Secondly, using the mild Itô representation (3.115) we obtain

$\displaystyle\mathcal{H}_{I;Bb}$
$\displaystyle=\sum_{k=0}^{\infty}\sum_{j=j_{\delta}}^{N}e^{-\varepsilon(T-r_{j})}\int_{r_{j-1}}^{r_{j}}\langle
E^{\delta}(r_{j},r)B^{j}(r)\sqrt{Q}e_{k},E^{\delta}(r_{j},r)B^{j}(r)\sqrt{Q}e_{k}\rangle_{H^{k}}\,\mathrm{d}r$
(3.123)
$\displaystyle\qquad\quad+2\sum_{j=j_{\delta}}^{N}e^{-\varepsilon(T-r_{j})}\int_{r_{j-1}}^{r_{j}}\langle
E^{\delta}(r_{j},r)Y_{\delta}^{j}(r),E^{\delta}(r_{j},r)B^{j}(r)\,\mathrm{d}W_{r}^{Q}\rangle_{H^{k}}.$

Both terms can be treated using Lemma 3.14 again, yet now with
$G(r)=G_{I;Bb}(r)$, where

$G_{I;Bb}(r)=Y_{\delta}^{j}(r)=Z_{\delta}(r)-E^{\delta}(r,r_{j-1})Z_{\delta}(r_{j-1}),\quad
r_{j-1}\leq r\leq r_{j}.$ (3.124)

Indeed, the latter satisfies the estimate

$\displaystyle e^{-\varepsilon(T-r)}\|G_{I;Bb}(r)\|_{H^{k}}^{2}$
$\displaystyle\leq
2e^{-\varepsilon(T-r)}\|Z_{\delta}(r)\|_{H^{k}}^{2}+2Me^{\varepsilon\delta}e^{-\varepsilon(T-r_{j-1})}\|Z_{\delta}(r_{j-1})\|_{H^{k}}^{2}$
(3.125) $\displaystyle\leq 2(Me^{\varepsilon\delta}+1)\sup_{0\leq t\leq
T}e^{-\varepsilon(T-t)}\|Z_{\delta}(t)\|_{H^{k}}^{2}.$

This completes the proof. ∎

######  Lemma 3.16.

Consider the setting of Proposition 3.11 together with the partition $\pi$.
Then there exists a constant $K>0$ that does not depend on $T$ so that for any
integer $p\geq 1$ we have the estimate

$\displaystyle\mathbb{E}[\mathcal{H}_{II}]^{p}$ $\displaystyle\leq
p^{p/2}K^{p}\mathbb{E}\sup_{0\leq t\leq
T}\Big{[}e^{-\varepsilon(T-t)}\|Z_{\delta}(t)\|_{H^{k}}^{2}\Big{]}^{p}$
(3.126)
$\displaystyle\qquad+p^{p/2}K^{p}\mathbb{E}\Big{[}\int_{0}^{T}e^{-\varepsilon(T-r)}\|B(r)\|_{HS(\mathcal{W}_{Q};H^{k})}^{2}\,\mathrm{d}r\Big{]}^{p}.$

###### Proof.

We first note that

$\mathcal{H}_{II}=\sum_{j=j_{\delta}}^{N}\int_{r_{j-1}}^{r_{j}}e^{-\varepsilon(T-s)}\int_{0}^{s}\langle
E^{\delta}(s,r_{j-1})Z_{\delta}(r_{j-1}),E^{\delta}(s,r)B^{j}(r)\mathrm{d}W_{r}^{Q}\rangle_{s;\delta;k+1}\mathrm{d}s.$
(3.127)

Reversing the order of integration yields

$\displaystyle\mathcal{H}_{II}$
$\displaystyle=\sum_{j=j_{\delta}}^{N}\int_{r_{j-1}}^{r_{j}}\int_{r}^{r_{j}}\
e^{-\varepsilon(T-s)}\langle
E^{\delta}(s,r_{j-1})Z_{\delta}(r_{j-1}),E^{\delta}(s,r)B^{j}(r)[\,\cdot\,]\rangle_{s;\delta;k+1}\mathrm{d}s\,\mathrm{d}W_{r}^{Q}$
(3.128)
$\displaystyle=\sum_{j=j_{\delta}}^{N}\int_{r_{j-1}}^{r_{j}}\mathcal{J}^{\delta}({r_{j},r})\big{[}E^{\delta}(r,r_{j-1})Z_{\delta}(r_{j-1}),B^{j}(r)[\,\cdot\,]\big{]}\mathrm{d}W_{r}^{Q}.$

The bound follows by applying Lemma 3.14 with $G(r)$ as in (3.121). ∎

######  Lemma 3.17.

Consider the setting of Proposition 3.11 together with the partition $\pi$.
Then there exists a constant $K>0$ that does not depend on $T$ so that for any
integer $p\geq 1$ we have the estimate

$\displaystyle\mathbb{E}[\mathcal{H}_{III}]^{p}$ $\displaystyle\leq
p^{p/2}K^{p}\mathbb{E}\sup_{0\leq t\leq
T}\Big{[}e^{-\varepsilon(T-t)}\|Z_{\delta}(t)\|_{H^{k}}^{2}\Big{]}^{p}$
(3.129)
$\displaystyle\qquad+p^{p/2}K^{p}\mathbb{E}\Big{[}\int_{0}^{T}e^{-\varepsilon(T-r)}\|B(r)\|_{HS(\mathcal{W}_{Q};H^{k})}^{2}\,\mathrm{d}r\Big{]}^{p}.$

###### Proof.

Applying the mild Itô representation (3.115), we see that

$\displaystyle\mathcal{H}_{III}$
$\displaystyle=\sum_{k=0}^{\infty}\sum_{j=j_{\delta}}^{N}\int_{r_{j-1}}^{r_{j}}e^{-\varepsilon(t-s)}\int_{r_{j-1}}^{s}\langle
E^{\delta}(s,r)B^{j}(r)\sqrt{Q}e_{k},E^{\delta}(s,r)B^{j}(r)\sqrt{Q}e_{k}\rangle_{s;\delta;k+1}\mathrm{d}r\,\mathrm{d}s$
(3.130)
$\displaystyle\qquad+2\sum_{j=j_{\delta}}^{N}\int_{r_{j-1}}^{r_{j}}e^{-\varepsilon(t-s)}\int_{r_{j-1}}^{s}\langle
E^{\delta}(s,r)Y_{\delta}^{j}(r),E^{\delta}(s,r)B^{j}(r)\mathrm{d}W_{r}^{Q}\rangle_{s;\delta;k+1}\,\mathrm{d}s.$

Reversing the order of integration gives us

$\displaystyle\mathcal{H}_{III}$
$\displaystyle=\sum_{k=0}^{\infty}\sum_{j=j_{\delta}}^{N}\int_{r_{j-1}}^{r_{j}}\int_{r}^{r_{j}}e^{-\varepsilon(t-s)}\langle
E^{\delta}(s,r)B^{j}(r)\sqrt{Q}e_{k},E^{\delta}(s,r)B^{j}(r)\sqrt{Q}e_{k}\rangle_{s;\delta;k+1}\,\mathrm{d}s\,\mathrm{d}r$
(3.131)
$\displaystyle\qquad+2\sum_{j=j_{\delta}}^{N}\int_{r_{j-1}}^{r_{j}}\int_{r}^{r_{j}}e^{-\varepsilon(t-s)}\langle
E^{\delta}(s,r)Y_{\delta}^{j}(r),E^{\delta}(s,r)B^{j}(r)[\,\cdot\,]\rangle_{s;\delta;k+1}\,\mathrm{d}s\,\mathrm{d}W_{r}^{Q},$

which can hence be written as

$\displaystyle\mathcal{H}_{III}$
$\displaystyle=\sum_{k=0}^{\infty}\sum_{j=j_{\delta}}^{N}\int_{r_{j-1}}^{r_{j}}\mathcal{J}^{\delta}(r_{j},r)[B^{j}(r)\sqrt{Q}e_{k},B^{j}(r)\sqrt{Q}e_{k}]\,\mathrm{d}r$
(3.132)
$\displaystyle\qquad\qquad\qquad\qquad\qquad+2\sum_{j=j_{\delta}}^{N}\int_{r_{j-1}}^{r_{j}}\mathcal{J}^{\delta}(r_{j},r)\big{[}Z^{j}(r),B^{j}(r)[\,\cdot\,]\big{]}\,\mathrm{d}W_{r}^{Q}.$

The bound now follows by appealing to Lemma 3.14, with $G(r)$ as in (3.124). ∎

###### Proof of Proposition 3.11.

We first consider adapted finite-rank step processes $B$ that take values in
$HS(\mathcal{W}_{Q};H^{k+3})$. Following step 3 in the proof of [70, Thm.
6.4], the representation (3.88) can be used to show that
$Z_{\delta}(\cdot,\omega)\to\mathcal{Z}^{\perp}[B](\cdot,\omega)$ holds in
$C([0,T];H^{k+1})$, as $\delta\downarrow 0$, for almost any $\omega\in\Omega$.
Moreover, for these $\omega\in\Omega$ we have the convergence

$I_{\delta}(\omega)\to\mathcal{I}^{\perp}_{B}(\omega),$ (3.133)

as $\delta\downarrow 0,$ since we can find a constant $K_{\omega}>0$ so that
$\sup_{0\leq t\leq T}\|Z_{\delta}(t,\omega)\|_{H^{k+1}}\leq K_{\omega}$ holds
for all small $\delta>0$.

