Bigg(\left\{ \sup_{t_1,t_2 \in [-\rho,0],|t_1-t_2|\leq\eta} \|u^{\varepsilon}_{t}(t_1)-u^{\varepsilon}_t(t_2)\|\geq\delta_1 \right\}\Bigg)\nonumber\\ & \quad \leq \mathbb{P}(\{ \tau_{t-\rho}< t\}) +\mathbb{P}\Bigg(\left\{ \tau_{t-\rho} \ge t, \sup_{t_1,t_2 \in [-\rho,0],|t_1-t_2|\leq\eta} \|u^{\varepsilon}_{t}(t_1)-u^{\varepsilon}_t(t_2)\|\geq \delta_1 \right\}\Bigg)\nonumber\\ & \quad = \mathbb{P}(\{ \tau_{t-\rho}< t\}) +\mathbb{P}\Bigg(\left\{ \tau_{t-\rho} \ge t, \sup_{t_1 \in [-\rho,0],t_2\in [t_1,(t_1+\eta)\wedge0]} \|u^{\varepsilon}(t+t_1)-u^{\varepsilon}(t+t_2)\|\geq \delta_1 \right\}\Bigg)\nonumber\\ & \quad \leq \frac{\varepsilon'}{2} +\mathbb{P}\Bigg(\Bigg\{ \tau_{t-\rho} \ge t, \max_{0\leq k \leq \left[\frac{\rho}{\eta}\right]} \sup_{s\in [t-(k+1)\eta \wedge \rho,t-k\eta]} \|u^{\varepsilon}(s)-u^{\varepsilon}(t-(k+1)\eta\wedge \rho)\|\geq\frac{\delta_1}{3} \Bigg\}\Bigg)\nonumber\\ & \quad \leq \frac{\varepsilon'}{2} +\sum_{k=0}^{[{\rho}/{\eta}]} \mathbb{P}\Bigg(\left\{ \tau_{t- \rho} \ge t, \sup_{s\in [t-(k+1)\eta \wedge \rho,t-k\eta]} \|u^{\varepsilon}(s)-u^{\varepsilon}(t-(k+1)\eta\wedge \rho)\|\geq\frac{\delta_1}{3} \right\}\Bigg)\nonumber\\ & \quad \leq \frac{\varepsilon'}{2} +\sum_{k=0}^{[{\rho}/{\eta}]} \mathbb{P}\Bigg(\left\{ \tau_{t-(k+1)\eta \wedge \rho} \ge t, \sup_{s\in [t-(k+1)\eta \wedge \rho,t-k\eta]} \|u^{\varepsilon}(s)-u^{\varepsilon}(t-(k+1)\eta\wedge \rho)\|\geq\frac{\delta_1}{3} \right\}\Bigg)\nonumber\\ & \quad \leq \frac{\varepsilon'}{2} + \sum_{k=0}^{[{\rho}/{\eta}]} \mathbb{P}\Bigg(\left\{ \sup_{s\in [t-(k+1)\eta \wedge \rho,t-k\eta]} \|u^{\varepsilon}(s\wedge \tau_{t-(k+1)\eta \wedge \rho})-u^{\varepsilon}(t-(k+1)\eta\wedge \rho)\|\geq\frac{\delta_1}{3} \right\}\Bigg)\nonumber\\ & \quad \leq \frac{\varepsilon'}{2} +\left(\left[\frac{\rho}{\eta}\right]+1\right)\frac{3^{2p}2C_{0}\eta^{p}}{\delta_1^{2p}}. \end{align}

Let

\[ \eta_1=\left(\frac{\varepsilon' \delta_1^{2p}}{4(1+\rho)C_{0}3^{2p}}\right)^{{1}/{p-1}}\wedge 1\wedge\rho.\]

Then by (3.19) we obtain that for any $t\geq 2\rho$,

(3.20)\begin{equation} \mathbb{P}\left(\left\{ \sup_{t_1,t_2 \in [-\rho,0],|t_1-t_2|\leq\eta_1} \|u^{\varepsilon}_{t}(t_1)-u^{\varepsilon}_t(t_2)\|\geq\delta_1 \right\}\right)\leq \varepsilon'. \end{equation}

On the other hand, since $u^{\varepsilon }\in L^2(\Omega ;C([0,\,2\rho ];l^2))$, we find that there exists a constant $\eta _2=\eta _2(\delta _1,\,\varepsilon ')>0$ such that for any $\rho \leq t \leq 2\rho$,

\[ \mathbb{P}\left(\left\{ \sup_{t_1,t_2 \in [-\rho,0],|t_1-t_2|\leq\eta_2} \|u^{\varepsilon}_{t}(t_1)-u^{\varepsilon}_t(t_2)\|\geq\delta_1 \right\}\right)\leq \varepsilon',\]

which along with (3.20) yields that

\[ \mathbb{P}\left(\left\{ \sup_{t_1,t_2 \in [-\rho,0],|t_1-t_2|\leq\eta} \|u^{\varepsilon}_{t}(t_1)-u^{\varepsilon}_t(t_2)\| \geq \delta_1 \right\}\right) \leq \varepsilon', \quad \forall\ t\geq \rho,\]

where $\eta =\eta _1\wedge \eta _2$, as desired.

Proof. Hereafter, we denote by $C$ a generic positive constant independent of $E$, $T$ and $\varepsilon '$. Consider a smooth function $\theta :\mathbb {R}\rightarrow [0,\,1]$ satisfying

(3.21)\begin{equation} \theta(s)=0 \ {\rm for}\ |s|\leq1; \quad {\rm and} \quad \theta(s)=1 \ {\rm for}\ |s|\geq2. \end{equation}

Fixed $m \in \mathbb {N}$, denote by $\theta _m=(\theta ({n}/{m}))_{n\in \mathbb {Z}}$ and $\theta _m u = (\theta ({n}/{m})u_n)_{n\in \mathbb {Z}}$ for $u=(u_n)_{n\in \mathbb {Z}}\in l^2$. Then by (2.6) we have

(3.22)\begin{align} & d(\theta_mu^{\varepsilon}(t))+\left(i\theta_mAu^{\varepsilon}(t) +i\theta_m|u^{\varepsilon}(t)|^2u^{\varepsilon}(t) +\lambda\theta_m u^{\varepsilon}(t) \right) {\rm d}t \nonumber\\ & \quad = \theta_mf(u^{\varepsilon}(t-\rho))\,{\rm d}t+\theta_m g\,{\rm d}t \nonumber\\ & \quad \quad + \sqrt{\varepsilon}\sum\limits_{k \in \mathbb{N}}\left( \theta_mh_{k}+\theta_m\sigma_{k}(u^{\varepsilon}(t-\rho)) \right) {\rm d}W_k(t). \end{align}

Similar to (3.4), by (3.22) we get that for all $t\geq 0$,

(3.23)\begin{align} e^{\gamma t}\mathbb{E} [ \|\theta_mu^{\varepsilon}(t)\|^2 ] = & \mathbb{E}[\|\theta_mu^{0}\|^2] + (\gamma-2\lambda)\int_{0}^{t}e^{\gamma s}\mathbb{E}[\|\theta_mu^{\varepsilon}(s)\|^2]\,{\rm d}s \nonumber\\ & -2{\rm Re} \int_{0}^{t}e^{\gamma s}\mathbb{E}\left[\left(\theta_m^2u^{\varepsilon}(s),iAu^{\varepsilon}(s)\right)\right]\,{\rm d}s \nonumber\\ & -2{\rm Re} \int_{0}^{t}e^{\gamma s} \mathbb{E}\left[\left(\theta_m^2u^{\varepsilon}(s),i|u^{\varepsilon}(s)|^2u^{\varepsilon}(s)\right)\right] \,{\rm d}s \nonumber\\ & +2{\rm Re\ }\int_{0}^{t}e^{\gamma s} \mathbb{E}\left[\left(\theta_mu^{\varepsilon}(s),\theta_mf(u^{\varepsilon}(s-\rho))\right)\right]\,{\rm d}s \nonumber\\ & +2{\rm Re\ }\int_{0}^{t}e^{\gamma s}\mathbb{E}\left[\left(\theta_mu^{\varepsilon}(s),\theta_mg\right)\right]{\rm d}s \nonumber\\ & +\varepsilon\sum_{k\in \mathbb{N}}\int_{0}^{t}e^{\gamma s} \mathbb{E}\left[\left\| \theta_mh_k+\theta_m\sigma_k(u^{\varepsilon}(s-\rho)) \right\|^2\right] \,{\rm d}s. \end{align}

By the argument of (4.4)–(4.6) in [Reference Wang31], we have

(3.24)\begin{align} & -2{\rm Re} \int_{0}^{t}e^{\gamma s} \mathbb{E}\left[\left(\theta_m^2u^{\varepsilon}(s),iAu^{\varepsilon}(s)\right)\right]{\rm d}s \nonumber\\ & \quad = {-}2{\rm Re} \int_{0}^{t}e^{\gamma s} \mathbb{E}\left[\left(B(\theta_m^2u^{\varepsilon}(s)),iBu^{\varepsilon}(s)\right)\right]{\rm d}s \nonumber\\ & \quad \leq \frac{C}{m}\int_{0}^{t}e^{\gamma s}\mathbb{E}[\|u^{\varepsilon}(s)\|^2]\,{\rm d}s. \end{align}

For the fifth term on the right-hand side of (3.23), by assumption (A3) we obtain

