$\displaystyle\mathbbm{E}\left(\sup_{r\in[\theta_{j},\theta_{k}]}\left\|\bm{\Psi}^{m,n,M}_{r}\right\|_{U}^{2}\right)+\mathbbm{E}\int_{\theta_{j}}^{\theta_{k}}\left\|\hat{\bm{\Psi}}^{m,n}_{s}\right\|_{H}^{2}$
$\displaystyle\qquad\qquad\qquad\qquad\leq
c\mathbbm{E}\left(\left\|\bm{\Psi}^{m,n,M}_{\theta_{j}}\right\|_{U}^{2}\right)+c\mathbbm{E}\int_{\theta_{j}}^{\theta_{k}}\left[1+\left\|\hat{\bm{\Psi}}^{m}_{s}\right\|_{H}^{2}+\left\|\hat{\bm{\Psi}}^{n}_{s}\right\|_{H}^{2}\right]\left\|\hat{\bm{\Psi}}^{m,n}_{s}\right\|_{U}^{2}ds+\frac{c}{\sqrt{\lambda_{m}}}$

with similar seen in Proposition 4.7. An application of Lemma 6.3 then
provides that

$\mathbbm{E}\left(\sup_{r\in[0,T]}\left\|\bm{\Psi}^{m,n,M}_{r}\right\|_{U}^{2}\right)+\mathbbm{E}\int_{0}^{T}\left\|\hat{\bm{\Psi}}^{m,n}_{s}\right\|_{H}^{2}\leq
C\left[\left\|\bm{\Psi}^{m,n}_{0}\right\|_{U}^{2}+\frac{1}{\sqrt{\lambda_{m}}}\right]$

which gives (67). The next result, (68), follows from this as

$\left\|\bm{\Psi}^{m,n}_{0}\right\|_{U}^{2}=\left\|\left(\mathcal{P}_{n}-\mathcal{P}_{m}\right)\bm{\Psi}_{0}\right\|_{U}^{2}=\left\|\mathcal{P}_{n}\left[\left(I-\mathcal{P}_{m}\right)\bm{\Psi}_{0}\right]\right\|_{U}^{2}\leq\left\|\left(I-\mathcal{P}_{m}\right)\bm{\Psi}_{0}\right\|_{U}^{2}$

and the fact that the system $(a_{k})$ forms an orthogonal basis of $U$.

∎

###  4.5 Weak Equicontinuity

In this subsection, the remaining condition of Proposition 6.4, (83), is
verified.

######  Lemma 4.10.

Let $\theta$ be a stopping time and $(\delta_{j})$ a sequence of stopping
times which converge to $0$ $\mathbb{P}-a.s.$. Then

$\lim_{j\rightarrow\infty}\sup_{n\in\mathbb{N}}\mathbbm{E}\left(\left\|\bm{\Psi}^{n}\right\|_{UH,(\theta+\delta_{j})\wedge\tau^{M,T}_{n}}^{2}-\left\|\bm{\Psi}^{n}\right\|_{UH,\theta\wedge\tau^{M,T}_{n}}^{2}\right)=0.$

###### Proof.

We look at the energy identity satisfied by $\bm{\Psi}^{n}$ up until the
stopping time $\theta\wedge\tau^{M,T}_{n}$ and then
$(\theta+r)\wedge\tau^{M,T}_{n}$ for some $r\geq 0$. We have that

$\displaystyle\left\|\bm{\Psi}^{n}_{\theta\wedge\tau^{M,T}_{n}}\right\|_{U}^{2}=\left\|\bm{\Psi}^{n}_{0}\right\|_{U}^{2}$
$\displaystyle+2\int_{0}^{\theta\wedge\tau^{M,T}_{n}}\left\langle\mathcal{P}_{n}\mathcal{A}\left(s,\bm{\Psi}^{n}_{s}\right),\bm{\Psi}^{n}_{s}\right\rangle_{U}ds+\int_{0}^{\theta\wedge\tau^{M,T}_{n}}\sum_{i=1}^{\infty}\left\|\mathcal{P}_{n}\mathcal{G}_{i}\left(s,\bm{\Psi}^{n}_{s}\right)\right\|_{U}^{2}ds$
$\displaystyle+2\int_{0}^{\theta\wedge\tau^{M,T}_{n}}\left\langle\mathcal{P}_{n}\mathcal{G}\left(s,\bm{\Psi}^{n}_{s}\right),\bm{\Psi}^{n}_{s}\right\rangle_{U}d\mathcal{W}_{s}$

and similarly for $(\theta+r)\wedge\tau^{M,T}_{n}$, from which the difference
of the equalities gives

$\displaystyle\left\|\bm{\Psi}^{n}_{(\theta+r)\wedge\tau^{M,T}_{n}}\right\|_{U}^{2}$
$\displaystyle=\left\|\bm{\Psi}^{n}_{\theta\wedge\tau^{M,T}_{n}}\right\|_{U}^{2}+2\int_{\theta\wedge\tau^{M,T}_{n}}^{(\theta+r)\wedge\tau^{M,T}_{n}}\left\langle\mathcal{P}_{n}\mathcal{A}\left(s,\bm{\Psi}^{n}_{s}\right),\bm{\Psi}^{n}_{s}\right\rangle_{U}ds$
$\displaystyle+\int_{\theta\wedge\tau^{M,T}_{n}}^{(\theta+r)\wedge\tau^{M,T}_{n}}\sum_{i=1}^{\infty}\left\|\mathcal{P}_{n}\mathcal{G}_{i}\left(s,\bm{\Psi}^{n}_{s}\right)\right\|_{U}^{2}ds+2\int_{\theta\wedge\tau^{M,T}_{n}}^{(\theta+r)\wedge\tau^{M,T}_{n}}\left\langle\mathcal{P}_{n}\mathcal{G}\left(s,\bm{\Psi}^{n}_{s}\right),\bm{\Psi}^{n}_{s}\right\rangle_{U}d\mathcal{W}_{s}.$

Handling the projections as we did in (26), and invoking (6), we reduce to

$\displaystyle\left\|\bm{\Psi}^{n}_{(\theta+r)\wedge\tau^{M,T}_{n}}\right\|_{U}^{2}$
$\displaystyle-\left\|\bm{\Psi}^{n}_{\theta\wedge\tau^{M,T}_{n}}\right\|_{U}^{2}+\gamma\int_{\theta\wedge\tau^{M,T}_{n}}^{(\theta+r)\wedge\tau^{M,T}_{n}}\left\|\bm{\Psi}^{n}_{s}\right\|_{H}^{2}ds$
$\displaystyle\qquad\qquad\qquad\leq
c\int_{\theta\wedge\tau^{M,T}_{n}}^{(\theta+r)\wedge\tau^{M,T}_{n}}1ds+\int_{\theta\wedge\tau^{M,T}_{n}}^{(\theta+r)\wedge\tau^{M,T}_{n}}\left\langle\mathcal{G}\left(s,\bm{\Psi}^{n}_{s}\right),\bm{\Psi}^{n}_{s}\right\rangle_{U}d\mathcal{W}_{s}.$

where the constant $c$ again depends on $M$, through (37). Taking expectation,
the supremum over $r\in[0,\delta_{j}]$ and then using the Burkholder-Davis-
Gundy Inequality,

$\displaystyle\mathbbm{E}\left[\sup_{r\in[0,\delta_{j}]}\left\|\bm{\Psi}^{n}_{(\theta+r)\wedge\tau^{M,T}_{n}}\right\|_{U}^{2}-\left\|\bm{\Psi}^{n}_{\theta\wedge\tau^{M,T}_{n}}\right\|_{U}^{2}+\gamma\int_{\theta\wedge\tau^{M,T}_{n}}^{(\theta+\delta_{j})\wedge\tau^{M,T}_{n}}\left\|\bm{\Psi}^{n}_{s}\right\|_{H}^{2}ds\right]$
$\displaystyle\qquad\qquad\qquad\qquad\leq
c\mathbbm{E}\int_{\theta\wedge\tau^{M,T}_{n}}^{(\theta+\delta_{j})\wedge\tau^{M,T}_{n}}1ds+c\mathbbm{E}\left(\int_{\theta\wedge\tau^{M,T}_{n}}^{(\theta+\delta_{j})\wedge\tau^{M,T}_{n}}\sum_{i=1}^{\infty}\left\langle\mathcal{G}_{i}\left(s,\bm{\Psi}^{n}_{s}\right),\bm{\Psi}^{n}_{s}\right\rangle^{2}_{U}ds\right)^{\frac{1}{2}}.$

