$\widetilde{R}$ and so the treatment of the term $\mathcal{M}_{2}$ here is
carried out as in (4.36).
Thus we discuss here the term $\mathcal{M}_{3}$ that is treated differently.

Observe that using (2.18) we obtain for some $C(T,p,K_{3})>0$ independent of
$\widetilde{R}$,

$\displaystyle\widetilde{\bf
E}\left[\left\|\mathcal{M}_{3}(\tau_{\widetilde{R}}+\delta)-\mathcal{M}_{3}(\tau_{\widetilde{R}})\right\|_{V^{\prime}+X^{\prime}}\right]$
$\displaystyle\lesssim\widetilde{\bf
E}\int_{\tau_{\widetilde{R}}}^{\tau_{\widetilde{R}}+\delta}|F(\bar{\bar{u}}^{\widetilde{R}}(s))|_{X^{\prime}}\mathrm{d}s$
$\displaystyle\lesssim\delta^{\frac{1}{p}}\left[\widetilde{\bf
E}\int_{\tau_{\widetilde{R}}}^{\tau_{\widetilde{R}}+\delta}|F(\bar{\bar{u}}^{\widetilde{R}}(s))|^{\frac{p}{p-1}}_{X^{\prime}}\mathrm{d}s\right]^{\frac{p-1}{p}}$
$\displaystyle\lesssim\delta^{\frac{1}{p}}\left[\widetilde{\bf
E}\int_{\tau_{\widetilde{R}}}^{\tau_{\widetilde{R}}+\delta}\|\bar{\bar{u}}^{\widetilde{R}}(s)\|^{p}_{X}\mathrm{d}s\right]^{\frac{p-1}{p}}$
(6.6) $\displaystyle\leq C\delta^{\frac{1}{p}}.$

Hence the condition (4.31) is satisfied by $\bar{\bar{u}}^{\widetilde{R}}$
with $\Xi=V^{\prime}+X^{\prime}$, $\alpha=1$ and
$\epsilon=\min\\{\frac{1}{p},\frac{1}{4}\\}$ which in turn implies that,
$\mu_{u}^{\widetilde{R}}$, the laws of
$\\{\bar{\bar{u}}^{\widetilde{R}}\\}_{\widetilde{R}>1}$ are tight in the space
$\mathcal{D}([0,T];V^{\prime}+X^{\prime})$ endowed with the Skorohod topology.
Hence, finally applying the procedure in Proposition 4.9 together with
Proposition 6.1, we finish the proof of Proposition 6.2. ∎

Thus, using Prohorov’s theorem we obtain the existence of a measure
$\mu^{\infty}_{2}$ on $\Upsilon_{1}=H\times\mathcal{Z}_{1}\times
C([0,T];U)\times\mathcal{N}^{\\#*}_{[0,\infty)\times E}$ such that the
sequence $\mu^{\widetilde{R}}$, of the joint probability law of
$(\bar{\bar{u}}_{0}^{\widetilde{R}},\bar{{\bar{u}}}^{\widetilde{R}},\bar{{\bar{W}}}^{\widetilde{R}},\bar{\bar{\pi}}^{\widetilde{R}})$
indexed by some infinite set $\Lambda^{\prime}\subseteq\mathbb{N}$, converges
weakly to $\mu^{\infty}_{2}$ as $\widetilde{R}\rightarrow\infty$. Next we use
Theorem A.1 again to obtain the following result.

######  Theorem 6.3.

For $\widetilde{R}\in\Lambda^{\prime}$ there exist ${{\Upsilon}_{1}}$-valued
random variables
$(\widetilde{u}_{0}^{\widetilde{R}},\widetilde{{u}}^{\widetilde{R}},\widetilde{{W}}^{\widetilde{R}},\widetilde{\pi}^{\widetilde{R}})$
and
$(\widetilde{u}_{0},\widetilde{u},\widetilde{W},\widetilde{\pi})$ such that

  1. i)

The probability laws of
$(\widetilde{u}_{0}^{\widetilde{R}},\widetilde{{u}}^{\widetilde{R}},\widetilde{{W}}^{\widetilde{R}},\widetilde{\pi}^{\widetilde{R}})$
and $(\widetilde{u}_{0},\widetilde{u},\widetilde{W},\widetilde{\pi})$ are
$\mu^{\widetilde{R}}$ and $\mu_{2}^{\infty}$, respectively,

  2. ii)

$(\widetilde{W}^{\widetilde{R}},\widetilde{\pi}^{\widetilde{R}})=(\widetilde{{W}},\widetilde{{\pi}})$
everywhere on $\widetilde{\Omega}$,

  3. iii)

$(\widetilde{u}_{0}^{\widetilde{R}},\widetilde{{u}}^{\widetilde{R}},\widetilde{{W}}^{\widetilde{R}},\widetilde{\pi}^{\widetilde{R}})$
converges almost surely to
$(\widetilde{u}_{0},\widetilde{u},\widetilde{W},\widetilde{\pi})$ as
$\widetilde{R}\rightarrow\infty$ in the topology of ${\Upsilon}_{1}$:

(6.7)
$\displaystyle\widetilde{{u}}^{\widetilde{R}}_{0}\rightarrow\widetilde{{u}}_{0}\text{
in }H,\widetilde{{\bf P}}\textnormal{-a.s.},$ (6.8)
$\displaystyle\widetilde{{u}}^{\widetilde{R}}\rightarrow\widetilde{u}\text{ in
}L^{2}(0,T;H)\cap\mathcal{D}([0,T];V^{\prime}+X^{\prime}),\widetilde{\bf
P}\textnormal{-a.s.},$

  4. iv)

$\widetilde{\pi}$, is a time-homogeneous Poisson random measure over
$(\widetilde{\Omega},{\widetilde{\mathcal{F}}},\widetilde{{\bf P}})$ with
intensity measure $\mathrm{d}t\otimes\mathrm{d}\nu$.

  5. v)

$\widetilde{u}^{\widetilde{R}}_{0}=\widetilde{u}^{\widetilde{R}}(0)$
$\widetilde{\bf P}$-a.s. and
$(\widetilde{{u}}^{\widetilde{R}},\widetilde{W},\widehat{\widetilde{\pi}})$
satisfies the following equation, $\widetilde{\bf P}$\- a.s. for every
$t\in[0,T]$

(6.9)
$\begin{split}&\widetilde{{u}}^{\widetilde{R}}(t)+\int_{0}^{t}\left[\gamma
A(\widetilde{{u}}^{\widetilde{R}}(s))+\theta_{\widetilde{R}}(|\widetilde{u}^{\widetilde{R}}|)B(\widetilde{{u}}^{\widetilde{R}}(s),\widetilde{{u}}^{\widetilde{R}}(s))+F(\widetilde{{u}}^{\widetilde{R}}(s))\right]\mathrm{d}s\\\
&\qquad\qquad=\widetilde{{u}}^{\widetilde{R}}_{0}+\int_{0}^{t}G(\widetilde{{u}}^{\widetilde{R}}(s))\mathrm{d}\widetilde{W}(s)+\int_{(0,t]}\int_{E_{0}}K(\widetilde{{u}}^{\widetilde{R}}(s),\xi)\mathrm{d}\widehat{\widetilde{\pi}}(s,\xi),\end{split}$

with respect to the filtration
$(\widetilde{\mathcal{F}}^{\widetilde{R}}_{t})_{t\geq 0}$ defined similarly to
(4.45). (See also Remark 5.6.)

Henceforth we will restrict our analysis to
$\widetilde{R}\in\Lambda^{\prime}$. Comparing equations (5.1) and (6.9) we can
observe that $\widetilde{u}^{\widetilde{R}}$ satisfies the energy estimates
(6.1) as well. This further implies that $\widetilde{u}\in
L^{\infty}([0,T];L^{2}(\widetilde{\Omega},H))\cap
L^{2}([0,T]\times{\widetilde{\Omega}},\mathrm{d}t\otimes\mathrm{d}\widetilde{{\bf
P}};V)\cap
L^{p}([0,T]\times{\widetilde{\Omega}},\mathrm{d}t\otimes{\mathrm{d}\widetilde{\bf
P}};X)$ and that there exists a subsequence $\widetilde{R}\rightarrow\infty$
such that,

(6.10) $\displaystyle\widetilde{u}^{\widetilde{R}}$
$\displaystyle\rightharpoonup\widetilde{u}\text{ weakly in
}L^{2}([0,T]\times\widetilde{\Omega},\mathrm{d}t\otimes\mathrm{d}\widetilde{{\bf
P}};V)\cap
L^{p}([0,T]\times\widetilde{\Omega},\mathrm{d}t\otimes\mathrm{d}\widetilde{{\bf
P}};X),$ (6.11) $\displaystyle\widetilde{u}^{\widetilde{R}}$
$\displaystyle\rightharpoonup\widetilde{u}\text{ weak${}^{*}$ in
}L^{\infty}([0,T];L^{2}(\widetilde{\Omega},H)).$

Observe that due to (2.18) we also have, for a subsequence
$\widetilde{R}\rightarrow\infty$, the existence of $Y\in
L^{\frac{p}{p-1}}([0,T]\times\widetilde{\Omega},\mathrm{d}t\otimes\mathrm{d}\widetilde{{\bf
P}};X^{\prime})$ such that

(6.12) $\displaystyle F(\widetilde{u}^{\widetilde{R}})$
$\displaystyle\rightharpoonup Y\text{ weakly in
}L^{\frac{p}{p-1}}([0,T]\times\widetilde{\Omega},\mathrm{d}t\otimes\mathrm{d}\widetilde{{\bf
P}};X^{\prime}).$

Now we are in position to prove one of our main results, namely Theorem 3.3.
We define $(\widetilde{\mathcal{F}})_{t\geq 0}$ as in (4.45) and denote
$\widetilde{\mathscr{S}}:=({\widetilde{\Omega}},{\widetilde{{\mathcal{F}}}},(\widetilde{{\mathcal{F}}}_{t})_{t\geq
0},\widetilde{{\bf P}},\widetilde{W},\widetilde{\pi})$. In the remainder of
the section we will show that if ${\bf E}[|u_{0}|^{2}]<\infty$ and $p>2$, then
$(\widetilde{\mathscr{S}},\widetilde{u})$ is a martingale solution to
(3.1)-(3.2) (in the sense of Definition 3.1) with initial condition
$\widetilde{{u}}_{0}$, that has the same law as $u_{0}$. We will do this in
two steps.
Step 1: We will first show that $\widetilde{u}$ solves the following equation

(6.13) $\begin{split}&\widetilde{u}(t)+\int_{0}^{t}[\gamma
A(\widetilde{u}(s))+B(\widetilde{u}(s))+Y(s)]\mathrm{d}s\\\ &\hskip
144.54pt=\widetilde{u}_{0}+\int_{0}^{t}G(\widetilde{u}(s))\mathrm{d}\widetilde{W}(s)+\int_{(0,t]}\int_{E_{0}}K(\widetilde{u}(s-),\xi)\mathrm{d}\widehat{\widetilde{\pi}}(s,\xi),\end{split}$

where $Y$ was defined in (6.12).

For that purpose we first observe that the following convergence results hold
for $\phi\in D(A)$, as $\widetilde{R}\rightarrow\infty$:

(6.14)
$\displaystyle\gamma\int_{0}^{t}(A\widetilde{u}^{\widetilde{R}}(s),\phi)\mathrm{d}s\rightarrow\gamma\int_{0}^{t}(A\widetilde{u}(s),\phi)\mathrm{d}s\quad\text{
in }L^{1}(\widetilde{\Omega}\times[0,T]),$ (6.15)
$\displaystyle\int_{0}^{t}G(\widetilde{u}^{\widetilde{R}}(s))\mathrm{d}\widetilde{W}(s)\rightarrow\int_{0}^{t}G(\widetilde{u}(s))\mathrm{d}\widetilde{W}(s)\quad\text{
in }L^{1}(\widetilde{\Omega};L^{1}(0,T;H)),$ (6.16)
$\displaystyle\int_{(0,t]}\int_{E_{0}}K(\widetilde{u}^{\widetilde{R}}(s-),\xi)\mathrm{d}\widehat{\widetilde{\pi}}(s,\xi)\rightarrow\int_{(0,t]}\int_{E_{0}}{K(\widetilde{u}(s-),\xi)}{}\mathrm{d}\widehat{\bar{\pi}}(s,\xi)\text{
in $L^{1}(\widetilde{\Omega};L^{1}(0,T;H)).$}$

We obtain the above convergence results as in the proofs of (4.51), Lemma 4.14
and Lemma 4.15 respectively. In fact the above convergence result (6.14) holds
in $L^{2}(\widetilde{\Omega}\times[0,T])$ and (6.15)-(6.16) hold in
$L^{2}(\widetilde{\Omega};L^{2}(0,T;H))$. This is due to the availability of
the $L^{p}(\widetilde{\Omega};L^{p}(0,T;X))$ bounds, for $p>2$, found in (6.1)
which can be used to upgrade the estimates (4.60) and (4.4.1).