(3.25)\begin{align} & 2{\rm Re }\int_{0}^{t}e^{\gamma s} \mathbb{E}\left[\left(\theta_mu^{\varepsilon}(s),\theta_mf(u^{\varepsilon}(s-\rho))\right)\right] \,{\rm d}s \nonumber\\ & \quad \leq \sqrt{2}\beta_0 \int_{0}^{t}e^{\gamma s} \mathbb{E}\left[\|\theta_mu^{\varepsilon}(s)\|^2\right]\,{\rm d}s +\frac{1}{\sqrt{2}\beta_0}\int_{0}^{t}e^{\gamma s}\mathbb{E}[\|\theta_mf(u^{\varepsilon}(s-\rho))\|^2]\,{\rm d}s \nonumber\\ & \quad \leq \sqrt{2}\beta_0(1+e^{\gamma\rho}) \int_{0}^{t}e^{\gamma s}\mathbb{E}\left[\|\theta_mu^{\varepsilon}(s)\|^2\right]\,{\rm d}s +\frac{\sqrt{2}}{\beta_0\gamma}e^{\gamma t}\sum_{|n|\geq m}|\alpha_n|^2 \nonumber\\ & \quad \quad +\sqrt{2}\beta_0e^{\gamma\rho}\int_{-\rho}^{0}\mathbb{E}\left[\|\theta_m\xi(s)\|^2\right]\,{\rm d}s. \end{align}

By Young's inequality we have

(3.26)\begin{align} 2{\rm Re\ }\int_{0}^{t}e^{\gamma s}\mathbb{E}\left[\left(\theta_mu^{\varepsilon}(s),\theta_mg\right)\right]\,{\rm d}s & \leq \alpha_1 \int_{0}^{t}e^{\gamma s}\mathbb{E}\left[\|\theta_mu^{\varepsilon}(s)\|^2\right]\,{\rm d}s\nonumber\\ & \quad +\frac{1}{\alpha_1 \gamma}e^{\gamma t}\sum_{|n|\geq m}|g_n|^2. \end{align}

For the last term on the right-hand side of (3.23), by assumption (A4) we get

(3.27)\begin{align} & \varepsilon\sum_{k\in \mathbb{N}}\int_{0}^{t}e^{\gamma s} \mathbb{E}\left [\left \| \theta_mh_k+\theta_m\sigma_k(u^{\varepsilon}(s-\rho)) \right\|^2\right ]{\rm d}s \nonumber\\ & \quad \leq 2\sum_{k\in \mathbb{N}}\sum_{|n|\geq m}|h_{k,n}|^2\frac{e^{\gamma t}}{\gamma} +2\sum_{k\in \mathbb{N}}\int_{0}^{t}e^{\gamma s} \mathbb{E}\left[\left\|\theta_m\sigma_k(u^{\varepsilon}(s-\rho)) \right\|^2\right]{\rm d}s \nonumber\\ & \quad \leq 2\sum_{k\in \mathbb{N}}\sum_{|n|\geq m}|h_{k,n}|^2\frac{e^{\gamma t}}{\gamma} +4\sum_{k\in \mathbb{N}}\sum_{|n|\geq m}|\delta_{k,n}|^2\frac{e^{\gamma t}}{\gamma} +4\|\beta\|^2e^{\gamma \rho}\int_{-\rho}^{0}\mathbb{E}\left[\|\theta_m\xi(s)\|^2\right]{\rm d}s \nonumber\\ & \quad \quad +4\|\beta\|^2e^{\gamma \rho}\int_{0}^{t}e^{\gamma s} \mathbb{E}\left[\|\theta_mu^{\varepsilon}(s)\|^2\right]{\rm d}s. \end{align}

It follows from (3.23)–(3.27) that for all $t\geq 0$,

(3.28)\begin{align} & \mathbb{E} [ \|\theta_mu^{\varepsilon}(t)\|^2 ] \leq \mathbb{E}[\|\theta_mu^{0}\|^2]e^{-\gamma t} \nonumber \\ & \quad+ \left[ \gamma-2\lambda+\sqrt{2}\beta_0(1+e^{\gamma \rho})+\alpha_1 +4\|\beta\|^2e^{\gamma \rho} \right] \int_{0}^{t}e^{\gamma( s-t)}\mathbb{E}\left[\|\theta_mu^{\varepsilon}(s)\|^2\right]\,{\rm d}s\nonumber\\ & \quad+\frac{2}{\gamma}\sum_{|n|\geq m}\sum_{k\in \mathbb{N}}|h_{k,n}|^2 +\frac{4}{\gamma}\sum_{|n|\geq m}\sum_{k\in \mathbb{N}}|\delta_{k,n}|^2 +\frac{1}{\alpha_1 \gamma}\sum_{|n|\geq m}|g_n|^2 +\frac{\sqrt{2}}{\beta_0\gamma}\sum_{|n|\geq m}|\alpha_n|^2\nonumber\\ & \quad +\frac{C}{m}\int_{0}^{t}e^{\gamma(s-t)}\mathbb{E}[\|u^{\varepsilon}(s)\|^2]\,{\rm d}s +(\sqrt{2}\beta_0+4\|\beta\|^2)e^{\gamma(\rho-t)} \int_{-\rho}^{0}\mathbb{E}\left[\|\theta_m\xi(s)\|^2\right]\,{\rm d}s. \end{align}

Since $\|h\|^2 \vee \|\delta \|^2 \vee \|\alpha \|^2 \vee \|g\|^2< \infty$, we infer that there exists $m_4=m_4(\varepsilon ')\geq 0$ such that for all $m\geq m_4$,

(3.29)\begin{align} \sum_{|n|\geq m}\sum_{k\in \mathbb{N}}|h_{k,n}|^2 \vee \sum_{|n|\geq m}\sum_{k\in \mathbb{N}}|\delta_{k,n}|^2 \vee \sum_{|n|\geq m}|g_n|^2 \vee \sum_{|n|\geq m}|\alpha_n|^2 \leq \varepsilon'. \end{align}

For any $\varepsilon '>0$, since $E$ is compact in $L^2(\Omega ;l^2)\times L^2(\Omega ;L^2((-\rho,\,0);l^2))$, then it has a finite open cover of balls with radius ${\sqrt {\varepsilon '}}/{2}$, which is denoted by $\left \{B\left( (u^{j},\,\xi ^{j}) ,\, {\sqrt {\varepsilon '}}/{2}\right) \right \}_{j=1}^{l}$. Since $(u^{j},\,\xi ^{j}) \in L^2(\Omega ;l^2) \times L^2(\Omega ;L^2((-\rho,\,0);l^2))$ for $j=1,\,2,\,\cdots,\,l$, there exists $m_5=m_5(\varepsilon ',\,E)\geq m_4$ such that for all $m\geq m_5$ and $j=1,\,2,\,\cdots,\,l$,

\[ \sum_{|n|\geq m}\Bigg(\mathbb{E}[|u^{j}_n|^2] +\int_{-\rho}^{0}\mathbb{E}[|\xi_{n}^{j}(s)|^2]\,{\rm d}s \Bigg) \leq \frac{\varepsilon'}{4},\]

which implies for all $m\geq m_5$ and $(u^{0},\,\xi )\in E$,

\[ \sum_{|n|\geq m}\bigg(\mathbb{E}[|u^{0}_n(s)|^2] +\int_{-\rho}^{0}\mathbb{E}[|\xi_{n}(s)|^2]\,{\rm d}s \bigg) \leq \varepsilon'.\]

We therefore obtain that for all $m\geq m_5$ and $(u^{0},\,\xi )\in E$,

(3.30)\begin{equation} \int_{-\rho}^{0}\mathbb{E}\left[\|\theta_m\xi(s)\|^2\right]\,{\rm d}s \leq\int_{-\rho}^{0}\sum_{|n|\geq m}\mathbb{E}\left[|\xi_{n}(s)|^2\right]\,{\rm d}s \leq \varepsilon', \end{equation}

and

(3.31)\begin{equation} \mathbb{E}[\|\theta_mu^{0}\|^2]\leq \sum_{|n|\geq m}\mathbb{E}[|u^{0}_n|^2] \leq\varepsilon'.\end{equation}

On the other hand, by lemma 3.1 we know that there exists $m_6=m_6(\varepsilon ',\,E)\geq m_5$ such that for all $m\geq m_6$ and $t\geq 0$,

(3.32)\begin{equation} \frac{C}{m}\int_{0}^{t}e^{\gamma (s-t)}\mathbb{E}[\|u^{\varepsilon}(s)\|^2]\,{\rm d}s \leq \frac{C_1}{m}\int_{0}^{t}e^{\gamma (s-t)}\,{\rm d}s \leq \varepsilon', \end{equation}

where $C_1>0$ is a constant depending only on $E$. Substituting (3.29)-(3.32) into (3.28), we obtain for all $m\geq m_6$ and $t\geq 0$,

\[ \mathbb{E} [ \|\theta_mu^{\varepsilon}(t)\|^2 ] \leq 2\varepsilon' +\frac{2\varepsilon'}{\gamma} +\frac{4\varepsilon'}{\gamma} +\frac{\varepsilon'}{\alpha_1 \gamma} +\frac{\sqrt{2}\varepsilon'}{\beta_0\gamma} +(\sqrt{2}\beta_0+4\|\beta\|^2)e^{\gamma\rho}\varepsilon' \leq C\varepsilon',\]

which implies that for all $m\geq m_6$ and $t\geq 0$,

\[ \sum_{|n|\geq 2 m} \mathbb{E} [|u^{\varepsilon}_{n}(t)|^2] \leq \mathbb{E}[\|\theta_mu^{\varepsilon}(t)\|^2 ] \leq C\varepsilon',\]

as desired.

