text:
left[\sup_{0\leq s\leq
T}|\eta(s)|_{L^{p^{\ast}}}^{p^{\ast}_{0}}\right]\leq C,$

and for any $q_{2}\in[1,m_{0}]$,

(5.6)
$\left(\int_{0}^{T}|\phi_{\kappa}(h(\eta,\xi,t))\xi(t)|_{L^{m}}^{m_{0}}\,dt\right)\leq(2\kappa)^{1/\nu}.$

Hence, by Hölder inequality and the Sobolev embedding
$L^{1}({{\mathcal{O}}})\hookrightarrow
H^{-2}_{\frac{\gamma+1}{\gamma}}({{\mathcal{O}}})$, we have for fixed
$\omega\in\Omega$ and fixed $t\in[0,T]$,

$\displaystyle\left|\int_{{\mathcal{O}}}(-\Delta)^{-1}(\chi\phi_{\kappa}(h(\eta,\xi,t))\xi^{2}(t,x)\eta(t,x))w\,dx\right|$
$\displaystyle\leq$
$\displaystyle\chi|(-\Delta)^{-1}(\phi_{\kappa}(h(\eta,\xi,t))\xi^{2}(t)\eta(t))|_{L^{(\gamma+1)/\gamma}}|w|_{L^{\gamma+1}}$
$\displaystyle\leq$ $\displaystyle\chi
C|\phi_{\kappa}^{1/4}(h(\eta,\xi,t))\xi(t)|_{L^{m}}^{2}|\phi_{\kappa}^{1/2}(h(\eta,\xi,t))\eta(t)|_{L^{p^{\ast}}}|w|_{L^{\gamma+1}}$
$\displaystyle\leq$
$\displaystyle\chi\left[C|\phi_{\kappa}^{1/4}(h(\eta,\xi,t))\xi(t)|_{L^{m}}^{m_{0}}+C|\phi_{\kappa}^{1/2}(h(\eta,\xi,t))\eta(t)|_{L^{p^{\ast}}}^{p_{0}^{\ast}}\right]|w|_{L^{\gamma+1}}.$

Then, under the assumption (5.6), we shall verify the conditions of [42,
Theorem 5.1.3] for $\gamma>1$. First note that for fixed $t\in[0,T]$ and fixed
$\omega\in\Omega$, $A$ maps from $V$ to $V^{\ast}$. In particular, by the
calculations done in (5.2) we have for $\gamma>1$

$\displaystyle\left|\mathbin{{}_{V^{\ast}}\langle A(t,u),w\rangle_{V}}\right|$
$\displaystyle=$
$\displaystyle\left|\int_{{\mathcal{O}}}\left[r_{u}u^{[\gamma]}(x)w(x)+(-\Delta)^{-1}(\chi\phi_{\kappa}(h(\eta,\xi,t))\xi^{2}(t,x)\eta(t,x))w(x)\right]\,dx\right|$
$\displaystyle\leq$
$\displaystyle\left|r_{u}|u|_{L^{\gamma+1}}^{\gamma}|w|_{L^{\gamma+1}}+\chi|(-\Delta)^{-1}(\phi_{\kappa}(h(\eta,\xi,t))\xi^{2}(t)\eta(t))|_{L^{(\gamma+1)/\gamma}}|w|_{L^{\gamma+1}}\right|$
$\displaystyle\leq$
$\displaystyle\left[r_{u}|u|_{L^{\gamma+1}}^{\gamma}+C\chi|\phi_{\kappa}(h(\eta,\xi,t))\xi^{2}(t)\eta(t)|_{L^{1}}\right]|w|_{L^{\gamma+1}}$
$\displaystyle\leq$
$\displaystyle\left[r_{u}|u|_{L^{\gamma+1}}^{\gamma}+C\chi|\phi_{\kappa}^{1/4}(h(\eta,\xi,t))\xi(t)|_{L^{m}}^{2}|\phi_{\kappa}^{1/2}(h(\eta,\xi,t))\eta(t)|_{L^{p^{\ast}}}\right]|w|_{L^{\gamma+1}}$
$\displaystyle\leq$
$\displaystyle\left[r_{u}|u|_{L^{\gamma+1}}^{\gamma}+C\chi|\phi_{\kappa}^{1/4}(h(\eta,\xi,t))\xi(t)|_{L^{m}}^{m_{0}}+C\chi|\phi_{\kappa}^{1/2}(h(\eta,\xi,t))\eta(t)|_{L^{p^{\ast}}}^{p_{0}^{\ast}}\right]|w|_{L^{\gamma+1}},$

where only $\xi$, $\eta$ and $h(\eta,\xi,\cdot)$ depend on $t$ and $\omega$
and $C$ may change from line to line. Next, we verify Hypothesis (H1), (H2′),
(H3), and (H4′) of [42, Theorem 5.1.3].

  1. (H1):

For $\lambda\in\mathbb{R}$, $u_{1},u_{2},w\in L^{\gamma+1}({\mathcal{O}})$,
$t\in[0,T]$, $\omega\in\Omega$ consider the map

$\lambda\mapsto\langle A(t,u_{1}+\lambda u_{2}),w\rangle$

and show its hemicontinuity. Note, that we have

$\displaystyle\langle A(t,u_{1}+\lambda u_{2}),w\rangle$ $\displaystyle=$
$\displaystyle-r_{u}\int_{{\mathcal{O}}}(u_{1}(x)+\lambda
u_{2}(x))^{[\gamma]}w(x)\,dx$
$\displaystyle-\chi\int_{{\mathcal{O}}}(-\Delta)^{-1}[\phi_{\kappa}(h(\eta,\xi,t))\xi^{2}(t,x)\eta(t,x)]w(x)\,dx,$

where only $\xi$, $\eta$ and $h(\eta,\xi,\cdot)$ depend on $t$ and $\omega$.
For the first integral in the above identity, we prove hemicontinuity with the
same arguments as in [42, Example 4.1.11, p. 87]. The second integral is
independent from $\lambda$ and the assumption (H1) of [42, Theorem 5.1.3] is
satisfied.

  2. (H2′):

Let $u,w\in L^{\gamma+1}({\mathcal{O}})$, $t\in[0,T]$, $\omega\in\Omega$. Take
(A.1) into account and consider

$\displaystyle\mathbin{{}_{V^{\ast}}\langle
A(t,u)-A(t,w),u-w\rangle_{V}}+|\sigma_{1}u-\sigma_{1}w|^{2}_{L_{\textup{HS}}(H_{1},{{\mathcal{H}}})}$
$\displaystyle\leq$ $\displaystyle-
r_{u}\int_{{\mathcal{O}}}(u^{[\gamma]}(x)-w^{[\gamma]}(x))(u(x)-w(x))\,dx+C|u-w|_{{{\mathcal{H}}}}^{2}$
$\displaystyle\leq$
$\displaystyle-2^{1-\gamma}r_{u}|u-w|_{L^{\gamma+1}}^{\gamma+1}+C|u-w|_{{{\mathcal{H}}}}^{2}$
$\displaystyle\leq$ $\displaystyle C|u-w|_{{{\mathcal{H}}}}^{2},$

and (H2′) of [42, Theorem 5.1.3] is satisfied.

  3. (H3):

Let $u\in L^{\gamma+1}({\mathcal{O}})$, $t\in[0,T]$, $\omega\in\Omega$.
Straightforward calculations and taking into account Hypothesis 3.4 give,

$\displaystyle\mathbin{{}_{V^{\ast}}\langle
A(t,u),u\rangle_{V}}+|\sigma_{1}u|^{2}_{L_{\textup{HS}}(H_{1},{\mathcal{H}})}$
$\displaystyle\leq$
$\displaystyle-\int_{{\mathcal{O}}}\left[r_{u}u^{[\gamma]}(x)u(x)+\chi(-\Delta)^{-1}(\phi_{\kappa}(h(t,\eta,\xi))\xi^{2}(t,x)\eta(t,x))u(x)\right]\,dx+C|u|_{{\mathcal{H}}}^{2}$
$\displaystyle\leq$ $\displaystyle-
r_{u}|u|_{L^{\gamma+1}}^{\gamma+1}+C\chi|\phi^{1/4}_{\kappa}(h(\eta,\xi,t))\xi(t)|^{2}_{L^{m}}|\phi_{\kappa}^{1/2}(h(\eta,\xi,t))\eta(t)|_{L^{p^{\ast}}}|u|_{L^{\gamma+1}}+C|u|_{{\mathcal{H}}}^{2}$
$\displaystyle\leq$
$\displaystyle-\frac{r_{u}}{2}|u|_{L^{\gamma+1}}^{\gamma+1}+C\chi|\phi^{1/4}_{\kappa}(h(\eta,\xi,t))\xi(t)|^{2\frac{\gamma+1}{\gamma}}_{L^{m}}|\phi^{1/2}_{\kappa}(h(\eta,\xi,t))\eta(t)|_{L^{p^{\ast}}}^{\frac{\gamma+1}{\gamma}}+C|u|_{{\mathcal{H}}}^{2}$
$\displaystyle\leq$
$\displaystyle-\frac{r_{u}}{2}|u|_{L^{\gamma+1}}^{\gamma+1}+\underset{=:f(t)}{\underbrace{C\chi|\phi^{1/4}_{\kappa}(h(\eta,\xi,t))\xi(t)|_{L^{m}}^{m_{0}}+|\phi_{\kappa}^{1/2}(h(\eta,\xi,t))\eta(t)|_{L^{p^{\ast}}}^{p_{0}^{\ast}}}}+C|u|_{{\mathcal{H}}}^{2}.$

Due to the definition of the cut-off, i.e. (5.5) and (5.6), we know $f(t)$ is
bounded. As a consequence, (H3) of [42, Theorem 5.1.3] holds with $f$ as
defined above and $\theta:=r_{u}$.

