$\displaystyle\mathbbm{E}\left(\sum_{i=1}^{\infty}\int_{0}^{t}\left\|\mathcal{G}_{i}^{2}\bm{\Psi}^{n}_{s}\right\|_{X}ds\right)^{2}$
$\displaystyle\leq\mathbbm{E}\left(\sum_{i=1}^{\infty}c_{i}^{2}\int_{0}^{t}\left\|\bm{\Psi}^{n}_{s}\right\|_{H}ds\right)^{2}$
$\displaystyle\leq\mathbbm{E}\left(\sum_{i=1}^{\infty}c_{i}^{2}\int_{0}^{t}c\left\|\bm{\Psi}^{n}_{s}\right\|_{V}ds\right)^{2}$
$\displaystyle\leq\left(c\sum_{i=1}^{\infty}c_{i}^{2}\right)^{2}\mathbbm{E}\left[t\int_{0}^{t}\left\|\bm{\Psi}^{n}_{s}\right\|_{V}^{2}ds\right]$
$\displaystyle\leq\left(c\sum_{i=1}^{\infty}c_{i}^{2}\right)^{2}tn$

where $c$ is the constant from the embedding of $V\xhookrightarrow{}H$. Thus
defining

$Y_{k}:=\mathbbm{E}\left(\sum_{i=1}^{k}\int_{0}^{t}\left\|\mathcal{G}_{i}^{2}\bm{\Psi}^{n}_{s}\right\|_{X}ds\right)^{2}$

then $(Y_{k})$ is a real valued sequence monotone increasing and bounded
above, hence it is convergent so Cauchy. But

$\mathbbm{E}\left\|\sum_{i=m}^{k}\int_{0}^{t}\mathcal{G}_{i}^{2}\bm{\Psi}^{n}_{s}ds\right\|_{X}^{2}\leq\mathbbm{E}\left(\sum_{i=m}^{k}\int_{0}^{t}\left\|\mathcal{G}_{i}^{2}\bm{\Psi}^{n}_{s}\right\|_{X}ds\right)^{2}\leq
Y_{k}-Y_{m-1}$

where we’ve just said that this sequence $(Y_{k})$ is Cauchy, proving the
required convergence.

###### Proof of 2.3.1:.

We must show that

$\int_{0}^{t}\mathcal{G}\bm{\Psi}_{s}\circ
d\mathcal{W}_{s}=\int_{0}^{t}\mathcal{G}\bm{\Psi}_{s}d\mathcal{W}_{s}+\frac{1}{2}\int_{0}^{t}\sum_{i=1}^{\infty}\mathcal{G}_{i}^{2}\bm{\Psi}_{s}ds$

$a.s.$ in $X$ for all $t\geq 0$. So working with fixed arbitrary $t$ and
$\omega$, we choose any $n$ such that $\tau_{n}(\omega)\geq t$ and have that
at this $\omega$,

$\displaystyle\int_{0}^{t}\mathcal{G}\bm{\Psi}_{s}\circ d\mathcal{W}_{s}$
$\displaystyle=\sum_{i=1}^{\infty}\int_{0}^{t}\mathcal{G}_{i}\bm{\Psi}^{n}_{s}d\mathcal{W}_{s}+\frac{1}{2}\sum_{i=1}^{\infty}[\mathcal{G}_{i}\bm{\Psi}^{n},W^{i}]_{t}$
$\displaystyle=\int_{0}^{t}\mathcal{G}\bm{\Psi}_{s}\circ
d\mathcal{W}_{s}+\frac{1}{2}\sum_{i=1}^{\infty}[\mathcal{G}_{i}\bm{\Psi}^{n},W^{i}]_{t}$

where this limit is taken in $L^{2}(\Omega;X)$. Of course we have just shown
that

$\sum_{i=1}^{\infty}[\mathcal{G}_{i}\bm{\Psi}^{n},W^{i}]_{t}=\sum_{i=1}^{\infty}\int_{0}^{t}\mathcal{G}_{i}^{2}\bm{\Psi}^{n}_{s}ds$

is well defined in this topology, but we must show that it is equal to

$\int_{0}^{t}\sum_{i=1}^{\infty}\mathcal{G}_{i}^{2}\bm{\Psi}_{s}ds$

evaluated at this fixed $\omega$, for the pointwise limit in $U$ as it was
defined. Firstly note that

$\int_{0}^{t}\sum_{i=1}^{\infty}\mathcal{G}_{i}^{2}\bm{\Psi}_{s}(\omega)ds=\int_{0}^{t}\sum_{i=1}^{\infty}\mathcal{G}_{i}^{2}\bm{\Psi}^{n}_{s}(\omega)ds$

and by an application of the dominated convergence theorem with dominating
function

$\sum_{i=1}^{\infty}c_{i}^{2}\lVert\bm{\Psi}^{n}\rVert_{V}$

we can rewrite

$\int_{0}^{t}\sum_{i=1}^{\infty}\mathcal{G}_{i}^{2}\bm{\Psi}^{n}_{s}ds=\sum_{i=1}^{\infty}\int_{0}^{t}\mathcal{G}_{i}^{2}\bm{\Psi}^{n}_{s}ds$

as a limit $a.s.$ in $U$, and thus $a.s.$ in $X$. However the convergence in
$L^{2}(\Omega;X)$ implies that of a subsequence $a.s.$ in $X$, which clearly
agrees with the limit of the whole sequence $a.s.$ in $X$ thus giving the
result. ∎

This theorem has been stated for the strong solution 2.2.4, though we note
that all arguments follow in the corresponding local case by incorporating the
stopping time as done in the justification that the integrals in 2.2.3 are
well defined. The result is stated below.

######  Corollary 2.3.1.1.

If $(\bm{\Psi},\tau)$ is a local strong solution of the equation (44) then
$\bm{\Psi}$ satisfies the identity

$\bm{\Psi}_{t}=\bm{\Psi}_{0}+\int_{0}^{t\wedge\tau}\mathcal{Q}\bm{\Psi}_{s}ds+\int_{0}^{t\wedge\tau}\mathcal{G}\bm{\Psi}_{s}\circ
d\mathcal{W}_{s}$

$\mathbbm{P}-$ $a.s.$ in $X$ for all $t\geq 0$.

To conclude this discussion we should emphasise why we the identity (43) holds
only in $X$ and not in $U$, the space in which (44) is satisfied. The
evolution equation (46) allowed us to identify the semi-martingale structure
in $X$ as is this is where the integrals are constructed. We can, however,
construct the stochastic integral in (46) in $U$, which allows us to conclude
that the time integral is itself an element of $U$ (as
$\mathcal{G}_{i}\bm{\Psi}^{n}_{t},\mathcal{G}_{i}\bm{\Psi}^{n}_{0}$ are as
well). Unfortunately we cannot say that this is actually an integral in $U$
(just an integral in $X$ which is in turn an element of $U$) so poignantly we
cannot say that this is of finite variation in $U$, which would be necessary
when considering the cross-variation
$[\mathcal{G}_{i}\bm{\Psi}^{n},W^{i}]_{t}$ in $U$.

###  2.4 Time-Dependent Operators

We did not facilitate time dependence in the operators
$\mathcal{Q},\mathcal{G}$ as solely for the fact that if $\mathcal{G}$ was
time dependent then the conversion from Stratonovich to Itô Form would be much
more troublesome. There is no real additional difficulty in establishing a
framework for the Itô Form for time-dependent operators, so we briefly do so
now. There is no longer a need for the space $X$ so we work with the triple

$V\xhookrightarrow{}H\xhookrightarrow{}U$

and now the SPDE

$\bm{\Psi}_{t}=\bm{\Psi}_{0}+\int_{0}^{t}\mathcal{Q}(s,\bm{\Psi}_{s})ds+\int_{0}^{t}\mathcal{G}(s,\bm{\Psi}_{s})d\mathcal{W}_{s}.$
(47)

We require the assumptions now as:

######  Assumption 2.4.1.

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

######  Assumption 2.4.2.

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 $\phi\in V$ and
$t\in[0,\infty)$,

$\displaystyle\lVert\mathcal{Q}(t,\bm{\phi})\rVert_{U}$ $\displaystyle\leq
C_{t}\tilde{K}(\phi)\left[1+\lVert\bm{\phi}\rVert_{V}^{2}\right]$
$\displaystyle\lVert\mathcal{G}_{i}(t,\bm{\phi})\rVert^{2}_{H}$
$\displaystyle\leq C_{t}c_{i}(1+\lVert\phi\rVert_{V}^{2})$
$\displaystyle\sum_{i=1}^{\infty}c_{i}$ $\displaystyle<\infty$

Definitions of solutions in this framework, and a justification that the
integrals are well-defined, then follows largely in the same way as for (41)
so we omit the details here. What is slightly more delicate is the progressive
measurability of the process
$\mathcal{G}(\cdot,\bm{\Phi}_{\cdot})\mathbbm{1}_{\cdot\leq\tau}$ in
$\mathscr{L}^{2}(\mathfrak{U};H)$. From the measurability of $\mathcal{G}$ and
the progressive measurability of $\bm{\Phi}$ we have that for any fixed $t$,
the mapping

$\mathcal{G}(\cdot,\bm{\Phi}_{\cdot}):[0,t]\times[0,t]\times\Omega\rightarrow\mathscr{L}^{2}(\mathfrak{U};H)$

defined by

$(s,r,\omega)\mapsto\mathcal{G}(s,\bm{\Phi}_{r}(\omega))$

is $\mathcal{B}([0,t])\times\mathcal{B}([0,t])\times\mathcal{F}_{t}$
measurable, and hence the mapping

$(s,\omega)\mapsto\mathcal{G}(s,\bm{\Phi}_{s}(\omega))$

is $\mathcal{B}([0,t])\times\mathcal{F}_{t}$ measurable as is the indicator up
to the stopping time, so the product retains the required measurability. For
completeness we define the notion of a strong solution here.

######  Definition 2.4.3.

A process $\bm{\Psi}$ such that for $\mathbbm{P}-a.e.$ $\omega$,
$\bm{\Psi}_{\cdot}(\omega)\in C\left([0,T];H\right)\cap
L^{2}\left([0,T];V\right)$ for all $T>0$ with $\bm{\Psi}$ progressively
measurable in $V$, is said to be a strong solution of the equation (47) if the
identity (47) holds $\mathbbm{P}-a.s.$ in $U$ for all $t\geq 0$.

###  2.5 An Energy Equality

Having established this framework we introduce techniques to facilitate our
analysis in it. The Itô Formula is well regarded as one of the most useful
tools in stochastic analysis, and we formulate the infinite dimensional
version here through first looking at a specific energy equality. We shall
introduce a new setting in which our established solution framework falls,
with the understanding that we would like to apply this to solutions whilst
also using the results to deduce the existence of solutions when they are not
a priori known. The ideas of this subsection are just a mild extension of [13]
Theorem 4.2.5.

To prove this energy equality we shall rely on looking at partitions in time
over which some nice properties are satisfied, before taking the limit as the
increments go to zero. Towards this we recall the following lemma from [13]
Lemma 4.2.6.

######  Lemma 2.5.1.

Let $\mathcal{X}_{1}\xhookrightarrow{}\mathcal{X}_{2}$ be two Banach Spaces
with continuous embedding and suppose that for some $T>0$ and stopping time
$\tau$, $\bm{\Phi}:\Omega\times[0,T]\rightarrow\mathcal{X}_{2}$ is such that
for $\mathbbm{P}-a.e.$ $\omega$, $\bm{\Phi}_{\cdot}(\omega)\in
C\left([0,T];\mathcal{X}_{2}\right)$ and
$\bm{\Phi}_{\cdot}\mathbbm{1}_{\cdot\leq\tau}\in
L^{2}\left(\Omega\times[0,T];\mathcal{X}_{1}\right)$. Then for any
$A\subset[0,T]$ with $\lambda(A)=0$ there exists a sequence of partitions
$(I_{l})$ such that

  1. 1.

$I_{l}\cap A=\emptyset$;

  2. 2.

$I_{l}\subset I_{l+1}$;

  3. 3.

$I_{l}:=\left\\{0=t^{l}_{0}<t^{l}_{1}<\dots<t^{l}_{k_{l}}=T\right\\}$,
$\max_{j}|t^{l}_{j}-t^{l}_{j-1}|\rightarrow 0$ as $l\rightarrow\infty$;

  4. 4.

For $\mathbbm{P}-a.e.$ $\omega$ and every $t^{l}_{j}$ with $1\leq j\leq
k_{l-1}$,
$\bm{\Phi}_{t^{l}_{j}}(\omega)\mathbbm{1}_{t^{l}_{j}\leq\tau(\omega)}\in\mathcal{X}_{1}$;

  5. 5.

The processes $\hat{\bm{\Phi}}^{l},\tilde{\bm{\Phi}}^{l}$ defined at each
$t\in[0,T]$ and $\omega\in\Omega$ by

$\hat{\bm{\Phi}}^{l}_{t}(\omega):=\sum_{j=2}^{k_{l}}\mathbbm{1}_{[t^{l}_{j-1},t^{l}_{j}]}(t)\bm{\Phi}_{t^{l}_{j-1}}(\omega)\mathbbm{1}_{t^{l}_{j-1}\leq\tau(\omega)},\qquad\tilde{\bm{\Phi}}^{l}:=\sum_{j=1}^{k_{l-1}}\mathbbm{1}_{[t^{l}_{j-1},t^{l}_{j}]}(t)\bm{\Phi}_{t^{l}_{j}}(\omega)\mathbbm{1}_{t^{l}_{j}\leq\tau(\omega)}$

belong to $L^{2}\left(\Omega\times[0,T];\mathcal{X}_{1}\right)$ and both
converge to $\bm{\Phi}_{\cdot}\mathbbm{1}_{\cdot\leq\tau}$ in this space.