Proof. Let $\theta$ be the smooth cut-off function as given by (3.21). It follows from lemma 3.6 that for every $\varepsilon '>0$, there exists $N_1=N_1(\varepsilon ',\, E)>0$ such that for any $t\geq 0$ and $m \geq N_1$,

(3.33)\begin{equation} \mathbb{E}[\|\theta_{m}u^{\varepsilon}(t)\|^2] \leq \varepsilon'. \end{equation}

Applying Itô's formula to (3.22), we obtain for all $t\geq 0$, $r\in [t,\,t+T]$, $\varepsilon \in (0,\,1)$ and $m\geq N_1$,

(3.34)\begin{align} \|\theta_mu^{\varepsilon}(r) \|^2 = & \|\theta_mu^{\varepsilon}(t)\|^2-2{\rm Re}\int_{t}^{r} \left(iAu^{\varepsilon}(s),\theta_m^2u^{\varepsilon}(s)\right)\,{\rm d}s \nonumber\\ & -2{\rm Re}\int_{t}^{r} \left(i|u^{\varepsilon}(s)|^2u^{\varepsilon}(s),\theta_m^2u^{\varepsilon}(s)\right)\,{\rm d}s -2\lambda\int_{t}^{r}\|\theta_mu^{\varepsilon}(s)\|^2\,{\rm d}s \nonumber\\ & +2{\rm Re}\int_{t}^{r}\left(\theta_mf(u^{\varepsilon}(s-\rho)),\theta_m u^{\varepsilon}(s)\right)\,{\rm d}s \nonumber\\ & +2{\rm Re}\int_{t}^{r}\left(\theta_m g, \theta_m u^{\varepsilon}(s)\right){\rm d}s +\sum_{k\in \mathbb{N}}\int_{t}^{r}\|\theta_mh_k+\theta_{m}\sigma_k(u^{\varepsilon}(s-\rho))\|^2{\rm d}s\nonumber\\ & +2{\rm Re}\sum_{k\in \mathbb{N}}\int_{t}^{r} \left(\theta_{m}h_k+\theta_m\sigma_k(u^{\varepsilon}(s-\rho)),\theta_mu^{\varepsilon}(s)\right)\,{\rm d}W_k(s). \end{align}

For the second term on the right-hand side of (3.34), similar to (3.24), we have

(3.35)\begin{equation} -2{\rm Re}\int_{t}^{r} \left(iAu^{\varepsilon}(s),\theta_m^2u^{\varepsilon}(s)\right)\,{\rm d}s \leq \frac{C}{m}\int_{t}^{r}\|u^{\varepsilon}(s)\|^2\,{\rm d}s. \end{equation}

Then by (3.34)–(3.35) we get

(3.36)\begin{align} & \mathbb{E}\left[\sup_{t\leq r \leq t+T}\|\theta_mu^{\varepsilon}(r) \|^2\right] \leq\mathbb{E}[ \|\theta_mu^{\varepsilon}(t)\|^2] +\frac{C}{m}\int_{t}^{t+T}\mathbb{E}[\|u^{\varepsilon}(s)\|^2]\,{\rm d}s\nonumber\\ & \quad +2\int_{t}^{t+T}\mathbb{E}\left[\|\theta_mf(u^{\varepsilon}(s-\rho))\|\|\theta_mu^{\varepsilon}(s)\|\right]\,{\rm d}s +2\int_{t}^{t+T}\mathbb{E}[\|\theta_mg\|\|\theta_mu^{\varepsilon}(s)\|]\,{\rm d}s\nonumber\\ & \quad +\sum_{k\in \mathbb{N}}\int_{t}^{t+T} \mathbb{E}[\|\theta_mh_k+\theta_m\sigma_k(u^{\varepsilon}(s-\rho))\|^2]\,{\rm d}s\nonumber\\ & \quad +2\mathbb{E}\left[\sup_{t\leq r \leq t+T}\left|\sum_{k\in \mathbb{N}}\int_{t}^{r} \left(\theta_mh_k+\theta_m\sigma_k(u^{\varepsilon}(s-\rho)),\theta_mu^{\varepsilon}(s)\right)\,{\rm d}W_k(s)\right|\right]. \end{align}

For the second term on the right-hand side of (3.36), by lemma 3.1 we know that there exists $N_2=N_2(\varepsilon ',\,E) \geq N_1$ such that for $m\geq N_2$,

(3.37)\begin{equation} \frac{C}{m}\int_{t}^{t+T}\mathbb{E}[\|u^{\varepsilon}(s)\|^2]\,{\rm d}s\leq \varepsilon' T. \end{equation}

For the third term on the right-hand side of (3.36), by (A3) and (3.33) we have for $m\geq N_2$,

(3.38)\begin{align} & 2\int_{t}^{t+T}\mathbb{E}[\|\theta_mf(u^{\varepsilon}(s-\rho))\|\|\theta_mu^{\varepsilon}(s)\|]\,{\rm d}s \nonumber\\ & \quad \leq 2\beta_0^2\int_{t-\rho}^{t+T-\rho}\mathbb{E}[\|\theta_mu^{\varepsilon}(s)\|^2]\,{\rm d}s +2\sum_{|n|\geq m}|\alpha_n|^2\,T +\int_{t}^{t+T}\mathbb{E}[\|\theta_mu^{\varepsilon}(s)\|^2]\,{\rm d}s\nonumber\\ & \quad \leq (2\beta_0^2+1)\varepsilon' T +2\sum_{|n|\geq m}|\alpha_n|^2\,T +2\beta_0^2\int_{-\rho}^{0}\mathbb{E}[\|\theta_m\xi(s)\|^2]\,{\rm d}s. \end{align}

For the fourth term on the right-hand side of (3.36), by (3.33) and Young's inequality we get for $m\geq N_2$,

(3.39)\begin{equation} 2\int_{t}^{t+T}\mathbb{E}[\|\theta_mg\|\|\theta_mu^{\varepsilon}(s)\|]\,{\rm d}s \leq \sum_{|n|\geq {m}}|g_n|^2T+\varepsilon' T.\end{equation}

For the fifth term on the right-hand side of (3.36), by (A4) we have for $m\geq N_2$,

(3.40)\begin{align} & \sum_{k\in \mathbb{N}}\int_{t}^{t+T} \mathbb{E}[\|\theta_mh_k+\theta_m\sigma_k(u^{\varepsilon}(s-\rho))\|^2]\,{\rm d}s \nonumber \\ & \quad \leq 2\sum_{|n|\geq m}\sum_{k \in \mathbb{N}}|h_{k,n}|^2\,T +4\sum_{|n|\geq m}\sum_{k \in \mathbb{N}}|\delta_{k,n}|^2T \nonumber\\ & \quad \quad +4\|\beta\|^2\varepsilon' T +4\|\beta\|^2\int_{-\rho}^{0}\mathbb{E}[\|\theta_m\xi(s) \|^2]{\rm d}s. \end{align}

For the last term on the right-hand side of (3.36), by (A4) and the BDG inequality we obtain for $m\geq N_2$,

(3.41)\begin{align} & 2\mathbb{E} \left[ \sup_{t\leq r \leq t+T} \left| \sum_{k\in \mathbb{N}}\int_{t}^{r} \left( \theta_mh_k + \theta_m \sigma_k( u^{\varepsilon}(s-\rho) ), \theta_mu^{\varepsilon}(s) \right) \,{\rm d}W_k(s) \right| \right] \nonumber\\ & \quad \leq \frac{1}{2}\mathbb{E}[\sup_{t \leq r \leq t+T}\|\theta_mu^{\varepsilon}(s)\|^2] + C \sum_{|n|\geq m} \sum_{k \in \mathbb{N}} |h_{k,n}|^2\,T \nonumber\\ & \quad \quad + 2C \sum_{|n|\geq m}\sum_{k \in \mathbb{N}} |\delta_{k,n}|^2\,T + 2C \|\beta\|^2 \varepsilon' T + 2C \|\beta\|^2 \int_{-\rho}^{0} \mathbb{E}[\| \theta_m \xi(s) \|^2] \,{\rm d}s. \end{align}

It follows from (3.36)-(3.41) that

(3.42)\begin{align} & \mathbb{E}\left[\sup_{t\leq r \leq t+T}\|\theta_mu^{\varepsilon}(r) \|^2\right] \leq 4\varepsilon' T +2(2\beta_0^2+1)\varepsilon' T \nonumber\\ & \quad +4\sum_{|n|\geq m}|\alpha_n|^2\,T +2\sum_{|n|\geq m}|g_n|^2T+2\varepsilon'\nonumber\\ & \quad +2(C+2)\sum_{|n|\geq m}\sum_{k \in \mathbb{N}}|h_{k,n}|^2\,T +4(C+2)\sum_{|n|\geq m}\sum_{k \in \mathbb{N}}|\delta_{k,n}|^2T\nonumber\\ & \quad +4(C+2)\|\beta\|^2\varepsilon' T +\left[4(C+2)\|\beta\|^2 +4\beta_{0}^2\right] \int_{-\rho}^{0}\mathbb{E}[\|\theta_m\xi(s) \|^2]\,{\rm d}s. \end{align}

Similar to (3.29)–(3.30), we obtain that there exists $N_3=N_3(\varepsilon ',\,E)\geq N_2$ such that for all $m\geq N_3$,

\[ \sum_{|n|\geq m}\sum_{k\in \mathbb{N}}|h_{n,k}|^2 \vee \sum_{|n|\geq m}\sum_{k\in \mathbb{N}}|\delta_{n,k}|^2 \vee \sum_{|n|\geq m}|g_n|^2 \vee \sum_{|n|\geq m}|\alpha_n|^2 \leq \varepsilon',\]

and

\[ \int_{-\rho}^{0}\mathbb{E}[\|\theta_m\xi(s) \|^2]\,{\rm d}s\leq \varepsilon',\]

which along with (3.42) implies that for all $t \geq 0$ and $m\geq N_3$,

\[ \mathbb{E}\left[\sup_{t\leq r \leq t+T}\|\theta_mu^{\varepsilon}(r) \|^2\right]\leq C_T \varepsilon',\]

where $C_T>0$ depends only on T but not on $\varepsilon ',\, m$ or $E$. This completes this proof.