Applying Fatou’s lemma, we see that

$\mathbb{E}[\mathcal{I}^{\perp}_{B}]^{p}\leq\liminf_{\delta\downarrow
0}\mathbb{E}[I_{\delta}]^{p}.$ (3.134)

Clearly, the convergence of $Z_{\delta}$ to $\mathcal{Z}^{\perp}[B]$ also
holds $\mathbb{P}$-a.s. in $C([0,T];H^{k})$. Recalling the bound (3.89), where
the right hand side is contained in $L^{2p}(\Omega)$, we may apply the
dominated convergence theorem to show that

$\mathbb{E}\sup_{0\leq t\leq
T}\|Z_{\delta}(t)\|^{2p}_{H^{k}}\to\mathbb{E}\sup_{0\leq t\leq
T}\|\mathcal{Z}^{\perp}[B](t)\|_{H^{k}}^{2p},$ (3.135)

as $\delta\downarrow 0$, together with

$\mathbb{E}\sup_{0\leq t\leq
T}\big{[}e^{-\varepsilon(T-t)}\|Z_{\delta}(t)\|^{2}_{H^{k}}\big{]}^{p}\to\mathbb{E}\sup_{0\leq
t\leq
T}\big{[}e^{-\varepsilon(T-t)}\|\mathcal{Z}^{\perp}[B](t)\|_{H^{k}}^{2}\big{]}^{p}.$
(3.136)

The desired bound follows by appealing to (3.71) and combining Lemma’s 3.13,
3.15, 3.16 and 3.17. In a standard fashion, the result can be extended to
general processes $B$ satisfying (3.83), by means of Corollary 3.8. ∎

###  3.5 Bounding suprema with forward integrals logarithmically in time

In this section, we examine how fast the unweighted supremum over $[0,T]$ of
the convolution $\mathcal{Z}^{\perp}[B]$ grows when increasing $T$. For
instance, if we assume that $B$ is constant and take $\varepsilon=0$ in
(3.72), we obtain a prediction of order $T^{p}$. Our main result here states
that this can be improved to yield growth estimates for the expressions

$\mathcal{Z}^{\perp}[B](t)=\int_{0}^{t}E(t,s)P^{\perp}B(s)\mathrm{d}W_{s}^{-}\quad\text{and}\quad\mathcal{I}^{\perp}[B](t)=\int_{0}^{t}e^{-\varepsilon(t-s)}\|\mathcal{Z}^{\perp}[B](s)\|_{H^{k+1}}^{2}\mathrm{d}s$
(3.137)

that are logarithmic in $T$. This will be achieved by imposing the following a
priori pathwise bounds on $B$, which will arise in a natural fashion during
our stability analysis in §7.

  * (HB)

For the process
$B\in\mathcal{N}^{2}\big{(}[0,T];\mathbb{F};HS(\mathcal{W}_{Q};H^{k})\big{)}$
there exists a constant $\Theta_{*}>0$ such that the pathwise bounds

$\int_{0}^{t}e^{-\varepsilon(t-s)}\|B(s)\|_{HS(\mathcal{W}_{Q};H^{k})}^{2}\mathrm{d}s\leq\Theta_{*}^{2}\quad\text{and}\quad\|E_{\rm
tw}(1)B(t)\|_{HS(\mathcal{W}_{Q};H^{k})}^{2}\leq\Theta_{*}^{2}$ (3.138)

hold for all $0\leq t\leq T$.

Observe that (3.138) in (HB) implies that
$B\in\mathcal{N}^{p}\big{(}[0,T];\mathbb{F};HS(\mathcal{W}_{Q};H^{k})\big{)}$
for all $p\geq 2.$ We recall that the constant
$\mu=\min\\{\beta,\lambda_{1}k_{\nu}\\}>0$ is defined for the first time in
Lemma 3.2.

######  Proposition 3.18.

Suppose that (HE) holds. Assume that $T\geq 2$ is an integer and pick
$\varepsilon\in(0,2\mu)$. Then there exists a constant $K_{\rm gr}>0$ that
does not depend on $T$ so that for any process $B$ that satisfies (HB), and
every integer $p\geq 1$, we have the growth bound

$\mathbb{E}\sup_{0\leq t\leq
T}\left\|\mathcal{Z}^{\perp}[B](t)\right\|_{H^{k}(\mathcal{W}_{Q};H^{k})}^{2p}+\mathbb{E}\sup_{0\leq
t\leq T}\mathcal{I}^{\perp}[B](t)^{p}\leq K^{2p}_{\rm
gr}\Theta_{*}^{2p}(p^{p}+\log(T)^{p}).$ (3.139)

We will primarily follow the approach of [29, Sec. 3.1], but the forward
integrals require subtle modifications to the procedure at several points,
which we highlight below. In addition, our bound for $\mathcal{I}^{\perp}[B]$
is sharper than the related bound in [29] due to Proposition 3.11. The
approach relies on integral splittings, which are only available for finite-
rank processes. We will therefore first impose the following condition and use
a limiting argument in the final step to lift the restriction.

  * (HB*)

The process $B$ satisfies condition (HB) and, in addition, is a finite-rank
process that takes values in $HS(\mathcal{W}_{Q};H^{k+2})$.

Under this assumption, we can use (3.49) to make the splitting

$\mathcal{Z}^{\perp}[B](t)=\mathcal{E}_{B}^{\mathrm{lt}}(t)+\mathcal{E}_{B}^{\mathrm{sh}}(t),$
(3.140)

as in [29]. Here the long time (lt) and short time (sh) contributions are
given by

$\mathcal{E}_{B}^{\mathrm{lt}}(t)=\int_{0}^{t-1}E(t,s)P^{\perp}B(s)\mathrm{d}W_{s}^{-}\quad\text{and}\quad\mathcal{E}_{B}^{\mathrm{sh}}(t)=\int_{t-1}^{t}E(t,s)P^{\perp}B(s)\mathrm{d}W_{s}^{-},$
(3.141)

respectively, where we interpret the boundary $t-1$ as $\max\\{t-1,0\\}$ if
necessary. Both these terms need to be handled using separate techniques.

######  Lemma 3.19 (short-term bound).

Consider the setting of Proposition 3.18. Then there exists a constant $K_{\rm
sh}>0$ that does not depend on $T$ so that for any process $B$ that satisfies
(HB*), and every integer $p\geq 1$, we have the supremum bound

$\mathbb{E}\sup_{0\leq t\leq T}\|\mathcal{E}_{B}^{\rm
sh}(t)\|_{H^{k}}^{2p}\leq(p^{p}+\log(T)^{p})K_{\rm sh}^{2p}\Theta_{*}^{2p}.$
(3.142)

###### Proof.

The result is obtained by generalising [29, Lem. 3.4] in a completely
analogous way, followed by using the estimate on maximum expectations in
either Corollary B.3 or [29, Cor. 2.4]. ∎

For any pair $0\leq t_{1}\leq t_{2}\leq T$, we split the long term increment
into two parts:

$\mathcal{E}_{B}^{\mathrm{lt}}(t_{1})-\mathcal{E}_{B}^{\mathrm{lt}}(t_{2})=\mathcal{I}_{1}(t_{1},t_{2})+\mathcal{I}_{2}(t_{1},t_{2}),$
(3.143)

where

$\displaystyle\mathcal{I}_{1}(t_{1},t_{2})=\int_{0}^{t_{1}-1}[E(t_{2},s)-E(t_{1},s)]P^{\perp}B(s)\mathrm{d}W_{s}^{-},$
(3.144)
$\displaystyle\mathcal{I}_{2}(t_{1},t_{2})=\int_{t_{1}-1}^{t_{2}-1}E(t_{2},s)P^{\perp}B(s)\mathrm{d}W_{s}^{-}.$
(3.145)

Note that this is in line with [29]. The first integral can be analysed by
exploiting the regularity of the random evolution family $E(t,s)$ for $t-s\geq
1$. The second integral requires a supremum bound on the smoothened process
$E_{\rm tw}(1)B$, which motivates the condition in (3.138).

######  Lemma 3.20.

Consider the setting of Proposition 3.18 and assume that (HB*) holds. Then for
any $1\leq t_{1}\leq t_{2}\leq T$ and any integer $p\geq 1$ we have the bound

$\mathbb{E}\left\|\mathcal{I}_{1}(t_{1},t_{2})\right\|_{H^{k}}^{2p}\leq
p^{p}e^{\varepsilon
p}K_{\mathrm{dc}}^{2p}M^{4p}\Theta_{*}^{2p}\left|t_{2}-t_{1}\right|^{2p}.$
(3.146)

###### Proof.

Observe first that by Cauchy-Schwarz, we have

$\displaystyle\mathbb{E}\left\|\mathcal{I}_{1}\left(t_{1},t_{2}\right)\right\|_{H^{k}}^{2p}$
$\displaystyle\leq\mathbb{E}\left[\left\|\left[E\left(t_{2},t_{1}\right)-I\right]E(t_{1},t_{1}-1)\right\|_{\mathscr{L}\left(H^{k}\right)}^{2p}\left\|\int_{0}^{t_{1}-1}E(t_{1}-1,s)P^{\perp}B(s)\mathrm{d}W_{s}^{-}\right\|_{H^{k}}^{2p}\right]$
$\displaystyle\leq
M^{2p}\left|t_{2}-t_{1}\right|^{2p}\mathbb{E}\left\|\int_{0}^{t_{1}-1}E\left(t_{1}-1,s\right)P^{\perp}B(s)\mathrm{d}W_{s}^{-}\right\|_{H^{k}}^{2p}$
(3.147) $\displaystyle\leq
M^{4p}\left|t_{2}-t_{1}\right|^{2p}p^{p}K_{\mathrm{dc}}^{2p}\mathbb{E}\left[\int_{0}^{t_{1}-1}e^{-\varepsilon\left(t_{1}-1-s\right)}\|B(s)\|_{HS(\mathcal{W}_{Q};H^{k})}^{2}\mathrm{d}s\right]^{p}.$

The second inequality exploits the fact that
$\|(E(t,r)-I)E(r,r-1)\|_{\mathscr{L}(H^{k})}\leq M|t-r|$ holds, which is a
generalised version of the final estimate in [29, Lem 2.1]. The ultimate
inequality follows from (3.72) in Proposition 3.10. The assertion follows in
view of (3.138). ∎

We remind the reader that $E(t,s)=E(t,s,\omega)$ depends on $\omega\in\Omega.$
Consequently, the proof of the lemma below is slightly different as compared
to [29, Lem 3.5]. In particular, we need the expanded decomposition (3.22).