Next we will prove that,

(6.17)
$\displaystyle\widetilde{u}^{\widetilde{R}}\rightarrow\widetilde{u}\text{ in
}L^{2}(\widetilde{\Omega};L^{2}(0,T;H))\quad\text{as
$\widetilde{R}\rightarrow\infty$}.$

Since $p>2$, using (6.1) we observe that for some $C=C(T,p,K_{3})>0$
independent of $\widetilde{R}$, we have

$\displaystyle\sup_{\widetilde{R}\geq 1}\widetilde{{\bf
E}}|\widetilde{u}^{\widetilde{R}}|^{p}_{L^{2}(0,T;H)}$
$\displaystyle=\sup_{\widetilde{R}\geq 1}\widetilde{{\bf
E}}(\int_{0}^{T}|\widetilde{u}^{\widetilde{R}}|^{2}\mathrm{d}s)^{\frac{p}{2}}\lesssim\sup_{\widetilde{R}\geq
1}\widetilde{{\bf
E}}\int_{0}^{T}|\widetilde{u}^{\widetilde{R}}|^{p}\mathrm{d}s\lesssim\sup_{\widetilde{R}\geq
1}\widetilde{{\bf
E}}\int_{0}^{T}\|\widetilde{u}^{\widetilde{R}}\|_{X}^{p}\mathrm{d}s\lesssim
C.$

Combining this result with (6.8) and applying the Vitali convergence theorem,
we infer (6.17). Next we will prove that as $\widetilde{R}\rightarrow\infty$,

(6.18)
$\int_{0}^{t}(B_{\widetilde{R}}(\widetilde{u}^{\widetilde{R}},\widetilde{u}^{\widetilde{R}}),\phi)\mathrm{d}s\rightarrow\int_{0}^{t}(B(\widetilde{u},\widetilde{u}),\phi)\mathrm{d}s\quad\text{
in $L^{1}(\widetilde{\Omega}\times[0,T])$.}$

We begin the proof of (6.18) by observing that,

$\displaystyle\left|\int_{0}^{t}(B_{\widetilde{R}}(\widetilde{u}^{\widetilde{R}},\widetilde{u}^{\widetilde{R}})-B(\widetilde{u},\widetilde{u}),\phi)\mathrm{d}s\right|$
$\displaystyle\leq\int_{0}^{t}\Big{[}|\theta_{\widetilde{R}}(|\widetilde{u}^{\widetilde{R}}|)\left(B(\widetilde{u}^{\widetilde{R}}-\widetilde{u},\widetilde{u}^{\widetilde{R}})+B(\widetilde{u},\widetilde{u}^{\widetilde{R}}-\widetilde{u}),\phi\right)|$
$\displaystyle\hskip
72.26999pt+|\left(1-\theta_{\widetilde{R}}(|\widetilde{u}^{\widetilde{R}}|)\right)(B(\widetilde{u},\widetilde{u}),\phi)|\Big{]}\mathrm{d}s$
$\displaystyle:=I_{1}^{\widetilde{R}}(t)+I_{2}^{\widetilde{R}}(t).$

We use (2.6) to treat $I_{1}^{\widetilde{R}}$ as follows

$\displaystyle I_{1}^{\widetilde{R}}(t)$
$\displaystyle\leq\int_{0}^{t}|\widetilde{u}^{\widetilde{R}}-\widetilde{u}|\left(\|\widetilde{u}^{\widetilde{R}}\|+\|\widetilde{u}\|\right)\|\phi\|_{D(A)}\mathrm{d}s,$

which implies that for some $C=C(T,p)>0$ independent of $\widetilde{R}$,

(6.19) $\displaystyle\widetilde{{\bf
E}}\int_{0}^{T}I^{\widetilde{R}}_{1}\mathrm{d}t$ $\displaystyle\leq
C\|\phi\|_{D(A)}\left(\widetilde{{\bf
E}}\int_{0}^{T}|\widetilde{u}^{\widetilde{R}}-\widetilde{u}|^{2}\mathrm{d}s\right)^{\frac{1}{2}}\left(\widetilde{{\bf
E}}\int_{0}^{T}\|\widetilde{u}^{\widetilde{R}}\|^{2}+\|\widetilde{u}\|^{2}\mathrm{d}s\right)^{\frac{1}{2}}.$

Thanks to (6.1) and (6.17) we then obtain,

$\displaystyle\widetilde{{\bf E}}\int_{0}^{T}I^{\widetilde{R}}_{1}\mathrm{d}t$
$\displaystyle\leq C\|\phi\|_{D(A)}\left(\widetilde{{\bf
E}}\int_{0}^{T}|\widetilde{u}^{\widetilde{R}}-\widetilde{u}|^{2}\mathrm{d}s\right)^{\frac{1}{2}}\rightarrow
0\text{ as }\widetilde{R}\rightarrow\infty.$

Thus thinning the sequence $\widetilde{R}$ if required, we see that
$I^{\widetilde{R}}_{1}\rightarrow 0$ a.e. on $\widetilde{\Omega}\times[0,T]$
as $\widetilde{R}\rightarrow\infty$.

We invoke (6.17) to obtain that, for a subsequence,
$\widetilde{u}^{\widetilde{R}}\rightarrow\widetilde{u}$ a.e. in
$\Omega\times[0,T]$. Hence,

$\displaystyle|\theta_{\widetilde{R}}(|\widetilde{u}^{\widetilde{R}}|)-1|$
$\displaystyle\leq|\theta_{\widetilde{R}}(|\widetilde{u}^{\widetilde{R}}|)-\theta_{\widetilde{R}}(|\widetilde{u}|)|+|\theta_{\widetilde{R}}(|\widetilde{u}|)-1|$
$\displaystyle\leq||\widetilde{u}^{\widetilde{R}}|-|\widetilde{u}||+|\theta_{\widetilde{R}}(|\widetilde{u}|)-1|$
$\displaystyle\leq|\widetilde{u}^{\widetilde{R}}-\widetilde{u}|+\mathbbm{1}_{\\{|\widetilde{u}|>2\widetilde{R}\\}}\rightarrow
0\qquad\text{ a.e. on }\Omega\times[0,T].$

This is key in treating the term $I^{\widetilde{R}}_{2}$. Observe that (2.6)
gives,

$\displaystyle I^{\widetilde{R}}_{2}$
$\displaystyle\leq\int_{0}^{t}\left|1-\theta_{\widetilde{R}}(|\widetilde{u}^{\widetilde{R}}|)\right||(B(\widetilde{u},\widetilde{u}),\phi)|\mathrm{d}s\leq\|\phi\|_{D(A)}\int_{0}^{t}\left|1-\theta_{\widetilde{R}}(|\widetilde{u}^{\widetilde{R}}|)\right||\widetilde{u}|\|\widetilde{u}\|\mathrm{d}s.$

The continuous embedding $X\subset H$ thus implies that for some $C=C(T,p)>0$

$\displaystyle\widetilde{{\bf E}}\int_{0}^{T}I^{\widetilde{R}}_{2}\mathrm{d}t$
$\displaystyle\lesssim\|\phi\|_{D(A)}\left(\widetilde{{\bf
E}}\int_{0}^{T}\left|1-\theta_{\widetilde{R}}(|\widetilde{u}^{\widetilde{R}}|)\right|^{\frac{2p}{p-2}}\mathrm{d}s\right)^{\frac{p-2}{2p}}\left(\widetilde{{\bf
E}}\int_{0}^{T}\|\widetilde{u}\|_{X}^{p}\mathrm{d}s\right)^{\frac{1}{p}}\left(\widetilde{{\bf
E}}\int_{0}^{T}\|\widetilde{u}\|^{2}\mathrm{d}s\right)^{\frac{1}{2}}$
$\displaystyle\leq C\|\phi\|_{D(A)}\left(\widetilde{{\bf
E}}\int_{0}^{T}\left|1-\theta_{\widetilde{R}}(|\widetilde{u}^{\widetilde{R}}|)\right|^{\frac{2p}{p-2}}\mathrm{d}s\right)^{\frac{p-2}{2p}}$
$\displaystyle\rightarrow 0\text{ as }\widetilde{R}\rightarrow\infty\text{
using the Lebesgue dominated convergence theorem}.$

This leads to the convergence of $I^{\widetilde{R}}_{2}\rightarrow 0$ a.e. on
$\widetilde{\Omega}\times[0,T]$ for a subsequence
$\widetilde{R}\rightarrow\infty$. Hence we have shown that

(6.20)
$\displaystyle\int_{0}^{t}(B_{\widetilde{R}}(\widetilde{u}^{\widetilde{R}}(s),\widetilde{u}^{\widetilde{R}}(s))\mathrm{d}s\rightarrow\int_{0}^{t}(B(\widetilde{u}(s),\widetilde{u}(s)),\phi)\mathrm{d}s\qquad\text{a.e.
on $\widetilde{\Omega}\times[0,T]$ as $\widetilde{R}\rightarrow\infty$.}$

Next, we choose $1<q<\frac{2p}{2+p}<2$. Using (2.7) we obtain that for some
$C=C(\phi,T,p,q,K_{3})>0$ independent of $\widetilde{R}$

$\displaystyle\sup_{\widetilde{R}\geq 1}\widetilde{{\bf
E}}\int_{0}^{T}\bigg{|}\int_{0}^{t}(B_{\widetilde{R}}(\widetilde{u}^{\widetilde{R}},\widetilde{u}^{\widetilde{R}}),\phi)\mathrm{d}s\bigg{|}^{q}\mathrm{d}t$
$\displaystyle=\widetilde{{\bf
E}}\int_{0}^{T}\bigg{|}\int_{0}^{t}\theta_{\widetilde{R}}(|u^{\widetilde{R}}|)(B(\widetilde{u}^{\widetilde{R}},\widetilde{u}^{\widetilde{R}}),\phi)\mathrm{d}s\bigg{|}^{q}\mathrm{d}t$
$\displaystyle\lesssim T\|\phi\|_{D(A)}^{q}\widetilde{{\bf
E}}\left(\int_{0}^{T}|\widetilde{u}^{\widetilde{R}}|\|\widetilde{u}^{\widetilde{R}}\|\mathrm{d}s\right)^{q}$
$\displaystyle\lesssim
T^{\frac{2p-q(p+2)}{2p}+1}\|\phi\|_{D(A)}^{q}\left(\widetilde{{\bf
E}}\int_{0}^{T}\|\widetilde{u}^{\widetilde{R}}\|^{p}_{X}\mathrm{d}s\right)^{\frac{q}{p}}\left(\widetilde{{\bf
E}}\int_{0}^{T}\|\widetilde{u}^{\widetilde{R}}\|^{2}\mathrm{d}s\right)^{\frac{q}{2}}$
(6.21) $\displaystyle\leq C(T,\phi,p,q)\quad\text{thanks to
\eqref{energytilR}.}$

Thus using the Vitali convergence theorem with (6.20) and (6.21) we obtain
(6.18).
Accumulating all convergence results (6.14), (6.15), (6.16) and (6.18), along
with (6.7), (6.8) and (6.12), we see that for any $\phi\in D(A)$ and for any
measurable set $\mathcal{S}\subset\widetilde{\Omega}\times[0,T]$, we have

(6.22) $\widetilde{{\bf
E}}\int_{0}^{T}\chi_{\mathcal{S}}\bigg{[}(\widetilde{u}(s),\phi)\mathrm{d}s+\int_{0}^{t}\gamma(A\widetilde{u}(s),\phi)\mathrm{d}s+\int_{0}^{t}(B(\widetilde{u}(s),\widetilde{u}(s),\phi)\mathrm{d}s+\int_{0}^{t}(Y,\phi)\mathrm{d}s\bigg{]}\mathrm{d}t\\\
=\widetilde{{\bf
E}}\int_{0}^{T}\chi_{\mathcal{S}}\bigg{[}(\widetilde{u}_{0},\phi)\mathrm{d}s+\int_{0}^{t}(G(\widetilde{u}(s)),\phi)\mathrm{d}\widetilde{W}(s)+\int_{(0,t]}\int_{E_{0}}(K(\widetilde{u}(s-),\xi),\phi)\mathrm{d}\widehat{\widetilde{\pi}}(s,\xi)\bigg{]}\mathrm{d}t.$