Proof. By the Chebyshev inequality, we obtain that

\[ \mathbb{P} \left(\left\{ \sup_{s \in [t,t+T]} \sum_{|n|\geq m}|u^{\varepsilon}_n(s)|^2>\delta_2 \right\}\right) \leq \frac{1}{\delta_2}\mathbb{E}\left[\sup_{s \in [t,t+T]} \sum_{|n|\geq m}|u^{\varepsilon}_n(s)|^2\right],\]

which together with lemma 3.7 completes the proof.

Proof. By (4.5) we obtain that for all $t \in [0,\,T]$,

(4.7)\begin{align} \frac{d}{\,{\rm d}t}\|u_{v}(t)\|^2 \leq& -2\lambda\|u_{v}(t)\|^2 +2{\rm Re\ }\left(f(u_{v}(t-\rho)),u_{v}(t)\right)\nonumber\\ & +2{\rm Re\ }\left(g,u_{v}(t)\right)+2{\rm Re\ }\left(\sigma(u_{v}(t-\rho))v(t),u_{v}(t)\right). \end{align}

For the second term on the right-hand side of (4.7), by (2.2) we get

(4.8)\begin{align} 2{\rm Re\ }\left(f(u_{v}(t-\rho)),u_{v}(t)\right) & \leq \|u_{v}(t)\|^2+\|f(u_{v}(t-\rho))\|^2\nonumber\\ & \leq 2\beta^2_{0}\|u_{v}(t-\rho)\|^2+2\|\alpha\|^2+\|u_{v}(t)\|^2. \end{align}

For the last term on the right-hand side of (4.7), by (2.4) we have

(4.9)\begin{align} & 2{\rm Re\ }\left(\sigma(u_v(t-\rho))v(t),u_v(t)\right) \nonumber\\ & \quad \leq \|\sigma(u_v(t-\rho))\|_{L(H;l^2)}^2+\|v(t)\|^2_{H}\|u_v(t)\|^2 \nonumber\\ & \quad \leq \sum_{k \in \mathbb{N}}\|h_k+\sigma_{k}(u_v(t-\rho))\|^2 +\|v(t)\|_{H}^2\|u_v(t)\|^2 \nonumber\\ & \quad \leq 2\|h\|^2 +4\|\beta\|^2\|u_v(t-\rho)\|^2+4\|\delta\|^2+\|v(t)\|_{H}^2\|u_v(t)\|^2. \end{align}

By (4.7)–(4.9) and Young's inequality we obtain that for all $t \in [0,\,T]$,

\begin{align*} \frac{d}{\,{\rm d}t}\|u_{v}(t)\|^2 & \leq (2+\|v(t)\|_{H}^2) \|u_{v}(t)\|^2+(2\beta^2_{0}+4\|\beta\|^2)\|u_{v}(t-\rho)\|^2\\ & \quad +\|g\|^2+2\|\alpha\|^2+4\|\delta\|^2+2\|h\|^2, \end{align*}

which implies

(4.10)\begin{align} \|u_{v}(t)\|^2 & \leq \|u^{0}\|^2 +\int_{0}^{t}(2+\|v(s)\|_{H}^2+2\beta^2_{0}+4\|\beta\|^2)\|u_{v}(s)\|^2\,{\rm d}s\nonumber\\ & \quad +(2\beta^2_{0}+4\|\beta\|^2)\int_{-\rho}^{0}\|\xi(s)\|^2\,{\rm d}s +(\|g\|^2+2\|\alpha\|^2+4\|\delta\|^2+2\|h\|^2)T. \end{align}

By (4.10) we have for all $t \in [0,\,T]$,

(4.11)\begin{align} \sup_{r \in [0,t]}\|u_{v}(r)\|^2 & \leq \|u^{0}\|^2 +\int_{0}^{t}(2+\|v(s)\|_{H}^2+2\beta^2_{0}+4\|\beta\|^2)\sup_{r \in [0,s]}\|u_{v}(r)\|^2\,{\rm d}s\nonumber\\ & \quad +(2\beta^2_{0}+4\|\beta\|^2)\int_{-\rho}^{0}\|\xi(s)\|^2\,{\rm d}s\nonumber\\ & \quad +(\|g\|^2+2\|\alpha\|^2+4\|\delta\|^2+2\|h\|^2)T. \end{align}

By Gronwall's inequality, it follows from (4.11) that for all $t \in [0,\,T]$,

\begin{align*} & \sup_{r \in [0,t]} \|u_{v}(r)\|^2 \nonumber \\ & \quad \leq \left(\vphantom{\left.+ ( 2\beta^2_{0} + 4\|\beta\|^2 ) \int_{-\rho}^{0} \|\xi(s)\|^2 {\rm d}s \right)} \|u^{0}\|^2 + ( \|g\|^2 + 2 \|\alpha\|^2 + 4 \|\delta\|^2 + 2 \|h\|^2 ) T + ( 2\beta^2_{0} + 4\|\beta\|^2 ) \int_{-\rho}^{0} \|\xi(s)\|^2 \,{\rm d}s \right) \nonumber\\ & \quad \quad \cdot e^{\int^{T}_{0} ( 2 + \|v(s) \|_{H}^2 + 2 \beta_{0} + 4\|\beta\|^2 ) \,{\rm d}s} \nonumber\\ & \quad \leq \left(\vphantom{\left.\quad + ( 2\beta^2_{0} + 4\|\beta\|^2 ) \int_{-\rho}^{0} \|\xi(s)\|^2 \,{\rm d}s \right)} \|u^{0}\|^2 + ( \|g\|^2 + 2\|\alpha\|^2 + 4\|\delta\|^2 + 2\|h\|^2 )T +( 2\beta^2_{0} + 4\|\beta\|^2) \int_{-\rho}^{0} \|\xi(s)\|^2 \,{\rm d}s \right) \nonumber\\ & \quad \quad \cdot e^{ ( 2 + 2\beta_{0} + 4 \|\beta\|^2 ) T + \|v\|_{L^2([0,T];H)}^2 }, \nonumber \end{align*}

which completes the proof.

Proof. Note that system (4.5) on $[0,\,\rho ]$ is equivalent to the following system without delay:

(4.13)\begin{equation} \left\{ \begin{array}{@{}l} {\rm d}u_{v}(t)=-iAu_{v}(t)\,{\rm d}t-i|u_{v}(t)|^2u_{v}(t)\,{\rm d}t -\lambda u_{v}(t)\,{\rm d}t+f(\xi(t-\rho))\,{\rm d}t\\ \qquad \qquad+ g\,{\rm d}t+\sigma(\xi(t-\rho))v(t)\,{\rm d}t, \ t\in [0,\rho],\\ u_{v}(0)=u^{0}. \end{array} \right. \end{equation}

Let $F(t,\,u)=-iAu-i|u|^2u-\lambda u+f(\xi (t-\rho ))+g+\sigma (\xi (t-\rho ))v(t)$. By (2.2) and (2.4) we find that for every $R>0$, there exists $C_{R}>0$ depending only on $R$ such that for all $t \in [0,\,T]$ and $u \in l^2$ with $\|u\|\leq R$,

(4.14)\begin{equation} \|F(t,u)\| \leq C_{R} \left[ (1 + \|v(t)\|_{H}) \|\xi(t - \rho)\| + \|u\| + \|v(t)\|_{H} +1 \right], \end{equation}

and for all $t \in [0,\,T]$ and $u_1,\,u_2 \in l^2$ with $\|u_1\|\vee \|u_2\| \leq R$,

(4.15)\begin{equation} \|F(t,u_1)-F(t.u_2)\|\leq C_{R}\|u_1-u_2\|. \end{equation}

Hence, by (4.14)–(4.15) and lemma 4.6, system (4.13) has a unique solution $u_v$ defined on $[0,\,\rho ]$. Repeating this argument, one can extend the solution $u_v$ to the whole interval $[0,\,T]$.