We then use (7) and again control the $U$ norm by a constant, achieving that

$\mathbbm{E}\left[\sup_{r\in[0,\delta_{j}]}\left\|\bm{\Psi}^{n}_{(\theta+r)\wedge\tau^{M,T}_{n}}\right\|_{U}^{2}-\left\|\bm{\Psi}^{n}_{\theta\wedge\tau^{M,T}_{n}}\right\|_{U}^{2}+\int_{\theta\wedge\tau^{M,T}_{n}}^{(\theta+\delta_{j})\wedge\tau^{M,T}_{n}}\left\|\bm{\Psi}^{n}_{s}\right\|_{H}^{2}ds\right]\leq
c\mathbbm{E}\left(\delta_{j}+\delta_{j}^{\frac{1}{2}}\right)$ (72)

having also scaled out the $\gamma$. We now need to relate the expression on
the left hand side with what we are interested in, which is
$\left\|\bm{\Psi}^{n}\right\|_{UH,(\theta+\delta_{j})\wedge\tau^{M,T}_{n}}^{2}-\left\|\bm{\Psi}^{n}\right\|_{UH,\theta\wedge\tau^{M,T}_{n}}^{2}$.
We have that

$\displaystyle\left\|\bm{\Psi}^{n}\right\|_{UH,(\theta+\delta_{j})\wedge\tau^{M,T}_{n}}^{2}$
$\displaystyle-\left\|\bm{\Psi}^{n}\right\|_{UH,\theta\wedge\tau^{M,T}_{n}}^{2}$
$\displaystyle=\sup_{s\in[0,(\theta+\delta_{j})\wedge\tau^{M,T}_{n}]}\left\|\bm{\Psi}^{n}_{s}\right\|_{U}^{2}-\sup_{s\in[0,\theta\wedge\tau^{M,T}_{n}]}\left\|\bm{\Psi}^{n}_{s}\right\|_{U}^{2}+\int_{\theta\wedge\tau^{M,T}_{n}}^{(\theta+\delta_{j})\wedge\tau^{M,T}_{n}}\left\|\bm{\Psi}^{n}_{s}\right\|_{H}^{2}ds$

and claim

$\sup_{s\in[0,(\theta+\delta_{j})\wedge\tau^{M,T}_{n}]}\left\|\bm{\Psi}^{n}_{s}\right\|_{U}^{2}-\sup_{s\in[0,\theta\wedge\tau^{M,T}_{n}]}\left\|\bm{\Psi}^{n}_{s}\right\|_{U}^{2}\leq\sup_{r\in[0,\delta_{j}]}\left\|\bm{\Psi}^{n}_{(\theta+r)\wedge\tau^{M,T}_{n}}\right\|_{U}^{2}-\left\|\bm{\Psi}^{n}_{\theta\wedge\tau^{M,T}_{n}}\right\|_{U}^{2}.$
(73)

Indeed, we have that

$\sup_{s\in[0,(\theta+\delta_{j})\wedge\tau^{M,T}_{n}]}\left\|\bm{\Psi}^{n}_{s}\right\|_{U}^{2}\leq\sup_{s\in[0,\theta\wedge\tau^{M,T}_{n}]}\left\|\bm{\Psi}^{n}_{s}\right\|_{U}^{2}+\sup_{s\in[\theta\wedge\tau^{M,T}_{n},(\theta+\delta_{j})\wedge\tau^{M,T}_{n}]}\left\|\bm{\Psi}^{n}_{s}\right\|_{U}^{2}-\left\|\bm{\Psi}^{n}_{\theta\wedge\tau^{M,T}_{n}}\right\|_{U}^{2}$

as the left hand side must equal either
$\sup_{s\in[0,\theta\wedge\tau^{M,T}_{n}]}\left\|\bm{\Psi}^{n}_{s}\right\|_{U}^{2}$
or
$\sup_{s\in[\theta\wedge\tau^{M,T}_{n},(\theta+\delta_{j})\wedge\tau^{M,T}_{n}]}\left\|\bm{\Psi}^{n}_{s}\right\|_{U}^{2}$,
both of which are greater than the subtracted term
$\left\|\bm{\Psi}^{n}_{\theta\wedge\tau^{M,T}_{n}}\right\|_{U}^{2}$.
Appreciating that

$\sup_{s\in[\theta\wedge\tau^{M,T}_{n},(\theta+\delta_{j})\wedge\tau^{M,T}_{n}]}\left\|\bm{\Psi}^{n}_{s}\right\|_{U}^{2}=\sup_{r\in[0,\delta_{j}]}\left\|\bm{\Psi}^{n}_{(\theta+r)\wedge\tau^{M,T}_{n}}\right\|_{U}^{2}$

then yields the claim (73), which in combination with (72) grants that

$\displaystyle\mathbbm{E}\left[\left\|\bm{\Psi}^{n}\right\|_{UH,(\theta+\delta_{j})\wedge\tau^{M,T}_{n}}^{2}-\left\|\bm{\Psi}^{n}\right\|_{UH,\theta\wedge\tau^{M,T}_{n}}^{2}\right]\leq
c\mathbbm{E}\left(\delta_{j}+\delta_{j}^{\frac{1}{2}}\right).$

To conclude the proof we simply note that
$\delta_{j}+\delta_{j}^{\frac{1}{2}}$ is $\mathbbm{P}-a.s.$ monotone
decreasing (as $j\rightarrow\infty$) and convergent to $0$. The Monotone
Convergence Theorem thus justifies that the limit as $j\rightarrow\infty$ of
the right hand side is zero, so the result is shown.

∎

###  4.6 Existence and Uniqueness of Strong Solutions

We now have all of the ingredients needed to apply Proposition 6.4. In this
subsection, we fix $\bm{\Psi}$ to be the unique weak solution of the equation
(1) for the initial condition fixed in (58), known to exist from Theorem 3.5.

######  Proposition 4.11.

For any $R>0$, there exists a subsequence indexed by $(m_{j})$ and a constant
$M>1$ such that the stopping time

$\tau^{R,T}:=T\wedge\inf\left\\{s\geq
0:\sup_{r\in[0,s]}\left\|\bm{\Psi}_{r}\right\|_{U}^{2}+\int_{0}^{s}\left\|\bm{\Psi}_{r}\right\|_{H}^{2}dr\geq
R\right\\}$

satisfies the property $\tau^{R,T}\leq\tau^{M,T}_{m_{j}}$ for all $m_{j}$
$\mathbb{P}-a.s.$.

###### Proof.

This comes from a direct application of Lemma 6.4, for the Banach Space
$X_{t}:=C\left([0,t];U\right)\cap L^{2}\left([0,t];H\right)$ equipped with
norm $\left\|\cdot\right\|_{UH,t}$. The conditions (82) and (83) are given in
Proposition 4.9 and Lemma 4.10. The only component in need of proof is that
the limiting process $\bm{\Phi}$ we achieve from Lemma 6.4 is in fact
$\bm{\Psi}$. We are given that $(\bm{\Psi}^{m_{j}})$ converges to $\bm{\Psi}$
in $L^{2}\left([0,\tau^{M,T}_{\infty}];H\right)$ which is a stronger
convergence (up to $\tau^{M,T}_{\infty}$) than what we had in Theorem 2.15,
which was used to show that the limit process was a weak solution of the
equation (1). Hence in the same manner as Proposition 2.18, we see that
$\bm{\Phi}$ must be a local weak solution as well, up until the stopping time
$\tau^{M,T}_{\infty}$, so from the uniqueness of weak solutions we must have
that $\bm{\Phi}$ is in fact equal to $\bm{\Psi}$ on $[0,\tau^{M,T}_{\infty}]$,
which is sufficient for the result. ∎

This allows us to deduce the additional regularity required for $\bm{\Psi}$ to
be a strong solution.