Before moving on we take a moment to dissect this result. In the statement of
[13] there is no continuity assumption on $\bm{\Phi}$ and indeed this is
superfluous to requirement, however we want to make explicit that is genuinely
the process $\bm{\Phi}$ taken in item 5 and not some other representative of
an equivalence class. The assumptions are of course reminiscent of 2.2.3 and
just as was stressed there that the progressively measurable process in $V$
was not necessarily the continuous process itself but just a
$\mathbbm{P}\times\lambda-a.s.$ equivalent representation, we are reminded
again that elements of $L^{2}\left(\Omega\times[0,T];\mathcal{X}_{1}\right)$
are only an equivalence class of $\mathbbm{P}\times\lambda-a.s.$ equal
functions so the fact that we can fix the representation $\bm{\Phi}$ in
$\hat{\bm{\Phi}}^{l},\tilde{\bm{\Phi}}^{l}$ is significant.

We now fix a framework in which we conduct the analysis of this subsection. We
work with a triple of embedded Hilbert Spaces

$V\xhookrightarrow{}H\xhookrightarrow{}U$

where the embeddings are continuous, $V$ is assumed dense in $H$, and there
exists a continuous bilinear form $\langle\cdot,\cdot\rangle_{U\times
V}:U\times V\rightarrow\mathbb{R}$ such that for every $\phi\in H,\psi\in V$,

$\langle\phi,\psi\rangle_{U\times V}=\langle\phi,\psi\rangle_{H}.$

We suppose that for some $T>0$ and stopping time $\tau$:

  1. 1.

$\bm{\Psi}_{0}\in L^{2}(\Omega;H)$ is $\mathcal{F}_{0}-$measurable;

  2. 2.

$\eta\in L^{2}\left(\Omega;L^{2}([0,T];U)\right)$;

  3. 3.

$B\in\mathcal{I}^{H}(\mathcal{W})$;

  4. 4.

$\bm{\Psi}_{\cdot}\mathbbm{1}_{\cdot\leq\tau}\in
L^{2}\left(\Omega;L^{2}([0,T];V)\right)$ and is progressively measurable in
$V$;

  5. 5.

The identity

$\bm{\Psi}_{t}=\bm{\Psi}_{0}+\int_{0}^{t\wedge\tau}\eta_{s}ds+\int_{0}^{t\wedge\tau}B_{s}d\mathcal{W}_{s}$
(48)

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

######  Remark 3.

Once more, it is clear from assumption 5 that for $\mathbbm{P}-a.e.$ $\omega$,
$\bm{\Psi}(\omega)\in C\left([0,T];U\right)$ but the progressively measurable
process assumed in item 4 is only equivalent to this
$\bm{\Psi}_{\cdot}\mathbbm{1}_{\cdot\leq\tau}$
$\mathbbm{P}\times\lambda-a.s.$. It is a slight abuse of notation commonplace
in the literature and we shall follow suit without excessive clarification at
each use.

With this structure in place, we first look to deduce some improved regularity
on $\bm{\Psi}$. For this we fix an application of 2.5.1 relative to the
assumptions laid out above. We take $T$ and $\tau$ as in the assumptions,
$\mathcal{X}_{1}=V$, $\mathcal{X}_{2}=U$, $\bm{\Phi}=\bm{\Psi}$. From item 4
and the continuous embedding of $V$ into $H$ we have that for $\lambda-a.e.$
$t\in[0,T]$,
$\mathbbm{E}\left(\lVert\bm{\Psi}_{t}\rVert_{H}^{2}\right)<\infty$ and we take
$A$ to be the $\lambda-$zero set on which this does not hold. Then
$\hat{\bm{\Psi}}^{l},\tilde{\bm{\Psi}}^{l},I^{l}$ are defined as in 2.5.1 and
we define

$I:=\bigcup_{l\in\mathbbm{N}}I^{l}.$

######  Lemma 2.5.2.

We have that

$\mathbbm{E}\left(\sup_{t\in[0,T]}\lVert\bm{\Psi}_{t}\rVert_{H}^{2}\right)<\infty.$

###### Proof.

For $\mathbbm{P}-a.e.$ $\omega$ and every $s<t$ with $t\in
I\cap[0,\tau(\omega)]$ and $s\in I\cup\\{0\\}$, observe that

$\displaystyle\left\|\int_{s}^{t}B_{r}d\mathcal{W}_{r}\right\|_{H}^{2}-\left\|\bm{\Psi}_{t}-\bm{\Psi}_{s}-\int_{s}^{t}B_{r}d\mathcal{W}_{r}\right\|_{H}^{2}+2\left\langle\bm{\Psi}_{s},\int_{s}^{t}B_{r}d\mathcal{W}_{r}\right\rangle_{\mathcal{H}}$
$\displaystyle\qquad=\left\|\int_{s}^{t}B_{r}d\mathcal{W}_{r}\right\|_{H}^{2}-\left\|\bm{\Psi}_{t}-\bm{\Psi}_{s}\right\|_{H}^{2}-\left\|\int_{s}^{t}B_{r}d\mathcal{W}_{r}\right\|_{H}^{2}$
$\displaystyle\qquad\qquad\qquad+2\left\langle\bm{\Psi}_{t}-\bm{\Psi}_{s},\int_{s}^{t}B_{r}d\mathcal{W}_{r}\right\rangle_{\mathcal{H}}+2\left\langle\bm{\Psi}_{s},\int_{s}^{t}B_{r}d\mathcal{W}_{r}\right\rangle_{\mathcal{H}}$
$\displaystyle\qquad=2\left\langle\bm{\Psi}_{t},\int_{s}^{t}B_{r}d\mathcal{W}_{r}\right\rangle_{\mathcal{H}}-\left\|\bm{\Psi}_{t}-\bm{\Psi}_{s}\right\|_{H}^{2}$
$\displaystyle\qquad=2\left\langle\bm{\Psi}_{t},\bm{\Psi}_{t}-\bm{\Psi}_{s}-\int_{s}^{t}\eta_{r}dr\right\rangle_{\mathcal{H}}-\left\|\bm{\Psi}_{t}\right\|_{H}^{2}-\left\|\bm{\Psi}_{s}\right\|_{H}^{2}+2\left\langle\bm{\Psi}_{t},\bm{\Psi}_{s}\right\rangle_{H}$
$\displaystyle\qquad=\left\|\bm{\Psi}_{t}\right\|_{H}^{2}-\left\|\bm{\Psi}_{s}\right\|_{H}^{2}-2\left\langle\bm{\Psi}_{t},\int_{s}^{t}\eta_{r}dr\right\rangle_{\mathcal{H}}$
$\displaystyle\qquad=\left\|\bm{\Psi}_{t}\right\|_{H}^{2}-\left\|\bm{\Psi}_{s}\right\|_{H}^{2}-2\int_{s}^{t}\left\langle\eta_{r},\bm{\Psi}_{t}\right\rangle_{U\times
V}dr$

which we rewrite as the equality

$\displaystyle\left\|\bm{\Psi}_{t}\right\|_{H}^{2}-\left\|\bm{\Psi}_{s}\right\|_{H}^{2}$
$\displaystyle=2\int_{s}^{t}\left\langle\eta_{r},\bm{\Psi}_{t}\right\rangle_{U\times
V}dr+2\left\langle\bm{\Psi}_{s},\int_{s}^{t}B_{r}d\mathcal{W}_{r}\right\rangle_{\mathcal{H}}$
$\displaystyle+\left\|\int_{s}^{t}B_{r}d\mathcal{W}_{r}\right\|_{H}^{2}-\left\|\bm{\Psi}_{t}-\bm{\Psi}_{s}-\int_{s}^{t}B_{r}d\mathcal{W}_{r}\right\|_{H}^{2}.$

Using this equality, for any $l\in\mathbbm{N}$ and $t=t^{l}_{i}\in
I_{l}\cap(0,\tau(\omega)]/\\{T\\}$,

$\displaystyle\lVert\bm{\Psi}_{t}\rVert_{H}^{2}$
$\displaystyle-\lVert\bm{\Psi}_{0}\rVert_{H}^{2}=\sum_{j=0}^{i-1}\left(\lVert\bm{\Psi}_{t^{l}_{j+1}}\rVert_{H}^{2}-\lVert\bm{\Psi}_{t^{l}_{j}}\rVert_{H}^{2}\right)$
$\displaystyle=\sum_{j=0}^{i-1}\Bigg{(}2\int_{t^{l}_{j}}^{t^{l}_{j+1}}\left\langle\eta_{r},\bm{\Psi}_{t^{l}_{j+1}}\right\rangle_{U\times
V}dr+2\left\langle\bm{\Psi}_{t^{l}_{j}},\int_{t^{l}_{j}}^{t^{l}_{j+1}}B_{r}d\mathcal{W}_{r}\right\rangle_{\mathcal{H}}$
$\displaystyle\qquad\qquad+\left\|\int_{t^{l}_{j}}^{t^{l}_{j+1}}B_{r}d\mathcal{W}_{r}\right\|_{H}^{2}-\left\|\bm{\Psi}_{t^{l}_{j+1}}-\bm{\Psi}_{t^{l}_{j}}-\int_{t^{l}_{j}}^{t^{l}_{j+1}}B_{r}d\mathcal{W}_{r}\right\|_{H}^{2}\Bigg{)}$
$\displaystyle=2\sum_{j=0}^{i-1}\left(\int_{t^{l}_{j}}^{t^{l}_{j+1}}\left\langle\eta_{r},\bm{\Psi}_{t^{l}_{j+1}}\right\rangle_{U\times
V}dr+\int_{t^{l}_{j}}^{t^{l}_{j+1}}\left\langle
B_{r},\bm{\Psi}_{t^{l}_{j}}\right\rangle_{\mathcal{H}}d\mathcal{W}_{r}\right)$
$\displaystyle\qquad\qquad+\sum_{j=0}^{i-1}\left(\left\|\int_{t^{l}_{j}}^{t^{l}_{j+1}}B_{r}d\mathcal{W}_{r}\right\|_{H}^{2}-\left\|\bm{\Psi}_{t^{l}_{j+1}}-\bm{\Psi}_{t^{l}_{j}}-\int_{t^{l}_{j}}^{t^{l}_{j+1}}B_{r}d\mathcal{W}_{r}\right\|_{H}^{2}\right)$
$\displaystyle=2\sum_{j=0}^{i-1}\left(\int_{t^{l}_{j}}^{t^{l}_{j+1}}\left\langle\eta_{r},\tilde{\bm{\Psi}}^{l}_{r}\right\rangle_{U\times
V}dr\right)+2\sum_{j=1}^{i-1}\left(\int_{t^{l}_{j}}^{t^{l}_{j+1}}\left\langle
B_{r},\hat{\bm{\Psi}}^{l}_{r}\right\rangle_{\mathcal{H}}d\mathcal{W}_{r}\right)+2\int_{0}^{t^{l}_{1}}\left\langle
B_{r},\bm{\Psi}_{0}\right\rangle_{\mathcal{H}}d\mathcal{W}_{r}$
$\displaystyle\qquad\qquad+\sum_{j=0}^{i-1}\left(\left\|\int_{t^{l}_{j}}^{t^{l}_{j+1}}B_{r}d\mathcal{W}_{r}\right\|_{H}^{2}-\left\|\bm{\Psi}_{t^{l}_{j+1}}-\bm{\Psi}_{t^{l}_{j}}-\int_{t^{l}_{j}}^{t^{l}_{j+1}}B_{r}d\mathcal{W}_{r}\right\|_{H}^{2}\right)$
$\displaystyle=2\int_{0}^{t}\left\langle\eta_{r},\tilde{\bm{\Psi}}^{l}_{r}\right\rangle_{U\times
V}dr+2\int_{0}^{t}\left\langle
B_{r},\hat{\bm{\Psi}}^{l}_{r}\right\rangle_{H}d\mathcal{W}_{r}+2\int_{0}^{t^{l}_{1}}\left\langle
B_{r},\bm{\Psi}_{0}\right\rangle_{H}d\mathcal{W}_{r}$
$\displaystyle\qquad\qquad+\sum_{j=0}^{i-1}\left(\left\|\int_{t^{l}_{j}}^{t^{l}_{j+1}}B_{r}d\mathcal{W}_{r}\right\|_{H}^{2}-\left\|\bm{\Psi}_{t^{l}_{j+1}}-\bm{\Psi}_{t^{l}_{j}}-\int_{t^{l}_{j}}^{t^{l}_{j+1}}B_{r}d\mathcal{W}_{r}\right\|_{H}^{2}\right)$

where we have applied 1.6.13 and the associated remark thereafter. In
particular we have that

$\displaystyle\lVert\bm{\Psi}_{t}\rVert_{H}^{2}$
$\displaystyle\leq\lVert\bm{\Psi}_{0}\rVert_{H}^{2}+2\int_{0}^{t}\left\langle\eta_{r},\tilde{\bm{\Psi}}^{l}_{r}\right\rangle_{U\times
V}dr+2\int_{0}^{t}\left\langle
B_{r},\hat{\bm{\Psi}}^{l}_{r}\right\rangle_{H}d\mathcal{W}_{r}$
$\displaystyle+2\int_{0}^{t^{l}_{1}}\left\langle
B_{r},\bm{\Psi}_{0}\right\rangle_{H}d\mathcal{W}_{r}+\sum_{j=0}^{i-1}\left(\left\|\int_{t^{l}_{j}}^{t^{l}_{j+1}}B_{r}d\mathcal{W}_{r}\right\|_{H}^{2}\right).$