This finishes the proof of (6.13).
Step 2: It remains to verify that

(6.23) $\displaystyle Y=F(\widetilde{u}),\,\,\,\mathrm{d}\widetilde{{\bf
P}}\otimes\mathrm{d}t\textnormal{-a.e.}$

We will apply the Minty method one last time here to prove (6.23).
Using the exact argument from Step 2 in the proof of Theorem 4.13 that leads
to inequality (4.73), we obtain for any $\varphi\in
L^{\infty}([0,T],\mathrm{d}t;\mathbb{R})$ such that $\varphi\geq 0$, and any
$\phi\in
L^{p}([0,T]\times\widetilde{\Omega},\mathrm{d}t\otimes\mathrm{d}\widetilde{{\bf
P}};X)$

(6.24) $\displaystyle\widetilde{\bf
E}\int_{0}^{T}\varphi(t)\bigg{(}\int_{0}^{t}2\gamma(A\widetilde{u}(s)-A\phi(s),\widetilde{u}(s)-\phi(s))\mathrm{d}s+2(F(\phi(s))-Y,\phi(s)-\widetilde{u}(s))\mathrm{d}s$
$\displaystyle+\rho|\widetilde{u}(s)-\phi(s)|^{2}\mathrm{d}s\bigg{)}\mathrm{d}t\geq
0.$

As earlier, we choose $\phi={\widetilde{u}}\pm\lambda\zeta{\bf v}$ for
$\lambda\geq 0$, ${\bf v}\in X$ and $\zeta\in
L^{\infty}([0,T]\times\widetilde{\Omega},\mathrm{d}t\otimes\mathrm{d}\widetilde{{\bf
P}};\mathbb{R})$ and then divide both sides by $\lambda$. Letting
$\lambda\rightarrow 0$ we obtain

(6.25) $\widetilde{{\bf
E}}\left(\int_{0}^{T}\varphi(t)\left(\int_{0}^{t}\zeta(s)(F(\widetilde{u}(s))-Y,{\bf
v})\mathrm{d}s\right)\mathrm{d}t\right)=0.$

Since $\varphi$, $\zeta$ and ${\bf v}$ are all arbitrary, we conclude that

(6.26) $Y=F(\widetilde{u})\quad\mathrm{d}\widetilde{{\bf
P}}\otimes\mathrm{d}t\textnormal{-a.e.}$

Thus we have proven that for the stochastic basis
$(\widetilde{\Omega},\widetilde{\mathcal{F}},(\widetilde{\mathcal{F}}_{t})_{t\geq
0},\widetilde{{\bf P}},\widetilde{W},\widetilde{\pi})$,

(6.27) $\begin{split}&\widetilde{u}(t)+\int_{0}^{t}[\gamma
A(\widetilde{u}(s))+B(\widetilde{u}(s),\widetilde{u}(s))+F(\widetilde{u}(s))]\mathrm{d}s\\\
&\hskip
122.85876pt=\widetilde{u}_{0}+\int_{0}^{t}G(\widetilde{u}(s))\mathrm{d}\widetilde{W}(s)+\int_{(0,t]}\int_{E_{0}}K(\widetilde{u}(s-),\xi)\mathrm{d}\widehat{\widetilde{\pi}}(s,\xi),\end{split}$

holds $\widetilde{{\bf P}}$-a.s. for all $t\in[0,T]$, where
$\widetilde{u}_{0}$ has the same law as $u_{0}$. That is, we have finished the
proof of Theorem 3.3, namely the proof of the existence of a martingale
solution $\widetilde{u}$ to (3.1)-(3.2).

######  Remark 6.4.

As mentioned in an earlier remark following the equations (6.14)-(6.16), we
have better convergence results (6.15)-(6.16) available in the space
$L^{2}(\widetilde{\Omega};L^{2}(0,T;H))$. Thus we do not need to use an
argument like in (4.67) and we can directly pass to the limit
$\widetilde{R}\rightarrow\infty$ after applying the Itô formula using the
function $|\widetilde{u}|^{2}$ to the process in (6.9).

##  7\. Global pathwise solution of (3.1)-(3.2)

In this section we show that there exists a unique global pathwise solution to
(3.1)-(3.2), i.e. we show the existence of $u$ such that (3.4) is satisfied
and thus establish Theorem 3.4.
As explained below this proof relies on an application of a result of Gyöngy
and Krylov [28] which is the infinite dimensional version of the result of
Watanabe and Yamada [53] and for that purpose we start by proving the
uniqueness of pathwise solutions to (3.1)-(3.2).

###  7.1. Pathwise uniqueness

In this section we will prove the uniqueness of martingale solutions,
existence of which has been established in the previous section.

######  Proposition 7.1.

Let $u_{1}$ and $u_{2}$ be two martingale solutions of (3.1)-(3.2) relative to
the same stochastic basis $(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\geq
0},{\bf P},W,\pi)$. Then $u_{1},u_{2}$ are indistinguishable, that is if
$u_{1}(0)=u_{2}(0)$ a.s. then we have

(7.1) ${\bf P}[u_{1}(t)=u_{2}(t),\forall t\geq 0]=1.$

###### Proof.

Letting $u=u_{1}-u_{2}$ we write the equations (3.1) satisfied by $u_{1}$ and
$u_{2}$ and obtain by difference the equation satisfied by $u$. We then
obtain, by applying the Itô formula using the function $|u|^{2}$ to the
difference equation:

$\displaystyle|u(t)|^{2}+2\int_{0}^{t}$
$\displaystyle(\gamma(Au(s),u(s))+(F(u_{1}(s))-F(u_{2}(s)),u(s))+2(B(u_{1}(s))-B(u_{2}(s)),u(s)))\mathrm{d}s$
$\displaystyle=\int_{0}^{t}|G(u_{1}(s))-G(u_{2}(s))|_{L_{2}(U_{0},H)}^{2}\mathrm{d}s+2\int_{0}^{t}([G(u_{1}(s))-G(u_{2}(s))]\mathrm{d}W(s),u)$
$\displaystyle+2\int_{(0,t]}\int_{E_{0}}(u(s),K(u_{1}(s-),\xi)-K(u_{2}(s-),\xi))\mathrm{d}\widehat{\pi}(s,\xi)$
(7.2)
$\displaystyle+\int_{(0,t]}\int_{E_{0}}|K(u_{1}(s-),\xi)-K(u_{2}(s-),\xi)|^{2}\mathrm{d}\pi(s,\xi).$

Here we recall that our assumption (2.11) gives:

(7.3) $(F(u_{1})-F(u_{2}),u)\geq 0.$

Next we derive upper bounds for the term $B$. Observe that due to the
bilinearity of $B$ and the assumptions (2.5) and (2.8) we have

$\displaystyle(B(u_{1},u_{1})-B(u_{2},u_{2}),u)$
$\displaystyle=(\frac{1}{2}\left(B(u_{1}+u_{2},u_{1}-u_{2})+B(u_{1}-u_{2},u_{1}+u_{2})\right),u)$
$\displaystyle=\frac{1}{2}\left((B(u_{1}+u_{2},u),u)+(B(u,u_{1}+u_{2}),u)\right)$
$\displaystyle=-(B(u,u_{1}),u)$ (7.4)
$\displaystyle\lesssim|u|\|u_{1}\|_{X}\|u\|.$

We continue our analysis by introducing the stopping time

(7.5) $\tau^{M}=\inf\\{t\geq
0:\int_{0}^{t}\|u_{1}\|^{p}_{X}\mathrm{d}s>M\\},\quad M>0.$

Then inserting (7.4) and the monotonicity condition (7.3) into equation (7.2),
taking the supremum in time over $[0,t\wedge\tau^{M}]$, and finally taking
expectations, we obtain

$\displaystyle{\bf E}\sup_{s\in[0,t\wedge\tau^{M}]}|u(s)|^{2}$
$\displaystyle+\gamma{\bf
E}\int_{0}^{t\wedge\tau^{M}}\|u(s)\|^{2}\mathrm{d}s\leq\frac{2}{\gamma}{\bf
E}\int_{0}^{t\wedge\tau^{M}}\left(|u(s)|^{2}\|u_{1}\|_{X}^{2}+\frac{\gamma}{2}\|u(s)\|^{2}\right)\mathrm{d}s$
$\displaystyle+2{\bf
E}\sup_{s\in[0,t\wedge\tau^{M}]}\bigg{|}\int_{0}^{s}(G(r,u_{1}(r))-G(r,u_{2}(r))\mathrm{d}W(r),u)\bigg{|}$
$\displaystyle+{\bf
E}\int_{0}^{t\wedge\tau^{M}}||G(u_{1}(s))-G(u_{2}(s))||_{L_{2}(U_{0},H)}^{2}\mathrm{d}s$
$\displaystyle+2{\bf
E}\sup_{s\in[0,t\wedge\tau^{M}]}\bigg{|}\int_{(0,s]}\int_{E_{0}}(K(r,u_{1}(r-),\xi)-K(r,u_{2}(r-),\xi),u(r))\mathrm{d}\widehat{\pi}(r,\xi)\bigg{|}$
(7.6) $\displaystyle+{\bf
E}\int_{(0,t\wedge\tau^{M}]}\int_{E_{0}}|K(u_{1}(s-),\xi)-K(u_{2}(s-),\xi)|^{2}\mathrm{d}\pi(s,\xi).$

By using the Lipschitz growth assumption (2.10), we arrive at

$\displaystyle{\bf
E}\int_{0}^{t\wedge\tau^{M}}||G(u_{1}(s))-G(u_{2}(s))||_{L_{2}(U_{0},H)}^{2}\mathrm{d}s$
$\displaystyle\hskip 93.95122pt+{\bf
E}\int_{(0,t\wedge\tau^{M}]}\int_{E_{0}}|K(u_{1}(s-),\xi)-K(u_{2}(s-),\xi)|^{2}\mathrm{d}\pi(s,\xi)$
(7.7) $\displaystyle\hskip 93.95122pt\leq\rho{\bf
E}\int_{0}^{t\wedge\tau^{M}}(1+|u(s)|^{2})\mathrm{d}s.$

By utilizing the BDG inequality along with the hypothesis (2.9), for some
$C=C(\rho,T)>0$, we see that

$\displaystyle{\bf E}\sup_{s\in[0,t\wedge\tau^{M}]}$
$\displaystyle\bigg{|}\int_{0}^{s}([G(r,u_{1}(r))-G(r,u_{2}(r))]\mathrm{d}W(r),u)\bigg{|}$
$\displaystyle\lesssim{\bf
E}\left(\int_{0}^{t\wedge\tau^{M}}(G(r,u_{1}(r))-G(r,u_{2}(r)),u)^{2}\mathrm{d}s\right)^{\frac{1}{2}}$
$\displaystyle\lesssim{\bf
E}\left(\int_{0}^{t\wedge\tau^{M}}|u(s)|^{2}|G(u_{1}(s))-G(u_{2}(s))|^{2}\mathrm{d}s\right)^{\frac{1}{2}}$
$\displaystyle\lesssim{\bf
E}\left(\sup_{s\in[0,t\wedge\tau^{M}]}|u(s)|^{2}\int_{0}^{t\wedge\tau^{M}}|G(u_{1}(s))-G(u_{2}(s))|^{2}\mathrm{d}s\right)^{\frac{1}{2}}$
$\displaystyle\leq\frac{1}{4}{\bf
E}\sup_{s\in[0,t\wedge\tau^{M}]}|u(s)|^{2}+C{\bf
E}\int_{0}^{t\wedge\tau^{M}}|G(u_{1}(s))-G(u_{2}(s))|^{2}\mathrm{d}s$ (7.8)
$\displaystyle\leq\frac{1}{4}{\bf
E}\sup_{s\in[0,t\wedge\tau^{M}]}|u(s)|^{2}+C{\bf
E}\int_{0}^{t\wedge\tau^{M}}\rho|u(s)|^{2}\mathrm{d}s.$