  4. (H4′):

Let $u\in L^{\gamma+1}({{\mathcal{O}}})$, $t\in[0,T]$, $\omega\in\Omega$.
Then, (H4′) of [42, Theorem 5.1.3] holds by (5.2) with $\alpha:=\gamma+1$,
$\beta:=0$ and $f$ given above.

The rest of the proof follows by an application of [42, Theorem 5.1.3], noting
that

$f\in L^{1}([0,T]\times\Omega)$

by (5.5) and (5.6). ∎

######  Proposition 5.4.

Under the conditions of Theorem 5.3 and the additional conditions
$u_{0}(x)\geq 0$ for all $x\in{{\mathcal{O}}}$ and $\xi(t,x)\geq 0$,
$\eta(t,x)\geq 0$, for a.e. $(t,x)\in[0,T]\times{{\mathcal{O}}}$, the solution
$u_{\kappa}$ to (3.13) is a.e. nonnegative.

###### Proof.

For the nonnegativity of the solution to (3.13), we refer to the proof of
positivity of the stochastic porous medium equation, see Section 2.6 in [2]
and see also [1]. Mimicking the proof of nonnegativity in [2] and applying a
comparison principle [37] the nonnegativity of (3.13) can be shown. ∎

In the next proposition we are using variational methods to verify uniform
bounds of $u_{\kappa}$, $\kappa\in\mathbb{N}$.

######  Proposition 5.5.

Fix $p\geq 1$. Then, there exists a constant $C_{0}(p,T)>0$ such that for
every $u_{0}\in
L^{p+1}(\Omega,{\mathcal{F}}_{0},{\mathbb{P}};L^{p+1}({{\mathcal{O}}}))$, and
for every $\kappa\in\mathbb{N}$ and for every
$(\xi,\eta)\in\mathcal{X}_{\mathfrak{A}}(R_{0},R_{1},R_{2})$ such that
$u_{\kappa}$ solves (3.13), the following estimate

$\displaystyle\mathbb{E}\left[\sup_{0\leq s\leq
T}|u_{\kappa}(s)|_{L^{p+1}}^{p+1}\right]+\gamma
p(p+1)r_{u}\mathbb{E}\int_{0}^{T}\int_{{\mathcal{O}}}|u_{\kappa}(s,x)|^{p+\gamma-2}|\nabla
u_{\kappa}(s,x)|^{2}\,dx\,ds$
$\displaystyle{}+(p+1)\chi\mathbb{E}\int_{0}^{T}\int_{{\mathcal{O}}}u_{\kappa}^{[p]}(s,x)\phi_{\kappa}(h(\eta,\xi,s))\eta(s,x)\xi^{2}(s,x)\,dx\,ds$
$\displaystyle\leq$ $\displaystyle
C_{0}(p,T)\,\left(\mathbb{E}|u_{0}|_{L^{p+1}}^{p+1}+1\right)$

is valid.

###### Proof.

By Itô’s formula for the functional $u\mapsto|u|^{p+1}_{L^{p+1}}$, we get that
$u_{\kappa}$ satisfies for $t\in[0,T]$,

$\displaystyle|u_{\kappa}(t)|^{p+1}_{L^{p+1}}-|u_{0}|^{p+1}_{L^{p+1}}$
$\displaystyle\leq$
$\displaystyle(p+1)r_{u}\int_{0}^{t}\int_{\mathcal{O}}\Delta(u_{\kappa}^{[\gamma]}(s,x))u^{[p]}_{\kappa}(s,x)\,dx\,ds$
$\displaystyle-(p+1)\chi\int_{0}^{t}\int_{\mathcal{O}}\phi_{\kappa}(h(\eta,\xi,s))\eta(s,x)\xi^{2}(s,x)u^{[p]}_{\kappa}(s,x)\,dx\,ds$
$\displaystyle+M(t)+\frac{p}{2}(p+1)\int_{0}^{t}|u_{\kappa}(s)|^{p-1}_{L^{p+1}}|\sigma_{1}u_{\kappa}(s)|_{\gamma(H_{1},L^{p+1})}^{2}\,ds,$

where $t\mapsto M(t)$ is a local martingale. Integrating by parts, taking
expectation (we may as well apply the Burkholder-Davis-Gundy inequality, see
Section C) and rearranging yields,

$\displaystyle\mathbb{E}\left[|u_{\kappa}(s)|^{p+1}_{L^{p+1}}\right]+(p+1)r_{u}\mathbb{E}\int_{0}^{t}\int_{\mathcal{O}}\nabla(u_{\kappa}^{[\gamma]}(s,x))\cdot\nabla(u_{\kappa}^{[p]}(s,x))\,dx\,ds$
$\displaystyle+(p+1)\chi\mathbb{E}\int_{0}^{t}\int_{\mathcal{O}}\phi_{\kappa}(h(\eta,\xi,s))\eta(s,x)\xi^{2}(s,x)u^{[p]}_{\kappa}(s,x)\,dx\,ds$
$\displaystyle\leq$
$\displaystyle\mathbb{E}|u_{0}|^{p+1}_{L^{p+1}}+\frac{p}{2}(p+1)C(p)\mathbb{E}\int_{0}^{t}|u_{\kappa}(s)|^{p+1}_{L^{p+1}}\,ds+\mathbb{E}\int_{0}^{t}|\sigma_{1}u_{\kappa}(s)|_{\gamma(H_{1},L^{p+1})}^{(p+1)/(p-1)}\,ds,$

and hence by Remark 5.2 and Gronwall’s lemma,

$\displaystyle\mathbb{E}\left[\sup_{0\leq s\leq
T}|u_{\kappa}(s)|^{p+1}_{L^{p+1}}\right]+\gamma
p(p+1)r_{u}\mathbb{E}\int_{0}^{T}\int_{\mathcal{O}}|\nabla
u_{\kappa}(s,x)|^{2}|u(s,x)|^{\gamma+p-2}dx\,ds$
$\displaystyle+(p+1)\chi\mathbb{E}\int_{0}^{T}\int_{\mathcal{O}}\phi_{\kappa}(h(\eta,\xi,s))\eta(s,x)\xi^{2}(s,x)u^{[p]}_{\kappa}(s,x)\,dx\,ds$
$\displaystyle\leq$ $\displaystyle
C_{0}(p,T)\left(\mathbb{E}|u_{0}|^{p+1}_{L^{p+1}}+1\right).$

∎

We may remark that the above result now permits an application of Proposition
A.2. Note that the inequality becomes particularly useful, when
$u_{\kappa}\geq 0$ and $\eta\geq 0$.

###  5.3. Properties of the subsystem (3.14)

Given the couple
$(\eta,\xi)\in{{\mathcal{M}}}_{\mathfrak{A}}(\mathbb{Y},\mathbb{X})$, we
consider the solution $v_{\kappa}$ to the equation (3.14). First, we will In
the next proposition we investigate existence and uniqueness and the
regularity of the process $v_{\kappa}$.

######  Proposition 5.6.

Assume that the Hypotheses of Theorem 3.6 hold. Then, for any pair
$(\eta,\xi)\in\mathcal{X}_{\mathfrak{A}}(R_{0},R_{1},R_{2})\subset{{\mathcal{M}}}_{\mathfrak{A}}(\mathbb{Y},\mathbb{Z})$
and for any initial datum $v_{0}$ as in Hypothesis 3.4, there exists a
solution $v_{\kappa}$ to (3.14) on the time interval $[0,T]$ such that
$v_{\kappa}$ is continuous in $H^{\rho}_{2}({\mathcal{O}})$ and such that

$v_{\kappa}\in L^{\infty}(0,T;H^{\rho}_{2}({{\mathcal{O}}}))\cap
L^{2}(0,T;H^{\rho+1}_{2}({{\mathcal{O}}}))\quad{\mathbb{P}}\mbox{-a.s.}$

In addition, for all $\kappa\in\mathbb{N}$ there exists a constant
$C=C(T,\kappa)>0$ such that

(5.9)
$\displaystyle\mathbb{E}\|v_{\kappa}\|_{\mathbb{H_{\rho}}}^{m_{0}}\leq\mathbb{E}|v_{0}|_{H^{\rho}_{2}}^{m_{0}}+C(\kappa,T)R_{0}.$

###### Proof.

Let us consider the following system,

(5.10) $\displaystyle\qquad dw(t)$ $\displaystyle=$ $\displaystyle r_{v}\Delta
w(t)\,dt+\sigma_{2}w(t)\,dW_{2}(t),\quad w(0)=w_{0}\in
H^{\rho}_{2}({{\mathcal{O}}}).$

A solution to (5.10) is given by standard methods (see e.g. [15, Chapter 6] or
[42, Theorem 4.2.4]) such that for any $q\geq 1$,

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

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 $\delta-\rho+\frac{d}{2}>d$ give

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

Setting

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

we obtain by the Hölder inequality for $\delta/2<1-\frac{1}{\mu}$ and
$\frac{2\mu}{m_{0}}<1$

$\displaystyle\qquad\qquad\int_{0}^{T}|F(s)|_{L^{1}}^{\mu}\,ds$
$\displaystyle\leq$ $\displaystyle\sup_{0\leq s\leq
T}|\eta(s)|_{L^{p^{\ast}}}^{\mu}\int_{0}^{T}(\phi_{\kappa}(h(\eta,\xi,t)))^{\mu}|\xi(s)|_{L^{m}}^{2\mu}\,ds\leq(2\kappa)^{\frac{2\mu}{m_{0}\nu}}\sup_{0\leq
s\leq T}|\eta(s)|_{L^{p^{\ast}}}^{\mu}.$

Observe, since $\rho<2-\frac{4}{m}-\frac{d}{2}$, such a $\delta$ and $\mu$
exists. Taking expectation, we know due to the assumptions on $\eta$, that the
term can be bounded. 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

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

and similarly to before

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

∎

######  Proposition 5.7.

Under the conditions of Theorem 5.6 and the additional condition $v_{0}(x)\geq
0$ for all $x\in{{\mathcal{O}}}$ and $\eta(t,x)\geq 0$, $\xi(t,x)\geq 0$ for
a.e. $(t,x)\in[0,T]\times{{\mathcal{O}}}$, the solution $v_{\kappa}$ to (3.14)
is nonnegative a.e.