######  Proposition 4.12.

The process $\bm{\Psi}$ is progressively measurable in $V$ and such that for
$\mathbb{P}-a.e.$ $\omega$, $\bm{\Psi}_{\cdot}(\omega)\in
L^{\infty}\left([0,T];H\right)\cap L^{2}\left([0,T];V\right)$.

###### Proof.

Propositions 4.7 and 4.11 together imply that, for any $R>0$,

$\mathbbm{E}\left[\sup_{r\in[0,\tau^{R,T}]}\left\|\bm{\Psi}^{m_{j}}_{r}\right\|_{1}^{2}+\int_{0}^{\tau^{R,T}}\left\|\bm{\Psi}^{m_{j}}_{r}\right\|_{2}^{2}dr\right]\leq
C_{R}$

where $C_{R}$ now incorporates
$\left\|\bm{\Psi}_{0}\right\|^{2}_{L^{\infty}(\Omega;H)}$ and shows the
dependency on $M$ in terms of $R$. We also note use of (2) in this deduction.
In consequence the sequence of processes
$(\bm{\Psi}^{m_{j}}_{\cdot}\mathbbm{1}_{\cdot\leq\tau^{R,T}})$ is uniformly
bounded in both $L^{2}\left(\Omega;L^{\infty}\left([0,T];H\right)\right)$ and
$L^{2}\left(\Omega;L^{2}\left([0,T];V\right)\right)$. We can deduce the
existence of a subsequence which is weakly convergent in the Hilbert Space
$L^{2}\left(\Omega;L^{2}\left([0,T];V\right)\right)$ to some
$\check{\bm{\Phi}}$, but we may also identify
$L^{2}\left(\Omega;L^{\infty}\left([0,T];H\right)\right)$ with the dual space
of $L^{2}\left(\Omega;L^{1}\left([0,T];H\right)\right)$ and as such from the
Banach-Alaoglu Theorem we can extract a further subsequence which is
convergent to some $\hat{\bm{\Phi}}$ in the weak* topology. These limits imply
convergence to both $\check{\bm{\Phi}}$ and $\hat{\bm{\Phi}}$ in the weak
topology of $L^{2}\left(\Omega;L^{2}\left([0,T];H\right)\right)$. However, we
know that $(\bm{\Psi}^{m_{j}}_{\cdot}\mathbbm{1}_{\cdot\leq\tau^{R,T}})$
converges to $\bm{\Psi}_{\cdot}\mathbbm{1}_{\cdot\leq\tau^{R,T}}$
$\mathbb{P}-a.s.$ in $L^{2}\left([0,T];H\right)$ from Lemma 6.4, and from the
Dominated Convergence Theorem (domination comes easily from
$\mathbbm{1}_{\cdot\leq\tau^{R,T}}$, $\tau^{R,T}\leq\tau^{M,T}_{m_{j}}$) then
in fact the convergence holds in
$L^{2}\left(\tilde{\Omega};L^{2}\left([0,T];H\right)\right)$. Of course the
same convergence then holds in the weak topology, and by uniqueness of limits
in the weak topology then
$\bm{\Psi}_{\cdot}\mathbbm{1}_{\cdot\leq\tau^{R,T}}=\check{\bm{\Phi}}=\hat{\bm{\Phi}}$
as elements of $L^{2}\left(\Omega;L^{2}\left([0,T];H\right)\right)$, so they
agree $\mathbbm{P}\times\lambda-a.s.$. Thus for $\mathbbm{P}-a.e.$ $\omega$,
$\bm{\Psi}_{\cdot}(\omega)\mathbbm{1}_{\cdot\leq\tau^{R,T}(\omega)}\in
L^{\infty}\left([0,T];H\right)\cap L^{2}\left([0,T];V\right)$. At each such
$\omega$ the regularity of $\bm{\Psi}$ as a weak solution ensures that for
sufficiently large $R$ (dependent on $\omega$), $\tau^{R,T}(\omega)=T$.
Therefore for $\mathbb{P}-a.e.$ $\omega$ , $\bm{\Psi}_{\cdot}(\omega)\in
L^{\infty}\left([0,T];H\right)\cap L^{2}\left([0,T];V\right)$.

The progressive measurability is justified similarly; for any $t\in[0,T]$, we
can use the progressive measurability of $(\bm{\Psi}^{m_{j}})$ and hence
$(\bm{\Psi}^{m_{j}}_{\cdot}\mathbbm{1}_{\cdot\leq\tau^{R,T}})$ to instead
deduce $\bm{\Psi}_{\cdot}\mathbbm{1}_{\cdot\leq\tau^{R,T}}$ as the weak limit
in $L^{2}\left(\Omega\times[0,t];V\right)$ where $\Omega\times[0,t]$ is
equipped with the $\mathcal{F}_{t}\times\mathcal{B}\left([0,t]\right)$ sigma-
algebra. Therefore
$\bm{\Psi}_{\cdot}\mathbbm{1}_{\cdot\leq\tau^{R,T}}:\Omega\times[0,t]\rightarrow
V$ is measurable with respect to this product sigma-algebra which justifies
the progressive measurability of the truncated process. To carry this property
over to $\bm{\Psi}$, we recognise $\bm{\Psi}$ as the $\mathbb{P}\times\lambda$
almost everywhere limit of the sequence
$(\bm{\Psi}_{\cdot}\mathbbm{1}_{\cdot\leq\tau^{R,T}})$ as $R\rightarrow\infty$
over the product space $\Omega\times[0,t]$ equipped with product sigma-algebra
$\mathcal{F}_{t}\times\mathcal{B}\left([0,t]\right)$. Such a limit preserves
the measurability in this product sigma-algebra, justifying the progressive
measurability of $\bm{\Psi}$.

∎

Applying Lemma 4.6 in conjunction with Proposition 4.12 now proves Theorem 4.5
in the case of a bounded initial condition (58). Extending this result to a
general $\mathcal{F}_{0}-$measurable $\bm{\Psi}_{0}:\Omega\rightarrow H$ is
done exactly as we did for the weak solution in Subsection 3.3, so we do not
repeat these steps and conclude the proof of Theorem 4.5 here.

###  4.7 Continuity?

We close this section with a somewhat more informal discussion around the
pathwise continuity of the strong solution $\bm{\Psi}$ in $H$. One may well
have expected this to appear in the definition of the strong solution, though
it seems challenging to verify and additional assumptions that could
facilitate it are perhaps too restrictive. A first result is given now.

######  Lemma 4.13.

Let Assumption Sets 1, 2 and 3 hold. Suppose that there exists a continuous
bilinear form $\left\langle\cdot,\cdot\right\rangle_{U\times V}:U\times
V\rightarrow\mathbb{R}$ such that for every $f\in H$, $\phi\in V$,

$\left\langle f,\phi\right\rangle_{U\times V}=\left\langle
f,\phi\right\rangle_{H}.$

Then for $\mathbbm{P}-a.e.$ $\omega$, $\bm{\Psi}_{\cdot}(\omega)\in
C_{w}\left([0,T];H\right)$.

###### Proof.

Fixing arbitrary $\phi\in V$, $\bm{\Psi}$ satisfies the identity

$\left\langle\bm{\Psi}_{t},\phi\right\rangle_{U\times
V}=\left\langle\bm{\Psi}_{0},\phi\right\rangle_{U\times
V}+\left\langle\int_{0}^{t}\mathcal{A}(s,\bm{\Psi}_{s})ds,\phi\right\rangle_{U\times
V}+\left\langle\int_{0}^{t}\mathcal{G}(s,\bm{\Psi}_{s})d\mathcal{W}_{s},\phi\right\rangle_{U\times
V}$

$\mathbbm{P}-a.s.$ for all $t\in[0,T]$, and by continuity of the bilinear
form,

$\left\langle\bm{\Psi}_{t},\phi\right\rangle_{U\times
V}=\left\langle\bm{\Psi}_{0},\phi\right\rangle_{U\times
V}+\int_{0}^{t}\left\langle\mathcal{A}(s,\bm{\Psi}_{s}),\phi\right\rangle_{U\times
V}ds+\int_{0}^{t}\left\langle\mathcal{G}(s,\bm{\Psi}_{s}),\phi\right\rangle_{U\times
V}d\mathcal{W}_{s}.$