And similarly

$\displaystyle{\bf E}$
$\displaystyle\sup_{s\in[0,t\wedge\tau^{M}]}\bigg{|}\int_{(0,t]}\int_{E_{0}}(u(s),K(u_{1}(s-),\xi)-K(u_{2}(s-),\xi))\mathrm{d}\widehat{\pi}(s,\xi)\bigg{|}$
$\displaystyle\lesssim{\bf
E}\left(\int_{(0,t]}\int_{E_{0}}(u(s),K(u_{1}(s-),\xi)-K(u_{2}(s-),\xi))^{2}\mathrm{d}\pi(s,\xi)\right)^{\frac{1}{2}}$
$\displaystyle\lesssim{\bf
E}\left(\sup_{s\in[0,t\wedge\tau^{M}]}|u(s)|^{2}\bigg{[}\int_{(0,t]}\int_{E_{0}}|K(u_{1}(s-),\xi)-K(u_{2}(s-),\xi)|^{2}\mathrm{d}\pi(s,\xi)\bigg{]}\right)^{\frac{1}{2}}$
$\displaystyle\leq\frac{1}{4}{\bf
E}\sup_{s\in[0,t\wedge\tau^{M}]}|u(s)|^{2}+C{\bf
E}\bigg{[}\int_{0}^{t\wedge\tau^{M}}\int_{E_{0}}|K(u_{1}(s-),\xi)-K(u_{2}(s-),\xi)|^{2}\mathrm{d}\nu(\xi)\mathrm{d}s\bigg{]}$
(7.9) $\displaystyle\leq\frac{1}{4}{\bf
E}\sup_{s\in[0,t\wedge\tau^{M}]}|u(s)|^{2}+C{\bf
E}\int_{0}^{t\wedge\tau^{M}}\rho|u(s)|^{2}\mathrm{d}s.$

Collecting the estimates (7.7), (7.1) and (7.9) we find that

(7.10) $\displaystyle{\bf E}\sup_{s\in[0,t\wedge\tau^{M}]}|u(s)|^{2}\leq
C(\rho,T,\gamma){\bf
E}\int_{0}^{t\wedge\tau^{M}}|u(s)|^{2}\|u_{1}\|_{X}^{2}\mathrm{d}s.$

We observe that the process $\varphi(s):=\|u_{1}(s)\|_{X}^{2}$ satisfies

(7.11)
$\int_{0}^{t\wedge\tau^{M}}\varphi(s)\mathrm{d}s\leq\left(\int_{0}^{t\wedge\tau^{M}}\|u_{1}(s)\|^{p}_{X}\mathrm{d}s\right)^{\frac{2}{p}}\leq
M^{\frac{2}{p}},\qquad{\bf P}{-a.s.}$

We now apply the stochastic Grönwall lemma (see e.g. Lemma 5.3 in [26]) to
(7.10) and use the fact that $\tau^{M}\rightarrow\infty$ as
$M\rightarrow\infty$ almost surely, to deduce

(7.12) $\displaystyle{\bf E}\left(\sup_{t\in[0,T]}|u(t)|^{2}\right)=0.$

Since $T\geq 0$ is arbitrary, this completes the proof of Proposition 7.1. ∎

######  Remark 7.2.

Note that by fixing the stochastic basis in the proof above one can prove the
uniqueness of the pathwise solutions to (3.1)-(3.2) defined as in Definition
3.2.

###  7.2. Pathwise solution

In this section we apply the well known result by Gyöngy and Krylov in [28]
that extends the Yamada–Watanabe theorem (cf. [53]) to infinite dimension.
This theorem states that the existence of martingale solutions and the
pathwise uniqueness imply the existence of a pathwise solution to (3.1)-(3.2).

We can now finish the proof of Theorem 3.4 which follows a standard procedure.
Here we will briefly describe the key steps involved.

First using the following theorem in [28], it can be shown that
$\bar{\bar{u}}^{\widetilde{R}}$, the sequence of solutions to (5.1), converges
in probability as $\widetilde{R}\rightarrow\infty$ with respect to the
original stochastic basis $(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\geq
0},{\bf P},W,\pi)$.

######  Theorem 7.3.

Let $\mathcal{E}$ be a Polish space. A sequence of ${\mathcal{E}}$-valued
random variables $\\{\mathscr{Y}_{n}\\}_{n\geq 0}$, defined on te same
probability space, converges in probability if and only if for every
subsequence of joint probability laws of $\\{\mathscr{Y}_{n}\\}_{n\geq 0}$,
given by $\\{\mu_{m_{l}},\mu_{n_{l}}\\}_{l\geq 0}$, there exists a further
subsequence that converges weakly to a probability measure $\mu$ such that

(7.13)
$\displaystyle\mu(\\{(x,y)\in{\mathcal{E}}\times{\mathcal{E}}:x=y\\})=1.$

In preparation for using this theorem, define

(7.14) $\displaystyle\mathcal{Y}:=H\times\mathcal{Z}_{1}\times
H\times\mathcal{Z}_{1}\times
C([0,T];U)\times\mathcal{N}^{\\#*}_{[0,\infty)\times E},$

where $\mathcal{N}^{\\#*}_{[0,\infty)}$ denotes the space of counting measures
on $[0,\infty)\times E$ that are finite on bounded sets. Now, consider two
subsequences
$(\mathscr{Y}_{k}:=\bar{\bar{u}}^{\widetilde{R}_{k}})_{k=1}^{\infty}$ and
$(\mathscr{Y}_{k}^{\\#}:=\bar{\bar{u}}^{\widetilde{R}^{\\#}_{k}})_{k=1}^{\infty}$
and the double sequence
$({\mathscr{Y}_{k}}_{0},\mathscr{Y}_{k},{\mathscr{Y}^{\\#}_{k0}},{\mathscr{Y}^{\\#}_{k}},W,\pi)$.
Next we denote by
$\\{\mu^{\widetilde{R}_{k},\widetilde{R}^{\\#}_{k}}\\}_{k\geq 1}$ the sequence
of joint laws of the $\mathcal{Y}$-valued random variables. Then it can be
shown that $\\{\mu^{\widetilde{R}_{k},\widetilde{R}^{\\#}_{k}}\\}_{k\geq 1}$
is tight and thus weakly compact over $\mathcal{Y}$ . Then by Prohorov’s
theorem (see e.g., [7]), there exists a subsequence $k\rightarrow\infty$ such
that $\\{\mu^{\widetilde{R}_{k},\widetilde{R}^{\\#}_{k}}\\}_{k\geq 1}$
converges weakly to a probability measure $\widetilde{\nu}$ on $\mathcal{Y}$.
Then by another application of the Skorohod representation theorem we know
that there exists a filtered probability space
$(\widetilde{\Omega},\widetilde{\mathcal{F}},(\widetilde{\mathcal{F}}_{t})_{t\geq
0},\widetilde{{\bf P}})$ and $\mathcal{Y}$-random variables
$(\widetilde{\mathscr{Y}_{k}}_{0},\widetilde{\mathscr{Y}}_{k},\widetilde{{\mathscr{Y}}^{\\#}_{k0}},\widetilde{{\mathscr{Y}}^{\\#}_{k}},\widetilde{W},\widetilde{\pi})$
and
$(\widetilde{\mathscr{Y}}_{0},\widetilde{\mathscr{Y}},\widetilde{{\mathscr{Y}}^{\\#}_{0}},\widetilde{{\mathscr{Y}}^{\\#}},\widetilde{W},\widetilde{\pi})$
such that
$(\widetilde{\mathscr{Y}_{k}}_{0},\widetilde{\mathscr{Y}}_{k},\widetilde{W},\widetilde{\pi})$
and
$(\widetilde{\mathscr{Y}}_{0},\widetilde{\mathscr{Y}},\widetilde{W},\widetilde{\pi})$
and,
$(\widetilde{{\mathscr{Y}}^{\\#}_{k0}},\widetilde{{\mathscr{Y}}^{\\#}_{k}},\widetilde{W},\widetilde{\pi})$
and
$(\widetilde{{\mathscr{Y}}^{\\#}_{0}},\widetilde{{\mathscr{Y}}^{\\#}},\widetilde{W},\widetilde{\pi})$
satisfy the conclusions of Theorem 6.3.

Thus using the arguments in the proof of Theorem 3.3 following Theorem 6.3 we
conclude that $\widetilde{\mathscr{Y}}$ and $\widetilde{\mathscr{Y}}^{\\#}$
are two martingale solutions to (3.1)-(3.2) with initial conditions
$\widetilde{\mathscr{Y}}_{0}$ and $\widetilde{\mathscr{Y}}^{\\#}_{0}$
respectively with respect to the stochastic basis
$(\widetilde{\Omega},\widetilde{\mathcal{F}},(\widetilde{\mathcal{F}}_{t})_{t\geq
0},\widetilde{{\bf P}},\widetilde{W},\widetilde{\pi})$. Now since
$\widetilde{{\bf
P}}[\widetilde{\mathscr{Y}}_{0}=\widetilde{\mathscr{Y}}_{0}^{\\#}=u_{0}]=1$,
thanks to Proposition 7.1 we obtain $\widetilde{{\bf P}}$-a.s. that
$\widetilde{\mathscr{Y}}=\widetilde{\mathscr{Y}}^{\\#}$ for all $t\in[0,T]$.

Thus as mentioned above we use Theorem 7.3 to infer that the sequence of
$\mathcal{Z}_{1}$-valued random variables
$\\{\bar{\bar{u}}^{\widetilde{R}}\\}_{R\geq 1}$, defined on the original
stochasic basis $(\Omega,\mathcal{F},{\bf P})$, converges to $u$ in
probability as $\widetilde{R}\rightarrow\infty$. Then, along another
subsequence $\widetilde{R}\rightarrow\infty$ we have that the following
conditions hold ${\bf P}$-a.s.

(7.15) $\displaystyle\bar{\bar{u}}_{0}^{\widetilde{R}}$
$\displaystyle\rightarrow u_{0}\text{ in }H,$ (7.16)
$\displaystyle\bar{\bar{u}}^{\widetilde{R}}$ $\displaystyle\rightarrow u\text{
in }L^{2}(0,T;H)\cap\mathcal{D}([0,T];V^{\prime}+X^{\prime}).$

Applying the identical argument as in Section 6 we obtain that the limit $u$
is indeed the unique global pathwise solution to (3.1)-(3.2), i.e., $u$
satisfies (3.4).
This finishes the proof of Theorem 3.4.

##  8\. Examples

###  8.1. The Ladyzensaya-Smagorinsky equations

The first application of Theorem 3.4 that we want to present, corresponds to a
variation of the Navier-Stokes equations introduced by Ladyzhenskaya [33],
[34]; the equations also appears in relation with the Smagorinsky model of
turbulence [47]. For $p>2$ we look for a vector valued function $u$ and a
scalar valued function $p_{*}$ that satisfy:

(8.1) $\begin{cases}\mathrm{d}u+[(u\cdot\nabla)u-\text{div}((1+|\nabla
u|^{2})^{\frac{p-2}{2}}\nabla u)+\nabla p_{*}]\mathrm{d}t\\\ \hskip
144.54pt=G(u)\mathrm{d}W+\int_{E_{0}}K(u,\xi)\mathrm{d}\widehat{\pi}&\textnormal{in
}(0,T)\times\mathcal{O}\\\ \text{div }u=0&\textnormal{in
}(0,T)\times\mathcal{O}\\\ u=0&\textnormal{on
}(0,T)\times\partial\mathcal{O}\\\ u(0)=u_{0}&\textnormal{on
}\\{t=0\\}\times\mathcal{O},\end{cases}$

We consider $X=\\{\varphi\in(W^{1,p}_{0}(\mathcal{O}))^{d},\text{ div
}\varphi=0\\}$, $V=\\{\varphi\in(H^{1}_{0}(\mathcal{O}))^{d},\text{ div
}\varphi=0\\}$ and $H=\\{\varphi\in(L^{2}(\mathcal{O}))^{d},\text{ div
}\varphi=0,\varphi\cdot\nu=0\text{ on }\partial\mathcal{O}\\}$ where $\nu$ is
the unit outward normal on $\partial\mathcal{O}$. Here $F(u)\in X^{\prime}$ is
associated with the operator $-\text{div}((1+|\nabla
u|^{2})^{\frac{p-2}{2}}\nabla u)$ and is defined by
$(F(u),v)=\int_{\mathcal{O}}(1+|\nabla u|^{2})^{\frac{p-2}{p}}\nabla u\nabla
v\mathrm{d}x$ for $u,v\in X$. As described in Section 2 the bilinear term
$B:V\times V\rightarrow V^{\prime}$ is defined by
$(B(u,v),w)=\int_{\mathcal{O}}u_{i}\partial_{i}v_{j}w_{j}\mathrm{d}x$.
It is easy to see that the monotonicity condition (2.11) i.e.
$(F(u)-F(v),u-v)_{X,X^{\prime}}\geq 0,\,\,\forall u,v\in X$ is satisfied.
Additionally we also have for some $c_{p}^{*}>0$ that (see [33]),