Next, we are going to prove (4.12). By lemma 4.6, for $v_1,\,v_2\in L^2([0,\,T];H)$ with $\|v_1\|_{L^2([0,T];H)} \vee \|v_2\|_{L^2([0,T];H)} \leq R_1$ for some $R_1>0$ and $\|u^{0}\|^2 \vee \int _{-\rho }^{0}\|\xi (s)\|^2\,{\rm d}s \leq R_2$ for some $R_2>0$, there exists $K=K(R_1,\,R_2,\,T)>0$ such that $\sup _{t\in [0,T]}(\|u_{v_1}(t)\| + \|u_{v_{2}}(t)\|) \leq K$. By (4.5) we get for all $t\in [0,\,T]$,

(4.16)\begin{align} & \|u_{v_1}(t)-u_{v_2}(t)\|^2 \leq 2 \int_{0}^{t} \| u_{v_1}(s) - u_{v_2}(s) \| \| i|u_{v_1}(s)|u_{v_1}(s) - i|u_{v_2}(s)|u_{v_2}(s) \| \,{\rm d}s \nonumber\\ & \quad + 2 \int_{0}^{t} \|u_{v_1}(s) - u_{v_2}(s) \| \| f(u_{v_1}(s-\rho)) - f(u_{v_2}(s-\rho)) \|\,{\rm d}s \nonumber\\ & \quad + 2 \int_{0}^{t} \| \sigma(u_{v_1}(s-\rho)) v_1(s) - \sigma(u_{v_2}(s-\rho)) v_2(s) \| \| u_{v_1}(s) - u_{v_2}(s) \| \,{\rm d}s. \end{align}

By (2.3) and Young's inequality we have

(4.17)\begin{align} & 2\int_{0}^{t}\|u_{v_1}(s)-u_{v_2}(s)\| \|f(u_{v_1}(s-\rho))-f(u_{v_2}(s-\rho))\|\,{\rm d}s \nonumber\\ & \quad \quad +2\int_{0}^{t}\|u_{v_1}(s)-u_{v_2}(s)\| \|i|u_{v_1}(s)|u_{v_1}(s)-i|u_{v_2}(s)|u_{v_2}(s)\|\,{\rm d}s \nonumber\\ & \quad \leq (K_1+1)\int_{0}^{t} \|u_{v_1}(s)-u_{v_2}(s)\|^2\,{\rm d}s, \end{align}

where $K_1>0$ depends on $R_1$, $R_2$ and $T$. For the last term on the right-hand side of (4.16), by (2.5) we get that there exists $K_2=K_2(R_1,\,R_2,\,T)>0$ such that

(4.18)\begin{align} & 2\int_{0}^{t} \|\sigma(u_{v_1}(s-\rho))v_1(s)-\sigma(u_{v_2}(s-\rho))v_2(s)\| \|u_{v_1}(s)-u_{v_2}(s)\|\,{\rm d}s \nonumber\\ & \quad \leq 2\int_{0}^{t} \left\|\sigma\left(u_{v_1}(s-\rho)\right)v_1(s)-\sigma(u_{v_2}(s-\rho))v_1(s)\right\| \|u_{v_1}(s)-u_{v_2}(s)\|\,{\rm d}s\nonumber\\ & \quad \quad +2\int_{0}^{t} \|\sigma(u_{v_2}(s-\rho))(v_1(s)-v_2(s))\|\|u_{v_1}(s)-u_{v_2}(s)\|\,{\rm d}s \nonumber\\ & \quad \leq \int_{0}^{t}\sum_{k\in \mathbb{N}}\left\|\sigma_k(u_{v_1}(s-\rho))-\sigma_k(u_{v_2}(s-\rho))\right\|^2\,{\rm d}s \nonumber\\ & \quad \quad +\int_{0}^{t}\|v_1(s)\|_{H}^2\|u_{v_1}(s)-u_{v_2}(s)\|^2{\rm d}s \nonumber\\ & \quad \quad +2K\int_{0}^{t} \|\sigma(u_{v_2}(s-\rho))(v_1(s)-v_2(s))\|{\rm d}s \nonumber\\ & \quad \leq K_2\int_{0}^{t}\left\|u_{v_1}(s)-u_{v_2}(s)\right\|^2\,{\rm d}s+\int_{0}^{t}\|v_1(s)\|_{H}^2\|u_{v_1}(s)-u_{v_2}(s)\|^2{\rm d}s \nonumber\\ & \quad \quad +2K\int_{0}^ {t} \|\sigma(u_{v_2}(s-\rho))(v_1(s)-v_2(s))\|\,{\rm d}s. \end{align}

Since $\int _{-\rho }^{0}\|\xi (s)\|^2\,{\rm d}s \leq R_2$ and $\sup _{t\in [0,T]}\|u_{v_{2}}(t)\| \leq K$, we obtain that there exists $K_3=K_3(R_1,\,R_2,\,T)>0$ such that for all $t \in [0,\,T]$,

(4.19)\begin{align} & \int_{0}^{t} \|\sigma(u_{v_2}(s-\rho))(v_1(s)-v_2(s))\|\,{\rm d}s \nonumber\\ & \quad \leq\left(\int_{0}^{t} \|\sigma(u_{v_2}(s-\rho))\|_{L_2(H;l^2)}^2\,{\rm d}s\right)^{{1}/{2}} \left(\int_{0}^{t} \|v_1(s)-v_2(s)\|_{H}^2\,{\rm d}s\right)^{{1}/{2}}\nonumber\\ & \quad \leq K_3\|v_1-v_2\|_{L^2([0,T];H)}. \end{align}

It follows from (4.16)–(4.19) that

\begin{align*} \|u_{v_1}(t)-u_{v_2}(t)\|^2& \leq\int_{0}^{t}(K_1+K_2+1+\|v_1(s)\|^2_{H})\|u_{v_1}(s)-u_{v_2}(s)\|^2\,{\rm d}s\\ & \quad +2KK_3\|v_1-v_2\|_{L^2([0,T];H)}. \end{align*}

Then by Gronwall's inequality we know that for all $t \in [0,\,T]$,

\[ \|u_{v_1}(t)-u_{v_2}(t)\|^2\leq 2KK_3\|v_1-v_2\|_{L^2([0,T];H)}e^{(K_1+K_2+1)T+\|v_1\|^2_{L^2([0,T];H)}},\]

which completes the proof.

Proof. Note that the operator $\Gamma : L^2([0,\,T];H)\rightarrow C([0,\,T];l^2)$ is well-defined. In fact, by (2.1) and (2.4) we get for every $v \in L^2([0,\,T];H)$,

(4.21)\begin{align} & \int_{0}^{T}\|\sigma(\varphi(s-\rho))v(s)\|\,{\rm d}s \nonumber\\ & \quad \leq \int_{0}^{T}\|\sigma(\varphi(s-\rho))\|_{L(H;l^2)}\|v(s)\|_{H}\,{\rm d}s \nonumber\\ & \quad \leq \left(\int_{0}^{T}\|\sigma(\varphi(s-\rho))\|^2_{L_2(H;l^2)}\,{\rm d}s\right)^{{1}/{2}}\|v\|_{L^2([0,T];H)} \nonumber\\ & \quad \leq \left( 2\|h\|^2T +4\|\beta\|^2\|\varphi\|^2_{L^{\infty}([0,T];l^2)}T +4\|\delta\|^2T +4\|\beta\|^2\int_{-\rho}^{0}\|\varphi(s)\|^2{\rm d}s \right)^{{1}/{2}} \nonumber\\ & \qquad \cdot \|v\|_{L^2([0,T];H)} <\infty, \end{align}

which implies that $\Gamma (v)\in C([0,\,T];l^2)$ for all $v \in L^2([0,\,T];H)$. Moreover, by (4.21) we get that $\Gamma : L^2([0,\,T];H)\rightarrow C([0,\,T];l^2)$ is bounded. On the other hand, from (4.2) and (4.20) it is easy to see that $\Gamma : L^2([0,\,T];H)\rightarrow C([0,\,T];l^2)$ is linear. Since the operator $\Gamma : L^2([0,\,T];H)\rightarrow C([0,\,T];l^2)$ is strongly continuous and linear, one can deduce that $\Gamma$ is weakly continuous. Following the argument of [Reference Wang33, Lemma 4.3] with small modification and the Ascoli–Arzelà theorem, one can further show that $\Gamma$ is continuous from the weak topology of $L^{2}([0,\,T];H)$ to the strong topology of $C([0,\,T];l^2)$.

Proof. Since $v_n\rightarrow v$ weakly in $L^{2}([0,\,T];H)$, there exists a constant $N_1>0$ such that $\|v\|_{L^{2}([0,\,T];H)}$ $\leq N_1$ and $\|v_{n}\|_{L^{2}([0,\,T];H)}\leq N_1$ for all $n \in \mathbb {N}$. Then by lemma 4.6 there exists a constant $N_2= N_2(N_1,\, T,\,u^0,\,\xi )>0$ such that

\[ \sup_{t\in [0,T]}\left(\|u_{v_n}(t)\|\vee \|u_{v}(t)\|\right) \leq N_2,\quad \forall\ n\in \mathbb{N}.\]

By (4.5) we have

(4.22)\begin{align} \frac{d}{\,{\rm d}t}(u_{v_n}(t)-u_{v}(t))& ={-}iA(u_{v_n}(t)-u_{v}(t)) -i(|u_{v_n}(t)|^2u_{v_n}(t)-|u_{v}(t)|^2u_{v}(t))\nonumber\\ & \quad -\lambda (u_{v_n}(t)-u_{v}(t)) +f(u_{v_n}(t-\rho))-f(u_{v}(t-\rho))\nonumber\\ & \quad +\sigma(u_{v_n}(t-\rho))v_n(t)-\sigma(u_{v}(t-\rho))v(t). \end{align}

We set

\[ \Theta_n(t)=\int_{0}^{t}\sigma(u_{v}(s-\rho))(v_n(s)-v(s))\,{\rm d}s.\]

Since $v_n\rightarrow v$ weakly in $L^{2}([0,\,T];H)$, by lemma 4.8 we obtain

(4.23)\begin{align} \Theta_n(t)\rightarrow 0\ {\rm in}\ C([0,T];l^2), \ {\rm as} \ n\rightarrow \infty. \end{align}

Then by (4.22) and (4.23) one can show $u_{v_n} \to u_{v}$ strongly in $C([0,\,T]; l^2)$. The details are similar to [Reference Wang33] and hence omitted here.

