w(t)\,dt+\sigma_{2}w(t)\,dW_{2}(t),\quad w(0)=w_{0}\in
H^{\rho}_{2}({{\mathcal{O}}}).$

A solution to (5.23) is given by standard methods (see e.g. [15, Chapter 6] or
[42, Theorem 4.2.4]) such that for any $q\geq 1$,

(5.24) $\displaystyle\mathbb{E}\|w\|^{q}_{C([0,T];H_{2}^{\rho})}\leq
C_{1}(T)\quad\mbox{and}\quad\mathbb{E}\|w\|^{q}_{L^{2}([0,T];H^{\rho+1}_{2})}\leq
C_{2}(T).$

The solution to (3.14) is given by

$v_{\kappa}(t)=w(t)+\int_{0}^{t}e^{r_{v}\Delta(t-s)}F(s)\,ds,\quad t\in[0,T].$

The Minkowski inequality, the smoothing property of the heat semigroup and the
Sobolev embedding $L^{1}({{\mathcal{O}}})\hookrightarrow
H^{\rho-\delta}_{2}({{\mathcal{O}}})$ for $\delta-\rho+\frac{d}{2}>d$ and
$1-\frac{1}{\mu}>\delta/2$ give

$\displaystyle\left|\int_{0}^{t}e^{r_{v}\Delta(t-s)}F(s)\,ds\right|_{H^{\rho}_{2}}\leq\int_{0}^{t}\left|e^{r_{v}\Delta(t-s)}F(s)\right|_{H^{\rho}_{2}}\,ds$
$\displaystyle\leq$
$\displaystyle\int_{0}^{t}(t-s)^{-\frac{\delta}{2}}\left|F(s)\right|_{H^{\rho-\delta}_{2}}\,ds\leq
C(T)\,\left(\int_{0}^{t}\left|F(s)\right|_{L^{1}}^{\mu}\,ds\right)^{\frac{1}{\mu}}.$

In particular, we have $2-\frac{2}{\alpha}>\frac{d}{2}+\rho$ and $\alpha<2$.
This gives as condition for $\rho$, $\rho<1-\frac{d}{2}$. In addition, setting
$p=2$ we need $-\rho<\frac{2(2-\alpha)}{\alpha}$, which is not a restriction.
However, due the hypotheses we find some $\alpha<2$ such that the first
inequality is satisfied. Note, that due to the condition in Hypothesis 3.4,
there exists some $\mu=\alpha\geq 1$ such that $\rho$ satisfies the assumption
above and those of the Technical Lemma 5.15. Setting

$F=\phi_{\kappa}(h(\eta,\xi,\cdot))\eta\xi^{2},$

we obtain by applying Technical Lemma 5.15 that there exists some constant
$C(T)>0$ and $\delta_{1},\delta_{2}\in(0,1)$, $\delta_{1}+\delta_{2}=1$ such
that

$\displaystyle\mathbb{E}\left(\int_{0}^{T}|F(s)|_{L^{1}}^{\mu}\,ds\right)^{\frac{m_{0}}{\mu}}$
$\displaystyle\leq$ $\displaystyle C(T)R_{0}^{\delta_{1}}R_{1}^{\delta_{2}}.$

It remains to calculate the norm in
$L^{2}(0,T;H^{\rho+1}_{2}({{\mathcal{O}}}))$. By standard calculations (i.e.
applying the smoothing property and the Young inequality for convolution), we
get for $\frac{3}{2}\geq\frac{1}{\mu}+\frac{1}{\kappa}$ and $\delta\kappa/2<1$

$\displaystyle\left\|\int_{0}^{\cdot}e^{r_{v}\Delta(\cdot-s)}F(s)\,ds\right\|_{L^{2}(0,T;H_{2}^{\rho+1})}\leq
C\,\|F\|_{L^{\mu}(0,T;H_{2}^{\rho+1-\delta})}.$

The embedding $L^{1}({{\mathcal{O}}})\hookrightarrow
H^{\rho+1-\delta}_{2}({{\mathcal{O}}})$ for $\delta-(\rho+1)>\frac{d}{2}$,
gives

$\displaystyle\|F\|_{L^{\mu}(0,T;H_{2}^{\rho-1})}\leq\|F\|_{L^{\mu}(0,T;L^{1})},$

for and similarly to before we know by hypothesis 3.4, that there exists some
$\mu=\alpha$ such that $\delta-(\rho+1)>\frac{d}{2}$,
$\frac{3}{2}\geq\frac{1}{\mu}+\frac{1}{\kappa}$, $\delta\kappa/2<1$, and
$\alpha(p-1)/(p-\alpha)\leq m_{0}$. Therefore, we get by the Technical Lemma
5.15

$\displaystyle\mathbb{E}\left\|\int_{0}^{\cdot}e^{r_{v}\Delta(\cdot-s)}F(s)\,ds\right\|^{m_{0}}_{L^{2}(0,T;H_{2}^{\rho+1})}\leq
C(T)R_{0}^{\delta_{1}}R_{1}^{\delta_{2}}.$

∎

##  6\. Pathwise uniqueness of the solution

###### Proof of Theorem 3.9.

Let us remind, since $(u_{1},v_{1})$ and $(u_{2},v_{2})$ are solutions to the
system (3.6)–(3.7) with ${\mathbb{P}}(u_{1}(0)=u_{2}(0))=1$ and
${\mathbb{P}}(v_{1}(0)=v_{2}(0))=1$, we can write for $i=1,2$,

(6.1) $\displaystyle d{u}_{i}(t)$ $\displaystyle=$
$\displaystyle\Big{(}r_{u}\Delta u_{i}^{[\gamma]}(t)-\chi
u_{i}(t)v^{2}_{i}(t)\big{)}\Big{)}\,dt+\sigma_{1}u_{i}(t)dW_{1}(t)\,\quad
t>0,$ (6.2) $\displaystyle d{v}_{i}(t)$ $\displaystyle=$
$\displaystyle\big{(}r_{v}\Delta
v_{i}(t)+u_{i}(t)v^{2}_{i}(t)\Big{)}\,dt+\sigma_{2}v_{i}(t)dW_{2}(t)\,\quad
t>0.$

In the first step we will introduce a family of stopping times
$\\{\tau_{N}\;\colon\;N\in\mathbb{N}\\}$, and show that on the time interval
$[0,\tau_{N}]$ the solutions $u_{1}$ and $u_{2}$, respective, $v_{1}$ and
$v_{2}$, are indistinguishable. Here, in the second step, we will show that
${\mathbb{P}}\left(\tau_{N}<T\right)\to 0$ for $N\to\infty$. From this follows
that $u_{1}$ and $u_{2}$ are indistinguishable on the time interval $[0,T]$.