We want to use that $\left\langle\bm{\Psi}_{t},\phi\right\rangle_{U\times
V}=\left\langle\bm{\Psi}_{t},\phi\right\rangle_{H}$, but it is not yet clear
that $\bm{\Psi}_{t}\in H$ $\mathbbm{P}-a.s.$ for all $t\in[0,T]$. It can, in
fact, be verified by a very similar process to the proof of Proposition 4.12.
To show this let us fix any $t\in[0,T]$. For $\mathbbm{P}-a.e.$ $\omega$,
$\bm{\Psi}_{\cdot}(\omega)\in L^{\infty}\left([0,T];H\right)$ thus for a
countable dense subset $(t_{n})$ of $[0,T]$, $\bm{\Psi}_{t_{n}}(\omega)\in H$
and such that $\left\|\bm{\Psi}_{t_{n}}(\omega)\right\|_{H}\leq c$ independent
of $n$. We choose now any sequence $(t_{k})$ which converges to $t$. The
sequence $(\bm{\Psi}_{t_{k}}(\omega))$ is uniformly bounded in $H$, so admits
a weakly convergent subsequence to a limit $\psi$. However, as
$\bm{\Psi}_{\cdot}(\omega)\in C\left([0,T];U\right)$ then
$(\bm{\Psi}_{t_{k}}(\omega))$ is strongly hence weakly convergent to
$\bm{\Psi}_{t}(\omega)$ in $U$, so by uniqueness of limits in the weak
topology then $\bm{\Psi}_{t}(\omega)=\psi$ and in particular
$\bm{\Psi}_{t}(\omega)\in H$. Thus, $\mathbbm{P}-a.s.$ for all $t\in[0,T]$,

$\left\langle\bm{\Psi}_{t},\phi\right\rangle_{H}=\left\langle\bm{\Psi}_{0},\phi\right\rangle_{U\times
V}+\int_{0}^{t}\left\langle\mathcal{A}(s,\bm{\Psi}_{s}),\phi\right\rangle_{U\times
V}ds+\int_{0}^{t}\left\langle\mathcal{G}(s,\bm{\Psi}_{s}),\phi\right\rangle_{U\times
V}d\mathcal{W}_{s}$

so that $\left\langle\bm{\Psi}_{t},\phi\right\rangle_{H}$ is
$\mathbbm{P}-a.s.$ continuous. This is not quite enough for weak continuity,
as we instead require that $\left\langle\bm{\Psi}_{t},f\right\rangle_{H}$ is
$\mathbbm{P}-a.s.$ continuous for any $f\in H$, not just any $\phi\in V$. From
this point, though, we can extend the continuity to $f$ by the density of $V$
in $H$, identically to the end of the proof of Proposition 2.18. ∎

Pathwise continuity of $\bm{\Psi}$ in $H$ would now follow from only pathwise
continuity of the norm $\left\|\bm{\Psi}_{\cdot}\right\|_{H}$, see e.g. [20]
Lemma 1.5.3. A popular way to show such a property is through Kolmogorov’s
Continuity Theorem, by verifying that there exists a constant $C$ such that
for all $s<t\in[0,T]$,

$\mathbbm{E}\left(\left|\left\|\bm{\Psi}_{t}\right\|_{H}^{2}-\left\|\bm{\Psi}_{s}\right\|_{H}^{2}\right|^{4}\right)\leq
C(t-s)^{2}$

which is the typical choice of parameters to apply Kolmogorov’s Continuity
Theorem for a Brownian Motion and related SDEs. Our assumptions don’t quite
allow us to show this, even at the level of the Galerkin System: the issue
arises in the $V$-norm dependence in (57), which is necessary to have for
applications in the case of Navier Boundary Conditions as seen in Theorem 5.5.
There is, however, a simpler case where continuity can be deduced.

######  Lemma 4.14.

Let Assumption Sets 1, 2 and 3 hold. Suppose that there exists a continuous
bilinear form $\left\langle\cdot,\cdot\right\rangle_{U\times V}:U\times
V\rightarrow\mathbb{R}$ such that for every $f\in H$, $\phi\in V$,

$\left\langle f,\phi\right\rangle_{U\times V}=\left\langle
f,\phi\right\rangle_{H}.$

In addition, suppose that $\bar{H}$ can be taken as $H$. Then for
$\mathbbm{P}-a.e.$ $\omega$, $\bm{\Psi}_{\cdot}(\omega)\in
C\left([0,T];H\right)$.

###### Proof.

This is now a direct application of Proposition 6.2. ∎

##  5 Applications: Stochastic Navier-Stokes Equations

We present applications of our framework for the Navier-Stokes Equation under
Stochastic Advection by Lie Transport (SALT) proposed in [29], given by

$u_{t}=u_{0}-\int_{0}^{t}\mathcal{L}_{u_{s}}u_{s}\,ds+\nu\int_{0}^{t}\Delta
u_{s}\,ds+\int_{0}^{t}B(u_{s})\circ d\mathcal{W}_{s}-\nabla\rho_{t}$ (74)

where $u$ represents the fluid velocity, $\rho$ the pressure555The pressure
term is a semimartingale, and an explicit form for the SALT Euler Equation is
given in [54] Subsection 3.3, $\mathcal{L}$ represents the nonlinear term and
$B$ is a first order differential operator (the SALT Operator). A complete
introduction to this equation is given in [22], with full technical details
that are glossed over here, although we note that our results can be applied
for a variety of additive, multiplicative and transport noise structures. The
choice of SALT noise is particularly challenging and demonstrates the efficacy
of our framework. Intrinsic to this stochastic methodology is that $B$ is
defined relative to a collection of functions $(\xi_{i})$ which physically
represent spatial correlations. These $(\xi_{i})$ can be determined at coarse-
grain resolutions from finely resolved numerical simulations, and
mathematically are derived as eigenvectors of a velocity-velocity correlation
matrix (see [7, 8, 9]). Indeed, this operator $B$ is given by the actions of
its components $B_{i}$ on a vector field $\phi$ by

$B_{i}:\phi\mapsto\sum_{j=1}^{N}\left(\xi_{i}^{j}\partial_{j}\phi+\phi^{j}\nabla\xi_{i}^{j}\right)$

in $N$-dimensions, where the superscript denotes the $j^{\textnormal{th}}$
component mapping. We note this is the sum of a classical transport term and a
zeroth-order term. The equation (74) is posed on a smooth bounded domain
$\mathscr{O}\subset\mathbb{R}^{N}$, and we introduce the notation
$C^{\infty}_{0,\sigma}$ to be the subspace of
$C^{\infty}_{0}\left(\mathscr{O};\mathbb{R}^{N}\right)$ consisting of
divergence-free functions, which is to say those $\phi\in
C^{\infty}_{0}\left(\mathscr{O};\mathbb{R}^{N}\right)$ such that
$\sum_{j=1}^{N}\partial_{j}\phi^{j}=0$. Following this, we define
$L^{2}_{\sigma}$ as the closure of $C^{\infty}_{0,\sigma}$ in
$L^{2}\left(\mathscr{O};\mathbb{R}^{N}\right)$, and the Leray Projector
$\mathcal{P}$ as the orthogonal projection in
$L^{2}\left(\mathscr{O};\mathbb{R}^{N}\right)$ onto $L^{2}_{\sigma}$.

To study this equation more freely, we commit two manipulations of (74); the
first is to project via $\mathcal{P}$, and the second is to convert to Itô
Form. With this, we arrive at

$u_{t}=u_{0}-\int_{0}^{t}\mathcal{P}\mathcal{L}_{u_{s}}u_{s}\
ds+\nu\int_{0}^{t}\mathcal{P}\Delta
u_{s}\,ds+\frac{1}{2}\int_{0}^{t}\sum_{i=1}^{\infty}\mathcal{P}B_{i}^{2}u_{s}ds+\int_{0}^{t}\mathcal{P}B(u_{s})d\mathcal{W}_{s}$
(75)

which is now in the form of (1). We emphasise again that a thorough overview
of this process is given in [22]. Applications of our framework for different
dimensions and boundary conditions are given below.