###### Proof.

The heat semigroup, which is generated by the Laplace, maps nonnegative
functions to nonnegative functions. In this way we refer to the proof of
nonnegativity by Tessitore and Zabczyk [62]. The perturbation can be
incorporated by comparison results, see [37]. ∎

######  Proposition 5.8.

Assume that the Hypotheses of Theorem 3.6 hold. In addition let us assume that
we have for the couple
$(\eta,\xi)\in\mathcal{X}_{\mathfrak{A}}(R_{0},R_{1},R_{2})\subset{{\mathcal{M}}}_{\mathfrak{A}}(\mathbb{Y},\mathbb{Z})$.
Let $v_{\kappa}$ be a solution to

$dv_{\kappa}(t)=[r_{v}\Delta
v_{\kappa}(t)+\phi_{\kappa}(h(\eta,\xi,t))\,\eta(t)\xi^{2}(t)]\,dt+\sigma_{2}v_{\kappa}(t)\,dW_{2}(t),\quad
v_{\kappa}(0)=v_{0}\in H^{-\delta_{1}}_{m}({{\mathcal{O}}}).$

Then,

  1. (i)

there exists a number $r_{0}>0$ and a constant $\tilde{C}_{2}(T)>0$ such that
for any $r\leq r_{0}$, we have for all $\kappa\in\mathbb{N}$

$\displaystyle\mathbb{E}\left\|v_{\kappa}\right\|_{L^{m_{0}}(0,T;H^{r}_{m})}^{{m_{0}}}$
$\displaystyle\leq$
$\displaystyle\tilde{C}_{2}(T)\bigg{\\{}\mathbb{E}|v_{0}|^{{m_{0}}}_{H^{-\delta_{0}}_{m}}+4\kappa\bigg{\\}}.$

  2. (ii)

there exists a number $\alpha_{0}>0$ and a constant $C=C(T)>0$ such that for
any $\alpha\leq\alpha_{0}$ we have for all $\kappa\in\mathbb{N}$

$\displaystyle\mathbb{E}\left\|v_{\kappa}\right\|^{{m_{0}}}_{\mathbb{W}_{m_{0}}^{\alpha}(0,T;H^{-\sigma}_{m})}$
$\displaystyle\leq$ $\displaystyle
C(T)\bigg{\\{}\mathbb{E}|v_{0}|^{{m_{0}}}_{H^{-\delta_{0}}_{m}}+4\kappa\bigg{\\}}.$

###### Proof.

We start to show (i). First, we get by the analyticity of the semigroup for
$\delta_{0},\delta_{1}\geq 0$

$\displaystyle\mathbb{E}\left\|v_{\kappa}\right\|_{L^{m_{0}}(0,T;H^{r}_{m})}^{{m_{0}}}$
$\displaystyle\leq$
$\displaystyle\left[\int_{0}^{T}\left\\{t^{-2{m_{0}}\delta_{0}}\mathbb{E}|v_{0}|_{H^{-\delta_{0}}_{m}}^{m_{0}}+\mathbb{E}\left|\int_{0}^{t}e^{r_{v}\Delta(t-s)}\phi_{\kappa}(h(\eta,\xi,s))\,\eta(s)\xi^{2}(s)\,ds\right|_{H^{r}_{m}}^{m_{0}}\,\right.\right.$
$\displaystyle{}+\left.\left.\mathbb{E}\left|\int_{0}^{t}e^{r_{v}\Delta(t-s)}\sigma_{2}v_{\kappa}(s)dW_{2}(s)\right|_{H^{r}_{m}}^{m_{0}}\,\right\\}\,dt\right]$
$\displaystyle=:$ $\displaystyle(I+II+III).$

Since, by the hypotheses,

$2{m_{0}}\delta_{0}<1,$

the first term, i.e. $I$ is bounded. In particular, we have

(5.14) $\displaystyle I\leq
T^{\frac{q}{m_{0}}(1-2{m_{0}}\delta_{0})}\mathbb{E}|v_{0}|^{{m_{0}}}_{H^{-\delta_{0}}_{m}}.$

Let us continue with the second term. The smoothing property of the semigroup
gives for any $\delta_{1}\geq 0$

$II\leq\mathbb{E}\left[\int_{0}^{T}\left(\int_{0}^{t}(t-s)^{-\frac{1}{2}(r+\delta_{1})}\phi_{\kappa}(h(\eta,\xi,s))\;\,\left|\eta(s)\xi^{2}(s)\right|_{H^{-\delta_{1}}_{m}}\,ds\right)^{m_{0}}\,dt\right].$

Using the Sobolev embedding $L^{1}({{\mathcal{O}}})\hookrightarrow
H^{-\delta_{1}}_{m}({{\mathcal{O}}})$, where $\delta_{1}\geq
d(1-\frac{1}{m})$, we get

$\displaystyle II$ $\displaystyle\leq$
$\displaystyle\mathbb{E}\left[\int_{0}^{T}\left(\int_{0}^{t}(t-s)^{-\frac{1}{2}(r+\delta_{1})}\phi_{\kappa}(h(\eta,\xi,s))\;\,\left|\eta(s)\xi^{2}(s)\right|_{L^{1}}\,ds\right)^{m_{0}}\,dt\right].$

Supposing $l\frac{\delta_{1}+r_{0}}{2}<1$ the Young inequality for
convolutions gives for

(5.15)
$\displaystyle\frac{1}{m_{0}}+1=\frac{1}{l}+\frac{1}{\mu}\quad\mbox{and}\quad\beta_{0}=\frac{1}{l}-\frac{1}{2}(r+\delta_{1})$

and therefore,

(5.16) $\displaystyle II$ $\displaystyle\leq$ $\displaystyle
C_{0}T^{\beta_{0}\frac{{m_{0}}}{m_{0}}}\,\mathbb{E}\left(\int_{0}^{T}\left|\phi_{\kappa}(h(\eta,\xi,s))\;\eta(s)\xi^{2}(s)\right|_{L^{1}}^{\mu}\,ds\right)^{\frac{{m_{0}}}{\mu}}.$

Which can be bounded by Hölder inequality as follows,

$\displaystyle\int_{0}^{T}\left|\phi_{\kappa}(h(\eta,\xi,s))\;\eta(s)\xi^{2}(s)\right|_{L^{1}}^{\mu}\,ds$
$\displaystyle\leq$ $\displaystyle\sup_{0\leq s\leq
T}|\eta(s)|_{L^{p^{\ast}}}^{\mu}\int_{0}^{T}\phi_{\kappa}(h(\eta,\xi,s))\;|\xi(s)|_{L^{m}}^{m_{0}}\,ds\leq
C(\kappa)R_{0}.$

Next, let us investigate $III$. We treat the stochastic term by applying [7,
Corollary 3.5 (ii)], from which it follows for $\tilde{\sigma}+r<1$ and
$\beta>0$

$\mathbb{E}\left[\int_{0}^{T}\left|\int_{0}^{t}e^{r_{v}\Delta(t-s)}\sigma_{2}v_{\kappa}(s)\,dW_{2}(s)\right|_{H^{r}_{m}}^{{m_{0}}}\,dt\right]^{\frac{q}{m_{0}}}\leq
T^{\beta}\mathbb{E}\left[\int_{0}^{T}|v_{\kappa}(t)|_{H^{-\tilde{\sigma}}_{m}}^{m_{0}}\,dt\right].$

Due to the Sobolev embedding and interpolation whenever

(5.17) $\tilde{\sigma}>\frac{d}{2}-\frac{d}{m}-\tilde{\rho}$

there exists some $\theta\in(0,1)$ such that

$|v_{\kappa}(t)|_{H^{-\tilde{\sigma}}_{m}}\leq|v_{\kappa}(t)|_{L^{m}}^{\theta}|v_{\kappa}(t)|_{H^{\tilde{\rho}}_{m}}^{1-\theta}\leq
C\,|v_{\kappa}(t)|_{L^{m}}^{\theta}|v_{\kappa}(t)|_{H^{\tilde{\rho}}_{2}}^{1-\theta}$

Thus, if $\tilde{\rho}$ satisfies (5.17), we get that

$\displaystyle\mathbb{E}\left[\int_{0}^{T}|v_{\kappa}(t)|_{H^{-\tilde{\sigma}}_{m}}^{m_{0}}\,dt\right]^{\frac{q}{m_{0}}}\leq
C\mathbb{E}\left\\{\left[\int_{0}^{T}|v_{\kappa}(t)|_{H^{\tilde{\rho}}_{2}}^{m_{0}}\,dt\right]^{1-\theta}\left[\int_{0}^{T}|v_{\kappa}(t)|_{L^{m}}^{m_{0}}\,dt\right]^{\theta}\right\\}^{\frac{q}{m_{0}}}$
$\displaystyle\leq$ $\displaystyle
C\mathbb{E}\left[\int_{0}^{T}|v_{\kappa}(t)|_{H^{\tilde{\rho}}_{2}}^{m_{0}}\,dt\right]^{(1-\theta)\frac{q}{m_{0}}}\left[\mathbb{E}\int_{0}^{T}|v_{\kappa}(t)|_{L^{m}}^{m_{0}}\,dt\right]^{\theta\frac{q}{m_{0}}}$
$\displaystyle\leq$ $\displaystyle
C(\varepsilon)\mathbb{E}\left[\int_{0}^{T}|v_{\kappa}(t)|_{H^{\tilde{\rho}}_{2}}^{m_{0}}\,dt\right]^{\frac{q}{m_{0}}}+\varepsilon\mathbb{E}\|v_{\kappa}\|_{L^{m_{0}}(0,T;L^{m})}^{q}.$

Collecting everything together, choosing $\varepsilon>0$ sufficiently small
and subtracting
$\varepsilon\mathbb{E}\|v_{\kappa}\|_{L^{m_{0}}(0,T;L^{m})}^{m_{0}}$ on both
sides, (i) follows.