######  Lemma 3.21.

Consider the setting of Proposition 3.18 and assume that (HB*) holds. Then for
any $1\leq t_{1}\leq t_{2}\leq T$ and any integer $p\geq 1$ we have the bound

$\mathbb{E}\left\|\mathcal{I}_{2}(t_{1},t_{2}\right)\|_{H^{k}}^{2p}\leq
p^{p}K_{\mathrm{dc}}^{2p}\|P^{\perp}\|_{\mathscr{L}(H^{k})}^{2p}M^{2p}\Theta_{*}^{2p}\left|t_{2}-t_{1}\right|^{p}.$
(3.148)

###### Proof.

An application of Cauchy-Schwarz yields

$\displaystyle\mathbb{E}\|$
$\displaystyle\mathcal{I}_{2}(t_{1},t_{2})\|_{H^{k}}^{2p}$ (3.149)
$\displaystyle\leq\|P^{\perp}\|_{\mathscr{L}(H^{k})}^{2p}\mathbb{E}\left[\left\|E(t_{2},t_{2}-1)\int_{t_{1}-1}^{t_{2}-1}E(t_{2}-1,s)B(s)\mathrm{d}W_{s}^{-}\right\|_{H^{k}}\right]^{2p}$
$\displaystyle\leq\|P^{\perp}\|_{\mathscr{L}(H^{k})}^{2p}\mathbb{E}\|H(t_{2},t_{2}-1)\|_{\mathscr{L}(H^{k})}^{2p}\mathbb{E}\left[\left\|\int_{t_{1}-1}^{t_{2}-1}E(t_{2}-1,s)E_{\rm
tw}(1)B(s)\mathrm{d}W_{s}^{-}\right\|_{H^{k}}\right]^{2p}$
$\displaystyle\leq\|P^{\perp}\|_{\mathscr{L}(H^{k})}^{2p}M^{2p}p^{p}K_{\mathrm{dc}}^{2p}\left|t_{2}-t_{1}\right|^{p}\mathbb{E}\left[\sup_{t_{1}-1\leq
s\leq t_{2}-1}\|E_{\rm
tw}(1)B(s)\|_{HS(\mathcal{W}_{Q};H^{k})}^{2}\right]^{p}.$

By applying (3.138), we hence arrive at the stated bound.∎

The previous two results give a handle on small increments
$\left|t_{2}-t_{1}\right|\leq 1$. For larger increments, one simply exploits
the decay of the random evolution family.

######  Lemma 3.22.

Consider the setting of Proposition 3.18 and assume that (HB*) holds. Then for
any $0\leq t\leq T$ and any integer $p\geq 1$ we have the bound

$\mathbb{E}\left\|\mathcal{E}_{B}^{\mathrm{lt}}(t)\right\|_{H^{k}}^{2p}\leq
p^{p}e^{\varepsilon p}K_{\mathrm{dc}}^{2p}M^{2p}\Theta_{*}^{2p}.$ (3.150)

###### Proof.

Using the decay estimate (3.72) in Proposition 3.10, it follows that

$\displaystyle\mathbb{E}\left\|\mathcal{E}_{B}^{\mathrm{lt}}(t)\right\|_{H^{k}}^{2p}$
$\displaystyle\leq\|E(t,t-1)\|_{\mathscr{L}(H^{k})}^{2p}\mathbb{E}\left\|\int_{0}^{t-1}E(t-1,s)P^{\perp}B(s)\mathrm{d}W_{s}^{-}\right\|_{H^{k}}$
(3.151) $\displaystyle\leq
M^{2p}p^{p}K_{\mathrm{dc}}^{2p}\mathbb{E}\left[\int_{0}^{t-1}e^{-\varepsilon(t-1-s)}\|B(s)\|_{HS(\mathcal{W}_{Q};H^{k})}^{2}\mathrm{d}s\right]^{p}.$

The assertion again follows from (3.138). ∎

######  Corollary 3.23.

Consider the setting of Proposition 3.18 and assume that (HB*) holds. Then for
any $0\leq t_{1}\leq t_{2}\leq T$ and any integer $p\geq 1$ we have the bound

$\mathbb{E}\left\|\mathcal{E}_{B}^{\mathrm{lt}}\left(t_{1}\right)-\mathcal{E}_{B}^{\mathrm{lt}}\left(t_{2}\right)\right\|_{H^{k}}^{2p}\leq
2^{2p}p^{p}e^{\varepsilon
p}K_{\mathrm{dc}}^{2p}\|P^{\perp}\|_{\mathscr{L}(H^{k})}^{2p}M^{4p}\Theta_{*}^{2p}\min\left\\{\left|t_{2}-t_{1}\right|^{1/2},1\right\\}^{2p}.$
(3.152)

###### Proof.

This follows from the standard inequality $(a+b)^{2p}\leq
2^{2p-1}\left(a^{2p}+b^{2p}\right)$ and a combination of the estimates from
Lemma 3.20, Lemma 3.21, and Lemma 3.22. ∎

As in [29], we will proceed by using the results from Dirksen [13], which is
again based on the chaining principle developed by Talagrand [91].

######  Lemma 3.24 (long-term bound).

Consider the setting of Proposition 3.18. Then there exists a constant
$K_{\mathrm{lt}}>0$ that does not depend on $T$ so that for any process $B$
that satisfies (HB*), and any integer $p\geq 1$, we have the supremum bound

$\mathbb{E}\sup_{0\leq t\leq
T}\left\|\mathcal{E}_{B}^{\mathrm{lt}}(t)\right\|_{H^{k}}^{2p}\leq\left(p^{p}+\log(T)^{p}\right)K_{\mathrm{lt}}^{2p}\Theta_{*}^{2p}.$
(3.153)

###### Proof.

Upon writing
$d_{\max}=2\sqrt{e}K_{\mathrm{dc}}M^{2}\|P^{\perp}\|_{\mathscr{L}(H^{k})}\Theta_{*}$
together with

$d\left(t_{1},t_{2}\right)=d_{\max}\min\left\\{\sqrt{\left|t_{2}-t_{1}\right|},1\right\\},$
(3.154)

an application of [29, Lemma 2.2] to Corollary 3.23 provides the bound

$P\left(\left\|\mathcal{E}_{B}^{\mathrm{lt}}\left(t_{1}\right)-\mathcal{E}_{B}^{\mathrm{lt}}\left(t_{2}\right)\right\|_{H^{k}}>\vartheta\right)\leq
2\exp\left[-\frac{\vartheta^{2}}{2d\left(t_{1},t_{2}\right)^{2}}\right].$
(3.155)

Furthermore, recall that the map $[0,T]\ni t\mapsto\mathcal{E}_{B}^{\rm
lt}(t)\in H^{k}$ has continuous paths $\mathbb{P}$-almost surely, as a
consequence of Proposition 3.5. This suffices to establish the equality

$\mathbb{E}\sup_{0\leq t\leq T}\|\mathcal{E}_{B}^{\rm
lt}(t)\|_{H^{k}}^{2p}=\textstyle\sup_{K\subset[0,T],|K|<\infty}\mathbb{E}\sup_{t\in
K}\|\mathcal{E}_{B}^{\rm lt}(t)\|_{H^{k}}^{2p},$ (3.156)

consequently allowing us to use the results in [13], since these are only
stated for suprema over sets of finite cardinality. In particular, for any
integer $p\geq 1$, there is a constant $C_{\rm ch}>0$ such that

$\mathbb{E}\sup_{0\leq t\leq T}\|\mathcal{E}_{B}^{\rm
lt}(t)\|_{H^{k}}^{2p}\leq C_{\rm
ch}^{2p}\left[\int_{0}^{\infty}\sqrt{\log(N(T,d,u))}\,\mathrm{d}u\right]^{2p}+C_{\rm
ch}^{2p}d_{\rm max}^{2p}p^{p},$ (3.157)

where $N(T,d,u)$ denotes the smallest number of intervals of length at most
$u>0$, in the metric $d$, required to cover the interval $[0,T].$ This follows
by choosing $\alpha=2$ in [13, eq. (3.2)], together with [13, eq. (2.3)], and
by applying the final inequality in the proof of [13, Thm. 3.2].

Conform to the proof in [29, Lem. 3.9], the Dudley entropy integral can be
bounded by

$\displaystyle\int_{0}^{\infty}\sqrt{\log(N(T,d,u))}\,\mathrm{d}u\leq d_{\rm
max}(\sqrt{2\log(T)}+\sqrt{\pi})\leq 4d_{\rm max}\sqrt{\log(T)},$ (3.158)

as $T\geq 2.$ The desired estimate now follows directly. ∎

###### Proof of Proposition 3.18.

Combining the short time result in Lemma 3.19 with the long time result in
Lemma 3.24 yields the supremum bound for $\mathcal{Z}^{\perp}[B]$ for
processes that satisfy (HB*). A standard limiting argument using Corollary 3.8
then generalises the bound to processes that satisfy (HB).