Let us remind that $\rho_{0}:=\delta_{0}\in(-\frac{1}{\gamma},0)$,
$\rho+\frac{1}{2}\geq\max(1,\frac{d}{2})$,
$\frac{d}{2}<\frac{\gamma}{\gamma+1}+\rho$, and
$\hat{\delta}<\frac{1}{\gamma}$

#### Step I:

Let us introduce the stopping times $\\{\tau_{N}\;\colon\;N\in\mathbb{N}\\}$
as follows:

$\displaystyle\tau_{N,i}^{1}$ $\displaystyle:=$ $\displaystyle\inf\\{t\geq
0:\|\mathbbm{1}_{[0,t]}v_{i}\|_{\mathbb{H}_{\rho}}\geq N\\}\wedge T\quad
i=1,2,$ $\displaystyle\tau_{N,i}^{2}$ $\displaystyle:=$
$\displaystyle\inf\\{t\geq
0:\|\mathbbm{1}_{[0,t]}u_{i}\|_{L^{2\gamma}(0,T;H^{\hat{\delta}}_{2})}\geq
N\\}\wedge T\quad i=1,2,$

and $\tau_{N}:=\min_{i=1,2}(\tau_{N,i}^{1},\tau_{N,i}^{2})$.

The aim is to show that $(u_{1},v_{1})$ and $(u_{2},v_{2})$ are
indistinguishable on the time interval $[0,\tau_{N}]$.

######  Remark 6.1.

Note, due to the assumptions in Theorem 3.9 and by Corollary A.3, and
Proposition 5.13 we know that there exists a constant $C>0$ such that for any
solutions $(u,v)$ to (3.6)–(3.7) we have
$\mathbb{E}\|v\|_{\mathbb{H}_{\rho}}^{m_{0}}\leq C$ and
$\mathbb{E}\|u\|_{L^{2\gamma}(0,T;H^{\hat{\delta}}_{2})}^{2\gamma}\leq C$.

Fix $N\in\mathbb{N}$. To get uniqueness on $[0,\tau_{N}]$ we first stop the
original solution processes at time $\tau_{N}$ and extend the processes
$(u_{1},v_{1})$ and $(u_{2},v_{2})$ by other processes to the whole interval
$[0,T]$. For this purpose, let $(y_{1},z_{1})$ be solutions to

(6.5) $\displaystyle\left\\{\begin{array}[]{rcl}dy_{1}(t)&=&r_{u}\Delta
y_{1}^{[\gamma]}(t)+\sigma_{1}y_{1}(t)\,d\theta_{\tau_{N}}\circ W_{1}(t),\quad
t\in[\tau_{N}\wedge T,T],\\\ dz_{1}(t)&=&r_{v}\Delta
z_{1}(t)-z_{1}(t)+\sigma_{2}z_{1}(t)\,d\theta_{\tau_{N}}\circ W_{2}(t),\quad
t\in[\tau_{N}\wedge T,T],\end{array}\right.$

with initial data $y_{1}(0):=u_{1}(\tau_{N})$, $z_{1}(0):=v_{1}(\tau_{N})$,
and let $(y_{2},z_{2})$ be solutions to

(6.8) $\displaystyle\left\\{\begin{array}[]{rcl}dy_{2}(t)&=&r_{u}\Delta
y_{2}^{[\gamma]}(t)+\sigma_{1}y_{2}(t)\,d\theta_{\tau_{N}}\circ W_{1}(t),\quad
t\in[\tau_{N}\wedge T,T],\\\ dz_{2}(t)&=&r_{v}\Delta
z_{2}(t)-z_{2}(t)+\sigma_{2}z_{2}(t)\,d\theta_{\tau_{N}}\circ W_{2}(t),\quad
t\in[\tau_{N}\wedge T,T],\end{array}\right.$

with initial data $y_{2}(0):=u_{2}(\tau_{N})$ and $z_{2}(0):=v_{2}(\tau_{N})$.
Here, $\theta$ denotes the shift operator, i.e. $\theta_{\tau}\circ
W_{i}(t)=W_{i}(t+\tau)-W_{i}(\tau)$, $i=1,2$. Since $(u_{1},v_{1})$ and
$(u_{2},v_{2})$ are continuous in $H^{-1}_{2}({{\mathcal{O}}})\times
L^{2}({{\mathcal{O}}})$, $(u_{1}(\tau_{N}),v_{1}(\tau_{N}))$ and
$(u_{2}(\tau_{N}),v_{1}(\tau_{N}))$ are well defined and belong
${\mathbb{P}}$–a.s. to $H^{-1}_{2}({{\mathcal{O}}})\times
H^{\rho_{0}}_{2}({{\mathcal{O}}})$. By [2, Theorem 2.5.1], we know that there
exists a unique solutions $y_{1}$ and $y_{2}$ to (6.5) belonging
${\mathbb{P}}$-a.s. to $C([0,T];H^{-1}_{2}({{\mathcal{O}}}))$. Since
$(e^{t(r_{v}\Delta-\operatorname{Id})})_{t\geq 0}$ is an analytic semigroup on
$H^{\rho}_{2}({{\mathcal{O}}})$, the existence of unique solutions $z_{1}$ and
$z_{2}$ to (6.5) and (6.8) belonging ${\mathbb{P}}$-a.s. to
$C([0,T];H^{\rho_{0}}_{2}({{\mathcal{O}}}))$ can be shown by standard methods,
cf. [15]. Now, let us define two processes $(\bar{u}_{1},\bar{v}_{1})$ and
$(\bar{u}_{2},\bar{v}_{2})$ which are equal to $(u_{1},v_{1})$ and
$(u_{2},v_{2})$ on the time interval $[0,\tau_{N})$ and follow the processes
$(y_{1},z_{1})$ and $(y_{2},z_{2})$ afterwards. In particular, let

$\bar{u}_{1}(t)=\begin{cases}u_{1}(t)&\mbox{ for }0\leq t<\tau_{N},\\\
y_{1}(t)&\mbox{ for }\tau_{N}\leq t\leq
T,\end{cases}\qquad\bar{v}_{1}(t)=\begin{cases}v_{1}(t)&\mbox{ for }0\leq
t<\tau_{N},\\\ z_{1}(t)&\mbox{ for }\tau_{N}\leq t\leq T,\end{cases}$

and

$\bar{u}_{2}(t)=\begin{cases}u_{2}(t)&\mbox{ for }0\leq t<\tau_{N},\\\
y_{2}(t)&\mbox{ for }\tau_{N}\leq t\leq
T.\end{cases}\qquad\bar{v}_{2}(t)=\begin{cases}v_{2}(t)&\mbox{ for }0\leq
t<\tau_{N},\\\ z_{2}(t)&\mbox{ for }\tau_{N}\leq t\leq T.\end{cases}$

Note, that $(u_{1},v_{1})$ and $(u_{2},v_{2})$ solve the equation
corresponding to (3.6)–(3.7), that is, for $i=1,2$,

(6.9) $\displaystyle d{u}_{i}(t)$ $\displaystyle=$
$\displaystyle\Big{(}r_{u}\Delta u_{i}^{\gamma}(t)-\chi
u_{i}(t)v^{2}_{i}(t)\Big{)}\,dt+\sigma_{1}u_{i}(t)dW_{1}(t)\,\quad t>0,$
(6.10) $\displaystyle d{v}_{i}(t)$ $\displaystyle=$
$\displaystyle\big{(}r_{v}\Delta
v_{i}(t)+u_{i}(t)v^{2}_{i}(t)\Big{)}\,dt+\sigma_{2}v_{i}(t)dW_{2}(t)\,\quad
t>0.$