The rest of the proof is devoted to item (ii). Note, that for $s<t$

$\displaystyle
v_{\kappa}(t)-v_{\kappa}(s)=(e^{r_{v}\Delta(t-s)}-\text{Id})v_{\kappa}(s)$
$\displaystyle{}+\int_{s}^{t}e^{r_{v}\Delta(t-\tilde{s})}\phi_{\kappa}(h(\eta,\xi,s))\;\eta(\tilde{s})\xi^{2}(\tilde{s})\,d\tilde{s}+\int_{0}^{s}e^{r_{v}\Delta(t-\tilde{s})}v_{\kappa}(\tilde{s})\,dW_{2}(\tilde{s}).$

Substituting it in the definition of
${\mathbb{W}}^{\alpha}_{m_{0}}([0,T];H^{-\sigma}_{m}({{\mathcal{O}}}))$, see
Appendix B, we can write

$\displaystyle\int_{0}^{T}\int_{0}^{T}\frac{{|v_{\kappa}(t)-v_{\kappa}(s)|_{H^{-\sigma}_{m}}^{m_{0}}}}{{|t-s|^{1+\alpha{m_{0}}}}}\,ds\,dt$
$\displaystyle\leq$ $\displaystyle
2\int_{0}^{T}\int_{0}^{t}\frac{|v_{\kappa}(t)-v_{\kappa}(s)|_{H^{-\sigma}_{m}}^{m_{0}}}{|t-s|^{1+\alpha{m_{0}}}}\,ds\,dt$
$\displaystyle\leq$ $\displaystyle
2\int_{0}^{T}\int_{0}^{t}\frac{|[e^{r_{v}\Delta(t-s)}-\text{Id}]v_{\kappa}(s)|_{H^{-\sigma}_{m}}^{m_{0}}}{|t-s|^{1+\alpha{m_{0}}}}\,ds\,dt$
$\displaystyle{}+2\int_{0}^{T}\int_{0}^{t}\frac{|\int_{\tilde{s}}^{t}e^{r_{v}\Delta(t-s)}\phi_{\kappa}(h(\eta,\xi,s))\;\eta(s)\xi^{2}(s)\,ds|_{H^{-\sigma}_{m}}^{m_{0}}}{|t-\tilde{s}|^{1+\alpha{m_{0}}}}\,d\tilde{s}\,dt$
$\displaystyle{}+2\int_{0}^{T}\int_{0}^{t}\frac{|\int_{s}^{t}e^{r_{v}\Delta(t-\tilde{s})}u(\tilde{s})\,dW(\tilde{s})|_{H^{-\sigma}_{m}}^{m_{0}}}{|t-s|^{1+\alpha{m_{0}}}}\,ds\,dt=:J+JJ+JJJ.$

First, let us note that by part (i) we know that there exists a number $r>0$
and a constant $\tilde{C}(T)>0$ such that

(5.18)
$\displaystyle\mathbb{E}\left[\int_{0}^{T}|v_{\kappa}(s)|^{m_{0}}_{H^{r}_{m}}\,ds\right]^{\frac{q}{m_{0}}}$
$\displaystyle\leq$
$\displaystyle\tilde{C}(T)\bigg{\\{}\mathbb{E}|v_{0}|^{{m_{0}}}_{H^{-\delta_{0}}_{m}}+4\kappa\bigg{\\}}.$

Let us consider the term $J$. Since

$|[e^{r_{v}\Delta(t-s)}-\text{Id}]x|_{H^{-\sigma}_{m}}=C\int_{0}^{t-s}\tilde{s}^{2(r+\sigma)-1}\,d\tilde{s}|x|_{H^{r}_{m}}\leq
C(t-s)^{2(r+\sigma)}|x|_{H^{r}_{m}},$

we can infer

$\displaystyle J$ $\displaystyle\leq$ $\displaystyle
2\int_{0}^{T}\int_{0}^{t}\frac{(t-s)^{2(r+\sigma){m_{0}}}|v_{\kappa}(s)|_{H^{r}_{m}}^{m_{0}}}{|t-s|^{1+\alpha{m_{0}}}}\,ds\,dt.$

The Young inequality for convolutions (with all exponents equal to $1$) gives

$\displaystyle J$ $\displaystyle\leq$
$\displaystyle\left(\int_{0}^{T}s^{{m_{0}}2(r+\delta)-1-\alpha{m_{0}}}\,ds\right)\cdot\left(\int_{0}^{T}|v_{\kappa}(s)|^{{m_{0}}}_{H^{r}_{m}}\,ds\right).$

In particular, since $2(r+\delta)>\alpha$, the right hand side is bounded by
estimate (5.18).

Now, we consider the second term $JJ$.

Jensen’s inequality gives for any $q>1$, $q^{\prime}:=\frac{q}{q-1}$,

$\displaystyle\left|\int_{\tilde{s}}^{t}e^{r_{v}\Delta(t-s)}\phi_{\kappa}(h(\eta,\xi,s))\;\eta(s)\xi^{2}(s)\,ds\right|_{H^{-\sigma}_{m}}^{q}$
$\displaystyle\leq$
$\displaystyle(t-\tilde{s})^{\frac{q}{q^{\prime}}}\left(\int_{\tilde{s}}^{t}\left|\phi_{\kappa}(h(\eta,\xi,s))\;\eta(s)\xi^{2}(s)\right|^{q}_{H^{-\sigma}_{m}}\,ds\right).$

Hence,

$\displaystyle JJ$ $\displaystyle\leq$
$\displaystyle\int_{0}^{T}\int_{0}^{t}\frac{|\int_{\tilde{s}}^{t}e^{r_{v}\Delta(t-s)}\phi_{\kappa}(h(\eta,\xi,s))\;\eta(s)\xi^{2}(s)\,ds|_{H^{-\sigma}_{m}}^{m_{0}}}{|t-\tilde{s}|^{1+\alpha{m_{0}}}}\,d\tilde{s}\,dt$
$\displaystyle\leq$
$\displaystyle\int_{0}^{T}\int_{0}^{t}\frac{(t-\tilde{s})^{\frac{m_{0}}{{m_{0}}^{\prime}}}\int_{\tilde{s}}^{t}\phi_{\kappa}(h(\eta,\xi,s))\;|\eta(s)\xi^{2}(s)|^{m_{0}}_{H^{-\sigma}_{m}}\,ds}{|t-\tilde{s}|^{1+\alpha{m_{0}}}}\,d\tilde{s}\,dt$
$\displaystyle\leq$
$\displaystyle\left[\int_{0}^{T}\int_{0}^{t}(t-s)^{\frac{m_{0}}{{m_{0}}^{\prime}}-1-\alpha{m_{0}}}\,ds\,dt\right]\,\left[\int_{0}^{T}\phi_{\kappa}(h(\eta,\xi,s))\;|\eta(s)\xi^{2}(s)|_{H^{-\sigma}_{m}}^{m_{0}}\,ds\right].$

Taking into account that $\alpha<\frac{1}{{m_{0}}^{\prime}}$, Integration and
the Sobolev embedding gives

$\displaystyle JJ$ $\displaystyle\leq$
$\displaystyle\left[\int_{0}^{T}t^{\frac{m_{0}}{m_{0}^{\prime}}-\alpha{m_{0}}}\,dt\right]\int_{0}^{T}\phi_{\kappa}(h(\eta,\xi,s))\;|\eta(s)\xi^{2}(s)|^{m_{0}}_{H^{-\tilde{r}}_{m}}\,ds$
$\displaystyle\leq$ $\displaystyle
T^{\frac{m_{0}}{m_{0}^{\prime}}+1-\alpha{m_{0}}}\int_{0}^{T}\phi_{\kappa}(h(\eta,\xi,s))\;|\eta(s)\xi^{2}(s)|^{m_{0}}_{L^{1}}\,ds\leq
4T^{\frac{m_{0}}{m_{0}^{\prime}}+1-\alpha{m_{0}}}C(\kappa)R_{0}.$

It remains to give an estimate to the second term. We can show the assertion
by the same computations as in the part of the proof for (i), i.e. starting at
(5.16).