Turning towards the supremum bound for $\mathcal{I}^{\perp}[B]$, note that for
any $0\leq t\leq T,$ Proposition 3.11 implies that for every integer $p\geq 1$
we have

$\displaystyle\mathbb{E}\,\mathcal{I}^{\perp}[B](t)^{p}$ $\displaystyle\leq
K^{p}\,\mathbb{E}\big{[}\sup_{0\leq r\leq
t}\|\mathcal{Z}^{\perp}[B](r)\|_{H^{k}}^{2p}\big{]}$ (3.159)
$\displaystyle\qquad+p^{p/2}K^{p}\mathbb{E}\Big{[}\int_{0}^{t}e^{-\varepsilon(t-r)}\|B(r)\|_{HS(\mathcal{W}_{Q};H^{k})}^{2}\,dr\Big{]}^{p},$

for some $K>0$. Using the estimate (3.139) for $\mathcal{Z}^{\perp}[B]$ which
we have just established, we obtain

$\mathbb{E}\,\mathcal{I}^{\perp}[B](t)^{p}\leq
K^{p}\Theta_{*}^{2p}(p^{p}+\log(T)^{p})+p^{p/2}K^{p}\Theta_{*}^{2p}\leq
p^{p}\Theta_{1}^{p}+\Theta_{2}^{p},$ (3.160)

for some updated $K>0$, with $\Theta_{1}=2K\Theta_{*}^{2}$ and
$\Theta_{2}=\frac{1}{2}\Theta_{1}\log(T).$ Corollary B.3 now implies

$\mathbb{E}\max_{i\in\\{1,\ldots,T\\}}\mathcal{I}^{\perp}[B](i)^{p}\leq
K^{p}\Theta_{*}^{2p}(p^{p}+\log(T)^{p}),$ (3.161)

after updating $K>0$ again. To conclude the proof, it suffices to observe that

$\displaystyle\sup_{0\leq t\leq T}\mathcal{I}^{\perp}[B](t)$
$\displaystyle=\max_{i\in\\{1,\ldots,T\\}}\sup_{i-1\leq t\leq
i}\mathcal{I}^{\perp}[B](t)\leq
e^{\varepsilon}\max_{i\in\\{1,\ldots,T\\}}\mathcal{I}^{\perp}[B](i)$ (3.162)

holds for any integer $T\geq 2$. ∎

##  4 Bounds on nonlinearities

In this section, we obtain estimates on the nonlinearities that we encounter
throughout this paper. We start in §4.1 with some useful preparatory bounds,
proceeding in §4.2 with estimates that hold in
$L^{2}(\mathcal{D};\mathbb{R}^{n})$. As a consequence of our pointwise global
Lipschitz assumptions, we will be able to extract more information than was
possible in [27, 30]. We proceed in §4.3 and §4.4 with $H^{k}$-based estimates
for the terms appearing in our stochastic and deterministic expressions,
respectively. Finally, in §4.5 we consider the low dimensional setting $1\leq
d\leq 4$ under the cubic growth condition (Hf-Cub). We remind the reader that
the full list of functions can be found in Appendix A. Recall that we
abbreviate the notation for function spaces only if the domain is
$\mathcal{D}=\mathbb{R}\times\mathbb{T}^{d-1}$ and the codomain is
$\mathbb{R}^{n}.$

###  4.1 Preliminaries

Let us start by considering $H^{k}$-based estimates for the Nemytskii
operators that we use. For this, we consider a sufficiently smooth function
$\Theta:\mathbb{R}^{n}\to\mathbb{R}^{N}$ and assume that $\Phi$ is bounded and
sufficiently smooth on $\mathcal{D}=\mathbb{R}\times\mathbb{T}^{d-1}$. As an
illustration, we compute

$\displaystyle\partial_{x}[\Theta(\Phi+v_{A})-\Theta(\Phi+v_{B})]$
$\displaystyle=D\Theta(\Phi+v_{A})[\partial_{x}v_{A}-\partial_{x}v_{B}]$ (4.1)
$\displaystyle\qquad+\big{(}D\Theta(\Phi+v_{A})-D\Theta(\Phi+v_{B}))[\partial_{x}\Phi+\partial_{x}v_{B}].$

Since $\Phi$ is bounded, this provides the pointwise estimate

$|\partial_{x}[\Theta(\Phi+v_{A})-\Theta(\Phi+v_{B})]|\leq
K\Big{(}|\partial_{x}v_{A}-\partial_{x}v_{B}|+|v_{A}-v_{B}|(|\partial_{x}\Phi|+|\partial_{x}v_{B}|)\Big{)}$
(4.2)

under the assumption that $\Theta$ and $D\Theta$ are globally Lipschitz, which
automatically implies that $D\Theta$ is bounded. We can hence not expect
global Lipschitz bounds to hold in $H^{k}$ and the cross-terms will rapidly
become more involved as $k$ is increased. To control these expressions, we
recall that for any $k>d/2$ we can find a constant $K>0$ so that for any
bounded $\ell$-linear map $\Lambda:(\mathbb{R}^{n})^{\ell}\to\mathbb{R}^{N}$,
the bound

$\|\Lambda[\partial^{\alpha_{1}}v_{1},\ldots,\partial^{\alpha_{\ell}}v_{\ell}]\|_{L^{2}(\mathcal{D};\mathbb{R}^{N})}\leq
K|\Lambda|\|v_{1}\|_{H^{k}}\cdots\|v_{\ell}\|_{H^{k}}$ (4.3)

holds for any tuple $(v_{1},\ldots,v_{\ell})\in(H^{k})^{\ell}$, provided that
$|\alpha_{1}|+\ldots+|\alpha_{\ell}|\leq k$. This is related to the fact that
$H^{k}$ is an algebra under multiplication, i.e., $\|vw\|_{H^{k}}\leq
K\|v\|_{H^{k}}\|w\|_{H^{k}}$, for $k>d/2$.

######  Lemma 4.1.

Pick $k>d/2$, assume that $\Phi$ is bounded with $\Phi^{\prime}\in H^{k}$, and
consider a $C^{k}$-smooth function $\Theta:\mathbb{R}^{n}\to\mathbb{R}^{N}$
for which $D^{\ell}\Theta$ is globally Lipschitz for all $0\leq\ell\leq k$.
Then there exists a constant $K>0$ so that for each pair $v_{A},v_{B}\in
H^{k}$ we have the bound

$\|\Theta(\Phi+v_{A})-\Theta(\Phi+v_{B})\|_{H^{k}(\mathcal{D};\mathbb{R}^{N})}\leq
K(1+\|v_{A}\|_{H^{k}}^{k}+\|v_{B}\|_{H^{k}}^{k})\|v_{A}-v_{B}\|_{H^{k}}.$
(4.4)

###### Proof.

Consider a multi-index $\alpha\in\mathbb{Z}^{d}_{\geq 0}$ with $|\alpha|\leq
k$. We now claim that it is possible to write the spatial derivative
$\partial^{\alpha}[\Theta(\Phi+v_{A})-\Theta(\Phi+v_{B})]$ as a finite sum of
expressions of two types. The first type is given by

$\mathcal{I}_{I}=D^{\ell}\Theta(\Phi+v_{A})[\partial^{\beta_{1}}(v_{A}-v_{B}),\partial^{\beta_{2}}(\Phi+v_{\\#_{1}}),\ldots,\partial^{\beta_{\ell}}(\Phi+v_{\\#_{\ell}})],$
(4.5)

with $\\#_{i}\in\\{A,B\\}$ and multi-indices
$\\{\beta_{i}\\}_{i=1}^{\ell}\in\mathbb{Z}^{d}_{\geq 0}$ that satisfy
$|\beta_{i}|\geq 1$, for each $1\leq i\leq\ell\leq|\alpha|$, together with
$|\beta_{1}|+\ldots+|\beta_{\ell}|=|\alpha|$. The second type is given by

$\mathcal{I}_{II}=\Big{(}D^{\ell}\Theta(\Phi+v_{A})-D^{\ell}\Theta(\Phi+v_{B})\Big{)}\Big{[}\partial^{\beta_{1}}(\Phi+v_{B}),\dots,\partial^{\beta_{\ell}}(\Phi+v_{B})\Big{]},$
(4.6)

with the same conditions on $\\{\beta_{i}\\}_{i=1}^{\ell}$. This can be
readily verified with induction.

Using (4.3) together with the global Lipschitz properties of $\Theta$, which
automatically imply that $D^{\ell}\Theta$ is bounded for $1\leq\ell\leq k$, we
obtain the bounds

$\begin{array}[]{lcl}\|\mathcal{I}_{I}\|_{L^{2}(\mathcal{D};\mathbb{R}^{N})}&\leq&K\|v_{A}-v_{B}\|_{H^{k}}\big{[}1+\|v_{A}\|^{k-1}_{H^{k}}+\|v_{B}\|^{k-1}_{H^{k}}\big{]},\\\\[5.69046pt]
\|\mathcal{I}_{II}\|_{L^{2}(\mathcal{D};\mathbb{R}^{N})}&\leq&K\|v_{A}-v_{B}\|_{\infty}\big{[}1+\|v_{B}\|^{k}_{H^{k}}\big{]}.\end{array}$
(4.7)

Both terms can be absorbed in (4.4) in view of the Sobolev embedding
$H^{k}\hookrightarrow L^{\infty}$. ∎

######  Lemma 4.2.

Pick $k>d/2$, assume that $\Phi$ is bounded with $\Phi^{\prime}\in H^{k+1}$,
and consider a $C^{k+1}$-smooth function
$\Theta:\mathbb{R}^{n}\to\mathbb{R}^{N}$ for which $D^{\ell}\Theta$ is
globally Lipschitz for all $0\leq\ell\leq k+1$. Then there exists a constant
$K>0$ so that for each pair $v_{A},v_{B}\in H^{k+1}$ we have the bound