#### Step II:

Let $\alpha=\rho+\frac{1}{2}$ (which implies $\alpha\geq\max(1,\frac{d}{2})$
due to the assumption on $\rho$) and $\frac{d}{2}>\delta>-\rho_{0}$. Our goal
is to show that $(u_{1},v_{1})$ and $(u_{2},v_{2})$ are indistinguishable on
the interval $[0,\tau_{N}]$. Applying the Itô formula while setting

$\Phi(t):=\mathbb{E}\left[\sup_{0\leq s\leq
t}|\bar{u}_{1}(s)-\bar{u}_{2}(s)|^{2}_{H^{-1}_{2}}\right]$

over $[0,\tau_{N}]$ and taking into account that $(\bar{u}_{1},v_{1})$ and
$(\bar{u}_{2},v_{2})$ are solutions to system (3.6)–(3.7) we obtain by
standard calculations for $0\leq{t}\leq\tau_{N}$

(6.11)
$\displaystyle\mathbb{E}\Big{[}|\bar{u}_{1}({{t}})-\bar{u}_{2}({{t}})|^{2}_{H^{-1}_{2}}\Big{]}+r_{u}\mathbb{E}\Big{[}\int_{0}^{{t}}|\bar{u}_{1}(s)-\bar{u}_{2}(s)|^{\gamma+1}_{L^{\gamma+1}}ds\Big{]}$
$\displaystyle\leq\chi\mathbb{E}\Big{[}\Big{|}\int_{0}^{{t}}\int_{{{\mathcal{O}}}}(-\nabla)^{-1}\Big{(}\bar{u}_{1}(s,x)\bar{v}^{2}_{1}(s,x)-\bar{u}_{2}(s,x)\bar{v}^{2}_{2}(s,x)\Big{)}(-\nabla)^{-1}\big{(}\bar{u}_{1}(s,x)-\bar{u}_{2}(s,x)\big{)}dx\,ds\Big{|}\Big{]}$
$\displaystyle{}+C\mathbb{E}\int_{0}^{t}|\bar{u}_{1}({s})-\bar{u}_{2}({s})|_{H^{-1}_{2}}^{2}\,ds.$

Furthermore, we obtain

$\displaystyle\mathbb{E}\Big{[}\Big{|}\int_{0}^{{t}}\int_{{{\mathcal{O}}}}(-\nabla)^{-1}\Big{(}u_{1}(s,x)\bar{v}^{2}_{1}(s,x)-\bar{u}_{2}(s,x)\bar{v}^{2}_{2}(s,x)\Big{)}(-\nabla)^{-1}\big{(}\bar{u}_{1}(s,x)-\bar{u}_{2}(s,x)\big{)}dx\,ds\Big{|}\Big{]}$
$\displaystyle\leq\mathbb{E}\Big{[}{\Big{|}\int_{0}^{{t}}\int_{{{\mathcal{O}}}}(-\nabla)^{-1}\big{(}\bar{u}_{1}(s,x)-\bar{u}_{2}(s,x)\big{)}\bar{v}^{2}_{1}(s,x)(-\nabla)^{-1}\big{(}\bar{u}_{1}(s,x)-\bar{u}_{2}(s,x)\big{)}dx\,ds\Big{|}}\Big{]}$
$\displaystyle\quad+\mathbb{E}\Big{[}{\Big{|}\int_{0}^{{t}}\int_{{{\mathcal{O}}}}(-\nabla)^{-1}\Big{[}\bar{u}_{2}(s,x)\bar{v}_{2}(s,x)\big{(}\bar{v}_{1}(s,x)-\bar{v}_{2}(s,x)\big{)}\Big{]}(-\nabla)^{-1}\big{(}\bar{u}_{1}(s,x)-\bar{u}_{2}(s,x)\big{)}dx\,ds\Big{|}}\Big{]}$
$\displaystyle=:I_{1}(t)+I_{2}(t).$

Now we have by Theorem 1-(iii) [52, p. 190] and the identities on Proposition
[52, p. 14] with $\alpha\geq\max(1,\frac{d}{2})$

$\displaystyle|(u_{1}-u_{2})\bar{v}_{2}^{2}|_{H^{-1}_{2}}\leq|u_{1}-u_{2}|_{H^{-1}_{2}}|v_{2}^{2}|_{H^{\alpha}_{2}}$

Since $H^{\alpha}_{2}({{\mathcal{O}}})$ is a Banach algebra (see Theorem
2-(18) [52, p. 192]), we have

$\displaystyle|(u_{1}-u_{2})v_{2}^{2}|_{H^{-1}_{2}}\leq|u_{1}-u_{2}|_{H^{-1}_{2}}|v_{2}|^{2}_{H^{\alpha}_{2}}.$

Hence we get by applying Young for $\varepsilon>0$

$\displaystyle I_{1}(t)$ $\displaystyle\leq$
$\displaystyle\mathbb{E}\int_{0}^{t}|\bar{u}_{1}(s)-\bar{u}_{2}(s)|_{H^{-1}_{2}}^{2}|\bar{v}_{2}(s)|^{2}_{H^{\alpha}_{2}}\,ds$
$\displaystyle\leq$
$\displaystyle\varepsilon\mathbb{E}\int_{0}^{t}|\bar{u}_{1}(s)-\bar{u}_{2}(s)|_{H^{-1}_{2}}^{2}|\bar{v}_{2}(s)|^{4}_{H^{\alpha}_{2}}\,ds+C(\varepsilon)\mathbb{E}\int_{0}^{t}|\bar{u}_{1}(s)-\bar{u}_{2}(s)|_{H^{-1}_{2}}^{2}\,ds$
$\displaystyle\leq$ $\displaystyle\varepsilon\mathbb{E}\left[\sup_{0\leq
s\leq{t}}|\bar{u}_{1}(s)-\bar{u}_{2}(s)|_{H^{-1}_{2}}^{2}\right]\times\mathbb{E}\int_{0}^{t}|\bar{v}_{2}(s)|^{4}_{H^{\alpha}_{2}}\,ds+C(\varepsilon)\mathbb{E}\int_{0}^{t}|\bar{u}_{1}(s)-\bar{u}_{2}(s)|_{H^{-1}_{2}}^{2}\,ds$

Since $\alpha=\rho+\frac{1}{2}$, we have
$\|v_{2}\|_{L^{4}(0,T;H^{\alpha}_{2})}\leq\|v_{2}\|_{\mathbb{H}_{\rho}}$.
Next, note that
$|u_{2}(v_{1}^{2}-v^{2}_{2})|\leq|u_{2}v_{1}(v_{1}-v_{2})|+|u_{2}v_{2}(v_{1}-v_{2})|$.
Similarly as above we get for $\delta-\delta_{1}>\max(0,\frac{d}{2}-1)$ and
$\delta_{2}=\delta<\frac{1}{\gamma}$ and $\delta_{3}>\frac{d}{2}$ (see Theorem
1-(iii) [52, p. 190], Theorem 2 [52, p. 200] and the identities on Proposition
[52, p. 14]) for $i=1,2$