Next, let us investigate $JJJ$. Here, again by [7, Corollary 3.5 (ii)], we get
for some $\beta>0$ that

$\mathbb{E}\left|\int_{s}^{t}e^{r_{v}\Delta(t-\tilde{s})}\sigma_{2}v_{\kappa}(\tilde{s})\,dW_{2}(\tilde{s})\right|^{{m_{0}}}_{H^{-\sigma}_{m}}\leq(t-s)^{\beta}\,\mathbb{E}\left(\int_{s}^{t}|v_{\kappa}(\tilde{s})|^{m_{0}}_{H^{-\tilde{\sigma}}_{m}}\,d\tilde{s}\right).$

In this way we get

$\displaystyle\mathbb{E}|JJJ|$ $\displaystyle\leq$ $\displaystyle
2\int_{0}^{T}\int_{0}^{t}\frac{|t-s|^{m_{0}-2}}{|t-s|^{1+\alpha{m_{0}}}}\mathbb{E}\int_{s}^{t}|v_{\kappa}(\tilde{s})|^{m_{0}}_{H^{-\tilde{\sigma}}_{m}}\,d\tilde{s}\,ds\,dt$
$\displaystyle\leq$ $\displaystyle
2\int_{0}^{T}\int_{0}^{t}|t-s|^{m_{0}(1-\alpha)-3}\left(\mathbb{E}\int_{s}^{t}|v_{\kappa}(\tilde{s})|^{m_{0}}_{H^{-\tilde{\sigma}}_{m}}\,d\tilde{s}\right)ds\,dt$

Applying the Hölder inequality, we show that

$\displaystyle\mathbb{E}|JJJ|$ $\displaystyle\leq$ $\displaystyle
C(T)\,\mathbb{E}\left(\int_{0}^{T}|v_{\kappa}(s)|^{m_{0}}_{H^{-\tilde{\sigma}}_{m}}\,ds\right).$

Collecting everything together gives (ii). ∎

######  Technical Proposition 5.9.

Let us assume that
$(\eta,\xi)\in\mathcal{X}_{\mathfrak{A}}(R_{0},R_{1},R_{2})\subset{{\mathcal{M}}}_{\mathfrak{A}}(\mathbb{Y},\mathbb{Z})$.
Then, for any ${\alpha}\in[1,\frac{m_{0}}{2})$, $q^{\ast}\in(1,m_{0}]$, and
$\kappa\in\mathbb{N}$ there exists some real numbers
$\delta_{1},\delta_{2}\in(0,1]$ and constants
$C_{1}(\kappa,R_{0}),C_{2}(\kappa,R_{0})>0$ such that we have

$\displaystyle\mathbb{E}\left\|\eta_{1}\xi_{1}^{2}-\eta_{2}\xi^{2}_{2}\right\|_{L^{\alpha}(0,T;L^{1})}^{q^{\ast}}$
$\displaystyle\leq$ $\displaystyle
C_{1}(\kappa,R_{0})\times\left(\left(\mathbb{E}\|\xi_{1}-\xi_{2}\|_{\mathbb{Z}}^{m_{0}}\right)^{\delta_{1}}+\left(\mathbb{E}\|\eta_{1}-\eta_{2}\|_{L^{\gamma+1}(0,T;L^{\gamma+1})}^{\gamma+1}\right)^{\delta_{2}}\right)$
$\displaystyle{}+C_{2}(\kappa,R_{0})\times\left(\mathbb{E}\|\xi_{1}-\xi_{2}\|_{\mathbb{Z}}^{m_{0}}+\mathbb{E}\|\eta_{1}-\eta_{2}\|_{L^{\gamma+1}(0,T;L^{\gamma+1})}^{\gamma+1}\right).$

###### Proof.

Firstly, observe that due to the cutoff function, at integrating with respect
to the time we have to take into account in which interval $s$ belongs to. Let
$\tau_{j}^{u}:=\inf_{s>0}\\{h(\eta_{j},\xi_{j},s)\geq\kappa\\}$ and
$\tau_{j}^{o}:=\inf_{s>0}\\{h(\eta_{j},\xi_{j},s)\geq 2\kappa\\}$, $j=1,2$.
Distinguishing the different cases, and noting that
$\operatorname{Lip}(\phi_{\kappa})\leq 2\kappa^{-1}$, we obtain

$\displaystyle\left|\phi_{\kappa}(h(\eta_{1},\xi_{1},s))\;\eta_{1}(s)\xi_{1}^{2}(s)-\phi_{\kappa}(h(\eta_{2},\xi_{2},s))\;\eta_{2}(s)\xi^{2}_{2}(s)\right|_{L^{1}}^{\alpha}\phantom{\Bigg{|}}$
$\displaystyle\leq$
$\displaystyle\begin{cases}|\eta_{1}(s)\xi_{1}^{2}(s)-\eta_{2}(s)\xi_{2}^{2}(s))|_{L^{1}}^{\alpha}&\mbox{
if }0\leq s\leq T\wedge\underline{\tau}^{u},\\\\[8.61108pt]
\left|\eta_{1}(s)\xi_{1}^{2}(s)-\eta_{2}(s)\xi^{2}_{2}(s)\right|_{L^{1}}^{\alpha}\\\
{}\times\operatorname{Lip}(\phi_{\kappa})^{\alpha}\big{(}\|\eta_{1}-\eta_{2}\|_{L^{\gamma+1}(0,T;L^{\gamma+1})}^{\nu}+\|\xi_{1}-\xi_{2}\|_{L^{m_{0}}(0,T;L^{m})}^{\nu}\big{)}&\\\\[8.61108pt]
\quad\quad{}+2|\eta_{1}(s)\xi_{1}^{2}(s)-\eta_{2}(s)\xi_{2}^{2}(s))|_{L^{1}}^{\alpha}&\mbox{
if }T\wedge\underline{\tau}^{u}<s\leq
T\wedge\overline{\tau}^{o},\\\\[8.61108pt] 0&\mbox{ if }T\geq
s>T\wedge\overline{\tau}^{o},\end{cases}$

where $\underline{\tau}^{u}:=\min(\tau_{1}^{u},\tau_{2}^{u})$ and
$\overline{\tau}^{o}:=\max(\tau_{1}^{o},\tau_{2}^{o})$. Therefore, we may
integrate over $[0,T]$.

Secondly, observe that for any $n\in\mathbb{N}$ and any $a,b\geq 0$, we have

$\displaystyle|a-b|$ $\displaystyle\leq$
$\displaystyle\left|\sum_{k=1}^{n}a^{\frac{k-1}{n}}(a^{\frac{1}{n}}-b^{\frac{1}{n}})b^{\frac{n-k}{n}}\right|$
$\displaystyle\leq$
$\displaystyle|a-b|^{\frac{1}{n}}\sum_{k=1}^{n}a^{\frac{k-1}{n}}b^{\frac{n-k}{n}}\leq
C|a-b|^{\frac{1}{n}}(n-1)\left(a^{\frac{n-1}{n}}+b^{\frac{n-1}{n}}\right),$

where the last inequality follows from an application of Young’s inequality.
The difference can be split in the following way for
$s\in[0,\overline{\tau}^{o}\wedge T]$

$\displaystyle\left|\xi^{2}_{1}(s,x)\eta_{1}(s,x)-\xi_{2}^{2}(s,x)\eta_{2}(s,x)\right|$
$\displaystyle\leq$
$\displaystyle|\xi_{1}(s,x)-\xi_{2}(s,x)|\xi_{1}(s,x)+\xi_{2}(s,x)||\eta_{1}(s,x)|+|\xi_{1}^{2}(s,x)+\xi_{2}^{2}(s,x)||\eta_{1}(s,x)-\eta_{2}(s,x)|.$

Let $n\in\mathbb{N}$ be that large such that

$\frac{1}{n(\gamma+1)}+\frac{n-1}{n}\frac{1}{p^{\ast}}+\frac{1}{m}\leq
1,\quad\frac{1}{n(\gamma+1)}+\frac{1}{m_{0}}\leq\frac{1}{\alpha},\quad\mbox{and}\quad\frac{1}{n(\gamma+1)}+\frac{1}{p^{\ast}}\leq\frac{1}{\alpha}.$

Therefore, we can write

$\displaystyle\int_{0}^{\overline{\tau}^{o}\wedge
T}\left|\eta_{1}(s)\xi_{1}^{2}(s)-\eta_{2}(s)\xi^{2}_{2}(s)\right|_{L^{1}}^{\alpha}\,ds$
$\displaystyle\leq$ $\displaystyle\int_{0}^{\overline{\tau}^{o}\wedge
T}\left(\left|\eta_{1}(s)-\eta_{2}(s)\right|_{L^{\gamma+1}}^{\frac{1}{n}}\left|\eta_{1}(s)-\eta_{2}(s)\right|_{L^{p^{\ast}}}^{\frac{n-1}{n}}\left|\xi^{2}_{1}(s)\right|_{L^{m}}\right)^{\alpha}\,ds$
$\displaystyle\leq$ $\displaystyle\left(\int_{0}^{\overline{\tau}^{o}\wedge
T}\left|\eta_{1}(s)-\eta_{2}(s)\right|^{\gamma+1}_{L^{\gamma+1}}\,ds\right)^{\frac{\alpha}{(\gamma+1)n}}\sup_{0\leq
s\leq
T}\left|\eta_{1}(s)+\eta_{2}(s)\right|_{L^{p^{\ast}}}^{\alpha\frac{n-1}{n}}\left(\int_{0}^{\overline{\tau}^{o}\wedge
T}\left|\xi^{2}_{1}(s)\right|_{L^{m}}^{m_{0}}\,ds\right)^{\frac{\alpha}{m_{0}}}.$