Our goal is to show that

$\mathbbm{E}\left(\sup_{t\in
I_{l}\cap(0,\tau]/\\{T\\}}\lVert\bm{\Psi}_{t}\rVert_{H}^{2}\right)\leq c$

for some constant $c$ independent of $l$. To this end, observe that

$\displaystyle\mathbbm{E}\left(\sup_{t\in
I_{l}\cap(0,\tau]/\\{T\\}}\lVert\bm{\Psi}_{t}\rVert_{H}^{2}\right)$
$\displaystyle\leq\mathbbm{E}\left(\lVert\bm{\Psi}_{0}\rVert_{H}^{2}\right)$
$\displaystyle+2\mathbbm{E}\left(\int_{0}^{T\wedge\tau}\left|\left\langle\eta_{r},\tilde{\bm{\Psi}}^{l}_{r}\right\rangle_{U\times
V}\right|dr\right)+2\mathbbm{E}\left(\sup_{t\in[0,T]}\left|\int_{0}^{t\wedge\tau}\left\langle
B_{r},\hat{\bm{\Psi}}^{l}_{r}\right\rangle_{H}d\mathcal{W}_{r}\right|\right)$
$\displaystyle+2\mathbbm{E}\left(\sup_{t\in[0,T]}\left|\int_{0}^{t\wedge\tau}\left\langle
B_{r},\bm{\Psi}_{0}\right\rangle_{H}d\mathcal{W}_{r}\right|\right)$
$\displaystyle+\mathbbm{E}\left[\sup_{t\in
I_{l}\cap(0,\tau]/\\{T\\}}\sum_{j=0}^{i-1}\left(\left\|\int_{t^{l}_{j}}^{t^{l}_{j+1}}B_{r}d\mathcal{W}_{r}\right\|_{H}^{2}\right)\right].$

We shall treat each term individually. Firstly we have that

$\displaystyle
2\mathbbm{E}\left(\int_{0}^{T\wedge\tau}\left|\left\langle\eta_{r},\tilde{\bm{\Psi}}^{l}_{r}\right\rangle_{U\times
V}\right|dr\right)$
$\displaystyle\leq\mathbbm{E}\left(\int_{0}^{T\wedge\tau}\lVert\eta_{r}\rVert_{U}^{2}+\lVert\tilde{\bm{\Psi}}^{l}_{r}\rVert_{V}^{2}dr\right)$
$\displaystyle\leq\mathbbm{E}\left(\int_{0}^{T\wedge\tau}\lVert\eta_{r}\rVert_{U}^{2}dr\right)+\max_{m\leq
L}\mathbbm{E}\left(\int_{0}^{T\wedge\tau}\lVert\tilde{\bm{\Psi}}^{m}_{r}\rVert_{V}^{2}dr\right)+1$

where $L$ is taken sufficiently large so that for all $m\geq L$,

$\mathbbm{E}\left(\int_{0}^{T\wedge\tau}\left|\lVert\tilde{\bm{\Psi}}^{m}_{r}\rVert_{V}^{2}-\lVert\bm{\Psi}_{r}\rVert_{V}^{2}\right|dr\right)\leq\frac{1}{2}.$

For the first stochastic integral we apply 1.6.9, seeing that

$\displaystyle
2\mathbbm{E}\left(\sup_{t\in[0,T]}\left|\int_{0}^{t\wedge\tau}\left\langle
B_{r},\hat{\bm{\Psi}}^{l}_{r}\right\rangle_{H}d\mathcal{W}_{r}\right|\right)$
$\displaystyle\leq c\mathbbm{E}\left(\int_{0}^{T\wedge\tau}\left\|\left\langle
B_{r},\hat{\bm{\Psi}}^{l}_{r}\right\rangle_{H}\right\|^{2}_{\mathscr{L}^{2}(\mathfrak{U};\mathbb{R})}dr\right)^{\frac{1}{2}}$
$\displaystyle\leq
c\mathbbm{E}\left(\int_{0}^{T\wedge\tau}\left\|B_{r}\right\|^{2}_{\mathscr{L}^{2}(\mathfrak{U};H)}\left\|\hat{\bm{\Psi}}^{l}_{r}\right\|_{H}^{2}dr\right)^{\frac{1}{2}}$
$\displaystyle\leq
c\mathbbm{E}\left(\sup_{r\in[0,T\wedge\tau]}\left\|\hat{\bm{\Psi}}^{l}_{r}\right\|_{H}^{2}\int_{0}^{T\wedge\tau}\left\|B_{r}\right\|^{2}_{\mathscr{L}^{2}(\mathfrak{U};H)}dr\right)^{\frac{1}{2}}$
$\displaystyle=c\mathbbm{E}\left(\sup_{r\in[0,T\wedge\tau]}\left\|\hat{\bm{\Psi}}^{l}_{r}\right\|_{H}^{2}\right)^{\frac{1}{2}}\left(\int_{0}^{T\wedge\tau}\left\|B_{r}\right\|^{2}_{\mathscr{L}^{2}(\mathfrak{U};H)}dr\right)^{\frac{1}{2}}$
$\displaystyle=\frac{1}{2}\mathbbm{E}\left(\sup_{r\in[0,T\wedge\tau]}\left\|\hat{\bm{\Psi}}^{l}_{r}\right\|_{H}^{2}\right)+c\mathbbm{E}\left(\int_{0}^{T\wedge\tau}\left\|B_{r}\right\|^{2}_{\mathscr{L}^{2}(\mathfrak{U};H)}dr\right)$
$\displaystyle=\frac{1}{2}\mathbbm{E}\left(\sup_{t\in
I_{l}\cap(0,\tau]/\\{T\\}}\lVert\bm{\Psi}_{t}\rVert_{H}^{2}\right)+c\mathbbm{E}\left(\int_{0}^{T\wedge\tau}\left\|B_{r}\right\|^{2}_{\mathscr{L}^{2}(\mathfrak{U};H)}dr\right)$

where $c$ here is a generic constant changing from line to line, coming
initially from the constant in 1.6.9 which is independent of $l$. Putting this
together we now see that

$\displaystyle\frac{1}{2}\mathbbm{E}\left(\sup_{t\in
I_{l}\cap(0,\tau]/\\{T\\}}\lVert\bm{\Psi}_{t}\rVert_{H}^{2}\right)$
$\displaystyle\leq\mathbbm{E}\left(\lVert\bm{\Psi}_{0}\rVert_{H}^{2}\right)+\mathbbm{E}\left(\int_{0}^{T\wedge\tau}\lVert\eta_{r}\rVert_{U}^{2}dr\right)$
$\displaystyle+\max_{m\leq
L}\mathbbm{E}\left(\int_{0}^{T\wedge\tau}\lVert\tilde{\bm{\Psi}}^{m}_{r}\rVert_{V}^{2}dr\right)+1+c\mathbbm{E}\left(\int_{0}^{T\wedge\tau}\left\|B_{r}\right\|^{2}_{\mathscr{L}^{2}(\mathfrak{U};H)}dr\right)$
$\displaystyle+2\mathbbm{E}\left(\sup_{t\in[0,T]}\left|\int_{0}^{t}\left\langle
B_{r},\bm{\Psi}_{0}\right\rangle_{H}d\mathcal{W}_{r}\right|\right)$
$\displaystyle+\mathbbm{E}\left[\sup_{t\in
I_{l}\cap(0,\tau]/\\{T\\}}\sum_{j=0}^{i-1}\left(\left\|\int_{t^{l}_{j}}^{t^{l}_{j+1}}B_{r}d\mathcal{W}_{r}\right\|_{H}^{2}\right)\right].$

For the stochastic integral involving the initial condition we can treat this
identically to generate the bound

$2\mathbbm{E}\left(\sup_{t\in[0,T]}\left|\int_{0}^{t\wedge\tau}\left\langle
B_{r},\bm{\Psi}_{0}\right\rangle_{H}d\mathcal{W}_{r}\right|\right)\leq\mathbbm{E}\left(\lVert\bm{\Psi}_{0}\rVert_{H}^{2}\right)+c\mathbbm{E}\left(\int_{0}^{T\wedge\tau}\left\|B_{r}\right\|^{2}_{\mathscr{L}^{2}(\mathfrak{U};H)}dr\right).$

As for the final term, it is clear that the supremum over all partitions can
be bounded by taking the partition for $t=t^{l}_{k_{l}}=T$. Thus

$\displaystyle\mathbbm{E}\left[\sup_{t\in
I_{l}\cap(0,\tau]/\\{T\\}}\sum_{j=0}^{i-1}\left(\left\|\int_{t^{l}_{j}}^{t^{l}_{j+1}}B_{r}d\mathcal{W}_{r}\right\|_{H}^{2}\right)\right]$
$\displaystyle\leq\mathbbm{E}\left[\sum_{j=0}^{k_{l}-1}\left(\left\|\int_{t^{l}_{j}}^{t^{l}_{j+1}}B_{r}d\mathcal{W}_{r}\right\|_{H}^{2}\right)\right]$
$\displaystyle=\sum_{j=0}^{k_{l}-1}\mathbbm{E}\left(\left\|\int_{t^{l}_{j}}^{t^{l}_{j+1}}B_{r}d\mathcal{W}_{r}\right\|_{H}^{2}\right)$
$\displaystyle=\sum_{j=0}^{k_{l}-1}\mathbbm{E}\left(\int_{t^{l}_{j}}^{t^{l}_{j+1}}\left\|B_{r}\right\|_{\mathscr{L}^{2}(\mathfrak{U};H)}^{2}dr\right)$
$\displaystyle=\mathbbm{E}\left(\int_{0}^{T}\left\|B_{r}\right\|_{\mathscr{L}^{2}(\mathfrak{U};H)}^{2}dr\right)$

having applied 1.6.4. In total then we have that

$\displaystyle\mathbbm{E}\left(\sup_{t\in
I_{l}\cap(0,\tau]/\\{T\\}}\lVert\bm{\Psi}_{t}\rVert_{H}^{2}\right)$
$\displaystyle\leq
4\mathbbm{E}\left(\lVert\bm{\Psi}_{0}\rVert_{H}^{2}\right)+2\mathbbm{E}\left(\int_{0}^{T\wedge\tau}\lVert\eta_{r}\rVert_{U}^{2}dr\right)$
$\displaystyle+2\max_{m\leq
L}\mathbbm{E}\left(\int_{0}^{T\wedge\tau}\lVert\tilde{\bm{\Psi}}^{m}_{r}\rVert_{V}^{2}dr\right)+2+c\mathbbm{E}\left(\int_{0}^{T}\left\|B_{r}\right\|^{2}_{\mathscr{L}^{2}(\mathfrak{U};H)}dr\right)$

which is a finite bound independent of $l$. As $I_{l}\subset I_{l+1}$ then the
sequence

$\sup_{t\in I_{l}\cap(0,\tau]/\\{T\\}}\lVert\bm{\Psi}_{t}\rVert_{H}^{2}$

is $\mathbbm{P}-a.s.$ monotone increasing in $l$, so we can apply the Monotone
Convergence Theorem to see that

$\mathbbm{E}\left(\sup_{t\in
I\cap(0,\tau]/\\{T\\}}\lVert\bm{\Psi}_{t}\rVert_{H}^{2}\right)=\lim_{l\rightarrow\infty}\mathbbm{E}\left(\sup_{t\in
I_{l}\cap(0,\tau]/\\{T\\}}\lVert\bm{\Psi}_{t}\rVert_{H}^{2}\right)<\infty.$
(49)

Thus for $\mathbbm{P}-a.e.$ $\omega$, we have that

$\sup_{t\in
I\cap(0,\tau(\omega)]/\\{T\\}}\lVert\bm{\Psi}_{t}(\omega)\rVert_{H}^{2}=\tilde{c}<\infty$

and $\bm{\Psi}_{\cdot}(\omega)\in C([0,T];U)$. We fix such an $\omega$ and any
$t\in[0,\tau(\omega)]$. As the mesh of the partitions go to zero then there
there is a sequence of times $(t_{n})$ in $I\cap(0,\tau(\omega)]/\\{T\\}$ such
that $t_{n}\longrightarrow t$. The sequence
$\left(\bm{\Psi}_{t_{n}}(\omega)\right)$ is uniformly bounded in $H$ so admits
a weakly convergent subsequence in this space, to a limit which we call
$\psi$. From the continuous embedding of $H$ into $U$ this weak convergence
also holds in $U$, but from the continuity of $\bm{\Psi}_{\cdot}(\omega)$ in
$U$ we have that $\left(\bm{\Psi}_{t_{n}}(\omega)\right)$ converges strongly
and therefore weakly to $\bm{\Psi}_{t}(\omega)$ in $U$. By the uniqueness of
limits in the weak topology, we conclude that $\bm{\Psi}_{t}(\omega)=\psi$ and
thus belongs to $H$. Moreover the weak limit preserves the boundedness in $H$,
so $\lVert\bm{\Psi}_{t}(\omega)\rVert_{H}\leq\tilde{c}.$ Therefore

$\sup_{t\in
I\cap(0,\tau(\omega)]/\\{T\\}}\lVert\bm{\Psi}_{t}(\omega)\rVert_{H}^{2}=\sup_{t\in[0,\tau(\omega)]}\lVert\bm{\Psi}_{t}(\omega)\rVert_{H}^{2}$

for our fixed $\omega$ in a full measure set, and thus $\mathbbm{P}-a.s.$. We
also know that $\bm{\Psi}_{\cdot}=\bm{\Psi}_{\cdot\wedge\tau}$ from the
identity (48), so this equality extends to

$\sup_{t\in
I\cap(0,\tau(\omega)]/\\{T\\}}\lVert\bm{\Psi}_{t}(\omega)\rVert_{H}^{2}=\sup_{t\in[0,T]}\lVert\bm{\Psi}_{t}(\omega)\rVert_{H}^{2}.$

Combining this with (49) concludes the proof. ∎

Having now justified that for $\mathbbm{P}-a.e.$ $\omega$
$\bm{\Psi}_{t}(\omega)\in H$, we move on to prove weak continuity in this
space.

######  Lemma 2.5.3.

For $\mathbbm{P}-a.e.$ $\omega$, $\bm{\Psi}_{\cdot}(\omega)$ is weakly
continuous in $H$.