(6.12) $\displaystyle\hskip
25.83325pt|u_{2}v_{i}(v_{1}-v_{2})|_{H^{-1}_{2}}\leq
C_{1}|v_{1}-v_{2}|_{H^{-\delta_{1}}_{2}}|u_{2}v_{i}|_{H^{\delta}_{2}}\leq
C_{2}|v_{1}-v_{2}|_{H^{-\delta_{1}}_{2}}|u_{2}|_{H^{\delta_{2}}_{2}}|v_{i}|_{H^{\delta_{3}}_{2}}.$

By Young inequality, we know that for
$\delta_{3}\leq\frac{\gamma}{\gamma+1}+\rho$ and $\rho_{0}>-\frac{1}{\gamma}$
and for any $\varepsilon>0$ there exists a constant $C(\varepsilon)>0$ such
that

$\displaystyle I_{2}(t)$ $\displaystyle\leq$ $\displaystyle
C(\varepsilon)\mathbb{E}\int_{0}^{t}|\bar{v}_{1}(s)-\bar{v}_{2}(s)|^{2}_{H^{\rho_{0}}_{2}}\,ds+\varepsilon\mathbb{E}\left(\int_{0}^{t}|\bar{u}_{1}(s)|_{H^{\delta_{2}}_{2}}^{2\gamma}\,ds\right)^{\frac{1}{\gamma}}\,$
$\displaystyle{}\times\mathbb{E}\left(\int_{0}^{t}\Big{(}|\bar{v}_{1}(s)|^{2\frac{\gamma+1}{\gamma}}_{H^{\delta_{3}}_{2}}+|\bar{v}_{2}(s)|^{2\frac{\gamma+1}{\gamma}}_{H^{\delta_{3}}_{2}}\big{)}\,\,ds\right)^{\frac{\gamma}{\gamma+1}}\mathbb{E}\left[\sup_{0\leq
s\leq t}|\bar{v}_{1}(s)-\bar{v}_{2}(s)|^{2}_{H^{\rho_{0}}_{2}}\right]$
$\displaystyle\leq$ $\displaystyle
C(\varepsilon)\mathbb{E}\int_{0}^{t}|\bar{v}_{1}(s)-\bar{v}_{2}(s)|^{2}_{H^{\rho_{0}}_{2}}\,ds$
$\displaystyle{}+\mathbb{E}\|\bar{u}_{1}\|_{L^{2\gamma}(0,T;H^{\delta_{2}}_{2})}\left(\mathbb{E}\|\bar{v}_{1}\|_{\mathbb{H}_{\rho}}+\mathbb{E}\|\bar{v}_{2}\|_{\mathbb{H}_{\rho}}\right)\mathbb{E}\left[\sup_{0\leq
s\leq t}|\bar{v}_{1}(s)-\bar{v}_{2}(s)|^{2}_{H^{\rho_{0}}_{2}}\right].$

The third term in (6.11) can be handled by the Burkholder-Davis-Gundy
inequality. Since $t\leq\tau_{N}$ and $\delta_{2}<\frac{1}{\gamma}$ we know
that for any $\varepsilon>0$ there exists a constant $C(\varepsilon)>0$ such
that

$\displaystyle\Phi(t)$ $\displaystyle\leq$
$\displaystyle\varepsilon_{1}N^{2}\mathbb{E}\left[\sup_{0\leq s\leq
t}|\bar{v}_{1}(s)-\bar{v}_{2}(s)|^{2}_{H^{\rho_{0}}_{2}}\right]+\mathbb{E}\int_{0}^{t}\sup_{0\leq
r\leq s}|\bar{v}_{1}(r)-\bar{v}_{2}(r)|^{2}_{H^{\rho_{0}}_{2}}\,ds$
$\displaystyle+\varepsilon N\mathbb{E}\left[\sup_{0\leq
s\leq{t}}|\bar{u}_{1}(s)-\bar{u}_{2}(s)|_{H^{-1}_{2}}^{2}\right]+C(\varepsilon)\mathbb{E}\int_{0}^{t}|\bar{u}_{1}(s)-\bar{u}_{2}(s)|_{H^{-1}_{2}}^{2}\,ds.$

#### Step III:

Let us set

$\displaystyle\Psi(t)$ $\displaystyle:=$
$\displaystyle\mathbb{E}\left[\sup_{0\leq s\leq
t}\|\bar{v}_{1}(s)-\bar{v}_{2}(s)\|_{H^{\rho_{0}}_{2}}^{2}\right]+\mathbb{E}\|\bar{v}_{1}-\bar{v}_{2}\|_{L^{2}(0,t;H^{\rho_{0}+1}_{2})}^{2}.$

Then we get by standard calculations for $\tilde{\delta}<1$

$\displaystyle\mathbb{E}\left[\sup_{0\leq s\leq
t}\|\bar{v}_{1}(s)-\bar{v}_{2}(s)\|_{H^{\rho_{0}}_{2}}^{2}\right]$
$\displaystyle{}\quad+\mathbb{E}\|\bar{v}_{1}-\bar{v}_{2}\|_{L^{2}(0,t;H_{2}^{\rho_{0}+1})}^{2}$
$\displaystyle\leq
C\mathbb{E}\|\bar{u}_{1}\bar{v}_{1}^{2}-\bar{u}_{2}\bar{v}_{2}^{2}\|_{L^{2}(0,t;H_{2}^{\rho_{0}-\tilde{\delta}})}^{2}$
$\displaystyle{}\quad+\mathbb{E}\left[\sup_{0\leq s\leq
t}\left|\int_{0}^{t}\sigma_{2}\left(\bar{v}_{1}({s})-\bar{v}_{2}({s})\right)dW_{2}(s)\right|^{2}_{H^{\rho_{0}}_{2}}\right].$

Next, since $\tilde{\delta}$ can be chosen such that
$\rho_{0}-\tilde{\delta}\leq-1$ we can use (6.12) and obtain for
$\delta-\delta_{1}>\max(0,\frac{d}{2}-1)$ and
$\delta_{2}=\delta<\frac{1}{\gamma}$ and $\delta_{3}>\frac{d}{2}$

$\displaystyle|u_{1}v_{1}^{2}-u_{1}v_{2}^{2}|_{H^{\rho_{0}-\tilde{\delta}}_{2}}$
$\displaystyle\leq$ $\displaystyle
C_{1}|v_{1}-v_{2}|_{H^{-\delta_{1}}_{2}}|u_{2}|_{H^{\delta_{2}}_{2}}\left(|v_{1}|_{H^{\delta_{3}}_{2}}+|v_{2}|_{H^{\delta_{3}}_{2}}\right).$

For $\alpha>\min(1+\rho_{0}-\delta+\frac{d}{2},1,\frac{d}{2})$ and
$\tilde{\delta}<1$ we have by Theorem 1 [52, p. 190] and Theorem 2-(18) [52,
p. 192]