###  5.1 No-Slip Boundary Condition

Let us impose the boundary condition $u=0$ on $\partial\mathscr{O}$; this is
the so-called no-slip boundary condition. We define the space
$W^{1,2}_{\sigma}$ as the closure of $C^{\infty}_{0,\sigma}$ in
$W^{1,2}\left(\mathscr{O};\mathbb{R}^{N}\right)$, and $W^{2,2}_{\sigma}$ as
the intersection of $W^{2,2}\left(\mathscr{O};\mathbb{R}^{N}\right)$ with
$W^{1,2}_{\sigma}$. Indeed functions in $W^{1,2}_{\sigma}$ satisfy both the
divergence-free and zero-trace conditions, so the incompressibility and
boundary constraints are embedded into this function space. The functional
framework of Subsection 1.3 is satisfied for the spaces

$V:=W^{2,2}_{\sigma},\qquad H:=W^{1,2}_{\sigma},\qquad U:=L^{2}_{\sigma}$

and the system $(a_{k})$ of eigenfunctions of the Stokes Operator
$-\mathcal{P}\Delta$. The mappings $\mathcal{P}\mathcal{L}$,
$\mathcal{P}\Delta$ are understood from $W^{1,2}_{\sigma}$ into
$\left(W^{1,2}_{\sigma}\right)^{*}$ by the duality pairings for $f,g\in
W^{1,2}_{\sigma}$ of

$\displaystyle\left\langle\mathcal{P}\mathcal{L}_{f}f,g\right\rangle_{\left(W^{1,2}_{\sigma}\right)^{*}\times
W^{1,2}_{\sigma}}$
$\displaystyle=\left\langle\mathcal{L}_{f}f,g\right\rangle_{L^{6/5}\times
L^{6}}$ $\displaystyle\left\langle\mathcal{P}\Delta
f,g\right\rangle_{\left(W^{1,2}_{\sigma}\right)^{*}\times W^{1,2}_{\sigma}}$
$\displaystyle=-\left\langle f,g\right\rangle_{1}$

where $\left\langle\cdot,\cdot\right\rangle_{1}$ represents the homogeneous
$W^{1,2}$ inner product, which is to say without the $L^{2}$ component. As a
direct application of Theorem 2.7, we can obtain the following.

######  Theorem 5.1.

Let $N=2$ or $3$, $u_{0}\in L^{\infty}\left(\Omega;L^{2}_{\sigma}\right)$ be
$\mathcal{F}_{0}-$measurable, $(\xi_{i})\in W^{1,2}_{\sigma}\cap W^{2,\infty}$
with $\sum_{i=1}^{\infty}\left\|\xi_{i}\right\|_{W^{2,\infty}}^{2}<\infty$.
Then there exists a filtered probability space
$\left(\tilde{\Omega},\tilde{\mathcal{F}},(\tilde{\mathcal{F}}_{t}),\tilde{\mathbbm{P}}\right)$,
a Cylindrical Brownian Motion $\tilde{\mathcal{W}}$ over $\mathfrak{U}$ with
respect to
$\left(\tilde{\Omega},\tilde{\mathcal{F}},(\tilde{\mathcal{F}}_{t}),\tilde{\mathbbm{P}}\right)$,
an $\mathcal{F}_{0}-$measurable $\tilde{u}_{0}:\tilde{\Omega}\rightarrow U$
with the same law as $u_{0}$, and a progressively measurable process
$\tilde{u}$ in $W^{1,2}_{\sigma}$ such that for $\tilde{\mathbb{P}}-a.e.$
$\tilde{\omega}$, $\tilde{u}_{\cdot}(\omega)\in
L^{\infty}\left([0,T];L^{2}_{\sigma}\right)\cap
C_{w}\left([0,T];L^{2}_{\sigma}\right)\cap
L^{2}\left([0,T];W^{1,2}_{\sigma}\right)$ and

$\displaystyle\tilde{u}_{t}=\tilde{u}_{0}-\int_{0}^{t}\mathcal{P}\mathcal{L}_{\tilde{u}_{s}}\tilde{u}_{s}\
ds+\nu\int_{0}^{t}\mathcal{P}\Delta\tilde{u}_{s}\,ds+\frac{1}{2}\int_{0}^{t}\sum_{i=1}^{\infty}\mathcal{P}B_{i}^{2}\tilde{u}_{s}ds+\int_{0}^{t}\mathcal{P}B(\tilde{u}_{s})d\tilde{\mathcal{W}}_{s}$

holds $\tilde{\mathbb{P}}-a.s.$ in $\left(W^{1,2}_{\sigma}\right)^{*}$ for all
$t\in[0,T]$.

In fact we can do better in 2D, as an application of Theorem 3.5.

######  Theorem 5.2.

Let $N=2$, $u_{0}:\Omega\rightarrow L^{2}_{\sigma}$ be
$\mathcal{F}_{0}-$measurable, $(\xi_{i})\in W^{1,2}_{\sigma}\cap W^{2,\infty}$
with $\sum_{i=1}^{\infty}\left\|\xi_{i}\right\|_{W^{2,\infty}}^{2}<\infty$.
Then there exists a progressively measurable process $u$ in $W^{1,2}_{\sigma}$
such that for $\mathbb{P}-a.e.$ $\tilde{\omega}$, $u_{\cdot}(\omega)\in
C\left([0,T];L^{2}_{\sigma}\right)\cap
L^{2}\left([0,T];W^{1,2}_{\sigma}\right)$ and (75) holds $\mathbb{P}-a.s.$ in
$\left(W^{1,2}_{\sigma}\right)^{*}$ for all $t\in[0,T]$.

These results are proven in [23] Theorems 1.9 and 1.10, stated slightly
differently although Lemma 3.9 connects the definitions. Whilst these results
are not new, they are very recent and our framework provides an efficient way
to obtain them. More than that, we can invoke Theorem 4.5 to obtain the
existence of strong solutions of the equation (75). This is a new result, and
full details of this application are to come in a following paper with Hao
Tang.

######  Theorem 5.3.

Let $N=2$, $u_{0}:\Omega\rightarrow W^{1,2}_{\sigma}$ be
$\mathcal{F}_{0}-$measurable, $(\xi_{i})\in L^{2}_{\sigma}\cap W^{3,2}_{0}\cap
W^{3,\infty}$ with
$\sum_{i=1}^{\infty}\left\|\xi_{i}\right\|_{W^{3,\infty}}^{2}<\infty$. Then
there exists a progressively measurable process $u$ in $W^{2,2}_{\sigma}$ such
that for $\mathbb{P}-a.e.$ $\omega$, $u_{\cdot}(\omega)\in
C_{w}\left([0,T];W^{1,2}_{\sigma}\right)\cap
L^{\infty}\left([0,T];W^{1,2}_{\sigma}\right)\cap
L^{2}\left([0,T];W^{2,2}_{\sigma}\right)$ and (75) holds $\mathbb{P}-a.s.$ in
$L^{2}_{\sigma}$ for all $t\in[0,T]$.

We note here the use of Lemma 4.13 and the necessity of considering an
extension $\bar{H}$ of $H$ in Assumption Set 3, Subsection 4.1. The Leray
Projector $\mathcal{P}$ does not preserve the zero-trace property, so
$\mathcal{P}B_{i}$ does not map from $W^{2,2}_{\sigma}$ into
$W^{1,2}_{\sigma}$, but instead an extended space $\bar{W}^{1,2}_{\sigma}$
defined as the intersection of $W^{1,2}$ with $L^{2}_{\sigma}$. This is again
a Hilbert Space with $\left\langle\cdot,\cdot\right\rangle_{1}$ inner product;
see [21] Subsection 1.2. Moreover, we cannot directly consider the energy of
the strong solution as the triplet
$W^{2,2}_{\sigma}\xhookrightarrow{}\bar{W}^{1,2}_{\sigma}\xhookrightarrow{}L^{2}_{\sigma}$
does not satisfy the duality relation as needed in Proposition 6.2, so use of
Proposition 6.4 to directly achieve blow-up in the energy norm of weak
solutions is instrumental in achieving the existence of the global solution.