###### Proof.

We fix $t\in[0,T]$ but now take any sequence of times $(t_{n})$ such that
$t_{n}\rightarrow t$. For $\mathbbm{P}-a.e.$ $\omega$ and any given $\phi\in
H$ we must justify that

$\lim_{n\rightarrow\infty}\langle\bm{\Psi}_{t_{n}}(\omega)-\bm{\Psi}_{t}(\omega),\phi\rangle_{H}=0.$
(50)

For any $\varepsilon>0$, from the density of $V$ in $H$ there exists
$\phi^{k}\in V$ such that

$\lVert\phi-\phi^{k}\rVert_{H}<\frac{\varepsilon}{4\lVert\bm{\Psi}(\omega)\rVert_{L^{\infty}([0,T];H)}}$

and then from the continuity in $U$ there exists an $N\in\mathbbm{N}$
sufficiently large such that for all $n\geq N$,

$\lVert\bm{\Psi}_{t_{n}}(\omega)-\bm{\Psi}_{t}(\omega)\rVert_{U}<\frac{\varepsilon}{2\lVert\phi^{k}\rVert_{V}}.$

Putting this together,

$\displaystyle\lvert\langle\bm{\Psi}_{t_{n}}(\omega)-\bm{\Psi}_{t}(\omega),\phi\rangle_{H}\rvert$
$\displaystyle\leq\lvert\langle\bm{\Psi}_{t_{n}}(\omega)-\bm{\Psi}_{t}(\omega),\phi-\phi^{k}\rangle_{H}\rvert+\lvert\langle\bm{\Psi}_{t_{n}}(\omega)-\bm{\Psi}_{t}(\omega),\phi^{k}\rangle_{H}\rvert$
$\displaystyle\leq\lVert\bm{\Psi}_{t_{n}}(\omega)-\bm{\Psi}_{t}(\omega)\rVert_{H}\lVert\phi-\phi^{k}\rVert_{H}+\lvert\langle\bm{\Psi}_{t_{n}}(\omega)-\bm{\Psi}_{t}(\omega),\phi^{k}\rangle_{U\times
V}\rvert$ $\displaystyle\leq
2\lVert\bm{\Psi}(\omega)\rVert_{L^{\infty}([0,T];H)}\lVert\phi-\phi^{k}\rVert_{H}+\lVert\bm{\Psi}_{t_{n}}(\omega)-\bm{\Psi}_{t}(\omega)\rVert_{U}\lVert\phi^{k}\rVert_{V}$
$\displaystyle<\varepsilon$

as required.

∎

######  Proposition 2.5.4.

The equality

$\displaystyle\lVert\bm{\Psi}_{t}\rVert^{2}_{H}=\lVert\bm{\Psi}_{0}\rVert^{2}_{H}+\int_{0}^{t\wedge\tau}\bigg{(}2\langle\eta_{s},\bm{\Psi}_{s}\rangle_{U\times
V}+\lVert
B_{s}\rVert^{2}_{\mathscr{L}^{2}(\mathcal{U};H)}\bigg{)}ds+2\int_{0}^{t\wedge\tau}\langle
B_{s},\bm{\Psi}_{s}\rangle_{H}d\mathcal{W}_{s}$ (51)

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

###### Proof.

The proof now directly follows from claim $c$ onwards in the proof of Theorem
4.2.5 of [13].

∎

We wish to relax the integrability constraints over $\Omega$, in accordance
with 2.2.3. In the same setting of the Hilbert Spaces $V,H,U$, we impose the
new assumptions for some $T>0$ and stopping time $\tau$:

  1. 1.

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

  2. 2.

For $\mathbbm{P}-a.e.$ $\omega$, $\eta(\omega)\in L^{2}([0,T];U)$;

  3. 3.

$B\in\bar{\mathcal{I}}^{H}(\mathcal{W})$;

  4. 4.

For $\mathbbm{P}-a.e.$ $\omega$,
$\bm{\Psi}_{\cdot}(\omega)\mathbbm{1}_{\cdot\leq\tau(\omega)}\in
L^{2}([0,T];V)$ and $\bm{\Psi}_{\cdot}\mathbbm{1}_{\cdot\leq\tau}$ is
progressively measurable in $V$;

  5. 5.

The identity

$\bm{\Psi}_{t}=\bm{\Psi}_{0}+\int_{0}^{t\wedge\tau}\eta_{s}ds+\int_{0}^{t\wedge\tau}B_{s}d\mathcal{W}_{s}$
(52)

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

We restate 2.5.4 for the new setting.

######  Proposition 2.5.5.

The equality

$\displaystyle\lVert\bm{\Psi}_{t}\rVert^{2}_{H}=\lVert\bm{\Psi}_{0}\rVert^{2}_{H}+\int_{0}^{t\wedge\tau}\bigg{(}2\langle\eta_{s},\bm{\Psi}_{s}\rangle_{U\times
V}+\lVert
B_{s}\rVert^{2}_{\mathscr{L}^{2}(\mathcal{U};H)}\bigg{)}ds+2\int_{0}^{t\wedge\tau}\langle
B_{s},\bm{\Psi}_{s}\rangle_{H}d\mathcal{W}_{s}$ (53)

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

###### Proof.

The idea is simply to apply 2.5.4 for some truncated versions of the
processes. We consider the stopping times

$\displaystyle\tau^{1}_{n}$ $\displaystyle:=n\wedge\inf\\{0\leq
t<\infty:\int_{0}^{t}\lVert\eta_{s}\rVert_{U}^{2}ds\geq n\\}$
$\displaystyle\tau^{2}_{n}$ $\displaystyle:=n\wedge\inf\\{0\leq
t<\infty:\int_{0}^{t}\lVert
B_{s}\rVert_{\mathscr{L}^{2}(\mathfrak{U};H)}^{2}ds\geq n\\}$
$\displaystyle\tau^{3}_{n}$ $\displaystyle:=n\wedge\inf\\{0\leq
t<\infty:\int_{0}^{t}\lVert\bm{\Psi}_{s}\mathbbm{1}_{s\leq\tau}\rVert_{V}^{2}ds\geq
n\\}$ $\displaystyle\tau_{n}$
$\displaystyle:=\tau^{1}_{n}\wedge\tau^{2}_{n}\wedge\tau^{3}_{n}.$

Then for every $n$, we have that

$\displaystyle\bm{\Psi}_{0}\mathbbm{1}_{\lVert\bm{\Psi}_{0}\rVert_{H}\leq
n},\qquad\eta_{\cdot}\mathbbm{1}_{\cdot\leq\tau_{n}}\mathbbm{1}_{\lVert\bm{\Psi}_{0}\rVert_{H}\leq
n}$ $\displaystyle
B_{\cdot}\mathbbm{1}_{\cdot\leq\tau_{n}}\mathbbm{1}_{\lVert\bm{\Psi}_{0}\rVert_{H}\leq
n},\qquad\bm{\Psi}_{\cdot\wedge\tau_{n}}\mathbbm{1}_{\lVert\bm{\Psi}_{0}\rVert_{H}\leq
n}$

satisfy the previous assumptions of 1, 2, 3, 4. Moreover from (52) we have
that for any $t\in[0,T]$

$\displaystyle\bm{\Psi}_{t\wedge\tau_{n}}$
$\displaystyle=\bm{\Psi}_{0}+\int_{0}^{t\wedge\tau\wedge\tau_{n}}\eta_{s}ds+\int_{0}^{t\wedge\tau\wedge\tau_{n}}B_{s}d\mathcal{W}_{s}$
$\displaystyle=\bm{\Psi}_{0}+\int_{0}^{t\wedge\tau}\eta_{s}\mathbbm{1}_{s\leq\tau_{n}}ds+\int_{0}^{t\wedge\tau}B_{s}\mathbbm{1}_{s\leq\tau_{n}}d\mathcal{W}_{s}$

$\mathbbm{P}-a.s.$ and moreover

$\displaystyle\bm{\Psi}_{t\wedge\tau_{n}}\mathbbm{1}_{\lVert\bm{\Psi}_{0}\rVert_{H}\leq
n}$ $\displaystyle=\bm{\Psi}_{0}\mathbbm{1}_{\lVert\bm{\Psi}_{0}\rVert_{H}\leq
n}+\mathbbm{1}_{\lVert\bm{\Psi}_{0}\rVert_{H}\leq
n}\int_{0}^{t\wedge\tau}\eta_{s}\mathbbm{1}_{s\leq\tau_{n}}ds+\mathbbm{1}_{\lVert\bm{\Psi}_{0}\rVert_{H}\leq
n}\int_{0}^{t\wedge\tau}B_{s}\mathbbm{1}_{s\leq\tau_{n}}d\mathcal{W}_{s}$
$\displaystyle=\bm{\Psi}_{0}\mathbbm{1}_{\lVert\bm{\Psi}_{0}\rVert_{H}\leq
n}+\int_{0}^{t\wedge\tau}\eta_{s}\mathbbm{1}_{s\leq\tau_{n}}\mathbbm{1}_{\lVert\bm{\Psi}_{0}\rVert_{H}\leq
n}ds+\int_{0}^{t\wedge\tau}B_{s}\mathbbm{1}_{s\leq\tau_{n}}\mathbbm{1}_{\lVert\bm{\Psi}_{0}\rVert_{H}\leq
n}d\mathcal{W}_{s}$

having applied 1.6.14. Therefore we can apply 2.5.4 to see that the equality

$\displaystyle\lVert\bm{\Psi}_{t\wedge\tau_{n}}\mathbbm{1}_{\lVert\bm{\Psi}_{0}\rVert_{H}\leq
n}\rVert^{2}_{H}$
$\displaystyle=\lVert\bm{\Psi}_{0}\mathbbm{1}_{\lVert\bm{\Psi}_{0}\rVert_{H}\leq
n}\rVert^{2}_{H}+\int_{0}^{t\wedge\tau}2\langle\eta_{s}\mathbbm{1}_{s\leq\tau_{n}}\mathbbm{1}_{\lVert\bm{\Psi}_{0}\rVert_{H}\leq
n},\bm{\Psi}_{s\wedge\tau_{n}}\mathbbm{1}_{\lVert\bm{\Psi}_{0}\rVert_{H}\leq
n}\rangle_{U\times V}ds$ $\displaystyle+\int_{0}^{t\wedge\tau}\lVert
B_{s}\mathbbm{1}_{s\leq\tau_{n}}\mathbbm{1}_{\lVert\bm{\Psi}_{0}\rVert_{H}\leq
n}\rVert^{2}_{\mathscr{L}^{2}(\mathcal{U};H)}ds$
$\displaystyle+2\int_{0}^{t\wedge\tau}\langle
B_{s}\mathbbm{1}_{s\leq\tau_{n}}\mathbbm{1}_{\lVert\bm{\Psi}_{0}\rVert_{H}\leq
n},\bm{\Psi}_{s\wedge\tau_{n}}\mathbbm{1}_{\lVert\bm{\Psi}_{0}\rVert_{H}}\rangle_{H}d\mathcal{W}_{s}$

holds for all $t\in[0,T]$ $\mathbbm{P}-a.s.$. We rewrite this as

$\displaystyle\mathbbm{1}_{\lVert\bm{\Psi}_{0}\rVert_{H}\leq
n}\lVert\bm{\Psi}_{t\wedge\tau_{n}}\rVert^{2}_{H}$
$\displaystyle=\mathbbm{1}_{\lVert\bm{\Psi}_{0}\rVert_{H}\leq
n}\lVert\bm{\Psi}_{0}\rVert^{2}_{H}+\mathbbm{1}_{\lVert\bm{\Psi}_{0}\rVert_{H}\leq
n}\int_{0}^{t\wedge\tau\wedge\tau_{n}}2\langle\eta_{s},\bm{\Psi}_{s}\rangle_{U\times
V}ds$ $\displaystyle+\mathbbm{1}_{\lVert\bm{\Psi}_{0}\rVert_{H}\leq
n}\int_{0}^{t\wedge\tau\wedge\tau_{n}}\lVert
B_{s}\rVert^{2}_{\mathscr{L}^{2}(\mathcal{U};H)}ds$
$\displaystyle+\mathbbm{1}_{\lVert\bm{\Psi}_{0}\rVert_{H}\leq
n}2\int_{0}^{t\wedge\tau\wedge\tau_{n}}\langle
B_{s},\bm{\Psi}_{s}\rangle_{H}d\mathcal{W}_{s}.$

Therefore for any $t\in[0,T]$, $\mathbbm{P}-a.e.$ $\omega$ with $n$
sufficiently large so that $\lVert\bm{\Psi}_{0}(\omega)\rVert_{H}\leq n$ and
$t\leq\tau_{n}(\omega)$, the identity (52) holds. We can always find such a
large enough $n$, which completes the justification of this identity. The
continuity then follows identically as we have again from 2.5.4 that for every
$n$ and $\mathbbm{P}-a.e.$ $\omega$,
$\bm{\Psi}_{\cdot\wedge\tau_{n}}(\omega)\in C([0,T];H)$, and we are done. ∎

###  2.6 The General Itô Formula

The key consideration is the notion of derivatives, for which we use the
standard functional analytic choice of the Fréchet Derivative. For a Fréchet
differentiable function between Banach Spaces
$\mathcal{K}:\mathcal{Y}\rightarrow\mathcal{Z}$, recall that its derivative is
defined as a mapping
$\mathcal{K}^{\prime}:\mathcal{Y}\rightarrow\mathscr{L}(\mathcal{Y};\mathcal{Z})$
and subsequently its second derivative as
$\mathcal{K}^{\prime\prime}:\mathcal{Y}\rightarrow\mathscr{L}\big{(}\mathcal{Y};\mathscr{L}(\mathcal{Y};\mathcal{Z})\big{)}$.
In the case where $\mathcal{Y}=\mathcal{H}$ is a Hilbert Space and
$\mathcal{Z}=\mathbb{R}$, then

$\displaystyle\mathcal{K}^{\prime}$
$\displaystyle:\mathcal{H}\rightarrow\mathcal{H}^{*}$
$\displaystyle\mathcal{K}^{\prime\prime}$
$\displaystyle:\mathcal{H}\rightarrow\mathscr{L}(\mathcal{H};\mathcal{H}^{*})$

so it is commonplace to apply the Riesz Representation and make the
identification

$\displaystyle\mathcal{K}^{\prime}$
$\displaystyle:\mathcal{H}\rightarrow\mathcal{H}$ (54)
$\displaystyle\mathcal{K}^{\prime\prime}$
$\displaystyle:\mathcal{H}\rightarrow\mathscr{L}(\mathcal{H}).$ (55)

We will be considering functions $F:[0,\infty)\times U\rightarrow\mathbb{R}$
with partial derivatives $F_{t},F_{x}$ and $F_{xx}$ where the latter two are
understood in the sense of (54) and (55).