$\displaystyle|(u_{1}-u_{2})v_{1}^{2}|_{H^{\rho_{0}-\tilde{\delta}}_{2}}\leq|u_{1}-u_{2}|_{H^{-1}_{2}}|v_{1}^{2}|_{H^{\alpha}_{2}}\leq|u_{1}-u_{2}|_{H^{-1}_{2}}|v_{1}|^{2}_{H^{\alpha}_{2}}.$

The third term can be handled by an application of the Burkholder-Davis-Gundy
inequality. The Young inequality implies that for any
$\varepsilon_{1},\varepsilon_{2}>0$ there exist constants
$C(\varepsilon_{1}),C(\varepsilon_{2})>0$ such that we have

$\displaystyle\Psi(t)\leq\mathbb{E}|v_{1}(0)-v_{2}(0)|^{2}_{H^{\rho_{0}}_{2}}$
$\displaystyle{}+\mathbb{E}\left(\int_{0}^{t}|\bar{u}_{1}(s)|^{2}_{H^{\delta_{2}}_{2}}\,\left(|\bar{v}_{1}(s)|^{2}_{H^{\delta_{3}}_{2}}+|\bar{v}_{2}(s)|^{2}_{H^{\delta_{3}}_{2}}\right)|\bar{v}_{1}(s)-\bar{v}_{2}(s)|^{2}_{H^{\rho_{0}}_{2}}\,ds\right)$
$\displaystyle{}+\mathbb{E}\left(\int_{0}^{t}|\bar{u}_{1}(s)-\bar{u}_{2}(s)|^{2}_{H^{-1}_{2}}\,|\bar{v}_{1}(s)|^{4}_{H^{\alpha}_{2}}\,ds\right)$
$\displaystyle\leq$
$\displaystyle\mathbb{E}|v_{1}(0)-v_{2}(0)|^{2}_{H^{\rho_{0}}_{2}}$
$\displaystyle+\varepsilon_{1}\mathbb{E}\left(\int_{0}^{t}|\bar{u}_{1}(s)|^{2}_{H^{\delta_{2}}_{2}},\left(|\bar{v}_{1}(s)|^{2}_{H^{\delta_{3}}_{2}}+|\bar{v}_{2}(s)|^{2}_{H^{\delta_{3}}_{2}}\right)|\bar{v}_{1}(s)-\bar{v}_{2}(s)|^{2}_{H^{\rho_{0}}_{2}}\,ds\right)$
$\displaystyle{}+C(\varepsilon_{1})\mathbb{E}\left(\int_{0}^{t}|\bar{v}_{1}(s)-\bar{v}_{2}(s)|^{2}_{L^{2}}\,ds\right)$
$\displaystyle{}+\varepsilon_{2}\mathbb{E}\left(\int_{0}^{t}|\bar{u}_{1}(s)-\bar{u}_{2}(s)|^{2}_{H^{-1}_{2}}\,|\bar{v}_{1}(s)|^{4}_{H^{\alpha}_{2}}\,ds\right)+C(\varepsilon_{2})\mathbb{E}\left(\int_{0}^{t}|\bar{u}_{1}(s)-\bar{u}_{2}(s)|^{2}_{H^{-1}_{2}}\,\,ds\right).$
$\displaystyle I_{2}(t)$ $\displaystyle\leq$
$\displaystyle\mathbb{E}\left(\int_{0}^{t}|\bar{u}_{1}(s)|_{H^{\delta_{2}}_{2}}^{2\gamma}\,ds\right)^{\frac{1}{\gamma}}\,$
$\displaystyle{}\times\mathbb{E}\left(\int_{0}^{t}\Big{(}|\bar{v}_{1}(s)|^{2\frac{\gamma+1}{\gamma}}_{H^{\delta_{3}}_{2}}+|\bar{v}_{2}(s)|^{2\frac{\gamma+1}{\gamma}}_{H^{\delta_{3}}_{2}}\big{)}\,\,ds\right)^{\frac{\gamma}{\gamma+1}}\mathbb{E}\left[\sup_{0\leq
s\leq t}|\bar{v}_{1}(s)-\bar{v}_{2}(s)|^{2}_{H^{\rho_{0}}_{2}}\right]$
$\displaystyle\leq$
$\displaystyle\mathbb{E}\|\bar{u}_{1}\|_{L^{2\gamma}(0,T;H^{\delta_{2}}_{2})}\left(\mathbb{E}\|\bar{v}_{1}\|_{\mathbb{H}_{\rho}}+\mathbb{E}\|\bar{v}_{2}\|_{\mathbb{H}_{\rho}}\right)\mathbb{E}\left[\sup_{0\leq
s\leq t}|\bar{v}_{1}(s)-\bar{v}_{2}(s)|^{2}_{H^{\rho_{0}}_{2}}\right].$

Since $\rho_{0}\in(0,-1)$, $1-\frac{1}{l}=-\frac{\rho_{0}}{2}$ and
$\alpha=\rho+\frac{1}{2}$

$\displaystyle\Psi(t)$ $\displaystyle\leq$
$\displaystyle\mathbb{E}|v_{1}(0)-v_{2}(0)|^{2}_{H^{\rho_{0}}_{2}}+\varepsilon_{1}\mathbb{E}\|\mathbbm{1}_{[0,t]}\bar{u}_{1}\|_{L^{2\gamma}(0,T;H^{\delta_{2}}_{2})}\,\left(\mathbb{E}\|\bar{v}_{1}\|_{\mathbb{H}_{\rho}}+\mathbb{E}\|\bar{v}_{2}\|_{\mathbb{H}_{\rho}}\right)$
$\displaystyle\qquad\times\mathbb{E}\left[\sup_{0\leq s\leq
t}|\bar{v}_{1}(s)-\bar{v}_{2}(s)|^{2}_{H^{\rho_{0}}_{2}}\right]$
$\displaystyle{}+C(\varepsilon_{1})\mathbb{E}\left(\int_{0}^{t}|\bar{v}_{1}(s)-\bar{v}_{2}(s)|^{2}_{H^{\rho_{0}}_{2}}\,ds\right)$
$\displaystyle{}+\varepsilon_{2}\mathbb{E}\left[\sup_{0\leq s\leq
t}|\bar{u}_{1}(s)-\bar{u}_{2}(s)|^{2}_{H^{-1}_{2}}\right]\mathbb{E}\|\mathbbm{1}_{[0,t]}\bar{v}_{1}\|_{L^{4}(0,T;H^{\alpha}_{2})}^{2}$
$\displaystyle+C(\varepsilon_{2})\mathbb{E}\int_{0}^{t}|\bar{u}_{1}(s)-\bar{u}_{2}(s)|^{2}_{H^{-1}_{2}}\,\,ds$
$\displaystyle\leq$
$\displaystyle\mathbb{E}|v_{1}(0)-v_{2}(0)|^{2}_{H^{\rho_{0}}_{2}}+\varepsilon_{1}\mathbb{E}\|\mathbbm{1}_{[0,t]}\bar{u}_{1}\|_{L^{2\gamma}(0,T;H^{\delta_{2}}_{2})}\,\left(\mathbb{E}\|\bar{v}_{1}\|_{\mathbb{H}_{\rho}}+\mathbb{E}\|\bar{v}_{2}\|_{\mathbb{H}_{\rho}}\right)$
$\displaystyle\qquad\times\mathbb{E}\left[\sup_{0\leq s\leq
t}|\bar{v}_{1}(s)-\bar{v}_{2}(s)|^{2}_{H^{\rho_{0}}_{2}}\right]$
$\displaystyle{}+C(\varepsilon_{1})\mathbb{E}\left(\int_{0}^{t}|\bar{v}_{1}(s)-\bar{v}_{2}(s)|^{2}_{H^{\rho_{0}}_{2}}\,ds\right)$
$\displaystyle{}+\varepsilon_{2}\mathbb{E}\left[\sup_{0\leq s\leq
t}|\bar{u}_{1}(s)-\bar{u}_{2}(s)|^{2}_{H^{-1}_{2}}\right]\mathbb{E}\|\mathbbm{1}_{[0,t]}\bar{v}_{1}\|_{\mathbb{H}_{\rho}}^{2}$
$\displaystyle+C(\varepsilon_{2})\mathbb{E}\int_{0}^{t}\sup_{0\leq r\leq
s}|\bar{u}_{1}(r)-\bar{u}_{2}(r)|^{2}_{H^{-1}_{2}}\,\,ds.$