Proof. Let $\{u_{v_{n}}\!\}$ be any sequence in $\Xi _{a}$ and $\{v_{n}\} \subset S_{a} \subset L^2([0,\,T];H)$. Since $S_{a}$ is a polish space under the weak topology of $L^2([0,\,T]; H)$, there exist $v \in S_{a}$ and a subsequence $\{v_{n_{k}}\!\}$ such that $v_{n_{k}}\rightarrow v$ weakly. Then, by lemma 4.9 we know that $u_{v_{n_{k}}}\rightarrow u_{v}$ in $C([0,\,T];l^2)$, which implies that $\Xi _{a}$ is compact in $C([0,\,T];l^2)$.

Proof. Applying Itô's formula to (4.4), we obtain that for all $t \in [0,\,T]$,

(4.26)\begin{align} & \|u_{v}^{\varepsilon}(t)\|^2 \leq \|u^{0}\|^2 +2{\rm Re\ }\int_{0}^{t}\left(u_{v}^{\varepsilon}(s),f(u_{v}^{\varepsilon}(s-\rho))\right)\,{\rm d}s +2{\rm Re\ }\int_{0}^{t}\left(u_{v}^{\varepsilon}(s),g\right)\,{\rm d}s\nonumber\\ & \quad+2{\rm Re\ }\int_{0}^{t}\left(u_{v}^{\varepsilon}(s),\sigma(u_{v}^{\varepsilon}(s-\rho))v^{\varepsilon}(s)\right)\,{\rm d}s +\varepsilon \int_{0}^{t} \|\sigma(u_{v}^{\varepsilon}(s-\rho)) \|^2_{L_2(H;l^2)}\,{\rm d}s\nonumber\\ & \quad+2 \sqrt{\varepsilon}{\rm Re}\int_{0}^{t}\left(u_{v}^{\varepsilon}(s),\sigma(u_{v}^{\varepsilon}(s-\rho))\,{\rm d}W(s)\right). \end{align}

By (2.4) and the Hölder inequality we obtain

(4.27)\begin{align} & 2{\rm Re}\int_{0}^{t}\left(u_{v}^{\varepsilon}(s),\sigma(u_{v}^{\varepsilon}(s-\rho))v(s)\right)\,{\rm d}s \nonumber\\ & \quad \leq 2\sup_{0 \leq s \leq t}\|u_{v}^{\varepsilon}(s)\| \int_{0}^{t}\|\sigma(u_{v}^{\varepsilon}(s-\rho))\|_{L_2(H;l^2)}\|v(s)\|_{H}\,{\rm d}s \nonumber\\ & \quad \leq 2\sup_{0 \leq s \leq t}\|u_{v}^{\varepsilon}(s)\| \left(\int_{0}^{t}\|\sigma(u_{v}^{\varepsilon}(s-\rho))\|_{L_2(H;l^2)}^2\,{\rm d}s\right)^{{1}/{2}} \left(\int_{0}^{t}\|v(s)\|^2_{H}\,{\rm d}s\right)^{{1}/{2}} \nonumber\\ & \quad \leq \frac{1}{4}\sup_{0 \leq s \leq t}\|u_{v}^{\varepsilon}(s)\|^2 +4a\int_{0}^{t}\|\sigma(u_{v}^{\varepsilon}(s-\rho))\|_{L_2(H;l^2)}^2\,{\rm d}s \nonumber\\ & \quad \leq \frac{1}{4}\sup_{0 \leq s \leq t}\|u_{v}^{\varepsilon}(s)\|^2 +8aT\|h\|^2+16a\|\delta\|^2\,T +16a\|\beta\|^2\int_{0}^{t}\|u_{v}^{\varepsilon}(s) \|^2\,{\rm d}s\nonumber\\ & \quad \quad +16a\|\beta\|^2\int_{-\rho}^{0}\|\xi(s) \|^2\,{\rm d}s. \end{align}

It follows from (2.4), (4.8), (4.26)–(4.27) and Young's inequality that for all $t \in [0,\,T] \ {\rm and} \ \varepsilon \in (0,\,1),$

(4.28)\begin{align} & \frac{3}{4}\mathbb{E}\left[\sup_{0\leq s \leq t}\|u_{v}^{\varepsilon}(s)\|^2\right] \nonumber\\ & \quad \leq \|u^{0}\|^2 +(2\beta_0^2+4\|\beta\|^2+2 +16a\|\beta\|^2)\int_{0}^{t} \mathbb{E}\left[\sup_{0\leq r \leq s}\|u_{v}^{\varepsilon}(r)\|^2\right] \,{\rm d}s \nonumber\\ & \quad \quad +(2\beta_0^2\!+\!4\|\beta\|^2\!+\!16a\|\beta\|^2)\int_{-\rho}^{0}\|\xi(s)\|^2\,{\rm d}s \!+\!2\|\alpha\|^2T+4\|\delta\|^2T \nonumber\\ & \quad \quad +\|g\|^2T+2\|h\|^2T +8aT\|h\|^2+16a\|\delta\|^2\,T \nonumber\\ & \quad \quad +2\sqrt{\varepsilon}\mathbb{E}\left[\sup_{0\leq r\leq t} \left|{\rm Re} \int_{0}^{r}\left(u_{v}^{\varepsilon}(s),\sigma(u_{v}^{\varepsilon}(s-\rho))\,{\rm d}W(s)\right) \right|\right]. \end{align}

For the last term on the right-hand side of (4.28), by the Burkholder inequality we have for all $t \in [0,\,T] \ {\rm and} \ \varepsilon \in (0,\,1)$,

(4.29)\begin{align} & 2\sqrt{\varepsilon}\mathbb{E}\left[\sup_{0\leq r\leq t} \left|{\rm Re} \int_{0}^{r}\left(u_{v}^{\varepsilon}(s),\sigma(u_{v}^{\varepsilon}(s-\rho))\,{\rm d}W(s)\right) \right|\right] \nonumber\\ & \quad \leq 6\mathbb{E}\left[\left( \int_{0}^{t}\|u_{v}^{\varepsilon}(s)\|^2\| \| \sigma(u_{v}^{\varepsilon}(s-\rho)\|_{L_2(H;l^2)}^2\,{\rm d}s \right)^{{1}/{2}}\right] \nonumber\\ & \quad \leq \frac{1}{4}\mathbb{E}\left[\sup_{0\leq s \leq t}\|u_{v}^{\varepsilon}(s)\|^2\right] +36\int_{0}^{t} \|\sigma(u_{v}^{\varepsilon}(s-\rho)) \|^2_{L_2(H;l^2)}\,{\rm d}s \nonumber\\ & \quad \leq \frac{1}{4}\mathbb{E}\left[\sup_{0\leq s \leq t}\|u_{v}^{\varepsilon}(s)\|^2\right] +36\Bigg(2\|h\|^2T+4\|\delta\|^2T \nonumber\\ & \quad \quad +4\|\beta\|^2\int_{-\rho}^{0}\|\xi(s)\|^2\,{\rm d}s +4\|\beta\|^2\int_{0}^{t}\mathbb{E}\left[\sup_{0\leq r \leq s}\|u_{v}^{\varepsilon}(r)\|^2\right]\,{\rm d}s \Bigg). \end{align}

Then (4.25) follows from (4.28)–(4.29) and Gronwall's inequality.

Proof. Notice that $u_v=G(\int _{0}^{.}v(t)\,{\rm d}t)$ is the solution of (4.5) with the control $v$. Let $u^{\varepsilon }_{v^{\varepsilon }}= G^{\varepsilon }(W + {1}/{\sqrt {\varepsilon }}\int _{0}^{.} v^{\varepsilon }(t)\,{\rm d}t)$. Then $u^{\varepsilon }_{v^{\varepsilon }}$ is the solution of the following system:

(4.30)\begin{equation} \left\{ \begin{array}{@{}l} {\rm d}u^{\varepsilon}_{v^{\varepsilon}}(t) = - iAu^{\varepsilon}_{v^{\varepsilon}}(t)\,{\rm d}t-i|u^{\varepsilon}_{v^{\varepsilon}}(t)|^2u^{\varepsilon}_{v^{\varepsilon}}(t)\,{\rm d}t -\lambda u^{\varepsilon}_{v^{\varepsilon}}(t)\,{\rm d}t+f(u^{\varepsilon}_{v^{\varepsilon}}(t-\rho))\,{\rm d}t \\ \qquad \quad \ \ \ \ + g{\rm d}t+\sigma(u^{\varepsilon}_{v^{\varepsilon}}(t-\rho))v^{\varepsilon}(t){\rm d}t +\sqrt{\varepsilon}\sigma(u^{\varepsilon}_{v^{\varepsilon}}(t-\rho)){\rm d}W(t), \ t\in [0,T],\\ u^{\varepsilon}_{v^{\varepsilon}}(0) = u^{0}, \ u^{\varepsilon}_{v^{\varepsilon}}(s)=\xi(s), \ s \in(-\rho,0). \end{array} \right. \end{equation}

In order to show that $u^{\varepsilon }_{v^{\varepsilon }}$ converges to $u_v$ in $C([0,\,T];l^2)$ in distribution, we first establish the convergence of $u^{\varepsilon }_{v^{\varepsilon }}-u_{v^{\varepsilon }}$, where $u_{v^{\varepsilon }}=G(\int _{0}^{.}v^{\varepsilon }(t)\,{\rm d}t)$ is the solution of the following system:

(4.31)\begin{equation} \left\{ \begin{array}{@{}l} {\rm d}u_{v^{\varepsilon}}(t) = - iAu_{v^{\varepsilon}}(t)\,{\rm d}t - i |u_{v^{\varepsilon}}(t)|^2u_{v^{\varepsilon}}(t)\,{\rm d}t - \lambda u_{v^{\varepsilon}}(t)\,{\rm d}t + f(u_{v^{\varepsilon}}(t-\rho))\,{\rm d}t \\ \qquad \quad \ \ \ \ + g\,{\rm d}t+\sigma(u_{v^{\varepsilon}}(t-\rho))v^{\varepsilon}(t)\,{\rm d}t, \ t\in [0,T],\\ u_{v^{\varepsilon}}(0) = u^{0},\ u_{v^{\varepsilon}}(s)=\xi(s),\ s \in(-\rho,0). \end{array} \right. \end{equation}

Thus by (4.30)–(4.31) we have

(4.32)\begin{align} d(u^{\varepsilon}_{v^{\varepsilon}}(t)-u_{v^{\varepsilon}}(t)) = & {-}iA\left( u^{\varepsilon}_{v^{\varepsilon}}(t)-u_{v^{\varepsilon}}(t)\right)\,{\rm d}t -\lambda\left(u^{\varepsilon}_{v^{\varepsilon}}(t)-u_{v^{\varepsilon}}(t)\right){\rm d}t\nonumber\\ & -i \left( |u^{\varepsilon}_{v^{\varepsilon}}(t)|^2u^{\varepsilon}_{v^{\varepsilon}}(t) - |u_{v^{\varepsilon}}(t)|^2u_{v^{\varepsilon}}(t) \right) {\rm d}t \nonumber\\ & +\left(f(u^{\varepsilon}_{v^{\varepsilon}}(t-\rho))-f(u_{v^{\varepsilon}}(t-\rho))\right){\rm d}t \nonumber\\ & +\left( \sigma(u^{\varepsilon}_{v^{\varepsilon}}(t-\rho))v^{\varepsilon}(t) -\sigma(u_{v^{\varepsilon}}(t-\rho))v^{\varepsilon}(t) \right){\rm d}t \nonumber\\ & +\sqrt{\varepsilon}\sigma(u^{\varepsilon}_{v^{\varepsilon}}(t-\rho))\,{\rm d}W(t). \end{align}

For a given constant $M>0$, we define a stopping time $\tau ^{\varepsilon }$ by

\[ \tau^{\varepsilon}=\inf\{t\geq 0: \|u^{\varepsilon}_{v^{\varepsilon}}(t)\|\geq M \}\wedge T,\]

and the infimum of the empty set is taken to be $\infty$. Applying Itô's formula to (4.32) yields that for all $t \in [0,\,T]$,

(4.33)\begin{align} & \sup_{0\leq r \leq t}\|u^{\varepsilon}_{v^{\varepsilon}}(r\wedge \tau^{\varepsilon})-u_{v^{\varepsilon}}(r\wedge \tau^{\varepsilon})\|^2\nonumber\\ & \quad \leq 2\int_{0}^{t\wedge \tau^{\varepsilon}}\left\||u^{\varepsilon}_{v^{\varepsilon}}(s)|^2u^{\varepsilon}_{v^{\varepsilon}}(s) -|u_{v^{\varepsilon}}(s)|^2u_{v^{\varepsilon}}(s) \right\| \left\|u^{\varepsilon}_{v^{\varepsilon}}(s)-u_{v^{\varepsilon}}(s)\right\|\,{\rm d}s\nonumber\\ & \quad \quad+2\int_{0}^{t\wedge \tau^{\varepsilon}} \|f(u^{\varepsilon}_{v^\varepsilon}(s-\rho)) -f(u_{v^{\varepsilon}}(s-\rho))\| \|u^{\varepsilon}_{v^{\varepsilon}}(s)-u_{v^{\varepsilon}}(s)\|\,{\rm d}s\nonumber\\ & \quad \quad+2\int_{0}^{t\wedge \tau^{\varepsilon}}\| \sigma(u^{\varepsilon}_{v^{\varepsilon}}(s-\rho))v^{\varepsilon}(s) -\sigma(u_{v^{\varepsilon}}(s-\rho))v^{\varepsilon}(s)\| \|u^{\varepsilon}_{v^{\varepsilon}}(s)-u_{v^{\varepsilon}}(s)\|\,{\rm d}s\nonumber\\ & \quad \quad+2\sqrt{\varepsilon}\sup_{0\leq r\leq t}\left|\int_{0}^{r\wedge \tau^{\varepsilon}} \left(u^{\varepsilon}_{v^{\varepsilon}}(s)-u_{v^{\varepsilon}}(s), \sigma(u^{\varepsilon}_{v^{\varepsilon}}(s-\rho))\,{\rm d}W(s)\right) \right|\nonumber\\ & \quad \quad +\varepsilon\int_{0}^{t\wedge \tau^{\varepsilon}}\|\sigma(u^{\varepsilon}_{v^{\varepsilon}}(s-\rho))\|_{L_2(H;l^2)}^2\,{\rm d}s. \end{align}

For fixed $(u^{0},\,\xi )\in l^2\times L^2((-\rho,\,0);l^2)$ and $\{v^{\varepsilon }\}\subset \mathcal {A}_{a}$, by lemma 4.6 there exists a positive constant $C_3=C_3(a,\,u^{0},\,\xi,\,T)$ such that for all $\varepsilon \in (0,\,1)$, $\mathbb {P}$-almost surely,

(4.34)\begin{equation} \sup_{t \in [0,T]}\|u_{v^{\varepsilon}}(t)\|\leq C_3. \end{equation}

For the first term on the right-hand side of (4.33), by (4.34) we get

(4.35)\begin{align} & 2\int_{0}^{t\wedge \tau^{\varepsilon}}\left\||u^{\varepsilon}_{v^{\varepsilon}}(s)|^2u^{\varepsilon}_{v^{\varepsilon}}(s) - |u_{v^{\varepsilon}}(s)|^2u_{v^{\varepsilon}}(s) \right\| \left\|u^{\varepsilon}_{v^{\varepsilon}}(s)-u_{v^{\varepsilon}}(s)\right\|\,{\rm d}s\nonumber\\ & \quad \leq C_4 \int_{0}^{t\wedge \tau^{\varepsilon}}\|u^{\varepsilon}_{v^{\varepsilon}}(s)-u_{v^{\varepsilon}}(s)\|^2\,{\rm d}s\nonumber\\ & \quad \leq C_4 \int_{0}^{t}\sup_{0\leq r\leq s}\|u^{\varepsilon}_{v^{\varepsilon}}(r\wedge \tau^{\varepsilon}) -u_{v^{\varepsilon}}(r\wedge \tau^{\varepsilon})\|^2\,{\rm d}s, \end{align}

where $C_4>0$ depends on $a,\,u^{0},\,\xi,\,T$ and $M$. For the second term on the right-hand side of (4.33), by (2.3) we get

(4.36)\begin{align} & 2\int_{0}^{t\wedge \tau^{\varepsilon}}\|f(u^{\varepsilon}_{v^{\varepsilon}}(s-\rho)) -f(u_{v^{\varepsilon}}(s-\rho))\| \|u^{\varepsilon}_{v^{\varepsilon}}(s)-u_{v^{\varepsilon}}(s)\|\,{\rm d}t\nonumber\\ & \quad \leq \int_{0}^{t\wedge \tau^{\varepsilon}}\|f(u^{\varepsilon}_{v^{\varepsilon}}(s-\rho)) -f(u_{v^{\varepsilon}}(s-\rho))\|^2\,{\rm d}s +\int_{0}^{t\wedge \tau^{\varepsilon}}\|u^{\varepsilon}_{v^{\varepsilon}}(s)-u_{v^{\varepsilon}}(s)\|^2\,{\rm d}s\nonumber\\ & \quad \leq (C_5+1) \int_{0}^{t}\sup_{0 \leq r\leq s}\|u^{\varepsilon}_{v^{\varepsilon}}(r\wedge \tau^{\varepsilon}) -u_{v^{\varepsilon}}(r\wedge \tau^{\varepsilon})\|^2\,{\rm d}s, \end{align}

where $C_5>0$ depends on $a,\,u^{0},\,\xi,\,T$ and $M$. For the third term on the right-hand side of (4.33), by (2.5) we obtain

(4.37)\begin{align} & 2 \int_{0}^{t\wedge \tau^{\varepsilon}} \| \sigma(u^{\varepsilon}_{v^{\varepsilon}}(s-\rho)) v^{\varepsilon}(s) - \sigma(u_{v^{\varepsilon}}(s-\rho)) v^{\varepsilon}(s) \| \| u^{\varepsilon}_{v^{\varepsilon}}(s) - u_{v^{\varepsilon}}(s) \| \,{\rm d}s \nonumber\\ & \quad \leq \int_{0}^{t\wedge \tau^{\varepsilon}} \| v^{\varepsilon}(s) \|^2_{H} \| u^{\varepsilon}_{v^{\varepsilon}}(s) - u_{v^{\varepsilon}}(s) \|^2 \,{\rm d}s\nonumber\\ & \quad \quad+ \int_{0}^{t\wedge \tau^{\varepsilon}} \sum_{k\in\mathbb{N}} \| \sigma_{k}(u^{\varepsilon}_{v^{\varepsilon}}(s-\rho)) - \sigma_{k}(u_{v^{\varepsilon}}(s-\rho)) \|^2 \,{\rm d}s \nonumber\\ & \quad \leq \int_{0}^{t}( \|v^{\varepsilon}(s)\|^2_{H} + C_6) \sup_{0 \leq r \leq s} \| u^{\varepsilon}_{v^{\varepsilon}}(r\wedge \tau^{\varepsilon}) - u_{v^{\varepsilon}}(r\wedge \tau^{\varepsilon}) \|^2 \,{\rm d}s, \end{align}

where $C_6$ depends on $a,\,u^{0},\,\xi,\,T$ and $M$. For the last term on the right-hand side of (4.33), by (2.4) we get