(8.2) $\displaystyle(F(u)-F(v),u-v)\geq c_{p}^{*}\|u-v\|^{2}\qquad\forall
u,v\in X.$

Indeed define $w^{r}=ur+(1-r)v$ and ${f}(v)=(1+|\nabla
v|^{2})^{\frac{p-2}{2}}\nabla v$. Then we have that

$\displaystyle f(u)-f(v)$
$\displaystyle=\int_{0}^{1}\frac{\mathrm{d}f(w^{r})}{\mathrm{d}r}\mathrm{d}r$
$\displaystyle=\nabla(u-v)\int_{0}^{1}\left((p-2)(1+|\nabla
w^{r}|^{2})^{\frac{p-2}{2}}+(1+|\nabla
w^{r}|^{2})^{\frac{p-2}{2}}\right)\mathrm{d}r.$

This gives,

$\displaystyle(F(u)-F(v),u-v)=(f(u)-f(v),\nabla u-\nabla v)$
$\displaystyle=(\int_{0}^{1}\frac{\mathrm{d}f(w^{r})}{\mathrm{d}r}\mathrm{d}r,\nabla
u-\nabla v),$ $\displaystyle\geq(p-1)|\nabla u-\nabla v|^{2}.$

In fact it has also been shown that a stronger condition (F1’) is satisfied
(see e.g. [27]):
(F1’) (Strong monotonicity): For $p>2$ there exists a $c_{p}>0$ such that,

(8.3) $\displaystyle(F(u)-F(v),u-v)\geq c_{p}\|u-v\|^{p}_{X}\qquad\forall
u,v\in X.$

It is easy to see that the hypothesis (F2) is satisfied for any
$p\in\mathbb{R}$ (see e.g. [36]).
As proved in [49], the hypotheses (2.14) - (2.15) are satisfied for any $p>2$
and $k=1$ by considering the corresponding convex functional for any $v\in X$

(8.4) $\displaystyle{J}(v)=\int_{\mathcal{O}}\mathscr{J}_{1}(\nabla
v)dx=\int_{\mathcal{O}}\frac{1}{p}(1+|\nabla v|^{2})^{\frac{p}{2}}dx,$

where $\mathscr{J}_{1}:\mathbb{R}^{d^{2}}\rightarrow\mathbb{R}$ for
$x\in\mathbb{R}^{d^{2}}$ is given by,

(8.5) $\displaystyle\mathscr{J}_{1}(x)=\frac{1}{p}(1+|x|^{2})^{\frac{p}{2}}.$

This also leads to (F3) and (F4).
The cancellation property (2.5) is a classical one; a proof can be found e.g.
in [50], [51]. Next we will show that the term $B$ satisfies (2.6)-(2.7). In
the most restrictive case, in space dimensions $d=3$ we use the Ladyzenskaya
inequality to obtain

$\displaystyle|\langle B(v,u),w\rangle|=|\langle B(v,w),u\rangle|$
$\displaystyle\leq|v|_{(L^{2}(\mathcal{O}))^{d}}|\nabla
w|_{(L^{4}(\mathcal{O}))^{d}}|u|_{(L^{4}(\mathcal{O}))^{d}}$
$\displaystyle\lesssim|v||u|^{\frac{1}{4}}\|u\|^{\frac{3}{4}}\|w\|^{\frac{1}{4}}|Aw|^{\frac{3}{4}}$
(8.6) $\displaystyle\lesssim|v|\|u\||Aw|.$

When $d=3$ we also have the embeddings $H^{1}(\mathcal{O})\subset
L^{6}(\mathcal{O})$ and $H^{\frac{1}{2}}(\mathcal{O})\subset
L^{3}(\mathcal{O})$ (see, e.g. [9]) which together with the Gagliardo-
Nirenberg interpolation inequality give us,

$\displaystyle|\langle B(v,u),w\rangle|$
$\displaystyle\leq|v|_{(L^{3}(\mathcal{O}))^{3}}|\nabla
u|_{(L^{2}(\mathcal{O}))^{3}}|w|_{(L^{6}(\mathcal{O}))^{3}}$ (8.7)
$\displaystyle\lesssim|v|^{\frac{1}{2}}\|v\|^{\frac{1}{2}}\|u\|\|w\|.$

The case $d=2$ is easier to prove and so we omit the calculations.
We will next prove that the assumption (2.8) is satisfied.

Observe that for $p\geq d$ we have, $X\subset(L^{\infty}(\mathcal{O}))^{d}$
and hence we readily obtain

$\displaystyle|(B(u,u),w)|=$
$\displaystyle\lesssim|u||w|_{(L^{\infty}(\mathcal{O}))^{d}}|\nabla u|$
$\displaystyle\lesssim|u||w|_{X}\|u\|$ (8.8)
$\displaystyle\lesssim|u|^{2}\|w\|_{X}^{2}+\|u\|^{2}.$

In conclusion Theorem 3.4 applies to equations (8.1) when $p\geq d$ and $p>2$.

######  Remark 8.1.

We now show how by a slight change in the proof of Proposition 7.1 we can
cover other related situations. Indeed, we observe that the hypothesis (2.8)
in Theorem 3.4 was used only in Section 7.1 in the proof of Proposition 7.1 to
prove pathwise uniqueness. However thanks to the stronger monotonocity
properties (8.2) and (8.3) available for the operator $F$ we can extend and
thus improve the uniqueness result for values of $p$ smaller than $d$ in the
following way; note that we still asume $p>2$, so that we cover here the cases
$2<p\leq d$.

For example, consider the case when $d=3>p\geq\frac{7}{3}$. Observe that
$L^{p}(0,T;X)\cap L^{\infty}(0,T;H)\subset
L^{\mathcal{S}}(0,T;(L^{\mathcal{R}}(\mathcal{O}))^{3})$ if for some
$\theta\in(0,1)$ we can write $\frac{1}{\mathcal{S}}=\frac{1-\theta}{p}$ and
$\frac{1}{\mathcal{R}}=\frac{(1-\theta)(3-p)}{3p}+\frac{\theta}{2}$. Indeed,
for $\frac{7}{3}\leq p<3$ we can let $\theta=\frac{2(3p-7)}{5(p-2)}$. For this
$\theta$ we write
$\frac{1}{\sigma}:=1-\frac{1}{p}-\frac{1}{\mathcal{R}}=\frac{1}{\mathcal{S}}+\frac{\frac{3}{\mathcal{R}}}{6}>0$
which gives,

$\displaystyle|(B(u,u),w)|\leq|u|_{(L^{\sigma}(\mathcal{O}))^{d}}\|u\|_{X}|w|_{(L^{\mathcal{R}}(\mathcal{O}))^{3}}$
$\displaystyle\leq|u|^{\frac{2}{\mathcal{S}}}_{(L^{2}(\mathcal{O}))^{3}}|u|^{\frac{3}{\mathcal{R}}}_{(L^{6}(\mathcal{O}))^{3}}\|u\|_{X}|w|_{(L^{\mathcal{R}}(\mathcal{O}))^{3}}$
$\displaystyle\leq|u|^{\frac{2}{\mathcal{S}}}\|u\|^{\frac{3}{\mathcal{R}}}\|u\|_{X}|w|_{(L^{\mathcal{R}}(\mathcal{O}))^{3}}$
(8.9)
$\displaystyle\leq|u|^{2}|w|^{\mathcal{S}}_{(L^{\mathcal{R}}(\mathcal{O}))^{3}}+\|u\|^{2}+\|u\|^{p}_{X}.$

Hence, using (8.2) and (8.3), we obtain the following estimates in place of
(7.6) in the proof of Proposition 7.1,

$\displaystyle{\bf E}\sup_{s\in[0,t\wedge\tau^{M}]}|u(s)|^{2}$
$\displaystyle+c_{p}^{*}\,{\bf
E}\int_{0}^{t\wedge\tau^{M}}\|u(s)\|^{2}\mathrm{d}s+c_{p}\,{\bf
E}\int_{0}^{t\wedge\tau^{M}}\|u(s)\|_{X}^{p}\mathrm{d}s$
$\displaystyle\leq{\bf
E}\int_{0}^{t\wedge\tau^{M}}|u(s)|^{2}\|u_{1}\|^{\mathcal{S}}_{(L^{\mathcal{R}}(\mathcal{O}))^{3}}+\|u(s)\|^{2}+\|u(s)\|_{X}^{p}\mathrm{d}s$
$\displaystyle+2{\bf
E}\sup_{s\in[0,t\wedge\tau^{M}]}\bigg{|}\int_{0}^{s}(G(r,u_{1}(s))-G(r,u_{2}(s))\mathrm{d}W(s),u)\bigg{|}$
$\displaystyle+{\bf
E}\int_{0}^{t\wedge\tau^{M}}||G(u_{1}(s))-G(u_{2}(s))||_{L_{2}(U_{0},H)}^{2}\mathrm{d}s$
$\displaystyle+2{\bf
E}\sup_{s\in[0,t\wedge\tau^{M}]}\bigg{|}\int_{(0,t]}(\int_{E_{0}}u(s),K(u_{1}(s-),\xi)-K(u_{2}(s-),\xi))\mathrm{d}\widehat{\pi}(s,\xi)\bigg{|}$
(8.10) $\displaystyle+{\bf
E}\int_{[0,t\wedge\tau^{M}]}\int_{E_{0}}|K(u_{1}(s-),\xi)-K(u_{2}(s-),\xi)|^{2}\mathrm{d}\pi(s,\xi).$

Similarly instead of (7.5) we define

(8.11) $\tau^{M}=\inf\\{t\geq
0:\int_{0}^{t}\|u_{1}\|^{\mathcal{S}}_{(L^{\mathcal{R}}(\mathcal{O}))^{3}}>M\\},\quad
M>0.$

The rest of the proof of Proposition 7.1 follows identically.

Since all the hypotheses are satisfied, Theorem 3.4 applies producing the
existence of a unique global pathwise solution to (8.1) for any $p>2$ when
$d=2$ and any $p\geq\frac{7}{3}$ when $d=3$.

######  Remark 8.2.

In the deterministic setting, Ladyzenkaya in [34], showed that weak solutions
to (8.1) exist and are globally unique in time for any $p\geq 3$ for space
dimensions $d=2,3$ (see also Chapter 2, Section 5 in [36] and [22]).

######  Remark 8.3.

Note that Theorem 3.3 gives the existence (without uniqueness) of a martingale
solution to (8.1) for all $p>2$. The same is true for the case where the
monotone operator in (8.1) is replaced by the operator
$F(u)=-\sum^{d}_{i=1}\frac{\partial}{\partial x_{i}}\left(|\nabla
u|^{p-2}\frac{\partial u}{\partial x_{i}}\right)$. This is because we require
the presence of the linear term $A=-\Delta$ only while proving pathwise
uniqueness of martingale solutions (in obtaining the estimates (8.10)) which
eventually leads to the existence of a unique pathwise solution as described
in Section 7.2. See below Remark 8.4 for a comment on how the equicoercivity
result (4.7) is utilized in proving the results from earlier sections when the
term $A$ is absent.

###  8.2. Equations without the operator $A$ and with $B=0$

####  8.2.1. Evolution equations with $p$-Laplacian

Consider the following equations for some $p>2$:

(8.12) $\begin{cases}\mathrm{d}u-\left[\sum^{d}_{i=1}\frac{\partial}{\partial
x_{i}}\left(|\nabla u|^{p-2}\frac{\partial u}{\partial
x_{i}}\right)\right]\mathrm{d}t=G(u)\mathrm{d}W+\int_{E_{0}}K(u,\xi)\mathrm{d}\widehat{\pi}&\textnormal{in
}(0,T)\times\mathcal{O}\\\ u=0&\textnormal{on
}(0,T)\times\partial\mathcal{O}\\\ u(0)=u_{0}&\textnormal{on
}\\{t=0\\}\times\mathcal{O}.\end{cases}$

Here $X=(W^{1,p}_{0}(\mathcal{O}))^{d}$ and $H=(L^{2}(\mathcal{O}))^{d}$. Even
though the operator $A$ is absent from equations (8.12), the natural choice
for $V$ is the space $(H^{1}_{0}(\mathcal{O}))^{d}$. The convex functional $J$
in (2.13) associated with the monotone operator
$F(u)=-\sum^{d}_{i=1}\frac{\partial}{\partial x_{i}}\left[|\nabla
u|^{p-2}\frac{\partial u}{\partial x_{i}}\right]$ is given by

$J(v)=\frac{1}{p}\int_{\mathcal{O}}|\nabla v|^{p}\mathrm{d}x.$

We again refer to [49] for a proof that the conditions (2.14)-(2.15) are
satisfied by $\mathscr{J}_{1}(x)=\frac{1}{p}|x|^{p}$. Observe that when $B=0$,
even though we used a special eigenbasis dependent on the term $A$ in the
definition (4.1), the absence of the term $A$ does not hinder the application
of the Galerkin method. One can use an appropriate orthonormal basis of the
space $H$ to carry out the analysis in Section 4. Theorem 3.4 then gives the
existence of a unique pathwise solution to (8.12) for all $p\geq 2$.