$\begin{array}[]{lcl}\|\Theta(\Phi+v_{A})-\Theta(\Phi+v_{B})\|_{H^{k+1}(\mathcal{D};\mathbb{R}^{N})}&\leq&K(1+\|v_{A}\|_{H^{k}}^{k}+\|v_{B}\|_{H^{k}}^{k})\\\\[5.69046pt]
&&\qquad\qquad\times\big{(}1+\|v_{A}\|_{H^{k+1}}+\|v_{B}\|_{H^{k+1}}\big{)}\\\\[5.69046pt]
&&\qquad\qquad\times\|v_{A}-v_{B}\|_{H^{k}}\\\\[5.69046pt]
&&\qquad+K(1+\|v_{A}\|_{H^{k}}^{k}+\|v_{B}\|_{H^{k}}^{k})\\\\[5.69046pt]
&&\qquad\qquad\qquad\times\|v_{A}-v_{B}\|_{H^{k+1}}.\end{array}$ (4.8)

###### Proof.

Inspecting the terms $\mathcal{I}_{I}$ and $\mathcal{I}_{II}$ in (4.5) and
(4.6), but where now $|\alpha|=k+1$ is also allowed, we see that each term can
be covered by appealing to the bound in (4.3). It is only necessary to replace
one of the $v_{i}$ in this bound by the differentiated version
$\partial^{\gamma}v_{i}$ for some multi-index $\gamma$ that has a single
component, i.e., $|\gamma|=1$. This leads directly to the stated estimates. ∎

When we additionally have $\Theta(\Phi)\in H^{k}(\mathcal{D};\mathbb{R}^{N})$,
we may use (4.4) to obtain the bound

$\|\Theta(\Phi+v)\|_{H^{k}(\mathcal{D};\mathbb{R}^{N})}\leq
K\big{[}1+\|v\|_{H^{k}}^{k+1}\big{]},$ (4.9)

for all $v\in H^{k}$, possibly after increasing $K$. In the same fashion,
whenever $\Theta(\Phi)\in H^{k+1}(\mathcal{D};\mathbb{R}^{N})$, we may use
(4.8) to obtain

$\|\Theta(\Phi+v)\|_{H^{k+1}(\mathcal{D};\mathbb{R}^{N})}\leq
K\big{[}1+\|v\|_{H^{k}}^{k+1}\big{]}\big{[}1+\|v\|_{H^{k+1}}\big{]},$ (4.10)

for all $v\in H^{k+1}$. This will be the case when we consider the functions
$f$, $g$ and $h$.

######  Lemma 4.3.

Pick $k>d/2$, assume that $\Phi$ is bounded with $\Phi^{\prime}\in H^{k}$, and
consider a $C^{k+2}$-smooth function $\Theta:\mathbb{R}^{n}\to\mathbb{R}^{N}$
for which $D^{\ell}\Theta$ is globally Lipschitz for all $0\leq\ell\leq k+2$.
Then there exists a constant $K>0$ so that for every $v\in H^{k}$ we have the
bound

$\|\Theta(\Phi+v)-\Theta(\Phi)-D\Theta(\Phi)[v]\|_{H^{k}(\mathcal{D};\mathbb{R}^{N})}\leq
K(1+\|v\|_{H^{k}}^{k})\|v\|^{2}_{H^{k}}.$ (4.11)

###### Proof.

Note first that we have the pointwise representation

$\displaystyle\Theta(\Phi+v)-\Theta(\Phi)-D\Theta(\Phi)[v]$
$\displaystyle=\int_{0}^{1}\Big{(}D\Theta(\Phi+tv)[v]-D\Theta(\Phi)[v]\Big{)}\,\mathrm{d}t$
(4.12)
$\displaystyle=\int_{0}^{1}\left(t\int_{0}^{1}D^{2}\Theta(\Phi+stv)[v,v]\,\mathrm{d}s\right)\mathrm{d}t.$

Consider a multi-index $\alpha\in\mathbb{Z}^{d}_{\geq 0}$ with $|\alpha|\leq
k$. We now claim that the spatial derivative
$\partial^{\alpha}D^{2}\Theta(\Phi+stv)[v,v]$ can be written as a finite sum
of expressions of the form

$\mathcal{I}=D^{\ell+2}\Theta(\Phi+stv)\Big{[}\partial^{\beta_{1}}(\Phi+stv),\partial^{\beta_{2}}(\Phi+stv),\ldots,\partial^{\beta_{\ell+1}}v,\partial^{\beta_{\ell+2}}v\Big{]},$
(4.13)

with multi-indices $\\{\beta_{i}\\}_{i=1}^{\ell+2}\in\mathbb{Z}^{d}_{\geq 0}$
that satisfy $|\beta_{i}|\geq 1$, for each $1\leq i\leq\ell+2\leq|\alpha|$,
together with $|\beta_{1}|+\ldots+|\beta_{\ell+2}|=|\alpha|$. This can again
readily be verified using induction. Using the global Lipschitz properties, we
obtain the bound

$\|\mathcal{I}\|_{L^{2}}\leq K[1+\|v\|_{H^{k}}^{k}]\|v\|_{H^{k}}^{2},$ (4.14)

as desired. ∎

We now turn to preliminary observations regarding the Hilbert-Schmidt norms of
operators that map $L^{2}_{Q}$ into $H^{k}$, extending the previous results
obtained in [30] for $d=1$. Assuming that (Hq) is satisfied, we can (formally)
introduce the function $p:\mathcal{D}\to\mathbb{R}^{m\times m}$ by taking the
Fourier inverse of $\sqrt{\hat{q}}$. We first show that this map is well-
defined, confirming that indeed $\hat{p}=\sqrt{\hat{q}}$ and hence that $p$
can be interpreted as the convolution kernel for $\sqrt{Q}:L^{2}_{Q}\to
L^{2}$, i.e., $\sqrt{Q}v=p*v$ for $v\in L^{2}_{Q}$.

######  Lemma 4.4.

Pick $k\geq 0$ and assume that (Hq) is satisfied. Then $p$ is well-defined
and, in addition, we have $p\in H^{k}(\mathcal{D};\mathbb{R}^{m\times m})$.

###### Proof.

Pick a multi-index
$\alpha=(\alpha_{1},\ldots,\alpha_{d})\in\mathbb{Z}^{d}_{\geq 0}$ with
$|\alpha|\leq k$ and write
$\textbf{z}^{\alpha}=z_{1}^{\alpha_{1}}z_{2}^{\alpha_{2}}\cdots
z_{d}^{\alpha_{d}}$. Applying Cauchy-Schwartz yields

$\int_{\widehat{\mathcal{D}}}\boldsymbol{\xi}^{2\alpha}|\hat{q}(\boldsymbol{\xi})|\mathrm{d}\boldsymbol{\xi}=\int_{\widehat{\mathcal{D}}}\frac{\boldsymbol{\xi}^{2\alpha}}{(1+|\boldsymbol{\xi}|^{2})^{\ell/2}}(1+|\boldsymbol{\xi}|^{2})^{\ell/2}|\hat{q}(\boldsymbol{\xi})|\mathrm{d}\boldsymbol{\xi}\leq
K(\alpha,\ell)\|q\|_{H^{\ell}(\mathcal{D};\mathbb{R}^{m\times m})},$ (4.15)

in which we have introduced the expression

$K(\alpha,\ell)=\int_{\widehat{\mathcal{D}}}\frac{\boldsymbol{\xi}^{4\alpha}}{(1+|\boldsymbol{\xi}|^{2})^{\ell}}\mathrm{d}\boldsymbol{\xi}=\sum_{\xi\in\mathbb{Z}^{d-1}}\int_{\mathbb{R}}\frac{(\omega,\xi)^{4\alpha}}{(1+\omega^{2}+|\xi|^{2})^{\ell}}\mathrm{d}\omega.$
(4.16)

We now claim that $K(\alpha,\ell)<\infty$, since $\ell>2|\alpha|+d/2$ by
assumption (Hq). To see this, note that

$K(\alpha,\ell)\leq\int_{\mathbb{R}^{d}}\frac{\textbf{z}^{4\alpha}}{(1+|\mathbf{z}|^{2})^{\ell}}\mathrm{d}\mathbf{z}\leq\int_{\mathbb{R}^{d}}\frac{|\textbf{z}|^{4|\alpha|}}{(1+|\mathbf{z}|^{2})^{\ell}}\mathrm{d}\mathbf{z}.$
(4.17)

Introducing the spherical coordinate $r=|z|$ with the associated integration
factor $r^{d-1}$, we obtain

$\displaystyle\int_{\mathbb{R}^{d}}\frac{|\textbf{z}|^{4|\alpha|}}{(1+|\mathbf{z}|^{2})^{\ell}}\mathrm{d}\mathbf{z}$
$\displaystyle=V_{d}\int_{0}^{\infty}\frac{r^{4|\alpha|}}{(1+r^{2})^{\ell}}r^{d-1}\mathrm{d}r$
(4.18)
$\displaystyle=V_{d}\int_{0}^{\infty}\frac{u^{2|\alpha|+d/2-1}}{(1+u)^{\ell}}\mathrm{d}u$
$\displaystyle=V_{d}\frac{\Gamma(\ell-2|\alpha|-\tfrac{d}{2})\Gamma(2|\alpha|+\tfrac{d}{2})}{\Gamma(\ell)},$

where $\Gamma$ denotes the Gamma-function and where
$V_{d}={2\pi^{\frac{d}{2}}/\Gamma(\frac{d}{2})}$ indicates the volume of the
unit $d$-sphere.