Taking expectation gives

$\displaystyle\mathbb{E}\left\\{\int_{0}^{\overline{\tau}^{o}\wedge
T}\left|\eta_{1}(s)\xi_{1}^{2}(s)-\eta_{2}(s)\xi^{2}_{2}(s)\right|_{L^{1}}^{\alpha}\,ds\right\\}^{q^{\ast}}$
$\displaystyle\leq$ $\displaystyle
C(\kappa)\,\left\\{\mathbb{E}\int_{0}^{\overline{\tau}^{o}\wedge
T}\left|\eta_{1}(s)-\eta_{2}(s)\right|^{\gamma+1}_{L^{\gamma+1}}\,ds\right\\}^{\frac{{q^{\ast}}\alpha}{(\gamma+1)n}}\left\\{\mathbb{E}\sup_{0\leq
s\leq
T}\left|\eta_{1}(s)+\eta_{2}(s)\right|_{L^{p^{\ast}}}^{p_{0}^{\ast}}\right\\}^{\frac{{q^{\ast}}\alpha}{p_{0}^{\ast}}\frac{n-1}{n}}.$

Similarly we get by the Hölder inequality

$\displaystyle\left|\xi^{2}_{1}(s)\eta_{2}(s)-\xi_{2}^{2}\eta_{2}(s)\right|_{L^{1}}\leq\left|\xi_{1}(s)-\xi_{2}(s)\right|_{L^{m}}\left|\xi_{1}(s)+\xi_{2}(s)\right|_{L^{m}}|\eta_{2}(s)|_{L^{p^{\ast}}}.$

Integration over time gives

$\displaystyle\int_{0}^{\overline{\tau}^{o}\wedge
T}\left|\xi^{2}_{1}(s)\eta_{2}(s)-\xi_{2}^{2}\eta_{2}(s)\right|_{L^{1}}^{\alpha}\,ds$
$\displaystyle\leq$ $\displaystyle\sup_{0\leq s\leq
T}|\eta_{2}(s)|^{\alpha}_{L^{p^{\ast}}}\,\left(\int_{0}^{\overline{\tau}^{o}\wedge
T}\left|\xi_{1}(s)-\xi_{2}(s)\right|_{L^{m}}^{m_{0}}\,ds\right)^{\frac{\alpha}{m_{0}}}\left(\int_{0}^{\overline{\tau}^{o}\wedge
T}\left|\xi_{1}(s)+\xi_{2}(s)\right|_{L^{m}}^{m_{0}}\,ds\right)^{\frac{\alpha}{m_{0}}}.$

Taking expectation we get

$\displaystyle\left\\{\mathbb{E}\int_{0}^{\overline{\tau}^{o}\wedge
T}\left|\xi^{2}_{1}(s)\eta_{2}(s)-\xi_{2}^{2}\eta_{2}(s)\right|_{L^{1}}^{\alpha}\,ds\right\\}^{q^{\ast}}$
$\displaystyle\leq$ $\displaystyle C(\kappa)\left\\{\mathbb{E}\sup_{0\leq
s\leq
T}|\eta_{2}(s)|_{L^{p^{\ast}}}^{p^{\ast}_{0}}\right\\}^{\frac{{q^{\ast}}\alpha}{p_{0}^{\ast}}}\,\left\\{\mathbb{E}\int_{0}^{\overline{\tau}^{o}\wedge
T}\left|\xi_{1}(s)-\xi_{2}(s)\right|_{L^{m}}^{m_{0}}\,ds\right\\}^{\frac{{q^{\ast}}\alpha}{m_{0}}}.$

Finally, take into account that

$\displaystyle\mathbb{E}\Bigg{[}\left(\int_{0}^{\overline{\tau}^{o}\wedge
T}\left|\eta_{1}(s)\xi_{1}^{2}(s)-\eta_{2}(s)\xi^{2}_{2}(s)\right|_{L^{1}}^{\alpha}\,ds\right)$
$\displaystyle{}\times\big{(}\|\eta_{1}-\eta_{2}\|_{L^{\gamma+1}(0,T;L^{\gamma+1})}^{\nu}+\|\xi_{1}-\xi_{2}\|_{L^{m_{0}}(0,T;L^{m})}^{\nu}\big{)}\Bigg{]}^{q^{\ast}}$
$\displaystyle\leq$
$\displaystyle\mathbb{E}\Bigg{[}\left(\sum_{j=1,2}\sup_{0\leq s\leq
T}|\eta_{j}(s)|_{L^{p^{\ast}}}\int_{0}^{\overline{\tau}^{o}\wedge
T}|\xi_{j}(s)|_{L^{m}}^{m_{0}}\,ds\right)$
$\displaystyle{}\times\big{(}\|\eta_{1}-\eta_{2}\|_{L^{\gamma+1}(0,T;L^{\gamma+1})}^{\nu}+\|\xi_{1}-\xi_{2}\|_{L^{m_{0}}(0,T;L^{m})}^{\nu}\big{)}\Bigg{]}^{q^{\ast}}$
$\displaystyle\leq$ $\displaystyle
C(\kappa)\,\left\\{\mathbb{E}\sum_{j=1,2}\sup_{0\leq s\leq
T}|\eta_{j}(s)|_{L^{p^{\ast}}}^{p_{0}^{\ast}}\right\\}^{\frac{q^{\ast}}{p_{0}^{\ast}}}$
$\displaystyle{}\times\Big{(}\big{(}\mathbb{E}\|\eta_{1}-\eta_{2}\|_{L^{\gamma+1}(0,T;L^{\gamma+1})}^{\gamma+1}\big{)}^{\frac{\nu
q^{\ast}}{\gamma+1}}+\big{(}\mathbb{E}\|\xi_{1}-\xi_{2}\|_{L^{m_{0}}(0,T;L^{m})}^{m_{0}}\big{)}^{\frac{\nu
q^{\ast}}{m_{0}}}\Bigg{]},$

the assertion is shown.∎

The next proposition gives the continuity property of the operator
${\mathcal{V}}_{\kappa,{\mathfrak{A}}}$.

######  Proposition 5.10.

Let
$(\eta_{1},\xi_{1})\in\mathcal{X}_{\mathfrak{A}}(R_{0},R_{1},R_{2})\subset{{\mathcal{M}}}_{\mathfrak{A}}(\mathbb{Y},\mathbb{Z})$
and
$(\eta_{2},\xi_{2})\in\mathcal{X}_{\mathfrak{A}}(R_{0},R_{1},R_{2})\subset{{\mathcal{M}}}_{\mathfrak{A}}(\mathbb{Y},\mathbb{Z})$.
Let $v_{\kappa}^{(1)}$ and $v_{\kappa}^{(2)}$ be the solutions to

$dv_{\kappa}^{(j)}(t)=[r_{v}\Delta
v_{\kappa}^{(j)}(t)+\phi_{\kappa}(h(\eta_{j},\xi_{j},t)\,\eta_{j}(t)\xi_{j}^{2}(t)]\,dt+\sigma_{2}v_{\kappa}^{(j)}(t)\,dW_{2}(t),\,\,t\in(0,T],$

with $v_{\kappa}^{(j)}(0)=v_{0},j=1,2$. Then, under the assumptions of Theorem
3.6 there exists a constant $C=C(R_{0},\kappa)\,>0$ and real numbers
$\delta_{1},\delta_{2}>0$ such that

$\displaystyle\mathbb{E}\left\|v_{\kappa}^{(1)}-v_{\kappa}^{(2)}\right\|^{m_{0}}_{L^{m_{0}}(0,T;L^{m})}$
$\displaystyle\leq$ $\displaystyle
C(R_{0},\kappa)\,\left\\{\left[\mathbb{E}\left\|\xi_{1}-\xi_{2}\right\|^{{m_{0}}}_{L^{m_{0}}(0,T;L^{m})}\right]^{\delta_{1}}+\left[\mathbb{E}\left\|\eta_{1}-\eta_{2}\right\|^{\gamma+1}_{L^{\gamma+1}(0,T;L^{\gamma+1})}\right]^{\delta_{2}}\right\\}\,.$

###### Proof.

First, we get by the analyticity of the semigroup,

$\displaystyle\hskip
68.88867pt\mathbb{E}\left(\int_{0}^{T}\left|v_{\kappa}^{(1)}(t)-v_{\kappa}^{(2)}(t)\right|^{m_{0}}_{L^{m}}\,dt\right)$
$\displaystyle\leq$
$\displaystyle\mathbb{E}\left(\int_{0}^{T}\left|\int_{0}^{t}e^{r_{v}\Delta(t-s)}(\phi_{\kappa}(h(\eta_{1},\xi_{1},s))\eta^{1}(s)\xi_{1}^{2}(s)-\phi_{\kappa}(h(\eta_{2},\xi_{2},s))\eta_{2}(s)\xi_{2}^{2}(s))\,ds\right|_{L^{m}}^{m_{0}}\,dt\right)$
$\displaystyle{}+\mathbb{E}\left(\int_{0}^{T}\left|\int_{0}^{t}(\sigma_{2}v_{\kappa}^{(1)}(s)-\sigma_{2}v_{\kappa}^{(2)}(s))\,dW_{2}(s)\right|_{L^{m}}^{m_{0}}\,dt\right)$
$\displaystyle=:$ $\displaystyle I+II.$

Let us consider $I$. Here, we use the Sobolev embedding
$L^{1}({{\mathcal{O}}})\hookrightarrow H^{-\delta}_{m}({{\mathcal{O}}})$,
where $\delta\geq d(1-\frac{1}{m})$. Next, the Young inequality for
convolution gives for $l,\mu\geq 1$, with
$\frac{1}{m_{0}}+1=\frac{1}{l}+\frac{1}{\mu}$ and
$\frac{\delta}{2}<\frac{1}{l}$