###  5.2 Navier Boundary Conditions

In two spatial dimensions we can impose different boundary conditions for
(75), namely the Navier boundary conditions. These are defined on
$\partial\mathscr{O}$ by

$u\cdot\mathbf{n}=0,\qquad 2(Du)\mathbf{n}\cdot\mathbf{\iota}+\alpha
u\cdot\mathbf{\iota}=0$ (76)

where $\mathbf{n}$ is the unit outwards normal vector, $\mathbf{\iota}$ the
unit tangent vector, $Du$ is the rate of strain tensor
$(Du)^{k,l}:=\frac{1}{2}\left(\partial_{k}u^{l}+\partial_{l}u^{k}\right)$ and
$\alpha\in C^{2}(\partial\mathscr{O};\mathbb{R})$ represents a friction
coefficient which determines the extent to which the fluid slips on the
boundary relative to the tangential stress. These conditions were first
proposed by Navier in [44, 45], and have been derived in [42] from the kinetic
theory of gases and in [41] as a hydrodynamic limit. Furthermore these
conditions have proven viable for modelling rough boundaries as seen in [2,
18, 50]. To fit the framework of this paper we again have to embed the
boundary conditions into useful function spaces. We shall use the space
$\bar{W}^{1,2}_{\sigma}$, which was the intersection of $W^{1,2}$ with
$L^{2}_{\sigma}$, and contains the divergence-free and impermeable boundary
condition (that $u\cdot\mathbf{n}=0$). The remaining component of (76) has to
be included at the $W^{2,2}$ level, as we are concerned with the trace of a
derivative which needs more than $W^{1,2}$ regularity to be understood in the
usual sense. Thus, we define

$\bar{W}^{2,2}_{\alpha}:=\left\\{f\in
W^{2,2}(\mathscr{O};\mathbb{R}^{2})\cap\bar{W}^{1,2}_{\sigma}:2(Df)\mathbf{n}\cdot\iota+\alpha
f\cdot\iota=0\textnormal{ on }\partial\mathscr{O}\right\\}.$

Of course this space does not appear in the definition of a weak solution, so
it is perhaps unclear how the boundary conditions (76) inform the weak
solution. The answer comes from how to extend the Stokes Operator
$-\mathcal{P}\Delta$ to $\bar{W}^{1,2}_{\sigma}$. In [32] equation (5.1), c.f.
[21] Lemma 1.4, it is verified that for $\phi\in\bar{W}^{2,2}_{\alpha}$,
$f\in\bar{W}^{1,2}_{\sigma}$,

$\left\langle\mathcal{P}\Delta\phi,f\right\rangle_{L^{2}}=-\left\langle\phi,f\right\rangle_{1}+\left\langle(\kappa-\alpha)\phi,f\right\rangle_{L^{2}(\partial\mathscr{O};\mathbb{R}^{2})}$

where $\kappa:\partial\mathscr{O}\rightarrow\mathbb{R}$ represents the
curvature of the boundary. Therefore, for each $\alpha$ as in (76), we extend
the Stokes Operator from $\bar{W}^{2,2}_{\alpha}$ to $\bar{W}^{1,2}_{\sigma}$
as a mapping into $\left(\bar{W}^{1,2}_{\sigma}\right)^{*}$ by the duality
pairing for $g,f\in\bar{W}^{1,2}_{\sigma}$ of

$\left\langle\mathcal{P}\Delta
g,f\right\rangle_{\left(\bar{W}^{1,2}_{\sigma}\right)^{*}\times\bar{W}^{1,2}_{\sigma}}=-\left\langle
g,f\right\rangle_{1}+\left\langle(\kappa-\alpha)g,f\right\rangle_{L^{2}(\partial\mathscr{O};\mathbb{R}^{2})}.$

The nonlinear term requires no special attention to be understood in the weak
sense, similarly to the no-slip boundary condition. We equip
$\bar{W}^{2,2}_{\alpha}$ with the inner product $\left\langle
f,g\right\rangle_{2}:=\left\langle\mathcal{P}\Delta f,\mathcal{P}\Delta
g\right\rangle_{L^{2}}$ which is equivalent to the standard $W^{2,2}$ inner
product (see [21] Lemma 1.2), and $\bar{W}^{1,2}_{\sigma}$ with the
$\left\langle\cdot,\cdot\right\rangle_{1}$ inner product. Then as a direct
application of Theorem 3.5 we obtain the following.

######  Theorem 5.4.

Let $\alpha\in C^{2}(\partial\mathscr{O};\mathbb{R})$,
$u_{0}:\Omega\rightarrow L^{2}_{\sigma}$ be $\mathcal{F}_{0}-$measurable,
$(\xi_{i})\in W^{1,2}_{\sigma}\cap W^{2,\infty}$ with
$\sum_{i=1}^{\infty}\left\|\xi_{i}\right\|_{W^{2,\infty}}^{2}<\infty$. Then
there exists a progressively measurable process $u$ in
$\bar{W}^{1,2}_{\sigma}$ such that for $\mathbb{P}-a.e.$ $\tilde{\omega}$,
$u_{\cdot}(\omega)\in C\left([0,T];L^{2}_{\sigma}\right)\cap
L^{2}\left([0,T];\bar{W}^{1,2}_{\sigma}\right)$ and (75) holds
$\mathbb{P}-a.s.$ in $\left(\bar{W}^{1,2}_{\sigma}\right)^{*}$ for all
$t\in[0,T]$.

This result is given in [21] Theorem 1.14, and a strong existence result is
proven as Theorem 1.15 in the same paper. For this, we need to make some
adjustments; to verify Assumption 4.2 we cannot use the
$\left\langle\cdot,\cdot\right\rangle_{1}$ inner product for $H$ as this leads
to an uncontrollable boundary integral. Instead, we have to manufacture a more
reasonable boundary integral into our inner product. Thus, we instead equip
$\bar{W}^{1,2}_{\sigma}$ with

$\left\langle f,g\right\rangle_{H}:=\left\langle
f,g\right\rangle_{1}+\left\langle(\kappa-\alpha)f,g\right\rangle_{L^{2}(\partial\mathscr{O};\mathbb{R}^{2})}$

which is an inner product equivalent to the usual $W^{1,2}$ form when
$\alpha\geq\kappa$ everywhere on $\partial\mathscr{O}$. This requirement
appears in the result, obtainable through Theorem 4.5.

######  Theorem 5.5.

Let $\alpha\in C^{2}(\partial\mathscr{O};\mathbb{R})$ be such that
$\alpha\geq\kappa$, $u_{0}:\Omega\rightarrow\bar{W}^{1,2}_{\sigma}$ be
$\mathcal{F}_{0}-$measurable, $(\xi_{i})\in L^{2}_{\sigma}\cap W^{3,2}_{0}\cap
W^{3,\infty}$ with
$\sum_{i=1}^{\infty}\left\|\xi_{i}\right\|_{W^{3,\infty}}^{2}<\infty$. Then
there exists a progressively measurable process $u$ in
$\bar{W}^{2,2}_{\alpha}$ such that for $\mathbb{P}-a.e.$ $\tilde{\omega}$,
$u_{\cdot}(\omega)\in C\left([0,T];\bar{W}^{1,2}_{\sigma}\right)\cap
L^{2}\left([0,T];\bar{W}^{2,2}_{\alpha}\right)$ and (75) holds
$\mathbb{P}-a.s.$ in $L^{2}_{\sigma}$ for all $t\in[0,T]$.

We note use of Lemma 4.14 here, as in this case, the extension $\bar{H}$ of
$H$ in Assumption Set 3, Subsection 4.1, can simply be
$\bar{W}^{1,2}_{\sigma}$ itself. In addition, it should be noted that the
requirement $\alpha\geq\kappa$ is rather reasonable; at least heuristically,
as $\alpha$ grows large then $u\cdot\iota=0$ dominates the second identity of
(76), which would then result in the traditional no-slip condition. Given the
wide acceptance of the no-slip condition, deviation from it with Navier
boundary conditions is only expected for large $\alpha$. A rigorous result
regarding the convergence of solutions to the deterministic Navier-Stokes
equation with Navier boundary conditions to the no-slip solution for
$\alpha\rightarrow\infty$ is available in [32] Section 9.

##  6 Appendix

We state some key theorems used throughout the paper. Firstly is a well-
posedness result for an SPDE in a finite dimensional Hilbert Space (though
driven by a Cylindrical Brownian Motion) in the standard case of Lipschitz and
linear growth constraints.