Taking $\alpha=0$, we see that
$\boldsymbol{\xi}\mapsto\sqrt{\hat{q}(\boldsymbol{\xi})}$ is in
$L^{2}(\widehat{\mathcal{D}};\mathbb{R}^{m\times m})$. This allows us to take
the Fourier inverse, ensuring that $p$ is a well-defined map. Lastly,
Plancherel’s identity (see Appendix C) implies

$\displaystyle\|p\|_{H^{k}(\mathcal{D};\mathbb{R}^{m\times m})}^{2}$
$\displaystyle=\sum_{|\alpha|\leq
k}\left\|\partial^{\alpha}p\right\|_{L^{2}(\mathcal{D};\mathbb{R}^{m\times
m})}^{2}$ (4.19) $\displaystyle=\frac{1}{|\mathbb{T}|^{d-1}}\sum_{|\alpha|\leq
k}\left\|\boldsymbol{\xi}\mapsto\boldsymbol{\xi}^{\alpha}\hat{p}(\boldsymbol{\xi})\right\|_{L^{2}(\widehat{\mathcal{D}};\mathbb{R}^{m\times
m})}^{2}$ $\displaystyle=\frac{1}{|\mathbb{T}|^{d-1}}\sum_{|\alpha|\leq
k}\int_{\widehat{\mathcal{D}}}\boldsymbol{\xi}^{2\alpha}|\hat{q}(\boldsymbol{\xi})|\mathrm{d}\boldsymbol{\xi},$

confirming that indeed $p\in H^{k}(\mathcal{D};\mathbb{R}^{m\times m})$. ∎

######  Lemma 4.5.

Pick $k\geq 0$ and suppose that (Hq) is satisfied. Then there exists a
constant $K>0$ so that any $z\in H^{k}(\mathcal{D};\mathbb{R}^{n\times m})$
can be interpreted as a Hilbert-Schmidt operator from $L^{2}_{Q}$ into $H^{k}$
that acts via the pointwise multiplication $z[w](x,y)=z(x,y)w(x,y)$ and admits
the bound

$\|z\|_{HS(L^{2}_{Q};H^{k})}\leq K\|z\|_{H^{k}(\mathcal{D};\mathbb{R}^{n\times
m})}.$ (4.20)

###### Proof.

Without loss of generality we take $n=m=1$ to ease the notational complexity.
Consider an orthonormal basis $(e_{\ell})_{\ell\geq 0}$ for
$L^{2}(\mathcal{D};\mathbb{R})$ and introduce the functions

$p^{(\ell)}(x,y)=[\sqrt{Q}e_{\ell}](x,y)=\langle
p(x-\cdot,y-\cdot),e_{\ell}(\cdot,\cdot)\rangle_{L^{2}(\mathcal{D};\mathbb{R})},$
(4.21)

together with the notation

$z_{\ell}(x,y)=z[\sqrt{Q}e_{\ell}](x,y)=z(x,y)[\sqrt{Q}e_{\ell}](x,y)=z(x,y)p^{(\ell)}(x,y).$
(4.22)

By definition, we have

$\displaystyle\|z\|_{HS(L^{2}_{Q};H^{k}(\mathcal{D};\mathbb{R}))}^{2}$
$\displaystyle=\sum_{\ell=0}^{\infty}\|z_{\ell}\|_{H^{k}(\mathcal{D};\mathbb{R})}^{2}=\sum_{\ell=0}^{\infty}\sum_{|\alpha|\leq
k}\int_{\mathcal{D}}|\partial^{\alpha}z_{\ell}(x,y)|^{2}\,\mathrm{d}x\,\mathrm{d}y$
$\displaystyle=\sum_{\ell=0}^{\infty}\sum_{|\alpha|\leq
k}\int_{\mathcal{D}}\left|\sum_{\beta\leq\alpha}\frac{\alpha!}{\beta!(\alpha-\beta)!}\partial^{\beta}z(x,y)\partial^{\alpha-\beta}p^{(\ell)}(x,y)\right|^{2}.$

Because $(e_{\ell})_{\ell\geq 0}$ is an orthonormal basis, and due to the
translation invariance of the integral, we obtain
$\sum_{\ell=0}^{\infty}|\partial^{\alpha-\beta}p^{(\ell)}(x,y)|^{2}=\sum_{\ell=0}^{\infty}\langle\partial^{\alpha-\beta}p(x-\cdot,y-\cdot),e_{\ell}(\cdot,\cdot)\rangle_{L^{2}(\mathcal{D};\mathbb{R})}^{2}=\|\partial^{\alpha-\beta}p\|_{L^{2}(\mathcal{D};\mathbb{R})}^{2},$
which is independent of the coordinate $(x,y)\in\mathcal{D}$. This allows us
to compute

$\displaystyle\|z\|_{HS(L^{2}_{Q};H^{k}(\mathcal{D};\mathbb{R}))}^{2}$
$\displaystyle\leq(k!)^{3}\sum_{\ell=0}^{\infty}\sum_{|\alpha|\leq
k}\sum_{\beta\leq\alpha}\int_{\mathcal{D}}|\partial^{\alpha}z(x,y)|^{2}|\partial^{\alpha-\beta}p^{(\ell)}(x,y)|^{2}\,\mathrm{d}x\,\mathrm{d}y$
(4.23) $\displaystyle=(k!)^{3}\sum_{|\alpha|\leq
k}\sum_{\beta\leq\alpha}\int_{\mathcal{D}}|\partial^{\alpha}z(x,y)|^{2}\|\partial^{\alpha-\beta}p\|^{2}_{L^{2}(\mathcal{D};\mathbb{R})}\,\mathrm{d}x\,\mathrm{d}y$
$\displaystyle\leq(k!)^{4}\|p\|_{H^{k}(\mathcal{D};\mathbb{R})}^{2}\sum_{|\alpha|\leq
k}\int_{\mathcal{D}}|\partial^{\alpha}z(x,y)|^{2}\,\mathrm{d}x\,\mathrm{d}y$
$\displaystyle=(k!)^{4}\|p\|_{H^{k}(\mathcal{D};\mathbb{R})}^{2}\|z\|_{H^{k}(\mathcal{D};\mathbb{R})}^{2},$

which in view of Lemma 4.4 completes the proof. ∎

###  4.2 Bounds in $L^{2}$

The pointwise global Lipschitz bounds on our nonlinearities $f$, $g$ and $h$
will enable us to obtain estimates for the cut-offs $\chi_{h}$ and $\chi_{l}$
and the scalar functions $a_{\sigma}$, $b$ and $\kappa_{\sigma}$ that are
entirely based on $L^{2}$-norms. In fact, several of the estimates obtained in
previous work for $d=1$ do not rely on Sobolev embeddings and will carry over
conveniently to the present context. We start by stating a basic consequence
of the global Lipschitz bounds on $f$, $g$ and $h$.

######  Lemma 4.6.

Pick $k=0$ and suppose that (Hf-Lip), (HSt), (HCor) and (HPar) are satisfied.
Then there exists a constant $K>0$, which does not depend on the pair
$(\Phi,c)$, so that the following holds true. For any $v\in L^{2}$ we have the
bounds

$\begin{array}[]{lcl}\|f(\Phi+v)\|_{L^{2}(\mathcal{D};\mathbb{R}^{n})}&\leq&K[1+\|v\|_{L^{2}}],\\\\[5.69046pt]
\|g(\Phi+v)\|_{L^{2}(\mathcal{D};\mathbb{R}^{n\times
m})}&\leq&K[1+\|v\|_{L^{2}}],\\\\[5.69046pt]
\|h(\Phi+v)\|_{L^{2}(\mathcal{D};\mathbb{R}^{n})}&\leq&K[1+\|v\|_{L^{2}}],\\\\[5.69046pt]
\end{array}$ (4.24)

while for any pair $v_{A},v_{B}\in L^{2}$ we have the estimates

$\begin{array}[]{lcl}\|f(\Phi+v_{A})-f(\Phi+v_{B})\|_{L^{2}(\mathcal{D};\mathbb{R}^{n})}&\leq&K\|v_{A}-v_{B}\|_{L^{2}},\\\\[5.69046pt]
\|g(\Phi+v_{A})-g(\Phi+v_{B})\|_{L^{2}(\mathcal{D};\mathbb{R}^{n\times
m})}&\leq&K\|v_{A}-v_{B}\|_{L^{2}},\\\\[5.69046pt]
\|h(\Phi+v_{A})-h(\Phi+v_{B})\|_{L^{2}(\mathcal{D};\mathbb{R}^{n})}&\leq&K\|v_{A}-v_{B}\|_{L^{2}}.\end{array}$
(4.25)

###### Proof.

This follows immediately from the fact that $f$, $g$ and $h$ are all globally
Lipschitz, that $\Phi(x)$ approaches its spatial limits at an exponential rate
as $x\to\pm\infty$, and that $\mathbb{T}^{d-1}$ is bounded. ∎

######  Corollary 4.7.

Pick $k=0$ and suppose that (Hf-Lip), (HSt), (HCor) and (HPar) are satisfied.
Then there exists a constant $K>0$, which does not depend on the pair
$(\Phi,c)$, so that the following holds true. For any $v\in L^{2}$ we have the
bounds

$\begin{array}[]{lcl}\|g(\Phi+v)\|_{HS(L^{2}_{Q};L^{2})}&\leq&K[1+\|v\|_{L^{2}}],\end{array}$
(4.26)

while for any pair $v_{A},v_{B}\in L^{2}$ we have the estimates

$\begin{array}[]{lcl}\|g(\Phi+v_{A})-g(\Phi+v_{B})\|_{HS(L^{2}_{Q};L^{2})}&\leq&K\|v_{A}-v_{B}\|_{L^{2}}.\end{array}$
(4.27)

###### Proof.

The assertion follows from Lemmas 4.5 and 4.6. ∎

We proceed by considering the scalar cut-off functions $\chi_{h}$ and
$\chi_{l}$ defined in (A.5). In addition, we provide $L^{2}_{Q}$-estimates for
the auxiliary function

$\widetilde{\mathcal{K}}_{C}(u,\gamma)=\chi_{l}(u,\gamma)\chi_{h}(u,\gamma)Qg(u)^{\top}T_{\gamma}\psi_{\rm
tw},$ (4.28)

which turn out to be highly convenient for our analysis of $b$ and
$\mathcal{K}_{C}$ below.