######  Theorem 2.6.1.

Suppose $(\bm{\Psi},\tau)$ is a local strong solution of (47), and
$F:[0,\infty)\times U\rightarrow\mathbb{R}$ is such that $F_{t},F_{x}$ and
$F_{xx}$ are uniformly continuous on bounded subsets of $[0,T]\times U$. Then
$F(\cdot,\bm{\Psi}_{\cdot})$ satisfies the identity

$\displaystyle F(t,\bm{\Psi}_{t})=F(0,\bm{\Psi}_{0})$
$\displaystyle+\int_{0}^{t\wedge\tau}\bigg{(}F_{s}(s,\bm{\Psi}_{s})+\langle
F_{x}(s,\bm{\Psi}_{s}),\mathcal{Q}(s,\bm{\Psi}_{s})\rangle_{U}\bigg{)}ds$
$\displaystyle+\frac{1}{2}\int_{0}^{t\wedge\tau}\textnormal{Tr}\Big{(}F_{xx}(s,\bm{\Psi}_{s})(\mathcal{G}(s,\bm{\Psi}_{s}))(\mathcal{G}(s,\bm{\Psi}_{s}))^{*}\Big{)}ds$
$\displaystyle+\int_{0}^{t\wedge\tau}\langle
F_{x}(s,\bm{\Psi}_{s}),\mathcal{G}(s,\bm{\Psi}_{s})\rangle_{U}d\mathcal{W}_{s}$

$a.e.$ in $\mathbb{R}$ for every $t\in[0,T]$, where the trace term is simply a
composition of operators and the stochastic integral is understood component
wise similarly to 1.6.12.

One particularly important application of this Itô Formula is the so called
energy equality, through the function $F:h\mapsto\lVert h\rVert^{2}_{U}$. The
result is as follows:

######  Proposition 2.6.2.

Suppose $(\bm{\Psi},\tau)$ is a local strong solution of (47). Then
$\bm{\Psi}$ satisfies the energy equality

$\displaystyle\lVert\bm{\Psi}_{t}\rVert^{2}_{U}=\lVert\bm{\Psi}_{0}\rVert^{2}_{U}+\int_{0}^{t\wedge\tau}\bigg{(}2\langle\bm{\Psi}_{s},\mathcal{Q}(s,\bm{\Psi}_{s})\rangle_{U}+\lVert\mathcal{G}(s,\bm{\Psi}_{s})\rVert^{2}_{\mathscr{L}^{2}(\mathcal{U};U}\bigg{)}ds+2\int_{0}^{t\wedge\tau}\langle\bm{\Psi}_{s},\mathcal{G}(s,\bm{\Psi}_{s})\rangle_{U}d\mathcal{W}_{s}$

$a.e.$ in $\mathbb{R}$ for all $t\geq 0$.

We do not prove these results here, but instead refer to [7] Theorem 4.18 and
related discussions therein. Also shown there is the Itô Formula for processes
defined by a more general evolution equation in a Hilbert Space $\mathcal{H}$,

$\bm{\Phi}_{t}=\bm{\Phi}_{0}+\int_{0}^{t}\phi_{s}ds+\int_{0}^{t}B(s)d\mathcal{W}_{s}$
(56)

for any given $\bm{\Phi}_{0}:\Omega\rightarrow\mathcal{H}$
$\mathcal{F}_{0}-$measurable, $\phi:\Omega\times[0,t]\rightarrow\mathcal{H}$
progressively measurable and $a.s.$ Bochner Integrable and
$B\in\bar{\mathcal{I}}^{\mathcal{H}}(\mathcal{W})$. Then we have the
corresponding result to 2.6.1.

######  Theorem 2.6.3.

Suppose that $F$ is as in 2.6.1 for $\mathcal{H}$ replacing $U$ and
$\bm{\Phi}$ is defined by (56). Then $F(\cdot,\bm{\Phi}_{\cdot})$ satisfies
the identity

$\displaystyle F(t,\bm{\Phi}_{t})=F(0,\bm{\Phi}_{0})$
$\displaystyle+\int_{0}^{t\wedge\tau}\bigg{(}F_{s}(s,\bm{\Phi}_{s})+\langle
F_{x}(s,\bm{\Phi}_{s}),\phi_{s}\rangle_{\mathcal{H}}\bigg{)}ds$
$\displaystyle+\frac{1}{2}\int_{0}^{t\wedge\tau}\textnormal{Tr}\Big{(}F_{xx}(s,\bm{\Phi}_{s})(B(s))(B(s))^{*}\Big{)}ds$
$\displaystyle+\int_{0}^{t\wedge\tau}\langle
F_{x}(s,\bm{\Phi}_{s}),B(s)\rangle_{\mathcal{H}}d\mathcal{W}_{s}$

$a.e.$ in $\mathbb{R}$ for every $t\in[0,T]$.

###  2.7 The Case of Constant Multiplicative Noise

Many techniques in proving existence and uniqueness of an SPDE in this
framework rely on simplifying the equation to one where we can apply the
standard theory, and then constructing solutions in the original framework via
some appropriate limit of solutions to the simplified equations. As such we
shall briefly considered a special type of equation in this framework which
reduces the driving noise from something infinite dimensional to one
dimensional. For this we work again with an arbitrary Hilbert Space
$\mathcal{H}$.

######  Proposition 2.7.1.

Suppose that $\bm{\Psi}\in\bar{\mathcal{I}}^{\mathcal{H}}$ and that the
operator $\mathcal{G}:\mathfrak{U}\times\mathcal{H}\rightarrow\mathcal{H}$ is
such that

$\mathcal{G}_{i}:\phi\mapsto\lambda_{i}\phi$

for each $i$ with $\lambda_{i}\in\mathbb{R}$, and

$\sum_{i=1}^{\infty}\lambda_{i}^{2}<\infty.$ (57)

Then $\mathcal{G}\bm{\Psi}\in\bar{\mathcal{I}}^{\mathcal{H}}(\mathcal{W})$ and

$\int_{0}^{t}\mathcal{G}\bm{\Psi}_{s}d\mathcal{W}_{s}=\bigg{(}\sum_{i=1}^{\infty}\lambda_{i}^{2}\bigg{)}^{1/2}\int_{0}^{t}\bm{\Psi}_{s}dW_{s}$
(58)

where $W$ is a standard one dimensional Brownian Motion.

In order to prove the above, we use an intermediary lemma.

######  Lemma 2.7.2.

In the setting of 2.7.1 the infinite sum

$M_{s}:=\sum_{i=1}^{\infty}\lambda_{i}W^{i}_{s}$ (59)

is convergent in $L^{2}\big{(}\Omega;\mathbb{R})$ at every $s$, and the
limiting martingale has the representation

$M=\bigg{(}\sum_{i=1}^{\infty}\lambda_{i}^{2}\bigg{)}^{1/2}W$ (60)

where $W$ is a standard Brownian Motion.

###### Proof of 2.7.2.

Firstly let’s verify that the convergence in (59) does indeed hold, which is
immediate from observing that

$\sum_{i=1}^{\infty}\lambda_{i}W^{i}_{s}=\sum_{i=1}^{\infty}\int_{0}^{s}\lambda_{i}dW^{i}_{r}$

which is simply the stochastic integral

$\int_{0}^{s}P(r)d\mathcal{W}_{r}$

for the process $P\in\mathcal{I}^{\mathbb{R}}(\mathcal{W})$ defined by
$P_{e_{i}}(r)=\lambda_{i}$. So $M$ is a continuous genuine martingale, which
we show is of the form (60) through Levy’s Characterisation of Brownian
Motion. Indeed the quadratic variation process $[M]$ is given precisely as in
1.5.8, where our approximating sequence of martingales

$M^{n}=\sum_{i=1}^{n}\lambda_{i}W^{i}_{s}$

have quadratic variation

$[M^{n}]_{s}=\sum_{i=1}^{n}\lambda_{i}^{2}s$

which of course converges in $L^{1}\big{(}\Omega;\mathbb{R}\big{)}$ to the
infinite sum, from which we conclude

$[M]_{s}=\sum_{i=1}^{\infty}\lambda_{i}^{2}s.$

Therefore

$\Bigg{[}\frac{M}{\big{(}\sum_{i=1}^{\infty}\lambda_{i}^{2}\big{)}^{1/2}}\Bigg{]}_{s}=s$

and we immediately deduce the representation (60) from Levy’s
Characterisation.

∎

###### Proof of 2.7.1.

As
$\mathcal{G}:\mathcal{H}\rightarrow\mathcal{L}^{2}(\mathcal{U};\mathcal{H})$
is linear and bounded from

$\lVert\mathcal{G}\phi\rVert_{\mathcal{L}^{2}(\mathcal{U};\mathcal{H})}^{2}=\sum_{i=1}^{\infty}\lVert\lambda_{i}\phi\rVert_{\mathcal{H}}^{2}=\sum_{i=1}^{\infty}\lambda_{i}^{2}\lVert\phi\rVert_{\mathcal{H}}^{2}$

then it is continuous as a mapping between these spaces so preserves the
progressive measurability and evidently the required boundedness to deduce
that $\mathcal{G}\bm{\Psi}\in\bar{\mathcal{I}}^{\mathcal{H}}(\mathcal{W})$. To
show the identity (58) let’s rewrite

$\int_{0}^{t}\mathcal{G}\bm{\Psi}_{s}d\mathcal{W}_{s}=\lim_{n\rightarrow\infty}\sum_{i=1}^{n}\int_{0}^{t}\lambda_{i}\bm{\Psi}_{s}dW^{i}_{s}=\lim_{n\rightarrow\infty}\int_{0}^{t}\bm{\Psi}_{s}dM^{n}_{s}$

with notation $M^{n}$ as in 2.7.2 and understanding once more that this limit
is taken in $L^{2}\big{(}\Omega;\mathcal{H}\big{)}$ for the stopped integrals.
The localising stopping times $(\tau_{m})$ defined by

$\tau_{m}=m\wedge\inf\\{0\leq
t<\infty:\int_{0}^{t}\sum_{i=1}^{\infty}\lambda_{i}^{2}\lVert\bm{\Psi}_{s}\rVert_{\mathcal{H}}^{2}ds\geq
m\\}$

are precisely as in 1.6.7 and (11) for the left and right side of (58)
respectively. It is therefore sufficient to show that for any $m$,

$\lim_{n\rightarrow\infty}\int_{0}^{t}\bm{\Psi}_{s}\mathbbm{1}_{s\leq\tau_{m}}dM^{n}_{s}=\int_{0}^{t}\bm{\Psi}_{s}\mathbbm{1}_{s\leq\tau_{m}}dM_{s}$

having simply inserted the representation (60) into our required identity. In
other words we want that

$\mathbbm{E}\Big{|}\Big{|}\int_{0}^{t}\bm{\Psi}_{s}\mathbbm{1}_{s\leq\tau_{m}}dM_{s}-\int_{0}^{t}\bm{\Psi}_{s}\mathbbm{1}_{s\leq\tau_{m}}dM_{s}^{n}\Big{|}\Big{|}_{\mathcal{H}}^{2}\longrightarrow
0$

as $n\rightarrow\infty$ which is equivalent to the statement

$\mathbbm{E}\Big{|}\Big{|}\int_{0}^{t}\bm{\Psi}_{s}\mathbbm{1}_{s\leq\tau_{m}}d(M-M^{n})_{s}\Big{|}\Big{|}_{\mathcal{H}}^{2}\longrightarrow
0.$

The same arguments of 2.7.2 afford us that the martingale $M-M^{n}$ which is
given at each time $s$ by

$(M-M_{n})_{s}=\sum_{i=n+1}^{\infty}\lambda_{i}W^{i}_{s}$

has the representation

$M-M^{n}=\bigg{(}\sum_{i=n+1}^{\infty}\lambda_{i}^{2}\bigg{)}^{1/2}V$

for a standard Brownian Motion $V$. So we have that

$\displaystyle\mathbbm{E}\bigg{|}\bigg{|}\int_{0}^{t}\bm{\Psi}_{s}\mathbbm{1}_{s\leq\tau_{m}}d(M-M^{n})_{s}\bigg{|}\bigg{|}_{\mathcal{H}}^{2}$
$\displaystyle=\mathbbm{E}\bigg{|}\bigg{|}\bigg{(}\sum_{i=n+1}^{\infty}\lambda_{i}^{2}\bigg{)}^{1/2}\int_{0}^{t}\bm{\Psi}_{s}\mathbbm{1}_{s\leq\tau_{m}}dV_{s}\bigg{|}\bigg{|}_{\mathcal{H}}^{2}$
$\displaystyle=\bigg{(}\sum_{i=n+1}^{\infty}\lambda_{i}^{2}\bigg{)}\mathbbm{E}\bigg{|}\bigg{|}\int_{0}^{t}\bm{\Psi}_{s}\mathbbm{1}_{s\leq\tau_{m}}dV_{s}\bigg{|}\bigg{|}_{\mathcal{H}}^{2}$
$\displaystyle=\bigg{(}\sum_{i=n+1}^{\infty}\lambda_{i}^{2}\bigg{)}\mathbbm{E}\int_{0}^{t}\lVert\bm{\Psi}_{s}\mathbbm{1}_{s\leq\tau_{m}}\rVert_{\mathcal{H}}^{2}ds$

having used the Itô Isometry 1.2.7.1. By definition of the stopping time the
integral is bounded uniformly in $\omega$ (a.e.) hence has finite expectation,
so we conclude that this approaches zero in the limit by finiteness of the
sum.