$\displaystyle I\leq
T^{\beta_{0}}\mathbb{E}\left(\int_{0}^{T}\left|\phi_{\kappa}(h(\eta_{1},\xi_{1},s))\;\eta^{1}(s)\xi_{1}^{2}(s)-\phi_{\kappa}(h(\eta_{2},\xi_{2},s))\;\eta_{2}(s)\xi_{2}^{2}(s)\right|^{\mu}_{L^{1}}\,ds\right).$

Note, the assumptions on $m_{0}$ and $m$ in the Hypothesis 3.4 gives the
existence of $l$ and $\mu$. Now, by the technical Proposition 5.9 we can infer
that there exists a constant $C=C(R_{0},\kappa)\,>0$ and
$\delta_{1},\delta_{2}>0$ such that

$\displaystyle I\leq
C(R_{0},\kappa)\,\left(\left(\mathbb{E}\|\xi_{1}-\xi_{2}\|_{L^{m_{0}}(0,T;L^{m})}^{m_{0}}\right)^{\delta_{1}}+\left(\mathbb{E}\|\eta_{1}-\eta_{2}\|_{L^{\gamma+1}(0,T;L^{\gamma+1})}^{\gamma+1}\right)^{\delta_{2}}\right).$

It remains to tackle the second term $II$. This can be done by standard
arguments using the Burkholder-Davis-Gundy inequality, see Section C. ∎

Next, we shall tackle the continuity of $u_{\kappa}$ with respect to $\eta$
and $\xi$.

######  Proposition 5.11.

Let the couples
$(\eta_{1},\xi_{1})\in\mathcal{X}_{\mathfrak{A}}(R_{0},R_{1},R_{2})\subset{{\mathcal{M}}}_{\mathfrak{A}}(\mathbb{Y},\mathbb{Z})$
and
$(\eta_{2},\xi_{2})\in\mathcal{X}_{\mathfrak{A}}(R_{0},R_{1},R_{2})\subset{{\mathcal{M}}}_{\mathfrak{A}}(\mathbb{Y},\mathbb{Z})$
be given. Let $u_{\kappa}^{(1)}$ and $u_{\kappa}^{(2)}$ be the solutions to

$du_{\kappa}^{(j)}(t)=[r_{u}\Delta(u_{\kappa}^{(j)})^{[\gamma]}(t)-\chi\phi_{\kappa}(h(\eta^{(j)},\xi^{(j)},t))\eta^{(j)}(t)(\xi^{(j)})^{2}(t)]\,dt+\sigma_{1}{u_{\kappa}}^{(j)}(t)\,dW_{1}(t),\quad
t\in(0,T],$

with $u_{\kappa}^{(j)}(0)=u_{0},j=1,2$. Then, under the assumptions of Theorem
3.6 there exist constants $C=C(\tilde{R},R_{1},R_{2})>0$ and
$c,\delta_{1},\delta_{2}>0$ such that

$\displaystyle\mathbb{E}\left[\sup_{0\leq s\leq
T}\left|u_{\kappa}^{(1)}(s)-u_{\kappa}^{(2)}(s)\right|^{2}_{H^{-1}_{2}}\right]+c\mathbb{E}\left\|u_{\kappa}^{(1)}-u_{\kappa}^{(2)}\right\|^{\gamma+1}_{L^{\gamma+1}(0,T;L^{\gamma+1})}$
$\displaystyle\leq$ $\displaystyle
C(R_{0},\kappa)\bigg{\\{}\left(\mathbb{E}\left\|\xi_{1}-\xi_{2}\right\|_{L^{m_{0}}(0,T;L^{m})}^{{2}}\right)^{\delta_{1}}+\left(\mathbb{E}\|\eta_{1}-\eta_{2}\|_{L^{\gamma+1}(0,T;L^{\gamma+1})}^{\gamma+1}\right)^{\delta_{2}}$
$\displaystyle{}+2\chi\left(\mathbb{E}\|\eta_{1}-\eta_{2}\|_{L^{\gamma+1}(0,T;L^{\gamma+1})}^{\gamma+1}+\mathbb{E}\|\xi_{1}-\xi_{2}\|_{L^{m_{0}}(0,T;L^{m})}^{m_{0}}\right)\bigg{\\}}.$

###### Proof.

Applying the Itô formula and integration by parts gives

$\displaystyle|u^{(1)}(t)-u^{(2)}(t)|_{H^{-1}_{2}}^{2}+2r_{u}\int_{0}^{t}\int_{\mathcal{O}}((u^{(1)})^{[\gamma]}(s,x)-(u^{(2)})^{[\gamma]}(s,x))(u^{(1)}(s,x)-u^{(2)}(s,x))\,dx\,ds$
$\displaystyle=$
$\displaystyle-2\chi\int_{0}^{t}\int_{\mathcal{O}}(\nabla^{-1}(u^{(1)}(s,x)-u^{(2)}(s,x)))$
$\displaystyle\qquad\qquad\qquad\times(\nabla^{-1}(\phi_{\kappa}(h(\eta_{1},\xi_{1},s))\eta_{1}(s,x)\xi_{1}^{2}(s,x)-\phi_{\kappa}(h(\eta_{2},\xi_{2},s))\eta_{2}(s,x)\xi_{2}^{2}(s,x))\,dx\,ds$
$\displaystyle+2\int_{0}^{t}\left\langle
u^{(1)}(s)-u^{(2)}(s),\sigma_{1}(u^{(1)}(s)-u^{(2)}(s))\,dW_{1}(s)\right\rangle$
$\displaystyle\qquad+\int_{0}^{t}|\sigma_{1}(u^{(1)}(s)-u^{(2)}(s))|_{L_{\text{HS}}(H_{1},H_{2}^{-1})}^{2}\,ds,$

where $\nabla^{-1}:=-(-\Delta)^{1/2}$. By the Young inequality, the
assumptions on $\sigma_{1}$ and Lemma A.1 we get for
$\varepsilon\in\left(0,2^{\frac{2-\gamma}{\gamma+1}}r_{u}^{\frac{1}{\gamma+1}}\right]$,

$\displaystyle\mathbb{E}\left[\sup_{0\leq s\leq
t}|u^{(1)}(s)-u^{(2)}(s)|_{H^{-1}_{2}}^{2}\right]+2^{2-\gamma}r_{u}\mathbb{E}\int_{0}^{t}|u^{(1)}(s)-u^{(2)}(s)|_{L^{\gamma+1}}^{\gamma+1}\,ds$
$\displaystyle\leq$
$\displaystyle\varepsilon^{\gamma+1}\mathbb{E}\int_{0}^{t}|u^{(1)}(s)-u^{(2)}(s)|_{L^{\gamma+1}}^{\gamma+1}\,ds$
$\displaystyle{}+\chi\mathbb{E}\int_{0}^{t}|(-\Delta)^{-1}(\phi_{\kappa}(h(\eta_{1},\xi_{1},s))\eta_{1}(s)\xi_{1}^{2}(s)-\phi_{\kappa}(h(\eta_{2},\xi_{2},s))\eta_{2}(s)\xi_{2}^{2}(s))|_{L^{\frac{\gamma+1}{\gamma}}}^{\frac{\gamma+1}{\gamma}}\,ds$
$\displaystyle+C\mathbb{E}\int_{0}^{t}|u^{(1)}(s)-u^{(2)}(s)|^{2}_{H_{2}^{-1}}\,ds$
$\displaystyle=:$ $\displaystyle I+II+III,$

where we have used the Burkholder-Davis-Gundy inequality.

The second term $II$ can be estimated by Technical Proposition 5.9 by noting
that $m_{0}>2\frac{\gamma+1}{\gamma}$. The embedding
$L^{1}({{\mathcal{O}}})\hookrightarrow
H^{-2}_{\frac{\gamma+1}{\gamma}}({{\mathcal{O}}})$ gives first

$\displaystyle II$ $\displaystyle\leq$
$\displaystyle\mathbb{E}\Big{\|}\phi_{\kappa}(h(\eta_{1},\xi_{1},\cdot))\eta_{1}\xi_{1}^{2}-\phi_{\kappa}(h(\eta_{2},\xi_{2},\cdot))\eta_{2}\xi_{2}^{2}\Big{\|}_{L^{\frac{\gamma+1}{\gamma}}(0,T;L^{\frac{\gamma+1}{\gamma}})}^{\frac{\gamma+1}{\gamma}}.$

Applying Technical Proposition 5.9 shows that there exists a constant
$C(\kappa)>0$ and $\delta_{1},\delta_{2}>0$ such that

$\displaystyle II$ $\displaystyle\leq$ $\displaystyle
C_{1}(\kappa,R_{0})\times\left(\left(\mathbb{E}\|\xi_{1}-\xi_{2}\|_{\mathbb{Z}}^{m_{0}}\right)^{\delta_{1}}+\left(\mathbb{E}\|\eta_{1}-\eta_{2}\|_{L^{\gamma+1}(0,T;L^{\gamma+1})}^{\gamma+1}\right)^{\delta_{2}}\right)$
$\displaystyle{}+C_{2}(\kappa,R_{0})\times\left(\mathbb{E}\|\xi_{1}-\xi_{2}\|_{\mathbb{Z}}^{m_{0}}+\mathbb{E}\|\eta_{1}-\eta_{2}\|_{L^{\gamma+1}(0,T;L^{\gamma+1})}^{\gamma+1}\right).$

Term $I$ cancels with the corresponding term on the left hand side. We may
apply Gronwall’s lemma in order to deal with the term $III$. ∎

######  Proposition 5.12.

For any initial condition $(u_{0},v_{0})$ satisfying Hypothesis 3.4,

  * (i)

there exists $r=r(T,\gamma)>0$ such that for any
$(\eta,\xi)\in\mathcal{X}_{\mathfrak{A}}(R_{0},R_{1},R_{2})$, we have

(5.20)
$\displaystyle\mathbb{E}\int_{0}^{T}|u_{\kappa}(s)|_{H_{\gamma+1}^{r}}^{\gamma+1}\,ds\leq
CR_{0};$

  * (ii)

we have that $t\mapsto u_{\kappa}(t)$ is $\mathbb{P}$-a.s. strongly continuous
in $H_{2}^{-1}$.