######  Remark 8.4.

For the equations (8.12), note that the equicoercivity result for $F^{R}$,
stated in (4.7) and derived in [49], is key in getting the energy estimates in
Lemma 4.1 and Lemma 5.1 independent of the parameter $R$. This is important
due to the absence of the term $A$ which usually provides estimates in the
space $L^{2}(\Omega;L^{2}(0,T;V))$. Note also that while proving uniqueness of
pathwise solutions, unlike in the equation discussed in Remark 8.3, the
absence of the term $B$ compensates for the absence of the term $A$ in the
derivation of the estimates (7.6).

#### 8.2.2.

Consider the following equations:

(8.13) $\begin{cases}\mathrm{d}u+{\Delta}\left[|{\Delta u}|^{p-2}{\Delta
u}\right]\mathrm{d}t=G(u)\mathrm{d}W+\int_{E_{0}}K(u,\xi)\mathrm{d}\widehat{\pi}&\textnormal{in
}(0,T)\times\mathcal{O}\\\ u=0,\,\,{\frac{\partial
u}{\partial\nu}=0}&\textnormal{on }(0,T)\times\partial\mathcal{O}\\\
u(0)=u_{0}&\textnormal{on }\\{t=0\\}\times\mathcal{O},\end{cases}$

where $\nu$ is the unit outward normal on $\partial\mathcal{O}$. Here we take
$X=W^{2,p}_{0}(\mathcal{O})$, $V=H^{2}_{0}(\mathcal{O})$ and
$H=L^{2}(\mathcal{O})$. The monotone, hemicontinuous operator
$F(u)={\Delta}\left[|{\Delta u}|^{p-2}{\Delta u}\right]$ is a variant of the
non-linear operator appearing in the classical porous medium equation (cf.
[4]) describing the flow of an ideal gas through a porous medium where the
scalar valued function $u$ represents density. Observe that the convex
functional $J$ associated with the aforementioned monotone operator is given
by

$\displaystyle J(v)=\frac{1}{p}\int_{\mathcal{O}}|\Delta v|^{p}\mathrm{d}x.$

Since we look for a scalar valued solution $u$ to (8.13), we prove for
$x\in\mathbb{R}^{d^{2}}$, that $\mathscr{J}_{2}(x)=|x|^{p}$ satisfies the
conditions (2.14)-(2.15), which were assumed in the general setting of a
vector valued solution to the SPDEs (1.1). We will omit the computations since
they are identical to the ones for $\mathscr{J}_{1}(x)$ in Section (8.2.1).
The assumptions (2.17)-(2.18) follow from (2.15). Observe here as well that
the absence of the term $A$ does not create any problem; see Remark 8.4. Thus
Theorem 3.4 applies and gives us the existence of a unique pathwise solution
to (8.13) for any $p\geq 2$.

###  8.3. Polynomial nonlinearity

We let $\mathcal{O}$ be an open, bounded subset of $\mathbb{R}^{d}$ with a
sufficiently smooth boundary. Now consider the following equations which are a
generalization of the stochastic Chafee-Infante equations.

(8.14) $\begin{cases}\mathrm{d}u+[-\Delta
u+P(u)]\mathrm{d}t=G(u)\mathrm{d}W+\int_{E_{0}}K(u,\xi)\mathrm{d}\widehat{\pi}&\textnormal{in
}(0,T)\times\mathcal{O}\\\ u=0&\textnormal{on
}(0,T)\times\partial\mathcal{O}\\\ u(0)=u_{0}&\textnormal{on
}\\{t=0\\}\times\mathcal{O},\end{cases}$

The operator $P$ can be a polynomial of odd order with a leading positive
coefficient: $P(u)=\sum^{2q+1}_{i=0}a_{i}u^{i}$ with $a_{2q+1}>0$. Also in
view of applications in [44] it could be a pseudo-polynomial of even order,
sum of a term $a_{2q}|u|^{2q-1}u,$ $a_{2q}>0$, and of lower order terms of
similar form.
The functional framework in Section 2 applies to the monotone operator
$F=-\Delta+P$ with $H=L^{2}(\mathcal{O}),V=H^{1}_{0}(\mathcal{O}),$
$X=L^{2q+2}(\mathcal{O})\cap H^{1}_{0}(\mathcal{O})$.

Notice that due to the absence, in these equations, of any additional
nonlinearity like the term $B$, the dimension $d\geq 1$ can be arbitrarily
large as opposed to the restriction $d\leq 3$ imposed in [20] (see also
Example 3 pg. 5698 in [14]). Note also that for the same reasons, we do not
require the strong monotonicity condition (8.3) and instead can get the
desired results as long as we have the condition (2.11):
$(F(u)-F(v),u-v)_{X,X^{\prime}}\geq 0$ satisfied for any $u,v\in X$. The
authors in [49] provided detailed calculations showing that this monotonicity
condition holds along with the other hypotheses (2.17)-(2.18) with $p=2q+2$.
Thus Theorem 3.4 gives the existence of a global pathwise solution to (8.14)
for any $q\geq 0$ and $d\geq 1$.

###  8.4. Other examples

In this section we consider the following equations.

(8.15) $\begin{cases}\mathrm{d}u-\frac{\partial}{\partial
x_{1}}\left[(|\frac{\partial u}{\partial x_{1}}|^{p-2})\frac{\partial
u}{\partial
x_{1}}\right]\mathrm{d}t=G(u)\mathrm{d}W+\int_{E_{0}}K(u,\xi)\mathrm{d}\widehat{\pi}&\textnormal{in
}(0,T)\times\mathcal{O}\\\ u\nu_{1}=0&\textnormal{on
}(0,T)\times\partial\mathcal{O}\\\ u(0)=u_{0}&\textnormal{on
}\\{t=0\\}\times\mathcal{O},\end{cases}$

where $\nu=(\nu_{1},...,\nu_{d})$ is the unit outward normal along
$\partial\mathcal{O}$. Here we consider the spaces $X=\\{u\in
L^{p}(\mathcal{O});\,\frac{\partial u}{\partial x_{1}}\in
L^{p}(\mathcal{O}),\,u\nu_{1}=0\text{ on }\partial\mathcal{O}\\}$ , $V=\\{u\in
L^{2}(\mathcal{O});\,\frac{\partial u}{\partial x_{1}}\in
L^{2}(\mathcal{O}),\,u\nu_{1}=0\text{ on }\partial\mathcal{O}\\}$ and
$H=L^{2}(\mathcal{O})$. The boundary condition $u\nu_{1}=0$ makes sense on
$\partial\mathcal{O}$, considering the vector $U=(u,0,...,0)$ which belongs to
$L^{2}(\mathcal{O})$, with div$(U)$ belonging also to $L^{2}(\mathcal{O})$,
when $u$ and $\frac{\partial u}{\partial x_{1}}\in L^{2}(\mathcal{O})$. We
then apply the classical trace theorem from Navier-Stokes equations theory,
see e.g. Theorem 1.2, Ch. I in [50]. The space $V$ is Hilbert under the scalar
product $(u,v)+(\frac{\partial u}{\partial x_{1}},\frac{\partial v}{\partial
x_{1}})$.
In this case the associated convex functional for any $v\in X$ is given by
$J(v)=\int_{\mathcal{O}}|\nabla v\cdot e_{1}|^{p}\mathrm{d}x$ and
$\mathscr{J}_{1}(x)=|x\cdot e_{1}|^{p}$ where
$e_{1}=(1,0,...,0),x\in\mathbb{R}^{d}$. With straightforward calculations it
can be seen that for $x\in\mathbb{R}^{d}$, a variant of the conditions
(2.14)-(2.15) is satisfied with $|x|_{l^{p}(\mathbb{R}^{d})}$ replaced by
$|x\cdot e_{1}|_{l^{p}(\mathbb{R})}$. The functions $f_{1}^{R}$ are defined as
follows:

(8.16) $\displaystyle f_{1}^{R}(x)=f_{1}(x)\mathbbm{1}_{\\{|x\cdot
e_{1}|_{p}<R\\}}+\left[f_{1}\left(\frac{Rx}{|x|_{p}}\right)+\left(1-\frac{R}{|x|_{p}}\right)xD^{2}\mathscr{J}_{1}\left(\frac{Rx}{|x|_{p}}\right)\right]\mathbbm{1}_{\\{|x\cdot
e_{1}|_{p}>R\\}},$

The rest of the analysis carries out identically as with the example described
in Section 8.2.1 due to the similarities between the monotone operators.
Observe that here the condition $X=W^{k,p}(\mathcal{O})\cap V$, which was
assumed only for the simplicity of our analysis, is not required. In general,
it suffices for the functions $\mathscr{J}_{i}$’s to have polynomial growth so
that the constructions defined in (4.3)-(4.4) approximate the monotone
operator $F$ and give us the desired existence result. Finally, an application
of Theorem 3.4 gives us the existence of a unique pathwise solution to the
SPDE (8.15) for any $p>2$.

##  Appendix A About the Skorohod representation theorem by Justin Cyr

Here we present a proof of the Skorohod representation theorem.222In this
article as well as in related articles [14], [15], [17] we relied on a
specialized version of the Skorohod theorem stated as Theorem C.1 in [11]. In
this appendix we provide, for the sake of completeness, an alternate proof of
Theorem C.1 in [11] due to Justin Cyr who has co-authored the articles [14],
[15], [17].

######  Theorem A.1.

Let $(\Omega,\mathcal{F},{\bf P})$ be a probability space and $U^{1},U^{2}$ be
two complete separable metric spaces. Let $\chi_{n}\colon\Omega\rightarrow
U^{1}\times U^{2}$, $n\in\mathbb{N}$, be a sequence of weakly convergent
random variables with laws $\left(\mu_{n}\right)_{n=1}^{\infty}$. For $i=1,2$
let $\pi_{i}\colon U^{1}\times U^{2}\rightarrow U^{i}$ denote the natural
projection, i.e.,

$\pi_{i}(\chi^{1},\chi^{2})={\chi^{i}},\qquad\forall(\chi^{1},\chi^{2})\in
U^{1}\times U^{2}.$

Finally, let us assume that the random variables
$\pi_{1}(\chi_{n})=\chi_{n}^{1}$ on $U^{1}$ share the same law, independent of
$n$.

Then, there exists a family of $U^{1}\times U^{2}$-valued random variables
$\left(\bar{\chi}_{n}\right)_{n=1}^{\infty}$ on the probability space
$(\widetilde{\Omega},\widetilde{\mathcal{F}},\widetilde{{\bf
P}}):=([0,1)\times[0,1),\textnormal{ Borel sets},\textnormal{Lebesgue
measure})$ and a random variable $\bar{\chi}_{\infty}$ on
$(\widetilde{\Omega},\widetilde{\mathcal{F}},\widetilde{{\bf P}})$ such that
the following statements hold:

  1. i)

The law of $\bar{\chi}_{n}$ is $\mu_{n}$ for every $n\in\mathbb{N}$,

  2. ii)

$\bar{\chi}_{n}\rightarrow\bar{\chi}_{\infty}$ in $U^{1}\times U^{2}$,
$\widetilde{{\bf P}}$-a.s.,

  3. iii)

$\bar{\chi}_{n}^{1}=\bar{\chi}_{\infty}^{1}$ everywhere on
$\widetilde{\Omega}$ for every $n\in\mathbb{N}$.

Note that in applications to martingale solutions for stochastic PDEs, $U^{1}$
is the noise space and $U^{2}$ is the solution space. A key part of the
extension, part $iii)$, says that the noise processes for the sequence on the
new probability space can be chosen in a way that does not depend on $n$.