∎

###  2.8 An Existence and Uniqueness Result in Finite Dimensions

Another such way that we look to simplify the equations in this framework to
apply the standard theory is through taking finite dimensional approximations
(which also motivated the previous result!). The scheme of application for
this is referred to as a Galerkin Scheme, used traditionally in the analysis
for highly non-trivial PDEs such as the Euler and Navier-Stokes Equations (see
[11, 12]) and indeed is used in the abstract stochastic solution method [1].

######  Theorem 2.8.1.

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
$\bm{\phi},\bm{\psi}\in\mathcal{H}$ and $t\in[0,\infty)$,

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

  * 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

$\bm{\Phi}_{t}=\bm{\Phi}_{0}+\int_{0}^{t}\mathscr{A}(s,\bm{\Phi}_{s})ds+\int_{0}^{t}\mathscr{G}(s,\bm{\Phi}_{s})d\mathcal{W}_{s}$
(61)

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

We remark that the operators satisfy the assumptions of 2.4.1, 2.4.2 for the
spaces $V=H=U:=\mathcal{H}$ and that the conclusion of this theorem is the
existence of a strong solution of (61) in the sense of 2.4.3.

###### Proof.

With the finite-dimesnionality of $\mathcal{H}$ in place, we first restrict
ourselves to finitely many Brownian Motions in our stochastic integral to make
things classical. Let’s define the operator
$\mathscr{G}^{k}:[0,T]\times\mathcal{H}\times\mathfrak{U}\rightarrow\mathcal{H}$
on the basis vectors $(e_{i})$ of $\mathfrak{U}$ by
$\mathscr{G}^{k}(\cdot,\cdot,e_{i})=\mathscr{G}(\cdot,\cdot,e_{i})$ for $i\leq
k$, and zero otherwise. We consider at first the equation

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

or equivalently,

$\bm{\Phi}^{k}_{t}=\bm{\Phi}^{k}_{0}+\int_{0}^{t}\mathscr{A}(s,\bm{\Phi}^{k}_{s})ds+\sum_{i=1}^{k}\int_{0}^{t}\mathscr{G}_{i}(s,\bm{\Phi}^{k}_{s})dW^{i}_{s}$

for $\bm{\Phi}^{k}_{0}:=\bm{\Phi}_{0}$. The existence and uniqueness of
solutions to this finite-dimensional system is then classical (for solutions
defined as in the theorem). Consider now solutions
$\bm{\Phi}^{j},\bm{\Phi}^{k}$ for $j<k$ arbitrary, which therefore satisfy the
difference equation

$\displaystyle\bm{\Phi}^{k}_{r}-\bm{\Phi}^{j}_{r}=\int_{0}^{r}\mathscr{A}(s,\bm{\Phi}^{k}_{s})-\mathscr{A}(s,\bm{\Phi}^{j}_{s})ds+\int_{0}^{r}\mathscr{G}^{k}(s,\bm{\Phi}^{k}_{s})-\mathscr{G}^{j}(s,\bm{\Phi}^{j}_{s})d\mathcal{W}_{s}$

for any $r\in[0,\infty)$. By applying the Itô Formula for the energy identity
2.6.2 but in the general case of 2.6.3, we see further that the identity

$\displaystyle\lVert\bm{\Phi}^{k}_{r}-\bm{\Phi}^{j}_{r}\rVert_{\mathcal{H}}^{2}$
$\displaystyle=2\int_{0}^{r}\langle\mathscr{A}(s,\bm{\Phi}^{k}_{s})-\mathscr{A}(s,\bm{\Phi}^{j}_{s}),\bm{\Phi}^{k}_{s}-\bm{\Phi}^{j}_{s}\rangle_{\mathcal{H}}ds$
$\displaystyle+\int_{0}^{r}\lVert\mathscr{G}^{k}(s,\bm{\Phi}^{k}_{s})-\mathscr{G}^{j}(s,\bm{\Phi}^{j}_{s})\rVert^{2}_{\mathscr{L}^{2}(\mathfrak{U};\mathcal{H})}ds+2\int_{0}^{r}\langle\mathscr{G}^{k}(s,\bm{\Phi}^{k}_{s})-\mathscr{G}^{j}(s,\bm{\Phi}^{j}_{s}),\bm{\Phi}^{k}_{s}-\bm{\Phi}^{j}_{s}\rangle_{\mathcal{H}}d\mathcal{W}_{s}$

holds $a.s.$. We use Cauchy-Schwartz to move to an inequality, and rewrite the
quadratic variation term to give us the bound

$\displaystyle\lVert\bm{\Phi}^{k}_{r}-\bm{\Phi}^{j}_{r}\rVert_{\mathcal{H}}^{2}$
$\displaystyle\leq\int_{0}^{r}\left(2\lVert\mathscr{A}(s,\bm{\Phi}^{k}_{s})-\mathscr{A}(s,\bm{\Phi}^{j}_{s})\rVert_{\mathcal{H}}\lVert\bm{\Phi}^{k}_{s}-\bm{\Phi}^{j}_{s}\rVert_{\mathcal{H}}+\sum_{i=1}^{j}\lVert\mathscr{G}_{i}(s,\bm{\Phi}^{k}_{s})-\mathscr{G}_{i}(s,\bm{\Phi}^{j}_{s})\rVert^{2}\right)ds$
$\displaystyle+\int_{0}^{r}\sum_{i=j+1}^{k}\lVert\mathscr{G}_{i}(s,\bm{\Phi}^{k}_{s})\rVert_{\mathcal{H}}^{2}ds+2\int_{0}^{r}\langle\mathscr{G}^{k}(s,\bm{\Phi}^{k}_{s})-\mathscr{G}^{j}(s,\bm{\Phi}^{j}_{s}),\bm{\Phi}^{k}_{s}-\bm{\Phi}^{j}_{s}\rangle_{\mathcal{H}}d\mathcal{W}_{s}.$

In one step now we bound the stochastic integral by its absolute value, take
the supremum over all such $r$ up to any arbitrary time $t\in[0,\infty)$ and
employ the Lipschitz assumption to see that

$\displaystyle\sup_{r\in[0,t]}\lVert\bm{\Phi}^{k}_{r}-\bm{\Phi}^{j}_{r}\rVert_{\mathcal{H}}^{2}$
$\displaystyle\leq
c\int_{0}^{t}\lVert\bm{\Phi}^{k}_{s}-\bm{\Phi}^{j}_{s}\rVert_{\mathcal{H}}^{2}ds$
$\displaystyle+\int_{0}^{t}\sum_{i=j+1}^{k}\lVert\mathscr{G}_{i}(s,\bm{\Phi}^{k}_{s})\rVert_{\mathcal{H}}^{2}ds+2\sup_{r\in[0,t]}\left|\int_{0}^{r}\langle\mathscr{G}^{k}(s,\bm{\Phi}^{k}_{s})-\mathscr{G}^{j}(s,\bm{\Phi}^{j}_{s}),\bm{\Phi}^{k}_{s}-\bm{\Phi}^{j}_{s}\rangle_{\mathcal{H}}d\mathcal{W}_{s}\right|$

for a generic constant $c$, allowed to depend on $t$. We want to take the
expectation here but have to be slightly careful in ensuring that the
expectation is finite; to this end we consider the stopping times

$\tau_{R}:=R\wedge\inf\\{s\geq
0:\max\\{\lVert\bm{\Phi}^{k}_{s}\rVert_{\mathcal{H}},\lVert\bm{\Phi}^{j}_{s}\rVert_{\mathcal{H}}\\}\geq
R\\}$

and the process defined for any fixed $R$ by

$\tilde{\bm{\Phi}}^{k}_{s}:=\bm{\Phi}^{k}_{s}\mathbbm{1}_{s\leq\tau_{R}},\qquad\tilde{\bm{\Phi}}^{j}_{s}:=\bm{\Phi}^{j}_{s}\mathbbm{1}_{s\leq\tau_{R}}.$

From the continuity of the processes $\bm{\Phi}^{k},\bm{\Phi}^{j}$ then
$(\tau_{R})$ is an $a.s.$ monotone increasing sequence convergent to infinity
and
$\max\\{\lVert\tilde{\bm{\Phi}}^{k}_{s}\rVert_{\mathcal{H}},\lVert\tilde{\bm{\Phi}}^{j}_{s}\rVert_{\mathcal{H}}\\}\leq
R$ for any $s\geq 0$. It is trivial that these processes satisfy the same
inequality

$\displaystyle\sup_{r\in[0,t]}\lVert\tilde{\bm{\Phi}}^{k}_{r}-\tilde{\bm{\Phi}}^{j}_{r}\rVert_{\mathcal{H}}^{2}$
$\displaystyle\leq
c\int_{0}^{t}\lVert\tilde{\bm{\Phi}}^{k}_{s}-\tilde{\bm{\Phi}}^{j}_{s}\rVert_{\mathcal{H}}^{2}ds$
$\displaystyle+\int_{0}^{t}\sum_{i=j+1}^{k}\lVert\mathscr{G}_{i}(s,\tilde{\bm{\Phi}}^{k}_{s})\rVert_{\mathcal{H}}^{2}ds+2\sup_{r\in[0,t]}\left|\int_{0}^{r}\langle\mathscr{G}^{k}(s,\tilde{\bm{\Phi}}^{k}_{s})-\mathscr{G}^{j}(s,\tilde{\bm{\Phi}}^{j}_{s}),\tilde{\bm{\Phi}}^{k}_{s}-\tilde{\bm{\Phi}}^{j}_{s}\rangle_{\mathcal{H}}d\mathcal{W}_{s}\right|$

and justify that the expectation of all terms involved is finite (indeed for
the stochastic integral, using the Lipschitz assumption and the boundedness of
$\tilde{\bm{\Phi}}^{k}_{\cdot}-\tilde{\bm{\Phi}}^{j}_{\cdot}$ then
$\langle\mathscr{G}^{k}(\cdot,\tilde{\bm{\Phi}}^{k}_{\cdot})-\mathscr{G}^{j}(\cdot,\tilde{\bm{\Phi}}^{j}_{\cdot}),\tilde{\bm{\Phi}}^{k}_{\cdot}-\tilde{\bm{\Phi}}^{j}_{\cdot}\rangle_{\mathcal{H}}\in\mathcal{I}^{\mathbb{R}}(\mathcal{W})$
so by 1.6.9 the expectation of this term is finite). We do now take the
expectation and apply 1.6.9 to give us the bound

$\displaystyle\mathbbm{E}\sup_{r\in[0,t]}\lVert\tilde{\bm{\Phi}}^{k}_{r}-\tilde{\bm{\Phi}}^{j}_{r}\rVert_{\mathcal{H}}^{2}$
$\displaystyle\leq
c\mathbbm{E}\int_{0}^{t}\lVert\tilde{\bm{\Phi}}^{k}_{s}-\tilde{\bm{\Phi}}^{j}_{s}\rVert_{\mathcal{H}}^{2}ds+\mathbbm{E}\int_{0}^{t}\sum_{i=j+1}^{k}\lVert\mathscr{G}_{i}(s,\tilde{\bm{\Phi}}^{k}_{s})\rVert_{\mathcal{H}}^{2}ds$
$\displaystyle+c\mathbbm{E}\left(\int_{0}^{t}\sum_{i=1}^{\infty}\langle\mathscr{G}_{i}^{k}(s,\tilde{\bm{\Phi}}^{k}_{s})-\mathscr{G}_{i}^{j}(s,\tilde{\bm{\Phi}}^{j}_{s}),\tilde{\bm{\Phi}}^{k}_{s}-\tilde{\bm{\Phi}}^{j}_{s}\rangle_{\mathcal{H}}^{2}ds\right)^{\frac{1}{2}}$

which we promptly reduce to

$\displaystyle\mathbbm{E}\sup_{r\in[0,t]}\lVert\tilde{\bm{\Phi}}^{k}_{r}-\tilde{\bm{\Phi}}^{j}_{r}\rVert_{\mathcal{H}}^{2}$
$\displaystyle\leq
c\mathbbm{E}\int_{0}^{t}\lVert\tilde{\bm{\Phi}}^{k}_{s}-\tilde{\bm{\Phi}}^{j}_{s}\rVert_{\mathcal{H}}^{2}ds+\mathbbm{E}\int_{0}^{t}\sum_{i=j+1}^{k}\lVert\mathscr{G}_{i}(s,\tilde{\bm{\Phi}}^{k}_{s})\rVert_{\mathcal{H}}^{2}ds$
$\displaystyle+c\mathbbm{E}\left(\int_{0}^{t}\left[\sum_{i=1}^{j}\lVert\mathscr{G}_{i}(s,\tilde{\bm{\Phi}}^{k}_{s})-\mathscr{G}_{i}(s,\tilde{\bm{\Phi}}^{j}_{s})\rVert^{2}+\sum_{i=j+1}^{k}\lVert\mathscr{G}_{i}(s,\tilde{\bm{\Phi}}^{k}_{s})\rVert_{\mathcal{H}}^{2}\right]\lVert\tilde{\bm{\Phi}}^{k}_{s}-\tilde{\bm{\Phi}}^{j}_{s}\rVert_{\mathcal{H}}^{2}ds\right)^{\frac{1}{2}}.$

Employing our Lipschitz assumption once more, followed by an application of
Young’s Inequality, we have that