######  Proposition 6.1.

Fix a finite-dimensional Hilbert Space $\mathcal{H}$. Suppose the following:

  * 1:

For any $T>0$, the operators
$\mathscr{A}:[0,T]\times\mathcal{H}\rightarrow\mathcal{H}$ and
$\mathscr{G}:[0,T]\times\mathcal{H}\rightarrow\mathscr{L}^{2}(\mathfrak{U};\mathcal{H})$
are measurable;

  * 2:

There exists a $C_{\cdot}:[0,\infty)\rightarrow\mathbb{R}$ bounded on $[0,T]$
for every $T$, and constants $c_{i}$ such that for every
$\bm{\phi},\psi\in\mathcal{H}$ and $t\in[0,\infty)$,

$\displaystyle\left\|\mathscr{A}(t,\bm{\phi})\right\|^{2}_{\mathcal{H}}$
$\displaystyle\leq
C_{t}\left[1+\left\|\bm{\phi}\right\|_{\mathcal{H}}^{2}\right]$
$\displaystyle\left\|\mathscr{G}_{i}(t,\bm{\phi})\right\|^{2}_{\mathcal{H}}$
$\displaystyle\leq
C_{t}c_{i}\left[1+\left\|\bm{\phi}\right\|_{\mathcal{H}}^{2}\right]$
$\displaystyle\sum_{i=1}^{\infty}c_{i}$ $\displaystyle<\infty$
$\displaystyle\left\|\mathscr{A}(t,\bm{\phi})-\mathscr{A}(t,\psi)\right\|^{2}_{\mathcal{H}}$
$\displaystyle+\sum_{i=1}^{\infty}\left\|\mathscr{G}_{i}(t,\bm{\phi})-\mathscr{G}_{i}(t,\psi)\right\|^{2}_{\mathcal{H}}\leq
C_{t}\left\|\bm{\phi}-\psi\right\|_{\mathcal{H}}^{2}$

  * 3:

$\bm{\Phi}_{0}\in L^{2}(\Omega;\mathcal{H})$.

Then there exists a process
$\bm{\Phi}:[0,\infty)\times\Omega\rightarrow\mathcal{H}$ such that for
$\mathbbm{P}-a.e.$ $\omega$, $\bm{\Phi}_{\cdot}(\omega)\in
C\left([0,T];\mathcal{H}\right)$ for every $T>0$, $\bm{\Phi}$ is progressively
measurable in $\mathcal{H}$ and the identity

$\bm{\Phi}_{t}=\bm{\Phi}_{0}+\int_{0}^{t}\mathscr{A}(s,\bm{\Phi}_{s})ds+\int_{0}^{t}\mathscr{G}(s,\bm{\Phi}_{s})d\mathcal{W}_{s}$
(77)

holds $\mathbbm{P}-a.s.$ in $\mathcal{H}$ for every $t\geq 0$.

Moreover, suppose that
$\bm{\Psi}:[0,\infty)\times\Omega\rightarrow\mathcal{H}$ is another process
such that for $\mathbbm{P}-a.e.$ $\omega$, $\bm{\Psi}_{\cdot}(\omega)\in
C\left([0,T];\mathcal{H}\right)$ for every $T>0$, $\bm{\Psi}$ is progressively
measurable in $\mathcal{H}$ and the identity (77) holds $\mathbbm{P}-a.s.$ in
$\mathcal{H}$ for every $t\geq 0$. Then for every $T\geq 0$,

$\mathbbm{P}\left(\left\\{\omega\in\Omega:\bm{\Phi}_{t}(\omega)=\bm{\Psi}_{t}(\omega)\quad\forall
t\in[0,T]\right\\}\right)=1.$

###### Proof.

See [20] Theorems 3.1.1 and 3.1.2.

∎

######  Proposition 6.2.

Let $\mathcal{H}_{1}\subset\mathcal{H}_{2}\subset\mathcal{H}_{3}$ be a triplet
of embedded Hilbert Spaces where $\mathcal{H}_{1}$ is dense in
$\mathcal{H}_{2}$, with the property that there exists a continuous
nondegenerate bilinear form
$\left\langle\cdot,\cdot\right\rangle_{\mathcal{H}_{3}\times\mathcal{H}_{1}}:\mathcal{H}_{3}\times\mathcal{H}_{1}\rightarrow\mathbb{R}$
such that for $\phi\in\mathcal{H}_{2}$ and $\psi\in\mathcal{H}_{1}$,

$\left\langle\phi,\psi\right\rangle_{\mathcal{H}_{3}\times\mathcal{H}_{1}}=\left\langle\phi,\psi\right\rangle_{\mathcal{H}_{2}}.$

Suppose that for some $T>0$ and stopping time $\tau$,

  1. 1.

$\bm{\Psi}_{0}:\Omega\rightarrow\mathcal{H}_{2}$ is
$\mathcal{F}_{0}-$measurable;

  2. 2.

$f:\Omega\times[0,T]\rightarrow\mathcal{H}_{3}$ is such that for
$\mathbbm{P}-a.e.$ $\omega$, $f(\omega)\in L^{2}([0,T];\mathcal{H}_{3})$;

  3. 3.

$B:\Omega\times[0,T]\rightarrow\mathscr{L}^{2}(\mathfrak{U};\mathcal{H}_{2})$
is progressively measurable and such that for $\mathbbm{P}-a.e.$ $\omega$,
$B(\omega)\in
L^{2}\left([0,T];\mathscr{L}^{2}(\mathfrak{U};\mathcal{H}_{2})\right)$;

  4. 4.

$\bm{\Psi}:\Omega\times[0,T]\rightarrow\mathcal{H}_{1}$ is such that for
$\mathbbm{P}-a.e.$ $\omega$,
$\bm{\Psi}_{\cdot}(\omega)\mathbbm{1}_{\cdot\leq\tau(\omega)}\in
L^{2}([0,T];\mathcal{H}_{1})$ and
$\bm{\Psi}_{\cdot}\mathbbm{1}_{\cdot\leq\tau}$ is progressively measurable in
$\mathcal{H}_{1}$;

  5. 5.

The identity

$\bm{\Psi}_{t}=\bm{\Psi}_{0}+\int_{0}^{t\wedge\tau}f_{s}ds+\int_{0}^{t\wedge\tau}B_{s}d\mathcal{W}_{s}$
(78)

holds $\mathbbm{P}-a.s.$ in $\mathcal{H}_{3}$ for all $t\in[0,T]$.

The the equality

$\displaystyle\left\|\bm{\Psi}_{t}\right\|^{2}_{\mathcal{H}_{2}}=\left\|\bm{\Psi}_{0}\right\|^{2}_{\mathcal{H}_{2}}+\int_{0}^{t\wedge\tau}\bigg{(}2\left\langle
f_{s},\bm{\Psi}_{s}\right\rangle_{\mathcal{H}_{3}\times\mathcal{H}_{1}}+\left\|B_{s}\right\|^{2}_{\mathscr{L}^{2}(\mathfrak{U};\mathcal{H}_{2})}\bigg{)}ds+2\int_{0}^{t\wedge\tau}\left\langle
B_{s},\bm{\Psi}_{s}\right\rangle_{\mathcal{H}_{2}}d\mathcal{W}_{s}$ (79)

holds for any $t\in[0,T]$, $\mathbbm{P}-a.s.$ in $\mathbb{R}$. Moreover for
$\mathbbm{P}-a.e.$ $\omega$, $\bm{\Psi}_{\cdot}(\omega)\in
C([0,T];\mathcal{H}_{2})$.