Taking into account the definition of $\tau_{N}$ we obtain

(6.13)
$\displaystyle\Psi(t)\leq\mathbb{E}|v_{1}(0)-v_{2}(0)|^{2}_{H^{\rho_{0}}_{2}}$
$\displaystyle{}+\varepsilon_{1}N^{2}\Psi(t)+\varepsilon_{2}N\Phi(t)+C(\varepsilon_{1})\int_{0}^{t}\Phi(s)\,ds+C(\varepsilon_{2})\int_{0}^{t}\Psi(s)\,ds.$

#### Step IV:

Noting that the estimates can be extended to the whole interval $[0,T]$ by
standard calculations. Next, collecting altogether we know, that for any
$\varepsilon_{1},\varepsilon_{2},\varepsilon_{3},\varepsilon_{4}>0$ there
exist constants
$C(\varepsilon_{1}),C(\varepsilon_{2}),C(\varepsilon_{3}),C(\varepsilon_{3})>0$
such that

$\displaystyle\Phi(t)+\Psi(t)\leq\mathbb{E}|u_{1}(0)-u_{2}(0)|^{2}_{H^{-1}_{2}}+\mathbb{E}|v_{1}(0)-\bar{v}_{2}(0)|^{2}_{H^{\rho_{0}}_{2}}$
$\displaystyle{}+\varepsilon_{1}N^{2}\Psi(t)+C(\varepsilon_{1})\int_{0}^{t}\Psi(s)\,ds+\varepsilon_{0}N\Phi(t)+C(\varepsilon_{0})\int_{0}^{t}\Phi(s)\,ds$
$\displaystyle{}+\varepsilon_{3}N^{2}\Psi(t)+\varepsilon_{4}N\Phi(t)+C(\varepsilon_{1})\int_{0}^{t}\Phi(s)\,ds+C(\varepsilon_{2})\int_{0}^{t}\Psi(s)\,ds.$

Taking the $\varepsilon_{1},\varepsilon_{2},\varepsilon_{3},\varepsilon_{4}>0$
accordingly, we know, that for any $N\in\mathbb{N}$ there exists a constant
$C(N)>0$ such that

$\displaystyle\Phi(t)+\Psi(t)\leq\mathbb{E}|u_{1}(0)-u_{2}(0)|^{2}_{H^{-1}_{2}}+\mathbb{E}|v_{1}(0)-\bar{v}_{2}(0)|^{2}_{H^{\rho_{0}}_{2}}$
$\displaystyle{}+C(N)\int_{0}^{t}\left(\Phi(s)+\Psi(s)\right)\,ds.$

An application of the Gronwall lemma and taking into account that
$v_{1}(0)=v_{2}(0)$, $u_{1}(0)=u_{2}(0)$ give that $\Phi(t)\leq 0$ and
$\Psi(t)\leq 0$.

#### Step V:

We show that ${\mathbb{P}}\left(\tau_{N}<T\right)\longrightarrow 0$ as
$N\to\infty$. Therefore,

$\displaystyle{\mathbb{P}}\left(\tau_{N}<T\right)$ $\displaystyle\leq$
$\displaystyle{\mathbb{P}}\left(\|\bar{u}_{1}\|_{L^{2\gamma}(0,T;H^{\hat{\delta}}_{2})}\geq
N\right)+{\mathbb{P}}\left(\|\bar{u}_{2}\|_{L^{2\gamma}(0,T;H^{\hat{\delta}}_{2})}\geq
N\right)$
$\displaystyle\qquad{}+{\mathbb{P}}\left(\|\bar{v}_{1}\|_{\mathbb{H}_{\rho}}\geq
N\right)+{\mathbb{P}}\left(\|\bar{v}_{2}\|_{\mathbb{H}_{\rho}}\geq N\right).$

Due to Remark 6.1, we can apply the Chebyshev inequality and get by the above
that

$\displaystyle{\mathbb{P}}\left(\tau_{N}<T\right)\leq\frac{C}{N^{2}},$

It follows that

${\mathbb{P}}\left(\tau_{N}\leq T\right)\longrightarrow 0,$

as $N\to\infty$. Hence, both the processes $u_{1}$ and $u_{2}$, and likewise
$v_{1}$ and $v_{2}$ are indistinguishable on $[0,T]$. This completes the proof
of Theorem 3.9. ∎

##  Appendix A Some useful inequalities

######  Lemma A.1.

Let $\gamma>1$, $x,y\in\mathbb{R}$. Then it holds that

(A.1) $(x^{[\gamma]}-y^{[\gamma]})(x-y)\geq 2^{1-\gamma}|x-y|^{\gamma+1},$

where $z^{[\gamma]}:=|z|^{\gamma-1}z$ for $z\in\mathbb{R}$.

###### Proof.

See e.g. [41, Lemma 3.1]. ∎

For the definition of the space $F^{s}_{p,q}$, we refer to [60, 63, 52]. It
translates to classical function spaces as in e.g. [60, Remark 2.1.1], in
particular, $F_{p,2}^{0}=L^{p}$, $1<p<\infty$ (Lebesgue spaces),
$F_{p,2}^{m}=W^{m}_{p}$, $m\in\mathbb{N}$ (Sobolev spaces), $1<p<\infty$ and
$F_{p,2}^{s}=H^{s}_{p}$, $s\in\mathbb{R}$, $1<p<\infty$ (fractional Sobolev
spaces).

Next, we shall record a variant of the Stroock-Varopoulos inequality together
with its proof. See e.g. [16, Lemma 3.6] for another version of this result.