$\displaystyle
c\left(\int_{0}^{t}\left[\sum_{i=1}^{j}\lVert\mathscr{G}_{i}(s,\tilde{\bm{\Phi}}^{k}_{s})-\mathscr{G}_{i}(s,\tilde{\bm{\Phi}}^{j}_{s})\rVert^{2}+\sum_{i=j+1}^{k}\lVert\mathscr{G}_{i}(s,\tilde{\bm{\Phi}}^{k}_{s})\rVert_{\mathcal{H}}^{2}\right]\lVert\tilde{\bm{\Phi}}^{k}_{s}-\tilde{\bm{\Phi}}^{j}_{s}\rVert_{\mathcal{H}}^{2}ds\right)^{\frac{1}{2}}$
$\displaystyle\qquad\qquad\leq
c\left(\int_{0}^{t}\left[\lVert\tilde{\bm{\Phi}}^{k}_{s}-\tilde{\bm{\Phi}}^{j}_{s}\rVert_{\mathcal{H}}^{2}+\sum_{i=j+1}^{k}\lVert\mathscr{G}_{i}(s,\tilde{\bm{\Phi}}^{k}_{s})\rVert_{\mathcal{H}}^{2}\right]\lVert\tilde{\bm{\Phi}}^{k}_{s}-\tilde{\bm{\Phi}}^{j}_{s}\rVert_{\mathcal{H}}^{2}ds\right)^{\frac{1}{2}}$
$\displaystyle\qquad\qquad\leq
c\left(\sup_{r\in[0,t]}\lVert\tilde{\bm{\Phi}}^{k}_{r}-\tilde{\bm{\Phi}}^{j}_{r}\rVert_{\mathcal{H}}^{2}\int_{0}^{t}\lVert\tilde{\bm{\Phi}}^{k}_{s}-\tilde{\bm{\Phi}}^{j}_{s}\rVert_{\mathcal{H}}^{2}+\sum_{i=j+1}^{k}\lVert\mathscr{G}_{i}(s,\tilde{\bm{\Phi}}^{k}_{s})\rVert_{\mathcal{H}}^{2}ds\right)^{\frac{1}{2}}$
$\displaystyle\qquad\qquad=c\left(\sup_{r\in[0,t]}\lVert\tilde{\bm{\Phi}}^{k}_{r}-\tilde{\bm{\Phi}}^{j}_{r}\rVert_{\mathcal{H}}^{2}\right)^{\frac{1}{2}}\left(\int_{0}^{t}\lVert\tilde{\bm{\Phi}}^{k}_{s}-\tilde{\bm{\Phi}}^{j}_{s}\rVert_{\mathcal{H}}^{2}+\sum_{i=j+1}^{k}\lVert\mathscr{G}_{i}(s,\tilde{\bm{\Phi}}^{k}_{s})\rVert_{\mathcal{H}}^{2}ds\right)^{\frac{1}{2}}$
$\displaystyle\qquad\qquad\leq\frac{1}{2}\sup_{r\in[0,t]}\lVert\tilde{\bm{\Phi}}^{k}_{r}-\tilde{\bm{\Phi}}^{j}_{r}\rVert_{\mathcal{H}}^{2}+\frac{c^{2}}{2}\int_{0}^{t}\left(\lVert\tilde{\bm{\Phi}}^{k}_{s}-\tilde{\bm{\Phi}}^{j}_{s}\rVert_{\mathcal{H}}^{2}+\sum_{i=j+1}^{k}\lVert\mathscr{G}_{i}(s,\tilde{\bm{\Phi}}^{k}_{s})\rVert_{\mathcal{H}}^{2}\right)ds$

and furthermore

$\displaystyle\mathbbm{E}\sup_{r\in[0,t]}\lVert\tilde{\bm{\Phi}}^{k}_{r}-\tilde{\bm{\Phi}}^{j}_{r}\rVert_{\mathcal{H}}^{2}$
$\displaystyle\leq
c\mathbbm{E}\int_{0}^{t}\lVert\tilde{\bm{\Phi}}^{k}_{s}-\tilde{\bm{\Phi}}^{j}_{s}\rVert_{\mathcal{H}}^{2}ds+c\mathbbm{E}\int_{0}^{t}\sum_{i=j+1}^{k}\lVert\mathscr{G}_{i}(s,\tilde{\bm{\Phi}}^{k}_{s})\rVert_{\mathcal{H}}^{2}ds.$

It is then a standard application of the Grönwall Inequality that

$\mathbbm{E}\sup_{r\in[0,t]}\lVert\tilde{\bm{\Phi}}^{k}_{r}-\tilde{\bm{\Phi}}^{j}_{r}\rVert_{\mathcal{H}}^{2}\leq
c\mathbbm{E}\int_{0}^{t}\sum_{i=j+1}^{k}\lVert\mathscr{G}_{i}(s,\tilde{\bm{\Phi}}^{k}_{s})\rVert_{\mathcal{H}}^{2}ds$
(62)

where the $c$ incorporates $e^{ct}$. Observe also that, through very similar
arguments just using the linear growth property instead of the Lipschitz one,
we have that

$\displaystyle\mathbbm{E}\sup_{r\in[0,t]}\lVert\tilde{\bm{\Phi}}^{k}_{r}\rVert_{\mathcal{H}}^{2}$
$\displaystyle\leq\mathbbm{E}\lVert\tilde{\bm{\Phi}}_{0}\rVert_{\mathcal{H}}^{2}+\mathbbm{E}\int_{0}^{t}\left(2\lVert\mathscr{A}(s,\tilde{\bm{\Phi}}^{k}_{s})\rVert_{\mathcal{H}}\lVert\tilde{\bm{\Phi}}^{k}_{s}\rVert_{\mathcal{H}}+\sum_{i=1}^{k}\lVert\mathscr{G}_{i}(s,\tilde{\bm{\Phi}}^{k}_{s})\rVert_{\mathcal{H}}^{2}\right)ds$
$\displaystyle\qquad\qquad\qquad\qquad+2\mathbbm{E}\sup_{r\in[0,t]}\left|\sum_{i=1}^{k}\int_{0}^{r}\langle\mathscr{G}_{i}(s,\tilde{\bm{\Phi}}^{k}_{s}),\tilde{\bm{\Phi}}^{k}_{s}\rangle_{\mathcal{H}}dW^{i}_{s}\right|$
$\displaystyle\leq\mathbbm{E}\lVert\tilde{\bm{\Phi}}_{0}\rVert_{\mathcal{H}}^{2}+c\mathbbm{E}\int_{0}^{t}(1+\lVert\tilde{\bm{\Phi}}^{k}_{s}\rVert_{\mathcal{H}})\lVert\tilde{\bm{\Phi}}^{k}_{s}\rVert_{\mathcal{H}}+1+\lVert\tilde{\bm{\Phi}}^{k}_{s}\rVert_{\mathcal{H}}^{2}ds$
$\displaystyle\qquad\qquad\qquad\qquad+c\mathbbm{E}\left(\int_{0}^{t}\left(1+\lVert\tilde{\bm{\Phi}}^{k}_{s}\rVert_{\mathcal{H}}^{2}\right)\lVert\tilde{\bm{\Phi}}^{k}_{s}\rVert_{\mathcal{H}}^{2}ds\right)^{\frac{1}{2}}$

to which we use that $\lVert\tilde{\bm{\Phi}}^{k}_{s}\rVert_{\mathcal{H}}\leq
1+\lVert\tilde{\bm{\Phi}}^{k}_{s}\rVert_{\mathcal{H}}^{2}$ to see that

$\displaystyle\mathbbm{E}\sup_{r\in[0,t]}\lVert\tilde{\bm{\Phi}}^{k}_{r}\rVert_{\mathcal{H}}^{2}\leq\mathbbm{E}\lVert\tilde{\bm{\Phi}}_{0}\rVert_{\mathcal{H}}^{2}+c\mathbbm{E}\int_{0}^{t}1+\lVert\tilde{\bm{\Phi}}^{k}_{s}\rVert_{\mathcal{H}}^{2}ds+c\mathbbm{E}\left(\int_{0}^{t}\left(1+\lVert\tilde{\bm{\Phi}}^{k}_{s}\rVert_{\mathcal{H}}^{2}\right)\lVert\tilde{\bm{\Phi}}^{k}_{s}\rVert_{\mathcal{H}}^{2}ds\right)^{\frac{1}{2}}$

and further

$\displaystyle\mathbbm{E}\sup_{r\in[0,t]}\lVert\tilde{\bm{\Phi}}^{k}_{r}\rVert_{\mathcal{H}}^{2}\leq
c\left[\mathbbm{E}\lVert\tilde{\bm{\Phi}}_{0}\rVert_{\mathcal{H}}^{2}+1\right]+c\mathbbm{E}\int_{0}^{t}\lVert\tilde{\bm{\Phi}}^{k}_{s}\rVert_{\mathcal{H}}^{2}ds$

as above, integrating the $1$ and adding it as a constant. Thus we have

$\mathbbm{E}\sup_{r\in[0,t]}\lVert\tilde{\bm{\Phi}}^{k}_{r}\rVert_{\mathcal{H}}^{2}\leq
c\left[\mathbbm{E}\lVert\tilde{\bm{\Phi}}_{0}\rVert_{\mathcal{H}}^{2}+1\right]=c\left[\mathbbm{E}\lVert\bm{\Phi}_{0}\rVert_{\mathcal{H}}^{2}+1\right]$

which is a bound uniform in $k$ and independent of $\tau_{R}$. Moreover for
each fixed $k$ we appreciate that the sequence of random variables

$\sup_{r\in[0,t]}\lVert\tilde{\bm{\Phi}}^{k}_{r}\rVert_{\mathcal{H}}^{2}$

is monotone increasing (indexed by $R$) and convergent to
$\sup_{r\in[0,t]}\lVert\bm{\Phi}^{k}_{r}\rVert_{\mathcal{H}}^{2}$ $a.s.$. Thus
we may apply the Monotone Convergence Theorem to this sequence of random
variables to see that

$\displaystyle\mathbbm{E}\sup_{r\in[0,t]}\lVert\bm{\Phi}^{k}_{r}\rVert_{\mathcal{H}}^{2}$
$\displaystyle=\lim_{R\rightarrow\infty}\mathbbm{E}\sup_{r\in[0,t]}\lVert\tilde{\bm{\Phi}}^{k}_{r}\rVert_{\mathcal{H}}^{2}$
$\displaystyle\leq
c\left[\mathbbm{E}\lVert\bm{\Phi}_{0}\rVert_{\mathcal{H}}^{2}+1\right].$

With this bound established we can revert back to (62), combining with the
boundedness of the $\mathscr{G}_{i}$ to deduce that

$\mathbbm{E}\int_{0}^{t}\sum_{i=j+1}^{k}\lVert\mathscr{G}_{i}(s,\bm{\Phi}^{k}_{s})\rVert_{\mathcal{H}}^{2}ds<\infty$

and clearly

$\mathbbm{E}\int_{0}^{t}\sum_{i=j+1}^{k}\lVert\mathscr{G}_{i}(s,\tilde{\bm{\Phi}}^{k}_{s})\rVert_{\mathcal{H}}^{2}ds\leq\mathbbm{E}\int_{0}^{t}\sum_{i=j+1}^{k}\lVert\mathscr{G}_{i}(s,\bm{\Phi}^{k}_{s})\rVert_{\mathcal{H}}^{2}ds$

so we can update (62) with the bound

$\mathbbm{E}\sup_{r\in[0,t]}\lVert\tilde{\bm{\Phi}}^{k}_{r}-\tilde{\bm{\Phi}}^{j}_{r}\rVert_{\mathcal{H}}^{2}\leq
c\mathbbm{E}\int_{0}^{t}\sum_{i=j+1}^{k}\lVert\mathscr{G}_{i}(s,\bm{\Phi}^{k}_{s})\rVert_{\mathcal{H}}^{2}ds$

to which we apply the same monotone convergence argument to deduce that

$\mathbbm{E}\sup_{r\in[0,t]}\lVert\bm{\Phi}^{k}_{r}-\bm{\Phi}^{j}_{r}\rVert_{\mathcal{H}}^{2}\leq
c\mathbbm{E}\int_{0}^{t}\sum_{i=j+1}^{k}\lVert\mathscr{G}_{i}(s,\bm{\Phi}^{k}_{s})\rVert_{\mathcal{H}}^{2}ds.$
(63)