(4.38)\begin{align} & \varepsilon \int_{0}^{t\wedge \tau^{\varepsilon}} \| \sigma ( u^{\varepsilon}_{v^{\varepsilon}}(s-\rho) ) \|_{L_2(H;l^2)}^2 \,{\rm d}s \nonumber\\ & \quad \leq 2 \varepsilon \int_{0}^{t\wedge \tau^{\varepsilon}} ( \|h\|^2 + 2 \|\beta\|^2 \| u^{\varepsilon}_{v^{\varepsilon}}(s-\rho) \|^2 + 2 \|\delta\|^2 ) \,{\rm d}s \nonumber\\ & \quad \leq 2 \varepsilon \|h\|^2\,T + 4\varepsilon \|\delta\|^2\,T + 4 \varepsilon \|\beta\|^2 M^2\,T + 4\varepsilon \|\beta\|^2 \int_{-\rho}^{0} \|\xi(s)\|^2 \,{\rm d}s. \end{align}

It follows from (4.33)–(4.38) that for all $t \in [0,\,T]$,

(4.39)\begin{align} & \sup_{0 \leq r \leq t} \| u^{\varepsilon}_{v^{\varepsilon}}(r\wedge \tau^{\varepsilon}) - u_{v^{\varepsilon}}(r\wedge \tau^{\varepsilon}) \|^2 \nonumber\\ & \quad \leq \int_{0}^{t} ( C_4 + C_5 + C_6 + 1 + \| v^{\varepsilon}(s) \|^2_{H} ) \sup_{0 \leq r \leq s} \| u^{\varepsilon}_{v^{\varepsilon}}(r \wedge \tau^{\varepsilon}) - u_{v^{\varepsilon}}(r \wedge \tau^{\varepsilon}) \|^2 \,{\rm d}s \nonumber\\ & \quad \quad + 2 \sqrt{\varepsilon} \sup_{0\leq r\leq T} \left| \int_{0}^{r\wedge \tau^{\varepsilon}} \left( u^{\varepsilon}_{v^{\varepsilon}}(s) - u_{v^{\varepsilon}}(s), \sigma( u^{\varepsilon}_{v^{\varepsilon}}(s-\rho) ) \,{\rm d}W(s) \right) \right| \nonumber\\ & \quad \quad+ 2 \varepsilon \|h\|^2\,T + 4 \varepsilon\|\delta\|^2\,T + 4 \varepsilon \|\beta\|^2 M^2\,T + 4 \varepsilon \|\beta\|^2 \int_{-\rho}^{0} \|\xi(s)\|^2 \,{\rm d}s. \end{align}

By (4.39) and Gronwall's inequality, we obtain that for all $t \in [0,\,T]$,

(4.40)\begin{align} & \sup_{0 \leq r \leq t} \| u^{\varepsilon}_{v^{\varepsilon}} (r\wedge \tau^{\varepsilon}) - u_{v^{\varepsilon}} (r\wedge \tau^{\varepsilon}) \|^2 \nonumber\\ & \quad \leq 2 C_7 \sqrt{\varepsilon} \sup_{0 \leq r \leq T} \left| \int_{0}^{r\wedge \tau^{\varepsilon}} \left( u^{\varepsilon}_{v^{\varepsilon}}(s) - u_{v^{\varepsilon}}(s) , \sigma( u^{\varepsilon}_{v^{\varepsilon}}(s-\rho) ) \,{\rm d}W(s) \right) \right| \nonumber\\ & \quad \quad + 2 C_7 \varepsilon \|h\|^2T + 4 C_7 \varepsilon \|\delta\|^2T + 4 C_{7}\varepsilon \|\beta\|^2 M^2T + 4 C_7 \varepsilon \|\beta\|^2 \int_{-\rho}^{0} \|\xi(s)\|^2 {\rm d}s, \end{align}

where $C_7=e^{(C_4+C_5+C_6+1)T+a}$. By Doob's maximal inequality and (4.34), we now estimate the first term on the right-hand side of (4.40),

(4.41)\begin{align} & 4\varepsilon \mathbb{E} \left[ \sup_{0\leq r\leq T} \left| \int_{0}^{r\wedge \tau^{\varepsilon}} \left( u^{\varepsilon}_{v^{\varepsilon}}(s) - u_{v^{\varepsilon}}(s) , \sigma(u^{\varepsilon}_{v^{\varepsilon}}(s-\rho)) \,{\rm d}W(s) \right) \right|^2 \right] \nonumber\\ & \quad \leq 16 \varepsilon \mathbb{E} \left[ \int_{0}^{T \wedge \tau^{\varepsilon}} \| u^{\varepsilon}_{v^{\varepsilon}}(s) - u_{v^{\varepsilon}}(s) \|^2 \| \sigma(u^{\varepsilon}_{v^{\varepsilon}}(s-\rho)) \|^2 _{L_2(H;l^2)} \,{\rm d}s \right] \nonumber\\ & \quad\leq 16 \varepsilon(M + C_3)^2 \mathbb{E} \left[ \int_{0}^{T\wedge \tau^{\varepsilon}} \|\sigma( u^{\varepsilon}_{v^{\varepsilon}}(s-\rho) ) \|^2_{L_2(H;l^2)} \,{\rm d}s \right] \nonumber\\ & \quad \leq 32 \varepsilon(M + C_3)^2 \left( \|h\|^2\,T + 2\|\delta\|^2\,T + 2 \|\beta\|^2 M^2\,T + 2 \|\beta\|^2 \int_{-\rho}^{0} \|\xi(s)\|^2 \,{\rm d}s \right). \end{align}

From (4.40)–(4.41), it follows that

(4.42)\begin{align} \lim_{\varepsilon \rightarrow 0} \sup_{0\leq r \leq T} \| u^{\varepsilon}_{v^{\varepsilon}} (r\wedge \tau^{\varepsilon}) - u_{v^{\varepsilon}} (r\wedge \tau^{\varepsilon}) \|^2 = 0 \ \ \ {\rm in \ probability}. \end{align}

Recalling the definition of $\tau ^{\varepsilon }$, by the Chebyshev inequality and lemma 4.11 we obtain

\[ \mathbb{P}( \tau^{\varepsilon} < T ) = \mathbb{P} \left( \sup_{t \in [0,T]} \|u^{\varepsilon}_{v^{\varepsilon}}(t) \| \geq M \right) \leq \frac{1}{M^2} \mathbb{E} \left[ \sup_{t \in [0,T]} \|u^{\varepsilon}_{v^{\varepsilon}}(t) \|^2 \right] \leq \frac{C_2}{M^2}.\]

Hence it follows that

(4.43)\begin{align} & \mathbb{P} \left( \sup_{0 \leq t \leq T} \| u^{\varepsilon}_{v^{\varepsilon}}(t) - u_{v^{\varepsilon}}(t) \| > \eta \right) \nonumber\\ & \quad \leq \mathbb{P} \left( \sup_{0 \leq t \leq T} \| u^{\varepsilon}_{v^{\varepsilon}}(t) - u_{v^{\varepsilon}}(t) \| > \eta, \tau^{\varepsilon} = T \right)\nonumber\\ & \quad \quad+ \mathbb{P} \left( \sup_{0 \leq t \leq T} \| u^{\varepsilon}_{v^{\varepsilon}}(t) - u_{v^{\varepsilon}}(t) \| > \eta, \tau^{\varepsilon} < T \right) \nonumber\\ & \quad \leq \mathbb{P} \left( \sup_{0 \leq t \leq T} \| u^{\varepsilon}_{v^{\varepsilon}}(t\wedge \tau^{\varepsilon}) - u_{v^{\varepsilon}}(t\wedge \tau^{\varepsilon}) \| > \eta \right) + \frac{C_2}{M^2}, \end{align}

which implies that

(4.44)\begin{align} \lim_{\varepsilon \rightarrow 0} \sup_{0 \leq t \leq T} \| u^{\varepsilon}_{v^{\varepsilon}}(t) - u_{v^{\varepsilon}}(t) \|^2 = 0 \ \ \ {\rm in\ probability}. \end{align}

Since $\{v^{\varepsilon }\}$ converges in distribution to $v$ as $S_a$-valued random variables, according to Skorokhod's representation theorem, there exist a probability space $(\widetilde {\Omega },\,\widetilde {F},\,\widetilde {P})$, and $S_a$-valued random variables $\{\widetilde {v}^{\varepsilon }\}$ and $\widetilde {v}$ with the same distribution as $\{v^{\varepsilon }\}$ and $v$, respectively, such that $\{\widetilde {v}^{\varepsilon }\}\rightarrow \widetilde {v}$ $\widetilde {P}$-almost surely in $S_a$. By lemma 4.9 we infer that $u_{\widetilde {v}^{\varepsilon }}\rightarrow u_{\widetilde {v}}$ $\widetilde {P}$-almost surely in $C([0,\,T];l^2)$. Then $u_{\widetilde {v}^{\varepsilon }}\rightarrow u_{\widetilde {v}}$ in $C([0,\,T];l^2)$ in distribution, and hence

(4.45)\begin{equation} u_{v^{\varepsilon}} \rightarrow u_{v} \ {\rm in} \ C([0,T];l^2) \ \ \ {\rm in\ distribution}, \end{equation}

which together with (4.44) implies the desired result.