###### Proof.

Let us prove (i) first. We know from Proposition 5.5 with $p=1$, that for any
$(\eta,\xi)\in\mathcal{X}_{\mathfrak{A}}(R_{0},R_{1},R_{2})$ we have

$\displaystyle\mathbb{E}\int_{0}^{T}\left|u_{\kappa}^{\left[\frac{\gamma-1}{2}\right]}(s)\nabla
u_{\kappa}(s)\right|^{2}_{L^{2}}\,ds\leq CR_{0}.$

On the other side, we know by Proposition A.2 that we have for any
$\theta<\frac{2}{p+\gamma}$,

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

where the LHS is bounded for $p=1$, noting that Proposition 5.5 with
$p=\gamma$ gives

$\mathbb{E}\left[\sup_{t\in[0,T]}|u_{\kappa}(t)|^{\gamma+1}_{L^{\gamma+1}}\right]\leq
C.$

The assertion (ii) follows from [42, Theorem 4.2.5]. ∎

###  5.4. The system (3.11)–(3.12) and uniform bounds on the stopping time

Within this section let us assume that
$\\{(\bar{u}_{\kappa},\bar{v}_{\kappa})\;\colon\;\kappa\in\mathbb{N}\\}$ the
family of stochastic processes constructed in Step III of the proof of Theorem
3.6.

######  Proposition 5.13.

Fix $p\geq 1$ and suppose $u_{0}\in L^{p+1}({{\mathcal{O}}})$. Then, there
exists a constant $C_{0}(p,T)>0$ such that we have for all
$\kappa\in\mathbb{N}$

$\displaystyle\mathbb{E}\left[\sup_{0\leq s\leq
T}|\bar{u}_{\kappa}(s)|_{L^{p+1}}^{p+1}\right]+\gamma
p(p+1)r_{u}\mathbb{E}\int_{0}^{T}\int_{{\mathcal{O}}}|\bar{u}_{\kappa}(s,x)|^{p+\gamma-2}|\nabla\bar{u}_{\kappa}(s,x)|^{2}\,dx\,ds$
$\displaystyle{}+(p+1)\chi\mathbb{E}\int_{0}^{T}\int_{{\mathcal{O}}}|\bar{u}_{\kappa}(s,x)|^{p+1}|\bar{v}_{\kappa}(s,x)|^{2}\,dx\,ds\leq
C_{0}(p,T)\,\left(\mathbb{E}|u_{0}|_{L^{p+1}}^{p+1}+1\right).$

###### Proof.

Follows from an application of Fatou’s lemma and nonnegativity of $u_{\kappa}$
and $v_{\kappa}$ in the proof of Proposition 5.5. ∎

######  Proposition 5.14.

Let $\rho<1-\frac{d}{2}$ and assume that the Hypotheses of Theorem 3.6 hold.
Secondly, let us assume that the constants $R_{0},R_{1}$ and $R_{2}$ are given
as in (3.29), (3.26), and (3.28), respectively. There exists a generic
constant ${C}(T)>0$, a number $l\geq\frac{2}{|\rho|}$ and
$\delta_{1},\delta_{2}\in(0,1)$ with $\delta_{1}+\delta_{2}=1$ such that for
all $\kappa\in\mathbb{N}$ we have

(5.21)
$\displaystyle\mathbb{E}\|\bar{v}_{\kappa}\|_{\mathbb{H_{\rho}}}^{m_{0}}$
$\displaystyle\leq$ $\displaystyle
C(T)\left(\mathbb{E}|v_{0}|_{H^{-\delta_{0}}_{m}}^{m_{0}}+R_{0}^{\delta_{1}}\,R_{1}^{\delta_{1}}\right).$

Before proving Proposition 5.14, we consider the following lemma, which will
be essential.

######  Technical Lemma 5.15.

For all $1\leq\alpha<2$, $-\rho<\frac{2(2-\alpha)}{\alpha}$, and for any
$r^{\ast}\geq 1$ there exist generic constants ${C}(T)>0$ and
$\delta_{1},\delta_{2}\in(0,1)$ with $\delta_{1}+\delta_{2}=1$, such that for
any pair of nonnegative processes
$\eta,\xi:[0,T]\times\Omega\times{{\mathcal{O}}}\rightarrow[0,+\infty)$ we
have

(5.22) $\displaystyle\hskip
8.61108pt\mathbb{E}\|\eta\xi^{2}\|^{r^{\ast}}_{L^{\alpha}(0,T;L^{1})}$
$\displaystyle\leq$ $\displaystyle
C(T)\left\\{\mathbb{E}\left(\int_{0}^{T}|\eta^{p}(s)\xi^{2}(s)|_{L^{1}}\,ds\right)^{\frac{r^{\ast}p}{\alpha}}\right\\}^{\delta_{1}}\left(\mathbb{E}\|\xi\|_{\mathbb{H_{\rho}}}^{r^{\ast}}\right)^{\delta_{2}}.$

###### Proof.

Let us set $\beta=\frac{1}{p}$, $1-\beta=\frac{p-1}{p}=\frac{1}{p^{\prime}}$.
Observe also that

$\eta(s)\xi^{2}(s)=\eta(s)\xi^{2\beta}(s)\,\xi^{2(1-\beta)}(s),\quad
s\in[0,T].$

Assume $\alpha<p$ and fix $\beta=\frac{1}{p}$, $q\alpha/p=1$ and
$q^{\prime}=\frac{p}{p-\alpha}$

$\displaystyle\left\|\eta\xi^{2}\right\|_{L^{\alpha}(0,T;L^{1})}^{r^{\ast}}$
$\displaystyle\leq$
$\displaystyle\left(\int_{0}^{T}\left(\int_{{\mathcal{O}}}\left|\eta^{p}(s,x)\xi^{2\beta
p}(s,x)\right|\,dx\right)^{\frac{\alpha}{p}}\left(\int_{{\mathcal{O}}}\left|\xi^{2(1-\beta)p^{\prime}}(s,x)\right|\,dx\right)^{\frac{\alpha}{p^{\prime}}}\,ds\right)^{\frac{{r^{\ast}}}{\alpha}}$
$\displaystyle\leq$
$\displaystyle\,\left(\int_{0}^{T}\int_{{\mathcal{O}}}\left|\eta^{p}(s,x)\xi^{{2}}(s,x)\right|\,dx\,ds\right)^{\frac{{r^{\ast}}}{q}}\left(\int_{0}^{T}\left|\xi(s)\right|_{L^{(1-\beta)p^{\prime}2}}^{\frac{\alpha
q^{\prime}}{p^{\prime}}}\,ds\right)^{\frac{{r^{\ast}}}{q^{\prime}}}.$
$\displaystyle\leq$
$\displaystyle\,\left\|\eta^{p}\xi^{{2}}\right\|_{L^{1}(0,T;L^{1})}^{\frac{{r^{\ast}}}{q}}\left\|\xi\right\|_{L^{\frac{\alpha
q^{\prime}}{p^{\prime}}}(0,T;L^{(1-\beta)p^{\prime}2}))}^{{r^{\ast}}\frac{\alpha}{p^{\prime}}}$
$\displaystyle\leq$
$\displaystyle\,\left\|\eta^{p}\xi^{{2}}\right\|_{L^{1}(0,T;L^{1})}^{\frac{{r^{\ast}}}{q}}\left\|\xi\right\|_{L^{\frac{\alpha(p-1)}{p-\alpha}}(0,T;L^{2})}^{{r^{\ast}}\frac{\alpha}{p^{\prime}}}.$

Here, we need $\alpha/p^{\prime}<1$ which gives $2\leq
p<\frac{\alpha}{\alpha-1}$. Also, if

$\frac{d}{2}-\rho<\frac{2({p-\alpha})}{\alpha(p-1)}+\frac{d}{2},$

then

$\displaystyle\left\|\eta\xi^{2}\right\|_{L^{\alpha}(0,T;L^{1})}^{r^{\ast}}$
$\displaystyle\leq$
$\displaystyle\,\left\|\eta^{p}\xi^{{2}}\right\|_{L^{1}(0,T;L^{1})}^{\frac{{r^{\ast}}}{q}}\left\|\xi\right\|_{\mathbb{H_{\rho}}}^{{r^{\ast}}\frac{\alpha}{p^{\prime}}}.$

Taking the expectation and applying the Hölder inequality where we have to
take into account that $\alpha/p^{\prime}<1$ we get the assertion. ∎

###### Proof of Proposition 5.14:.

Let us consider the following system for a locally integrable and
progressively measurable $t\mapsto F(t)$,

(5.23) $\displaystyle\qquad dw(t)$ $\displaystyle=$ $\displaystyle r_{v}\Delta
w(t)\,dt+\sigma_{2}w(t)\,dW_{2}(t),\quad w(0)=w_{0}\in
H^{\rho}_{2}({{\mathcal{O}}}).$

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


text:
}{\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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license: Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/

authors: Erika Hausenblas and Jonas M. T\"olle

submitter: Jonas M. T\"olle Dr. math.