###### Proof.

Let $\mu_{\infty}$ denote the weak limit of the sequence
$\left(\mu_{n}\right)_{n=1}^{\infty}$, which exists by hypothesis. The proof
begins by defining partitions of $U^{1}$ and $U^{2}$. Let
$\left(x_{i}\right)_{i=1}^{\infty}$ and $\left(y_{j}\right)_{j=1}^{\infty}$ be
countable dense subsets of $U^{1}$ and $U^{2}$, respectively, and let
$\left(r_{k}\right)_{k=1}^{\infty}$ be a sequence of positive numbers such
that $r_{k}\downarrow 0$ and333$B(x,r)$ denotes the closed ball of radius
$r>0$ centered at the point $x$ in a metric space. We use this notation in
both spaces $U^{1}$ and $U^{2}$. $\mu_{\infty}(\partial B(x_{i},r_{k})\times
U^{2})=0$ and $\mu_{\infty}(U^{1}\times\partial B(y_{j},r_{k}))=0$ for all
$i,j,k\in\mathbb{N}$. It is possible to choose $r_{k}$ with this property
because, for example, for each fixed $i$ there are at most countably many
$r>0$ such that $\mu_{\infty}(\partial B(x_{i},r)\times U^{2})>0$.

We will define a sequence $\left(\mathcal{O}^{k}\right)_{k=1}^{\infty}$ of
nested countable partitions of $U^{1}$ and a sequence
$\left(\mathcal{C}^{k}\right)_{k=1}^{\infty}$ of nested countable partitions
of $U^{2}$. For each fixed $k\in\mathbb{N}$ and multi-index
$\alpha\in\mathbb{N}^{k}$, define

$\mathcal{O}^{k}_{\alpha}:=\\{x\in U^{1}:\textnormal{for each }\ell\leq
k,\alpha_{\ell}\textnormal{ is the minimal index }i\textnormal{ such that
}x\in B(x_{i},r_{\ell})\\}.$

The proof in [11] uses an inductive construction to define an equivalent
partition, denoted $O^{k}_{i_{1},\ldots,i_{k}}$. The sets
$O^{k}_{i_{1},\ldots,i_{k}}$ defined in [11] are empty if $i_{1},\ldots,i_{k}$
is not monotone increasing but otherwise $O^{k}_{i_{1},\ldots,i_{k}}$ is equal
to the set $\mathcal{O}^{k}_{\alpha}$ defined above, when
$\alpha=(i_{1},\ldots,i_{k})$. In particular, the inductive construction shows
that the sets
$\mathcal{O}^{k}:=\\{\mathcal{O}^{k}_{\alpha}:\alpha\in\mathbb{N}^{k}\\}$ are
Borel-measurable. Since $\left(x_{i}\right)_{i=1}^{\infty}$ is dense in
$U^{1}$ it is clear that
$U^{1}=\bigcup_{\alpha\in\mathbb{N}^{k}}\mathcal{O}^{k}_{\alpha}$ for each
$k$. The minimality property ensures that
$\mathcal{O}^{k}_{\alpha}\cap\mathcal{O}^{k}_{\beta}=\varnothing$ if
$\alpha\neq\beta$. Thus, $\mathcal{O}^{k}$ is a partition of $U^{1}$.

Note that the partitions are nested. That is, each set
$\mathcal{O}^{k}_{\alpha}$ in partition $\mathcal{O}^{k}$ is itself
partitioned by the sets
$\\{\mathcal{O}^{k+m}_{(\alpha,\widetilde{\alpha})}:\widetilde{\alpha}\in\mathbb{N}^{m}\\}$
in partition $\mathcal{O}^{k+m}$.

We define nested countable partitions
$\mathcal{C}^{k}=\\{\mathcal{C}^{k}_{\beta}:\beta\in\mathbb{N}^{k}\\}$ of
$U^{2}$ for each $k\in\mathbb{N}$ by the same construction, thereby obtaining
nested countable partitions
$\mathcal{O}^{k}\times\mathcal{C}^{k}:=\\{\mathcal{O}^{k}_{\alpha}\times\mathcal{C}^{k}_{\beta}:\alpha,\beta\in\mathbb{N}^{k}\\}$
of $U^{1}\times U^{2}$. By our choice of the sequence
$\left(r_{k}\right)_{k=1}^{\infty}$ we have
$\mu_{\infty}(\partial(\mathcal{O}^{k}_{\alpha}\times\mathcal{C}^{k}_{\beta}))=0$
for every $k\in\mathbb{N}$ and $\alpha,\beta\in\mathbb{N}^{k}$.

For each measure in the sequence $\mu_{1},\mu_{2},\ldots$ and including
$\mu_{\infty}$, and for each partition $\mathcal{O}^{k}\times\mathcal{C}^{k}$
of $U^{1}\times U^{2}$ we define a nested countable partition of rectangles
for $\widetilde{\Omega}:=[0,1)\times[0,1)$. In a change from the proof in [11]
we do this construction for $\mu_{\infty}$ as well to obtain the limit
$\bar{\chi}_{\infty}$; so, we allow $n=\infty$ in the notation below.

######  Definition A.2.

For each $n\in\bar{\mathbb{N}}$ and $k\in\mathbb{N}$ define a partition
$I^{n,k}$ of the square $[0,1)\times[0,1)$ as follows. For
$\alpha,\beta\in\mathbb{N}^{k}$ we put

$\displaystyle I^{n,k}_{\alpha,\beta}$
$\displaystyle:=\Big{[}\mu_{n}^{1}\Big{(}\bigcup_{\alpha^{\prime}<\alpha}\mathcal{O}^{k}_{\alpha^{\prime}}\Big{)},\mu_{n}^{1}\Big{(}\bigcup_{\alpha^{\prime}\leq\alpha}\mathcal{O}^{k}_{\alpha^{\prime}}\Big{)}\Big{)}$
(A.1)
$\displaystyle\qquad\times\Big{[}\mu_{n}\Big{(}\mathcal{O}^{k}_{\alpha}\times\bigcup_{\beta^{\prime}<\beta}\mathcal{C}^{k}_{\beta^{\prime}}\
\Big{|}\ \mathcal{O}^{k}_{\alpha}\times
U^{2}\Big{)},\mu_{n}\Big{(}\mathcal{O}^{k}_{\alpha}\times\bigcup_{\beta^{\prime}\leq\beta}\mathcal{C}^{k}_{\beta^{\prime}}\
\Big{|}\ \mathcal{O}^{k}_{\alpha}\times U^{2}\Big{)}\Big{)}.$

Remarks about (A.1):

  * •

The set unions, e.g.
$\bigcup_{\alpha^{\prime}<\alpha}\mathcal{O}^{k}_{\alpha^{\prime}}$, are
indexed by lexicographic order on $\mathbb{N}^{k}$. These are finite unions
only when $\alpha$ is of the form $\alpha=(1,1,\ldots,1,a)$ for
$a\in\mathbb{N}$, otherwise they are countably infinite.

  * •

$\mu_{n}(A\ |\ B)$ denotes the conditional probability of $A$ given $B$ under
$\mu_{n}$.

  * •

$\mu^{1}_{n}$ denotes the marginal of $\mu_{n}$ on $U^{1}$, i.e.,
$\mu^{1}_{n}(A):=\mu_{n}(A\times U^{2})$ for $A\in\mathcal{B}(U^{1})$. In
other words, $\mu^{1}_{n}$ is the law of $\pi_{1}(\chi_{n})$.

For $k\in\mathbb{N}$ and each measure $\mu_{n}$ we are partitioning the unit
square into countably many rectangles
$I^{n,k}:=\\{I^{n,k}_{\alpha,\beta}:\alpha,\beta\in\mathbb{N}^{k}\\}$ with
horizontal lengths
$\mu_{n}^{1}(\mathcal{O}^{k}_{\alpha})=\mu_{n}(\mathcal{O}^{k}_{\alpha}\times
U^{2})$ and vertical lengths
$\mu_{n}(\mathcal{O}^{k}_{\alpha}\times\mathcal{C}^{k}_{\beta}\ |\
\mathcal{O}^{k}_{\alpha}\times U^{2})$. Note that the horizontal lengths
$\mu_{n}^{1}(\mathcal{O}^{k}_{\alpha})$ do not depend on $n$ (including
$n=\infty$), since we assume that the marginals $\mu^{1}_{n}$ are the same for
all $n\in\mathbb{N}$. The area of rectangle $I^{n,k}_{\alpha,\beta}$ is
$\mu_{n}(\mathcal{O}^{k}_{\alpha}\times\mathcal{C}^{k}_{\beta})$. The
rectangles $I^{n,k}$ cover the entire unit square $\widetilde{\Omega}$ because
$\mathcal{O}^{k}\times\mathcal{C}^{k}$ partitions $U^{1}\times U^{2}$ and
$\mu_{n}$ is a probability measure.

The partitions $I^{k,n}$ are nested as $k$ increases. That is, for
$k_{1}<k_{2}$ the rectangle $I^{n,k_{1}}_{\alpha,\beta}$ in partition
$I^{n,k_{1}}$ is itself partitioned by the rectangles
$\\{I^{n,k_{2}}_{(\alpha,\widetilde{\alpha}),(\beta,\widetilde{\beta})}:\widetilde{\alpha},\widetilde{\beta}\in\mathbb{N}^{k_{2}-k_{1}}\\}$
from partition $I^{n,k_{2}}$.

Now we define the $U^{1}\times U^{2}$ random variables
$\left(\bar{\chi}_{n}\right)_{n=1}^{\infty}$ and $\bar{\chi}_{\infty}$ on the
new probability space
$(\widetilde{\Omega},\widetilde{\mathcal{F}},\widetilde{{\bf P}})$.

First, for each $k\in\mathbb{N}$ and $\alpha\in\mathbb{N}^{k}$ we choose an
element $x^{k}_{\alpha}$ of the set $\mathcal{O}^{k}_{\alpha}$ from the
countable dense subset $\left(x_{i}\right)_{i=1}^{\infty}$. For instance, let
$i^{*}$ be the minimal index $i$ such that $x_{i}\in\mathcal{O}^{k}_{\alpha}$
and set $x^{k}_{\alpha}:=x_{i^{*}}$. Similarly, for each $k\in\mathbb{N}$ and
$\beta\in\mathbb{N}^{k}$ choose an element $y^{k}_{\beta}$ of
$\mathcal{C}^{k}_{\beta}$ from the countable dense set
$\left(y_{j}\right)_{j=1}^{\infty}$.

Second, for each $n\in\bar{\mathbb{N}}$ and $k\in\mathbb{N}$ we define a
$U^{1}\times U^{2}$-valued random variable
$Z_{n,k}\colon\widetilde{\Omega}\rightarrow U^{1}\times U^{2}$ by setting

$Z_{n,k}(\bar{\omega}):=(Z^{1}_{n,k}(\bar{\omega}),Z^{2}_{n,k}(\bar{\omega})):=(x^{k}_{\alpha},y^{k}_{\beta})\qquad\textnormal{if
}\bar{\omega}\in I^{k,n}_{\alpha,\beta}\ (\alpha,\beta\in\mathbb{N}^{k}).$

Each $Z_{n,k}$ is constant on a countable set of measurable rectangles and is
therefore measurable. The values of $Z_{n,k}$ do not depend on $n$; only the
rectangles depend on $n$ and even then only in the vertical axis.

Third, we pass to the limit as $k\rightarrow\infty$. We claim that the
sequence $\left(Z_{n,k}\right)_{k=1}^{\infty}$ is Cauchy in $U^{1}\times
U^{2}$ for each fixed $n\in\bar{\mathbb{N}}$. Indeed, consider
$Z_{n,k_{1}}(\bar{\omega})$ and $Z_{n,k_{2}}(\bar{\omega})$ for positive
integers $k_{1}<k_{2}$ and $\bar{\omega}\in\widetilde{\Omega}$. The key is
that $\bar{\omega}$ belongs to a rectangle of the form
$I^{n,k_{1}}_{\alpha,\beta}$ in partition $I^{n,k_{1}}$ and a rectangle of the
form $I^{n,k_{2}}_{(\alpha,\widetilde{\alpha}),(\beta,\widetilde{\beta})}$ in
partition $I^{n,k_{2}}$, where
$\widetilde{\alpha},\widetilde{\beta}\in\mathbb{N}^{k_{2}-k_{1}}$. The points
$Z^{1}_{n,k_{1}}(\bar{\omega})=x^{k_{1}}_{\alpha}$ and
$Z^{1}_{n,k_{2}}(\bar{\omega})=x^{k_{2}}_{(\alpha,\widetilde{\alpha})}$ both
belong to the set $\mathcal{O}^{k_{1}}_{\alpha}$ in $U^{1}$, since
$\mathcal{O}^{k_{2}}_{(\alpha,\widetilde{\alpha})}\subseteq\mathcal{O}^{k_{1}}_{\alpha}$;
likewise,
$y^{k_{1}}_{\beta},y^{k_{2}}_{(\beta,\bar{\beta})}\in\mathcal{C}^{k_{1}}_{\beta}$.
Since $\mathcal{O}^{k_{1}}_{\alpha}$ and $\mathcal{C}^{k_{1}}_{\beta}$ have
diameter bounded by $2r_{k_{1}}$, we obtain the following estimate, uniformly
in $\bar{\omega}$:444$d$ denotes a complete metric for the topology on
$U^{1}\times U^{2}$. We can take $d:=d^{1}+d^{2}$ where $d^{i}$ is a complete
metric on $U^{i}$.