Moreover

$\displaystyle\sup_{k>j}\mathbbm{E}\int_{0}^{t}\sum_{i=j+1}^{k}\lVert\mathscr{G}_{i}(s,\bm{\Phi}^{k}_{s})\rVert_{\mathcal{H}}^{2}ds$
$\displaystyle\leq
t\sup_{k>j}\mathbbm{E}\sup_{r\in[0,t]}\sum_{i=j+1}^{k}\lVert\mathscr{G}_{i}(r,\bm{\Phi}^{k}_{s})\rVert_{\mathcal{H}}^{2}ds$
$\displaystyle\leq
t\sup_{k>j}\mathbbm{E}\sup_{r\in[0,t]}\sum_{i=j+1}^{k}C_{t}c_{i}(1+\lVert\bm{\Phi}^{k}_{s}\rVert_{\mathcal{H}}^{2})ds$
$\displaystyle:=c\sup_{k>j}\mathbbm{E}\sup_{r\in[0,t]}\sum_{i=j+1}^{k}c_{i}(1+\lVert\bm{\Phi}^{k}_{s}\rVert_{\mathcal{H}}^{2})ds$
$\displaystyle=c\sup_{k>j}\sum_{i=j+1}^{k}c_{i}\mathbbm{E}\sup_{r\in[0,t]}(1+\lVert\bm{\Phi}^{k}_{s}\rVert_{\mathcal{H}}^{2})ds$
$\displaystyle\leq
c\sum_{i=j+1}^{\infty}c_{i}\sup_{k>j}\mathbbm{E}\sup_{r\in[0,t]}(1+\lVert\bm{\Phi}^{k}_{s}\rVert_{\mathcal{H}}^{2})ds$
$\displaystyle\leq
c\sum_{i=j+1}^{\infty}c_{i}\left(c\left[\mathbbm{E}\lVert\bm{\Phi}_{0}\rVert_{\mathcal{H}}^{2}+1\right]\right)$

which is a sequence in $j$ monotone decreasing to zero. Thus in view of (63),

$\lim_{j\rightarrow\infty}\sup_{k>j}\left[\mathbbm{E}\sup_{r\in[0,t]}\lVert\bm{\Phi}^{k}_{r}-\bm{\Phi}^{j}_{r}\rVert_{\mathcal{H}}^{2}\right]=0$

so the sequence $(\bm{\Phi}^{k})$ is Cauchy in
$L^{2}\left(\Omega;C\left([0,t];\mathcal{H}\right)\right)$ and as such we can
deduce the existence of a $\bm{\Phi}$ such that
$\bm{\Phi}^{k}\rightarrow\bm{\Phi}$ in this space (and hence, in
$L^{2}\left(\Omega;L^{2}\left([0,t];\mathcal{H}\right)\right))$ for every
$t\in[0,\infty)$, and thus $\bm{\Phi}$ is also the $\mathbbm{P}-a.s.$ limit of
a subsequence of the $(\bm{\Phi}^{k})$ in $C\left([0,t];\mathcal{H}\right)$.
This limit process inherits the progressive measurability (indeed it is
trivially adapted and has continuous paths in $\mathcal{H}$). It simply
remains to show that $\bm{\Phi}$ satisfies the identity (61), so we first
consider the $\mathbbm{P}-a.s.$ convergent subsequence $(\bm{\Phi}^{k_{l}})$.
Looking at the stochastic integral, we have that

$\displaystyle\mathbbm{E}\left\|\int_{0}^{t}\mathscr{G}(s,\bm{\Phi}_{s})d\mathcal{W}_{s}-\int_{0}^{t}\mathscr{G}^{k_{l}}(s,\bm{\Phi}^{k_{l}}_{s})d\mathcal{W}_{s}\right\|_{\mathcal{H}}^{2}$
$\displaystyle=\mathbbm{E}\left\|\int_{0}^{t}\mathscr{G}(s,\bm{\Phi}_{s})-\mathscr{G}^{k_{l}}(s,\bm{\Phi}^{k_{l}}_{s})d\mathcal{W}_{s}\right\|_{\mathcal{H}}^{2}$
$\displaystyle=\lim_{N\rightarrow\infty}\mathbbm{E}\left\|\sum_{i=1}^{N}\int_{0}^{t}\mathscr{G}_{i}(s,\bm{\Phi}_{s})-\mathscr{G}^{k_{l}}_{i}(s,\bm{\Phi}^{k_{l}}_{s})dW^{i}_{s}\right\|_{\mathcal{H}}^{2}$
$\displaystyle=\lim_{N\rightarrow\infty}\mathbbm{E}\int_{0}^{t}\sum_{i=1}^{N}\lVert\mathscr{G}_{i}(s,\bm{\Phi}_{s})-\mathscr{G}^{k_{l}}_{i}(s,\bm{\Phi}^{k_{l}}_{s})\rVert_{\mathcal{H}}^{2}ds$
$\displaystyle\leq\mathbbm{E}\int_{0}^{t}\sum_{i=1}^{\infty}\lVert\mathscr{G}_{i}(s,\bm{\Phi}_{s})-\mathscr{G}^{k_{l}}_{i}(s,\bm{\Phi}^{k_{l}}_{s})\rVert_{\mathcal{H}}^{2}ds$
$\displaystyle\leq\sum_{i=k_{l}+1}^{\infty}\mathbbm{E}\int_{0}^{t}c_{i}\lVert\bm{\Phi}_{s}-\bm{\Phi}^{k_{l}}_{s}\rVert_{\mathcal{H}}^{2}ds$
$\displaystyle\leq
c\mathbbm{E}\int_{0}^{t}\lVert\bm{\Phi}_{s}-\bm{\Phi}^{k_{l}}_{s}\rVert_{\mathcal{H}}^{2}ds$

so from the known
$L^{2}\left(\Omega;L^{2}\left([0,t];\mathcal{H}\right)\right)$ convergence, we
have that

$\lim_{k_{l}\rightarrow\infty}\int_{0}^{t}\mathscr{G}^{k_{l}}(s,\bm{\Phi}^{k_{l}}_{s})d\mathcal{W}_{s}=\int_{0}^{t}\mathscr{G}(s,\bm{\Phi}_{s})d\mathcal{W}_{s}$

with the limit in $L^{2}\left(\Omega;\mathcal{H}\right)$. We can thus extract
a further subsequence which we denote $(\bm{\Phi}^{k_{m}})$ such that this
limit holds $\mathbbm{P}-a.s.$ in $\mathcal{H}$, and is of course still such
that $(\bm{\Phi}^{k_{m}})\rightarrow\bm{\Phi}$ $\mathbbm{P}-a.s.$ in
$L^{2}\left([0,t];\mathcal{H}\right)$ (again a trivial consequence of the
$C\left([0,t];\mathcal{H}\right)$ convergence). Therefore for
$\mathbbm{P}-a.e.$ $\omega$,

$\displaystyle\left\|\int_{0}^{t}\mathscr{A}(s,\bm{\Phi}_{s}(\omega))ds-\int_{0}^{t}\mathscr{A}(s,\bm{\Phi}^{k_{m}}_{s}(\omega))ds\right\|_{\mathcal{H}}^{2}$
$\displaystyle\leq\int_{0}^{t}\lVert\mathscr{A}(s,\bm{\Phi}_{s}(\omega))-\mathscr{A}(s,\bm{\Phi}^{k_{m}}_{s}(\omega))\rVert_{\mathcal{H}}^{2}ds$
$\displaystyle\leq
c\int_{0}^{t}\lVert\bm{\Phi}_{s}-\bm{\Phi}^{k_{m}}_{s}\rVert_{\mathcal{H}}^{2}ds$

and so

$\lim_{k_{m}\rightarrow\infty}\int_{0}^{t}\mathscr{A}(s,\bm{\Phi}^{k_{m}}_{s}(\omega))ds=\int_{0}^{t}\mathscr{A}(s,\bm{\Phi}_{s}(\omega))ds$

with the limit in $\mathcal{H}$. Thus by taking the $\mathbbm{P}-a.s.$ limit
in $\mathcal{H}$ of the identity satisfied by $\bm{\Phi}^{k_{m}}$, we reach
(61) as required.

∎

######  Theorem 2.8.2.

Suppose $\bm{\Psi}$ is another strong solution of (61). Then for all
$t\in[0,\infty)$,

$\mathbbm{P}\left(\left\\{\omega\in\Omega:\bm{\Phi}_{t}(\omega)=\bm{\Psi}_{t}(\omega)\right\\}\right)=1.$

###### Proof.

The method of proof here is entirely contained in that for the existence just
seen. Indeed we look at the energy equality satisfied by the difference of the
solutions, which is

$\displaystyle\lVert\bm{\Phi}_{r}-\bm{\Psi}_{r}\rVert_{\mathcal{H}}^{2}$
$\displaystyle=2\int_{0}^{r}\langle\mathscr{A}(s,\bm{\Phi}_{s})-\mathscr{A}(s,\bm{\Psi}_{s}),\bm{\Phi}_{s}-\bm{\Psi}_{s}\rangle_{\mathcal{H}}ds$
$\displaystyle+\int_{0}^{r}\lVert\mathscr{G}(s,\bm{\Phi}_{s})-\mathscr{G}(s,\bm{\Psi}_{s})\rVert^{2}_{\mathscr{L}^{2}(\mathfrak{U};\mathcal{H})}ds+2\int_{0}^{r}\langle\mathscr{G}(s,\bm{\Phi}_{s})-\mathscr{G}(s,\bm{\Psi}_{s}),\bm{\Phi}_{s}-\bm{\Psi}_{s}\rangle_{\mathcal{H}}d\mathcal{W}_{s}.$

Following along the proof, we introduce

$\tau_{R}:=R\wedge\inf\\{s\geq
0:\max\\{\lVert\bm{\Phi}_{s}\rVert_{\mathcal{H}},\lVert\bm{\Psi}_{s}\rVert_{\mathcal{H}}\\}\geq
R\\}$

and the process defined for any fixed $R$ by

$\tilde{\bm{\Phi}}_{s}:=\bm{\Phi}_{s}\mathbbm{1}_{s\leq\tau_{R}},\qquad\tilde{\bm{\Psi}}_{s}:=\bm{\Psi}_{s}\mathbbm{1}_{s\leq\tau_{R}}.$

In this case we have the inequality

$\displaystyle\sup_{r\in[0,t]}\lVert\tilde{\bm{\Phi}}_{r}-\tilde{\bm{\Psi}}_{r}\rVert_{\mathcal{H}}^{2}\leq
c\int_{0}^{t}\lVert\tilde{\bm{\Phi}}_{s}-\tilde{\bm{\Psi}}_{s}\rVert_{\mathcal{H}}^{2}ds+2\sup_{r\in[0,t]}\left|\int_{0}^{r}\langle\mathscr{G}(s,\tilde{\bm{\Phi}}_{s})-\mathscr{G}(s,\tilde{\bm{\Psi}}_{s}),\tilde{\bm{\Phi}}_{s}-\tilde{\bm{\Psi}}_{s}\rangle_{\mathcal{H}}d\mathcal{W}_{s}\right|$

so following all of the same steps, simply now without the
$\sum_{i=j+1}^{k}\lVert\mathscr{G}_{i}(s,\tilde{\bm{\Phi}}^{k}_{s})\rVert^{2}_{\mathcal{H}}$
term, we deduce again that

$\mathbbm{E}\sup_{r\in[0,t]}\lVert\tilde{\bm{\Phi}}_{r}-\tilde{\bm{\Psi}}_{r}\rVert_{\mathcal{H}}^{2}\leq
0$

in analogy with (62). By the same monotone convergence argument, we have that

$\mathbbm{E}\sup_{r\in[0,t]}\lVert\bm{\Phi}_{r}-\bm{\Psi}_{r}\rVert_{\mathcal{H}}^{2}=0$

which gives the result.

∎

###  2.9 Applications

We have alluded quite heavily to the application of this framework for SALT
[4] derived SPDEs, which is done for now in [1] and to be expanded upon in
[2,3]. In [1] we establish an abstract solution method in the context of the
Hilbert Spaces

$V\xhookrightarrow{}H\xhookrightarrow{}U\xhookrightarrow{}X$

posing additional assumptions on the operators to the ones given in 2.4.1,
2.4.2. In particular this is applied to the Stratonovich SPDE

$u_{t}-u_{0}+\int_{0}^{t}\mathcal{L}_{u_{s}}u_{s}\,ds-\int_{0}^{t}\Delta
u_{s}\,ds+\int_{0}^{t}Bu_{s}\circ d\mathcal{W}_{s}+\nabla\rho_{t}=0$ (64)

where $B$ here is a differential (transport) operator and $\mathcal{L}$ is the
usual fluids non-linear term. Clearly this is a highly non-trivial SPDE, but
we show that this fits our framework for (43) to swiftly convert into the Itô
Form for which we then apply the general existence and uniqueness arguments in
the paper. The notion of $V-$valued solution in [1] is thus what corresponds
precisely and rigorously to a solution of the Stratonovich form as laid out in
this text. Indeed the general existence argument also used the result 2.8.1.

We hope that this very general framework can facilitate a much easier analysis
of many SPDEs in the future.

## References

  * 1 Goodair, D., 2022. Existence and Uniqueness of Maximal Solutions to a 3D Stochastic Navier-Stokes Equation on a Bounded Domain. arXiv preprint arXiv:2202.09242.
  * 2 Goodair, D., Crisan, D., Lang, O., Mensah, P.R., 2022. Existence and Uniqueness of Maximal Strong Solutions to Nonlinear SPDEs with Applications to Viscous Fluid Models. In preparation.
  * 3 Goodair, D., Crisan, D., 2022. Analytical Properties of a 3D Stochastic Navier-Stokes Equation on a Bounded Domain. In preparation.
  * 4 Holm, D.D., 2015. Variational principles for stochastic fluid dynamics. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 471(2176), p.20140963.
  * 5 Coffey, W. and Kalmykov, Y.P., 2012. The Langevin equation: with applications to stochastic problems in physics, chemistry and electrical engineering (Vol. 27). World Scientific.
  * 6 Black, F. and Scholes, M., 2019. The pricing of options and corporate liabilities. In World Scientific Reference on Contingent Claims Analysis in Corporate Finance: Volume 1: Foundations of CCA and Equity Valuation (pp. 3-21).
  * 7 Da Prato, G. and Zabczyk, J., 2014. Stochastic equations in infinite dimensions. Cambridge university press.
  * 8 Lototsky, S.V. and Rozovsky, B.L., 2017. Stochastic partial differential equations. New York: Springer.
  * 9 Karatzas, I. and Shreve, S., 2012. Brownian motion and stochastic calculus (Vol. 113). Springer Science and Business Media.
  * 10 Øksendal, B., 2003. Stochastic differential equations. In Stochastic differential equations (pp. 65-84). Springer, Berlin, Heidelberg.
  * 11 Temam, R., 1974. On the Euler equations of incompressible perfect fluids. Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi” Séminaire Goulaouic-Schwartz”, pp.1-14.
  * 12 Robinson, J.C., Rodrigo, J.L. and Sadowski, W., 2016. The three-dimensional Navier–Stokes equations: Classical theory (Vol. 157). Cambridge university press.
  * 13 Liu, W. and Röckner, M., 2015. Stochastic partial differential equations: an introduction (p. 272). Cham: Springer.