######  Lemma 4.8.

Pick $k=0$ and suppose that (HSt), (Hq) and (HPar) are satisfied. Then there
exists a constant $K>0$, which does not depend on the pair $(\Phi,c)$, so that
the following holds true. For any $v\in L^{2}$ and $\gamma\in\mathbb{R}$ we
have the bound

$\begin{array}[]{lcl}|\chi_{h}(\Phi+v,\gamma)|+|\chi_{l}(\Phi+v,\gamma)|+\|\widetilde{\mathcal{K}}_{C}(\Phi+v,\gamma)\|_{L^{2}_{Q}}&\leq&K.\end{array}$
(4.29)

In addition, for any pair $v_{A},v_{B}\in L^{2}$ and any pair
$\gamma_{A},\gamma_{B}\in\mathbb{R}$, we have the estimates

$\begin{array}[]{lcl}|\chi_{h}(\Phi+v_{A},\gamma_{A})-\chi_{h}(\Phi+v_{B},\gamma_{B})|&\leq&K\big{[}\|v_{A}-v_{B}\|_{L^{2}}+|\gamma_{A}-\gamma_{B}|\big{]},\\\\[5.69046pt]
|\chi_{l}(\Phi+v_{A},\gamma_{A})-\chi_{l}(\Phi+v_{B},\gamma_{B})|&\leq&K\big{[}\|v_{A}-v_{B}\|_{L^{2}}+(1+\|v_{A}\|_{L^{2}})|\gamma_{A}-\gamma_{B}|\big{]},\end{array}$
(4.30)

while the expression

$\begin{array}[]{lcl}\Delta_{AB}\widetilde{\mathcal{K}}_{C}&=&\widetilde{\mathcal{K}}_{C}(\Phi+v_{A},\gamma_{A})-\widetilde{\mathcal{K}}_{C}(\Phi+v_{B},\gamma_{B})\end{array}$
(4.31)

satisfies the bound

$\begin{array}[]{lcl}\|\Delta_{AB}\widetilde{\mathcal{K}}_{C}\|_{L^{2}_{Q}}&\leq&K\big{[}\|v_{A}-v_{B}\|_{L^{2}}+(1+\|v_{A}\|_{L^{2}})|\gamma_{A}-\gamma_{B}|\big{]}.\end{array}$
(4.32)

###### Proof.

The results for $\chi_{h}$ and $\chi_{l}$ follow directly from their
definitions in (A.5), as in [30, App. A]. Turning to
$\widetilde{\mathcal{K}}_{C}$, we note that for any $z\in
L^{2}(\mathcal{D};\mathbb{R}^{m\times n})$ and any $\psi\in
L^{\infty}(\mathbb{R};\mathbb{R}^{n})$ we have

$\|Qz\psi\|_{L^{2}_{Q}}^{2}=\langle
Qz\psi,z\psi\rangle_{L^{2}(\mathcal{D};\mathbb{R}^{m})}\leq\|q\|_{L^{1}(\mathcal{D};\mathbb{R}^{m\times
m})}\|z\|^{2}_{L^{2}(\mathcal{D};\mathbb{R}^{m\times
n})}\|\psi\|_{\infty}^{2},$ (4.33)

which upon taking $z=g(\Phi+v)^{\top}$ and $\psi=T_{\gamma}\psi_{\rm tw}$
shows that

$\|\widetilde{\mathcal{K}}_{C}(\Phi+v,\gamma)\|_{L^{2}_{Q}}\leq
K\|g(\Phi+v)\|_{L^{2}(\mathcal{D};\mathbb{R}^{n\times m})}$ (4.34)

for any $\gamma\in\mathbb{R}$. We now observe that whenever
$\chi_{h}(\Phi+v,\gamma)\neq 0$ holds, the function

$\tilde{v}=\Phi+v-T_{\gamma}\Phi_{\rm ref}$ (4.35)

necessarily satisfies $\|\tilde{v}\|_{L^{2}}\leq 3+\|\Phi_{0}-\Phi_{\rm
ref}\|_{L^{2}}$, which leads to the uniform a priori bound

$\|g(\Phi+v)\|_{L^{2}(\mathcal{D};\mathbb{R}^{n\times
m})}=\|g(T_{\gamma}\Phi_{\rm
ref}+\tilde{v})\|_{L^{2}(\mathcal{D};\mathbb{R}^{n\times m})}\leq C^{\prime}$
(4.36)

for some $C^{\prime}>0$ on account of the global Lipschitz smoothness of $g$
and the fact that the quantity $\|g(T_{\gamma}\Phi_{\rm
ref})\|_{L^{2}(\mathcal{D};\mathbb{R}^{n\times m})}$ is finite (and
independent of $\gamma$).

Turning to the Lipschitz bound (4.32) in $L^{2}_{Q}$, we first compute

$\displaystyle\Delta_{AB}\widetilde{\mathcal{K}}_{C}$
$\displaystyle=\big{[}\chi_{l}(\Phi+v_{A},\gamma_{A})-\chi_{l}(\Phi+v_{B},\gamma_{B})\big{]}Q\chi_{h}(\Phi+v_{A},\gamma_{A})g(\Phi+v_{A})^{\top}T_{\gamma_{A}}\psi_{\rm
tw}$ (4.37)
$\displaystyle\qquad+\chi_{l}(\Phi+v_{B},\gamma_{B})Q(\chi_{h}(\Phi+v_{A},\gamma_{A})-\chi_{h}(\Phi+v_{B},\gamma_{B}))g(\Phi+v_{A})^{\top}T_{\gamma_{A}}\psi_{\rm
tw}$
$\displaystyle\qquad+\chi_{l}(\Phi+v_{B},\gamma_{B})Q\chi_{h}(\Phi+v_{B},\gamma_{B})\big{(}g(\Phi+v_{A})^{\top}-g(\Phi+v_{B})^{\top}\big{)}T_{\gamma_{B}}\psi_{\rm
tw}$
$\displaystyle\qquad+\chi_{l}(\Phi+v_{B},\gamma_{B})Q\chi_{h}(\Phi+v_{B},\gamma_{B})g(\Phi+v_{B})^{\top}\big{(}T_{\gamma_{B}}\psi_{\rm
tw}-T_{\gamma_{A}}\psi_{\rm tw}\big{)}.$

Assuming without loss that $\chi_{h}(\Phi+v_{A},\gamma_{A})\neq 0$ (reverse
$A$ and $B$ otherwise), the result now follows by applying (4.30) and the
observations above, noting that $\psi_{\rm tw}^{\prime}$ is uniformly bounded.
∎

We are now ready to consider the function $\mathcal{K}_{C}$ defined in (A.19),
which can be written as

$\mathcal{K}_{C}(u,\gamma)=-\chi_{h}(u,\gamma)g(u)\widetilde{\mathcal{K}}_{C}(u,\gamma),$
(4.38)

together with the function $b$ defined in (A.11). Note that the computations
in [30, App. A] provide the convenient relationship

$\|{b}(\Phi+v,\gamma)\|^{2}_{HS(L^{2}_{Q};\mathbb{R})}=\chi_{h}(\Phi+v,\gamma)^{2}\|\widetilde{\mathcal{K}}_{C}(\Phi+v,\gamma)\|_{L^{2}_{Q}}^{2}.$
(4.39)

######  Corollary 4.9.

Pick $k=0$ and suppose that (HSt), (Hq) and (HPar) are satisfied. Then there
exists a constant $K>0$, which does not depend on the pair $(\Phi,c)$, so that
the following holds true. For any $v\in L^{2}$ and $\gamma\in\mathbb{R}$ we
have the bound

$\begin{array}[]{lcl}\|\mathcal{K}_{C}(\Phi+v,\gamma)\|_{L^{2}}+\|b(\Phi+v,\gamma)\|_{HS(L^{2}_{Q};\mathbb{R})}&\leq&K.\end{array}$
(4.40)

In addition, for any pair $v_{A},v_{B}\in L^{2}$ and any pair
$\gamma_{A},\gamma_{B}\in\mathbb{R}$, the expressions

$\begin{array}[]{lcl}\Delta_{AB}\mathcal{K}_{C}&=&\mathcal{K}_{C}(\Phi+v_{A},\gamma_{A})-\mathcal{K}_{C}(\Phi+v_{B},\gamma_{B}),\\\\[5.69046pt]
\Delta_{AB}b&=&b(\Phi+v_{A},\gamma_{A})-b(\Phi+v_{B},\gamma_{B})\end{array}$
(4.41)

satisfy the bounds

$\begin{array}[]{lcl}\|\Delta_{AB}\mathcal{K}_{C}\|_{L^{2}}&\leq&K\big{[}\|v_{A}-v_{B}\|_{L^{2}}+(1+\|v_{A}\|_{L^{2}})|\gamma_{A}-\gamma_{B}|\big{]},\\\\[5.69046pt]
\|\Delta_{AB}b\|_{HS(L^{2}_{Q};\mathbb{R})}&\leq&K\big{[}\|v_{A}-v_{B}\|_{L^{2}}+\big{(}1+\|v_{A}\|_{L^{2}}\big{)}|\gamma_{A}-\gamma_{B}|\big{]}.\end{array}$
(4.42)

###### Proof.

The estimate (4.40) follows directly from Lemma 4.8, the identities
(4.38)–(4.39), and from the general inequality

$\|\mathcal{K}_{C}(u,\gamma)\|_{L^{2}}\leq\|\chi_{h}(u,\gamma)g(u)\|_{HS(L^{2}_{Q};L^{2})}\|\widetilde{\mathcal{K}}_{C}(u,\gamma)\|_{L^{2}_{Q}}.$
(4.43)

The computations in [30, App. A] yield

$\displaystyle\|\Delta_{AB}b\|_{HS(L^{2}_{Q};\mathbb{R})}^{2}=\|\chi_{h}(\Phi+v_{A},\gamma_{A})\widetilde{\mathcal{K}}_{C}(\Phi+v_{A},\gamma_{A})-\chi_{h}(\Phi+v_{B},\gamma_{B})\widetilde{\mathcal{K}}_{C}(\Phi+v_{B},\gamma_{B})\|_{L^{2}_{Q}}^{2},$
(4.44)

which leads to the Lipschitz bound for $\Delta_{AB}b$ in $L^{2}$ after
appealing to Lemma 4.8. The estimate for $\Delta_{AB}\mathcal{K}_{C}$ follows
analogously. ∎

We now consider the function

$\kappa_{\sigma}(u,\gamma)=1+\frac{\sigma^{2}}{2}\|b(u,\gamma)\|^{2}_{HS(\mathcal{W}_{Q};\mathbb{R})},$
(4.45)

together with the associated quantities $\nu_{\sigma}^{(\vartheta)}$ defined
in (A.13). These are the final ingredient required towards analysing the
function $a_{\sigma}$ that describes the deterministic part of the evolution
of the phase.