###### Proof.

This is a minor extension of [51] Theorem 4.2.5, which is stated and justified
as Proposition 2.5.5. in [20]. This is necessary for us as it extends the
Gelfand Triple setting, removes the need for moment estimates and allows for
local solutions. ∎

######  Lemma 6.3 (Stochastic Grönwall).

Fix $t>0$ and suppose that $\bm{\phi},\bm{\psi},\bm{\eta}$ are real-valued,
non-negative stochastic processes. Assume, moreover, that there exists
constants $c^{\prime},\hat{c},\tilde{c}$ (allowed to depend on $t$) such that
for $\mathbbm{P}-a.e.$ $\omega$,

$\int_{0}^{t}\bm{\eta}_{s}(\omega)ds\leq c^{\prime}$ (80)

and for all stopping times $0\leq\theta_{j}<\theta_{k}\leq t$,

$\mathbbm{E}\left(\sup_{r\in[\theta_{j},\theta_{k}]}\bm{\phi}_{r}\right)+\mathbbm{E}\int_{\theta_{j}}^{\theta_{k}}\bm{\psi}_{s}ds\leq\hat{c}\mathbbm{E}\left(\left[\bm{\phi}_{\theta_{j}}+\tilde{c}\right]+\int_{\theta_{j}}^{\theta_{k}}\bm{\eta}_{s}\bm{\phi}_{s}ds\right)<\infty.$

Then there exists a constant $C$ dependent only on
$c^{\prime},\hat{c},\tilde{c},t$ such that

$\mathbbm{E}\sup_{r\in[0,t]}\bm{\phi}_{r}+\mathbbm{E}\int_{0}^{t}\bm{\psi}_{s}ds\leq
C\left[\mathbbm{E}(\bm{\phi}_{0})+\tilde{c}\right].$

###### Proof.

See [19] Lemma 5.3. ∎

######  Proposition 6.4.

Fix $T>0$. For $t\in[0,T]$ let $X_{t}$ denote a Banach Space with norm
$\left\|\cdot\right\|_{X,t}$ such that for all $s>t$,
$X_{s}\xhookrightarrow{}X_{t}$ and
$\left\|\cdot\right\|_{X,t}\leq\left\|\cdot\right\|_{X,s}$. Suppose that
$(\bm{\Psi}^{n})$ is a sequence of processes $\bm{\Psi}^{n}:\Omega\mapsto
X_{T}$, $\left\|\bm{\Psi}^{n}\right\|_{X,\cdot}$ is adapted and
$\mathbb{P}-a.s.$ continuous, $\bm{\Psi}^{n}\in
L^{2}\left(\Omega;X_{T}\right)$, and such that
$\sup_{n}\left\|\bm{\Psi}^{n}\right\|_{X,0}\in
L^{\infty}\left(\Omega;\mathbb{R}\right)$. For any given $M>1$ define the
stopping times

$\tau^{M,T}_{n}:=T\wedge\inf\left\\{s\geq
0:\left\|\bm{\Psi}^{n}\right\|_{X,s}^{2}\geq
M+\left\|\bm{\Psi}^{n}\right\|_{X,0}^{2}\right\\}.$ (81)

Furthermore suppose

$\lim_{m\rightarrow\infty}\sup_{n\geq
m}\mathbbm{E}\left[\left\|\bm{\Psi}^{n}-\bm{\Psi}^{m}\right\|^{2}_{X,\tau_{m}^{M,T}\wedge\tau_{n}^{M,T}}\right]=0$
(82)

and that for any stopping time $\gamma$ and sequence of stopping times
$(\delta_{j})$ which converge to $0$ $\mathbb{P}-a.s.$,

$\lim_{j\rightarrow\infty}\sup_{n\in\mathbb{N}}\mathbbm{E}\left(\left\|\bm{\Psi}^{n}\right\|_{X,(\gamma+\delta_{j})\wedge\tau^{M,T}_{n}}^{2}-\left\|\bm{\Psi}^{n}\right\|_{X,\gamma\wedge\tau^{M,T}_{n}}^{2}\right)=0.$
(83)

Then there exists a stopping time $\tau^{M,T}_{\infty}$, a process
$\bm{\Psi}:\Omega\mapsto X_{\tau^{M,T}_{\infty}}$ whereby
$\left\|\bm{\Psi}\right\|_{X,\cdot\wedge\tau^{M,T}_{\infty}}$ is adapted and
$\mathbb{P}-a.s.$ continuous, and a subsequence indexed by $(m_{j})$ such that

  * •

$\tau^{M,T}_{\infty}\leq\tau^{M,T}_{m_{j}}$ $\mathbb{P}-a.s.$,

  * •

$\lim_{j\rightarrow\infty}\left\|\bm{\Psi}-\bm{\Psi}^{m_{j}}\right\|_{X,\tau^{M,T}_{\infty}}=0$
$\mathbb{P}-a.s.$.

Moreover for any $R>0$ we can choose $M$ to be such that the stopping time

$\tau^{R,T}:=T\wedge\inf\left\\{s\geq
0:\left\|\bm{\Psi}\right\|_{X,s\wedge\tau^{M,T}_{\infty}}^{2}\geq R\right\\}$
(84)

satisfies $\tau^{R,T}\leq\tau^{M,T}_{\infty}$ $\mathbb{P}-a.s.$. Thus
$\tau^{R,T}$ is simply $T\wedge\inf\left\\{s\geq
0:\left\|\bm{\Psi}\right\|_{X,s}^{2}\geq R\right\\}$.

###### Proof.

See [21] Lemma 5.1, [20] Lemma 3.4.1. ∎

######  Lemma 6.5.

Let $\mathcal{H}_{1},\mathcal{H}_{2}$ be Hilbert Spaces such that
$\mathcal{H}_{1}$ is compactly embedded into $\mathcal{H}_{2}$, and for some
fixed $T>0$ let $(\bm{\Psi}^{n}):\Omega\times[0,T]\rightarrow\mathcal{H}_{1}$
be a sequence of measurable processes such that

$\sup_{n\in\mathbb{N}}\mathbbm{E}\int_{0}^{T}\left\|\bm{\Psi}^{n}_{s}\right\|^{2}_{\mathcal{H}_{1}}ds<\infty$
(85)

and for any $\varepsilon>0$,

$\lim_{\delta\rightarrow
0^{+}}\sup_{n\in\mathbb{N}}\mathbbm{P}\left(\left\\{\omega\in\Omega:\int_{0}^{T-\delta}\left\|\bm{\Psi}^{n}_{s+\delta}(\omega)-\bm{\Psi}^{n}_{s}(\omega)\right\|^{2}_{\mathcal{H}_{2}}ds>\varepsilon\right\\}\right)=0.$
(86)

Then the sequence of the laws of $(\bm{\Psi}^{n})$ is tight in the space of
probability measures over $L^{2}\left([0,T];\mathcal{H}_{2}\right)$.

###### Proof.

See [53] Lemma 5.2. ∎

######  Lemma 6.6.

Let $\mathcal{Y}$ be a reflexive separable Banach Space and $\mathcal{H}$ a
separable Hilbert Space such that $\mathcal{Y}$ is compactly embedded into
$\mathcal{\mathcal{H}}$, and consider the induced Gelfand Triple

$\mathcal{Y}\xhookrightarrow{}\mathcal{H}\xhookrightarrow{}\mathcal{Y}^{*}.$

For some fixed $T>0$ let $\bm{\Psi}^{n}:\Omega\rightarrow
C\left([0,T];\mathcal{H}\right)$ be a sequence of measurable processes such
that for every $t\in[0,T]$,

$\sup_{n\in\mathbb{N}}\mathbbm{E}\left(\sup_{t\in[0,T]}\left\|\bm{\Psi}^{n}_{t}\right\|_{\mathcal{H}}\right)<\infty$
(87)

and for any sequence of stopping times $(\gamma_{n})$ with
$\gamma_{n}:\Omega\rightarrow[0,T]$, and any $\varepsilon>0$,
$y\in\mathcal{Y}$,

$\lim_{\delta\rightarrow
0^{+}}\sup_{n\in\mathbb{N}}\mathbbm{P}\left(\left\\{\omega\in\Omega:\left|\left\langle\bm{\Psi}^{n}_{(\gamma_{n}+\delta)\wedge
T}-\bm{\Psi}^{n}_{\gamma_{n}},y\right\rangle_{\mathcal{H}}\right|>\varepsilon\right\\}\right)=0.$
(88)

Then the sequence of the laws of $(\bm{\Psi}^{n})$ is tight in the space of
probability measures over $\mathcal{D}\left([0,T];\mathcal{Y}^{*}\right)$.

###### Proof.

See [20] Lemma 3.3.2. ∎

Thanks: I would like to give my sincerest thanks to Dan Crisan for the regular
and extended discussions around the paper, his feedback on it, and overall
guidance during this process.

Acknowledgements: The author was supported by the Engineering and Physical
Sciences Research Council (EPSCR) Project 2478902.

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