(A.2) $d\big{(}Z_{n,k_{1}}(\bar{\omega}),Z_{n,k_{2}}(\bar{\omega})\big{)}\leq
4r_{k_{1}},\qquad\textnormal{ for all }k_{2}\geq k_{1},\
\widetilde{\omega}\in\widetilde{\Omega}.$

Therefore, $\left(Z_{n,k}\right)_{k=1}^{\infty}$ is Cauchy in $U^{1}\times
U^{2}$ for each fixed $n$. Since $U^{1}\times U^{2}$ is complete we can define
$U^{1}\times U^{2}$-valued random variables
$\bar{\chi}_{n}\colon\widetilde{\Omega}\rightarrow U^{1}\times U^{2}$ for each
$n\in\bar{\mathbb{N}}$ by

$\bar{\chi}_{n}(\bar{\omega}):=\lim_{k\rightarrow\infty}Z_{n,k}(\bar{\omega}).$

This limit exists for every $\bar{\omega}\in\widetilde{\Omega}$ and the
resulting function $\bar{\chi}_{n}$ is Borel-measurable.

It remains to show that the random variables
$\left(\bar{\chi}_{n}\right)_{n=1}^{\infty}$ and $\bar{\chi}_{\infty}$ have
the following properties:

  1. i)

The law of $\bar{\chi}_{n}$ is $\mu_{n}$ for every $n\in\bar{\mathbb{N}}$.

  2. ii)

The projection $\pi_{1}(\bar{\chi}_{n})=\bar{\chi}_{n}^{1}$ onto $U^{1}$ does
not depend on $n$ (including $n=\infty$).

  3. iii)

$\bar{\chi}_{n}\rightarrow\bar{\chi}_{\infty}$ in $U^{1}\times U^{2}$,
$\widetilde{{\bf P}}$-a.s.

We save the trickiest part, passing to the limit as $n\rightarrow\infty$, for
last.

For property i), note that for fixed $n\in\bar{\mathbb{N}}$, $k\in\mathbb{N}$
and $\alpha,\beta\in\mathbb{N}^{k}$ we have

$\widetilde{{\bf
P}}[\bar{\chi}_{n}\in\mathcal{O}^{k}_{\alpha}\times\mathcal{C}^{k}_{\beta}]=\textnormal{Area}(I^{n,k}_{\alpha,\beta})=\mu_{n}(\mathcal{O}^{k}_{\alpha}\times\mathcal{C}^{k}_{\beta}).$

Therefore, the law of $\bar{\chi}_{n}$ agrees with $\mu_{n}$ on the sets
$\Pi:=\\{\mathcal{O}^{k}_{\alpha}\times\mathcal{C}^{k}_{\beta}:k\in\mathbb{N},\alpha,\beta\in\mathbb{N}^{k}\\}$.
$\Pi$ is closed under finite intersections (i.e., $\Pi$ is a $\pi$-system) and
generates the Borel $\sigma$-algebra on $U^{1}\times U^{2}$. It follows from
the $\pi$-$\lambda$ theorem (e.g., Theorem 3.3 in [6]) that the law of
$\bar{\chi}_{n}$ is equal to $\mu_{n}$.

For property ii), recall that, by hypothesis, the endpoints of the horizontal
interval

$\Big{[}\mu_{n}^{1}\Big{(}\bigcup_{\alpha^{\prime}<\alpha}\mathcal{O}^{k}_{\alpha^{\prime}}\Big{)},\mu_{n}^{1}\Big{(}\bigcup_{\alpha^{\prime}\leq\alpha}\mathcal{O}^{k}_{\alpha^{\prime}}\Big{)}\Big{)}$

of rectangle $I^{n,k}_{\alpha,\beta}$ do not depend on $n$. The value of
$\beta$ for which $\bar{\omega}\in I^{n,k}_{\alpha,\beta}$ varies with $n$ but
the value of $\alpha$ does not. Therefore, as $n$ varies with $k$ fixed the
projection onto $U^{1}$ of
$Z_{n,k}(\bar{\omega})=(x^{k}_{\alpha},y^{k}_{\beta})$ remains unchanged.
Thus, after passing to the limit as $k\rightarrow\infty$ we see that
$\bar{\chi}^{1}_{n}$ does not depend on $n$.

For property iii) consider the set $B\subseteq\widetilde{\Omega}$ defined by

$B:=\bigcup_{k=1}^{\infty}\bigcap_{N=1}^{\infty}\bigcup_{n=N}^{\infty}\\{d(\bar{\chi}_{n},\bar{\chi}_{\infty})\geq
4r_{k}\\}.$

It is clear that $\bar{\chi}_{n}\rightarrow\bar{\chi}_{\infty}$ as
$n\rightarrow\infty$ on $B^{c}$. Therefore, we must show that $\widetilde{{\bf
P}}(B)=0$. For this it is sufficient to show that for every $k\in\mathbb{N}$
and every $\varepsilon>0$ there exists an $N\in\mathbb{N}$ such that

(A.3) $\widetilde{{\bf
P}}\Big{(}\bigcup_{n=N}^{\infty}\\{d(\bar{\chi}_{n},\bar{\chi}_{\infty})\geq
4r_{k}\\}\Big{)}\leq\varepsilon.$

For a given $k\in\mathbb{N}$ and $\varepsilon>0$ we can find a positive
integer $K$ and multi-indexes
$\alpha_{1},\beta_{1},\ldots,\alpha_{K},\beta_{K}$ in $\mathbb{N}^{k}$ such
that

(A.4)
$\mu_{\infty}\Big{(}\Big{(}\bigcup_{i=1}^{K}\mathcal{O}^{k}_{\alpha_{i}}\times\mathcal{C}^{k}_{\beta_{i}}\Big{)}^{c}\Big{)}<\frac{\varepsilon}{2},$

because
$\mu_{\infty}\Big{(}\bigcup_{\alpha,\beta\in\mathbb{N}^{k}}\mathcal{O}^{k}_{\alpha}\times\mathcal{C}^{k}_{\beta}\Big{)}=1$.
To shorten the notation, define

$I^{k}_{i}:=I^{\infty,k}_{\alpha_{i},\beta_{i}},\ 1\leq i\leq
K,\quad\textnormal{and}\quad
I^{k}_{0}:=\Big{(}\bigcup_{i=1}^{K}I^{\infty,k}_{\alpha_{i},\beta_{i}}\Big{)}^{c}.$

Now, if $d(\bar{\chi}_{n}(\bar{\omega}),\bar{\chi}_{\infty}(\bar{\omega}))\geq
4r_{k}$ then either $\bar{\omega}\in I^{k}_{0}$ (which has small probability
by (A.4)) or $\bar{\omega}$ belongs to $I^{\infty,k}_{\alpha_{i},\beta_{i}}$
for some $i=1,\ldots,K$. In the second case $\bar{\omega}$ cannot belong to
$I^{n,k}_{\alpha_{i},\beta_{i}}$, otherwise $\bar{\chi}_{n}(\bar{\omega})$ and
$\bar{\chi}_{\infty}(\bar{\omega})$ would both belong to the closure of
$\mathcal{O}^{k}_{\alpha_{i}}\times\mathcal{C}^{k}_{\beta_{i}}$, which has
radius $2r_{k}$. Therefore, we have

$\displaystyle\bigcup_{n=N}^{\infty}\\{d(\bar{\chi}_{n},\bar{\chi}_{\infty})\geq
4r_{k}\\}$ $\displaystyle\subseteq
I^{k}_{0}\cup\bigcup_{n=N}^{\infty}\bigcup_{i=1}^{K}I^{k}_{i}\setminus
I^{n,k}_{\alpha_{i},\beta_{i}}$
$\displaystyle=I^{k}_{0}\cup\Big{(}\bigcup_{i=1}^{K}I^{k}_{i}\setminus\Big{(}\bigcap_{n=N}^{\infty}I^{n,k}_{\alpha_{i},\beta_{i}}\Big{)}\Big{)}$

and the estimate

(A.5) $\widetilde{{\bf
P}}\Big{(}\bigcup_{n=N}^{\infty}\\{d(\bar{\chi}_{n},\bar{\chi}_{\infty})\geq
4r_{k}\\}\Big{)}\leq\frac{\varepsilon}{2}+\sum_{i=1}^{K}\widetilde{{\bf
P}}\Big{(}I^{k}_{i}\setminus\Big{(}\bigcap_{n=N}^{\infty}I^{n,k}_{\alpha_{i},\beta_{i}}\Big{)}\Big{)}.$

To estimate $\widetilde{{\bf
P}}\Big{(}I^{k}_{i}\setminus\Big{(}\bigcap_{n=N}^{\infty}I^{n,k}_{\alpha_{i},\beta_{i}}\Big{)}\Big{)}$
for each $i$, recall that all of the rectangles have the same horizontal
component of length $\mu_{\infty}^{1}(\mathcal{O}^{k}_{\alpha_{i}})$.
Therefore, we just need to estimate the differences between the endpoints of
the vertical components. We have

$\displaystyle\widetilde{{\bf P}}\Big{(}I^{k}_{i}$
$\displaystyle\setminus\Big{(}\bigcap_{n=N}^{\infty}I^{n,k}_{\alpha_{i},\beta_{i}}\Big{)}\Big{)}\leq$
$\displaystyle\mu_{\infty}^{1}(\mathcal{O}^{k}_{\alpha_{i}})\cdot\Big{[}\Big{|}\mu_{\infty}\Big{(}\mathcal{O}^{k}_{\alpha_{i}}\times\bigcup_{\beta^{\prime}<\beta_{i}}\mathcal{C}^{k}_{\beta^{\prime}}\
\Big{|}\ \mathcal{O}^{k}_{\alpha_{i}}\times U^{2}\Big{)}-\sup_{n\geq
N}\mu_{n}\Big{(}\mathcal{O}^{k}_{\alpha_{i}}\times\bigcup_{\beta^{\prime}<\beta_{i}}\mathcal{C}^{k}_{\beta^{\prime}}\
\Big{|}\ \mathcal{O}^{k}_{\alpha_{i}}\times U^{2}\Big{)}\Big{|}$
$\displaystyle+\Big{|}\mu_{\infty}\Big{(}\mathcal{O}^{k}_{\alpha_{i}}\times\bigcup_{\beta^{\prime}\leq\beta_{i}}\mathcal{C}^{k}_{\beta^{\prime}}\
\Big{|}\ \mathcal{O}^{k}_{\alpha_{i}}\times U^{2}\Big{)}-\inf_{n\geq
N}\mu_{n}\Big{(}\mathcal{O}^{k}_{\alpha_{i}}\times\bigcup_{\beta^{\prime}\leq\beta_{i}}\mathcal{C}^{k}_{\beta^{\prime}}\
\Big{|}\ \mathcal{O}^{k}_{\alpha_{i}}\times U^{2}\Big{)}\Big{|}\Big{]}.$

We chose the sequence $\left(r_{k}\right)_{k=1}^{\infty}$ in such a way that
the boundary of every set appearing above has measure $0$ under
$\mu_{\infty}$. Therefore it follows from the weak convergence
$\mu_{n}\Rightarrow\mu_{\infty}$ that we can choose $N\in\mathbb{N}$ large
enough so that the RHS above is bounded by $\varepsilon/(2K)$. Combining this
estimate with (A.5) yields (A.3) and establishes the desired $\widetilde{{\bf
P}}$-a.s. convergence. ∎

##  2\. Acknowledgment

This work was supported by the Research Fund of Indiana University.

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