######  Proposition A.2.

For any $\gamma>1$, $\theta\in(0,\frac{1}{\gamma})$, there exists a constant
$C(\gamma)>0$ such that

$\displaystyle\|\eta\|_{L^{2\gamma}(0,T;H^{\theta}_{2\gamma})}^{2\gamma}\leq
C(\gamma)\left(\int_{0}^{T}|\eta(s)^{[\gamma-1]}\nabla\eta(s)|_{L^{2}}^{2}\,ds+\int_{0}^{T}|\eta(s)|^{2\gamma}_{L^{2}}\,ds\right),$

where $z^{[\gamma-1]}:=|z|^{\gamma-2}z$ for $z\in\mathbb{R}$.

###### Proof.

By [52, p. 365], we have for any $p\in(1,\infty)$, $s\in(0,1)$, $\mu\in(0,1)$
and $\varepsilon\in(0,s\mu)$,

(A.2)
$||w|^{\mu}|_{H^{s\mu-\varepsilon}_{\frac{p}{\mu}}}=||w|^{\mu}|_{F^{s\mu-\varepsilon}_{\frac{p}{\mu},2}}\leq
C||w|^{\mu}|_{F_{\frac{p}{\mu},\frac{2}{\mu}}^{s\mu}}\leq
C|w|_{F_{p,2}^{s}}^{\mu}=C|w|_{H^{s}_{p}}^{\mu},\quad w\in
H^{s}_{p}({{\mathcal{O}}}).$

From (A.2) we know that for any $\gamma>1$, $p\in(1,\infty)$,
$\theta\in(0,\frac{1}{\gamma})$, $\varepsilon>0$, there exists a constant
$C>0$ such that

$|w|_{H^{\theta}_{p\gamma}}=||w^{\gamma}|^{\frac{1}{\gamma}}|_{H^{\theta}_{p\gamma}}\leq
C||w|^{\gamma}|^{\frac{1}{\gamma}}_{H^{\theta\gamma+\varepsilon}_{p}}.$

In particular, for any $0<\theta<\frac{1}{\gamma}$ and $p=2$, there exists a
constant $C>0$ such that

$|w|_{H^{\theta}_{2\gamma}}^{\gamma}\leq C||w|^{\gamma}|_{H^{1}_{2}}.$

Since we know by the chain rule,

$\int_{0}^{T}|\nabla|\eta(s)|^{\gamma}|_{L^{2}}^{2}\,ds=\gamma^{2}\int_{0}^{T}|\eta^{[\gamma-1]}(s)\nabla\eta(s)|_{L^{2}}^{2}\,ds$

we know that for any $\theta<\frac{1}{\gamma}$,

$\int_{0}^{T}|\eta(s)|_{H^{\theta}_{2\gamma}}^{2\gamma}\,ds\leq
C(\gamma)\left(\int_{0}^{T}|\eta^{\gamma-1}(s)\nabla\eta(s)|_{L^{2}}^{2}\,ds+\int_{0}^{T}|\eta(s)|^{2\gamma}_{L^{2}}\,ds\right).$

∎

Clearly, we have by the fractional Rellich-Kondrachov theorem that for any
$\varepsilon\in(0,\theta)$,

$|w|_{H^{\theta-\varepsilon}_{2}}^{\gamma}\leq
C|w|_{H^{\theta}_{2\gamma}}^{\gamma}.$

As $\theta\in(0,\gamma^{-1})$ is arbitrary, this yields the following result.

######  Corollary A.3.

For any $\gamma>1$, $\theta\in(0,\frac{1}{\gamma})$, there exists a constant
$C(\gamma)>0$ such that

$\displaystyle\|\eta\|_{L^{2\gamma}(0,T;H^{\theta}_{2})}^{2\gamma}\leq
C(\gamma)\left(\int_{0}^{T}|\eta(s)^{[\gamma-1]}\nabla\eta(s)|_{L^{2}}^{2}\,ds+\int_{0}^{T}|\eta(s)|^{2\gamma}_{L^{2}}\,ds\right),$

where $z^{[\gamma-1]}:=|z|^{\gamma-2}z$ for $z\in\mathbb{R}$.

We have the following embedding.

######  Proposition A.4.

Let $l_{1},l_{2}\in(2,\infty)$ with
$\frac{d}{2}-\rho\leq\frac{2}{l_{1}}+\frac{d}{l_{2}}$. Then there exists $C>0$
such that

$\|\xi\|_{L^{l_{1}}(0,T;L^{l_{2}})}\leq
C\|\xi\|_{\mathbb{H_{\rho}}},\quad\xi\in\mathbb{H_{\rho}},$

where $\mathbb{H_{\rho}}$ is defined in (3.8).

###### Proof.

First, let us note that for $\delta>0$ such that

(A.3) $\frac{1}{l_{2}}\geq\frac{1}{2}-\frac{\delta}{d},$

we have the embedding $H^{\delta}_{2}({{\mathcal{O}}})\hookrightarrow
L^{l_{2}}({{\mathcal{O}}})$, and therefore

$\|\xi\|_{L^{l_{1}}(0,T;L^{l_{2}})}\leq
C\|\xi\|_{L^{l_{1}}(0,T;H^{\delta}_{2})},\quad\xi\in\mathbb{X}.$

Due to interpolation, compare with [3, Theorem 5.1.2, p. 107 and Theorem
6.4.5, p. 152], we have

$\|\xi\|_{L^{l_{1}}(0,T;H^{\delta}_{2})}\leq
C\|\xi\|_{L^{\infty}(0,T;H^{\rho}_{2})}^{\theta}\|\xi\|_{L^{2}(0,T;H^{\rho+1}_{2})}^{1-\theta},\quad\xi\in\mathbb{H_{\rho}},$

for $\theta\in(0,1)$ with,

(A.4)
$\frac{1}{l_{1}}\leq\frac{1}{2}(1-\theta),\quad\delta\leq\theta\rho+(1-\theta)(\rho+1).$

Taking into account that we have

$\frac{d}{2}-\rho\leq\frac{2}{l_{1}}+\frac{d}{l_{2}},$

gives that $\delta$ and $\theta$ satisfying (A.3) and (A.4) exist and thus the
Young inequality for products yields the assertion. ∎

##  Appendix B The Aubin-Lions-Simon compactness theorem

Let $B$ be a separable Banach space, $I\subset[0,\infty)$ be an interval.
Recall that

$L^{p}(I;B)=\left\\{u:I\to B:\mbox{ $u$ measurable and
}\int_{I}|u(t)|_{B}^{p}\,dt<\infty\right\\}.$

Given $p\in(1,\infty)$, $\alpha\in(0,1)$, let ${\mathbb{W}}^{\alpha}_{p}(I;B)$
be the Sobolev space of all $u\in L^{p}(0,\infty;B)$ such that

$\int_{I}\int_{I\cap[t,t+1]}\;\frac{|u(t)-u(s)|_{B}^{p}}{|t-s|^{1+\alpha
p}}\,ds\,dt<\infty;$

equipped with the norm

$\left\|u\right\|_{{\mathbb{W}}^{\alpha}_{p}(I;B)}:=\left(\int_{I}\int_{I\cap[t,t+1]}\;\frac{|u(t)-u(s)|_{B}^{p}}{|t-s|^{1+\alpha
p}}\,ds\,dt\right)^{\frac{1}{p}}.$

######  Theorem B.1.

Let $B_{0}\subset B\subset B_{1}$ be Banach spaces, $B_{0}$ and $B_{1}$
reflexive, with compact embedding of $B_{0}$ to $B$. Let $p\in(1,\infty)$ and
$\alpha\in(0,1)$ be given. Let $X$ be the space

$X=L^{p}(0,T;B_{0})\cap{\mathbb{W}}^{\alpha}_{p}(0,T;B_{1}).$

Then the embedding of $X$ to $L^{p}(0,T;B)$ is compact.

###### Proof.

See [61, p. 86, Corollary 5] or [26, Theorem 2.1]. ∎

##  Appendix C On the Burkholder-Davis-Gundy inequality

We collect the exact form of the Burkholder-Davis-Gundy inequality needed
here. Given a cylindrical Wiener process $W$ on
$H^{\delta}_{2}({{\mathcal{O}}})$, $\delta>1$, over
$\mathfrak{A}=(\Omega,{\mathcal{F}},\mathbb{F},{\mathbb{P}})$, and a
progressively measurable process
$\xi\in\mathcal{M}^{2}_{\mathfrak{A}}(H^{\rho}_{2}({{\mathcal{O}}}))$,
$\rho\in[0,\frac{1}{2}]$, let us define $\\{Y(t):t\in[0,T]\\}$ by

$Y(t):=\int_{0}^{t}\xi(s)\,dW(s),\quad t\in[0,T].$

Here, for each $t\in[0,T]$, $\xi(t)$ is interpreted as a multiplication
operator acting on the elements of $H^{\delta}_{2}({\mathcal{O}})$, namely,
$\xi:H_{2}^{\delta}({\mathcal{O}})\ni\psi\mapsto\xi\psi\in{{\mathcal{S}}}^{\prime}({\mathcal{O}})$.
Since for any $\nu>\frac{1}{2}$ and for any $\varphi\in
H^{\nu}_{2}({{\mathcal{O}}})$ the product $\xi(t)\varphi$ belongs to
$H^{\rho}_{2}({{\mathcal{O}}})$ (see [52, p. 190]), we can view $\xi(t)$ as a
linear map from $H^{\nu}_{2}({{\mathcal{O}}})$ into
$H^{\rho}_{2}({{\mathcal{O}}})$. It is shown in [52, Theorem 1 (4), p. 190]
that

(C.1) $|\xi(t)\varphi|_{H^{\rho}_{2}}\leq
C|\xi(t)|_{H^{\rho}_{2}}|\varphi|_{H^{\nu}_{2}}$

where the constant $C>0$ is independent of $\varphi$. Consequently, for any
$p\geq 1$, $\delta>1$, and any $\rho\in[0,\frac{1}{2}]$

$\mathbb{E}\left[\sup_{t\in[0,T]}|Y(t)|^{p}_{H^{\rho}_{2}}\right]\leq
C\,\mathbb{E}\,\left[\int_{0}^{T}|\xi(t)|^{2}_{L_{\text{HS}}(H^{\delta}_{2},H^{\rho}_{2})}\,dt\right]^{\frac{p}{2}},$

where $|\cdot|_{L_{\text{HS}}(H^{\delta}_{2},H^{\rho}_{2})}$ denotes the
Hilbert–Schmidt norm from $H^{\delta}_{2}({{\mathcal{O}}})$ to
$H^{\rho}_{2}({{\mathcal{O}}})$. First, let us note that for $\delta>1$ there
exists a $\nu>\frac{1}{2}$ such that the embedding
$H^{\delta}_{2}({{\mathcal{O}}})\hookrightarrow H^{\nu}_{2}({{\mathcal{O}}})$
is a Hilbert–Schmidt operator. Using the fact that
$\\{\psi_{k}^{(\delta)}:k\in\mathbb{Z}\\}$ is an orthonormal basis of
$H^{\delta}_{2}({{\mathcal{O}}})$, and
$\psi_{k}^{(\delta)}=\lambda_{k}\psi_{k}$ we obtain, by using (C.1),

(C.2) $\displaystyle|\xi(s)|^{2}_{L_{\text{HS}}(H^{\delta}_{2},H^{\rho}_{2})}$
$\displaystyle=\sum_{k\in\mathbb{Z}}\left|\xi(s)\psi_{k}^{(\delta)}\right|^{2}_{H^{\rho}_{2}}\leq
C\sum_{k\in\mathbb{Z}}\left|\xi(s)\right|_{H^{\rho}_{2}}^{2}\left|\psi_{k}^{(\delta)}\right|^{2}_{H^{\nu}_{2}}$
$\displaystyle=C|\xi(s)|^{2}_{H^{\rho}_{2}}\sum_{k\in\mathbb{Z}}|\psi_{k}^{(\delta)}|^{2}_{H^{\nu}_{2}}.$

If the embedding $H^{\delta}_{2}({{\mathcal{O}}})\hookrightarrow
H^{\rho}_{2}({{\mathcal{O}}})$ is supposed to be a Hilbert–Schmidt, the right
hand side of (C.2) is finite and we obtain

(C.3) $\mathbb{E}\left[\sup_{t\in[0,T]}|Y(t)|^{p}_{H^{\rho}_{2}}\right]\leq
C\,\mathbb{E}\,\left[\int_{0}^{T}|\xi(t)|^{2}_{H^{\rho}_{2}}\,dt\right]^{\frac{p}{2}}.$

In case $\rho\geq\frac{1}{2}$, we use the fact that
$H^{\rho}_{2}({{\mathcal{O}}})$ is an algebra and obtain

(C.4) $\displaystyle|\xi(s)|^{2}_{L_{\text{HS}}(H^{\delta}_{2},H^{\rho}_{2})}$
$\displaystyle=\sum_{k\in\mathbb{Z}}\left|\xi(s)\psi_{k}^{(\delta)}\right|^{2}_{H^{\rho}_{2}}=|\xi(s)|^{2}_{H^{\rho}_{2}}\sum_{k\in\mathbb{Z}}\left|\psi_{k}^{(\delta)}\right|^{2}_{H^{\rho}_{2}}.$

If the embedding $H^{\delta}_{2}({{\mathcal{O}}})\hookrightarrow
H^{\rho}_{2}({{\mathcal{O}}})$ is a Hilbert–Schmidt operator, then the right
hand side of (C.4) is finite and we obtain for $\delta>\rho+1$

(C.5) $\mathbb{E}\left[\sup_{t\in[0,T]}|Y(t)|^{p}_{H^{\rho}_{2}}\right]\leq
C\,\mathbb{E}\,\left[\int_{0}^{T}|\xi(t)|^{2}_{H^{\rho}_{2}}\,dt\right]^{\frac{p}{2}}.$

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