######  Lemma 4.10.

Pick $k=0$ together with $\vartheta\in\\{-1,-1/2,1\\}$ and suppose that (HSt),
(Hq) and (HPar) are satisfied. Then there exists a constant $K>0$, which does
not depend on the pair $(\Phi,c)$, so that the following holds true. For any
$\sigma\geq 0$, any $v\in L^{2}$ and any $\gamma\in\mathbb{R}$ we have the
bound

$|\nu_{\sigma}^{(\vartheta)}(\Phi+v,\gamma)|\leq\sigma^{2}K,$ (4.46)

In addition, for any $\sigma\geq 0$, any pair $v_{A},v_{B}\in L^{2}$, and any
pair $\gamma_{A},\gamma_{B}\in\mathbb{R}$, we have the estimate

$|\nu_{\sigma}^{(\vartheta)}(\Phi+v_{A},\gamma_{A})-\nu_{\sigma}^{(\vartheta)}(\Phi+v_{B},\gamma_{B})|\leq\sigma^{2}K\big{[}\|v_{A}-v_{B}\|_{L^{2}}+\big{(}1+\|v_{A}\|_{L^{2}}\big{)}|\gamma_{A}-\gamma_{B}|\big{]}.$
(4.47)

###### Proof.

Using Corollary 4.9, the bounds can be established as in the proof of [27,
Lem. 3.9]. ∎

######  Lemma 4.11.

Pick $k=0$ and suppose that (Hf-Lip), (HSt), (HCor), (Hq) and (HPar) all hold.
Then there exists a constant $K_{a}>0$, which does not depend on the pair
$(\Phi,c)$, such that for any $v\in L^{2}$, any $\gamma\in\mathbb{R}$, and any
$0\leq\sigma\leq 1$, we have

$|a_{\sigma}(\Phi+v,\gamma;c)|\leq K_{a}(1+\|v\|_{L^{2}}).$ (4.48)

In addition, for any pair $v_{A},v_{B}\in L^{2}$, any
$\gamma_{A},\gamma_{B}\in\mathbb{R}$, and any $0\leq\sigma\leq 1$, we have

$\displaystyle|a_{\sigma}(\Phi+v_{A},\gamma_{A}$
$\displaystyle;c)-a_{\sigma}(\Phi+v_{B},\gamma_{B};c)|\leq$ (4.49)
$\displaystyle
K_{a}\big{[}1+\|v_{A}\|_{L^{2}}^{2}\big{]}\big{[}\|v_{A}-v_{B}\|_{L^{2}}+|\gamma_{A}-\gamma_{B}|\big{]}.$

###### Proof.

Recalling the definition (A.22) and introducing the functions

$\displaystyle\mathcal{E}_{I}(u,\gamma)$ $\displaystyle=\langle
f(u),T_{\gamma}\psi_{\rm tw}\rangle_{L^{2}}+\sigma^{2}\langle
h(u),T_{\gamma}\psi_{\rm tw}\rangle_{L^{2}},$ (4.50)
$\displaystyle\mathcal{E}_{II}(u,\gamma;c)$ $\displaystyle=c\langle
u,T_{\gamma}\psi_{\rm
tw}^{\prime}\rangle_{L^{2}}+\sigma^{2}\langle\mathcal{K}_{C}(u,\gamma;c),T_{\gamma}\psi_{\rm
tw}^{\prime}\rangle_{L^{2}},$ (4.51)
$\displaystyle\mathcal{E}_{III}(u,\gamma)$
$\displaystyle=\kappa_{\sigma}(u,\gamma)\langle u,T_{\gamma}\psi_{\rm
tw}^{\prime\prime}\rangle_{L^{2}},$ (4.52)

together with the shorthand notation

$\mathcal{E}(u,\gamma;c)=\mathcal{E}_{I}(u,\gamma)-\mathcal{E}_{II}(u,\gamma;c)+\mathcal{E}_{III}(u,\gamma),$
(4.53)

we see that

$a_{\sigma}(u,\gamma;c)=-\chi_{l}(u,\gamma)\mathcal{E}(u,\gamma;c).$ (4.54)

In particular, we have

$\displaystyle|a_{\sigma}(u_{A},\gamma_{A};c)-a_{\sigma}(u_{B},\gamma_{B};c)|\leq|\chi_{l}(u_{A}$
$\displaystyle,\gamma_{A})-\chi_{l}(u_{B},\gamma_{B})\|\mathcal{E}(u_{A},\gamma_{A};c)|$
(4.55)
$\displaystyle+|\chi_{l}(u_{B},\gamma_{B})\|\mathcal{E}(u_{A},\gamma_{A};c)-\mathcal{E}(u_{B},\gamma_{B};c)|.$

Note that (HPar) implies

$|c|+\|\Phi\|_{H^{1}(\mathbb{R};\mathbb{R}^{n})}\leq C_{1},$ (4.56)

which using Lemmas 4.6 and 4.10, Corollary 4.9, and the fact that $\psi_{\rm
tw}\in H^{2}(\mathbb{R};\mathbb{R}^{n})$, yields

$\displaystyle|\mathcal{E}(\Phi+v,\gamma;c)|$ $\displaystyle\leq
C_{2}(1+\|v\|_{L^{2}}).$ (4.57)

Together with the uniform bound (4.29) for $\chi_{l}$, we hence obtain (4.48)
as well as an estimate for the first term of (4.55) that can be absorbed in
(4.49).

We continue by only focusing on the term $\mathcal{E}_{III}$, remarking that
$\mathcal{E}_{I}$ and $\mathcal{E}_{II}$ can be handled in a similar fashion.
Note that

$\displaystyle|\mathcal{E}_{III}(\Phi+v_{A},\gamma_{A})-\mathcal{E}_{III}(\Phi+v_{B},\gamma_{B})|\leq$
(4.58)
$\displaystyle\qquad\quad|\kappa_{\sigma}(u_{A},\gamma_{A})-\kappa_{\sigma}(u_{B},\gamma_{B})\|\langle
u_{A},T_{\gamma_{A}}\psi_{\rm tw}^{\prime\prime}\rangle_{L^{2}}|$
$\displaystyle\qquad\qquad\quad+|\kappa_{\sigma}(u_{B},\gamma_{B})\|\langle
u_{A},T_{\gamma_{A}}\psi_{\rm tw}^{\prime\prime}\rangle_{L^{2}}-\langle
u_{B},T_{\gamma_{B}}\psi_{\rm tw}^{\prime\prime}\rangle_{L^{2}}|,$

where $|\kappa_{\sigma}(u,\gamma)|\leq C_{3}$ on account of (4.46). Appealing
to (4.47), we arrive at a bound for the first term of (4.58) that can be
absorbed in (4.49).

Since in fact $\psi_{\rm tw}\in H^{3}(\mathbb{R};\mathbb{R}^{n})$ holds, we
may use

$T_{\gamma_{A}}\psi_{\rm tw}-T_{\gamma_{B}}\psi_{\rm
tw}=\int_{0}^{\gamma_{B}-\gamma_{A}}T_{\gamma_{A}+s}\psi^{\prime}_{\rm
tw}\mathrm{d}s$ (4.59)

to conclude

$\|T_{\gamma_{A}}\psi_{\rm tw}-T_{\gamma_{B}}\psi_{\rm
tw}\|_{H^{2}(\mathbb{R};\mathbb{R}^{n})}\leq|\gamma_{A}-\gamma_{B}|\|\psi_{\rm
tw}\|_{H^{3}(\mathbb{R};\mathbb{R}^{n})}.$ (4.60)

In order to find an estimate for the second part of (4.58), it suffices to
compute

$\displaystyle|\langle u_{A},T_{\gamma_{A}}\psi_{\rm
tw}^{\prime\prime}\rangle_{L^{2}}-\langle u_{B},T_{\gamma_{B}}\psi_{\rm
tw}^{\prime\prime}\rangle_{L^{2}}|$ $\displaystyle\quad\quad\leq|\langle
u_{A}-u_{B},T_{\gamma_{B}}\psi_{\rm
tw}^{\prime\prime}\rangle_{L^{2}}|+|\langle u_{A},T_{\gamma_{A}}\psi_{\rm
tw}^{\prime\prime}-T_{\gamma_{B}}\psi_{\rm tw}^{\prime\prime}\rangle_{L^{2}}|$
(4.61) $\displaystyle\quad\quad\leq\|u_{A}-u_{B}\|_{L^{2}}\|\psi_{\rm
tw}\|_{H^{2}(\mathbb{R};\mathbb{R}^{n})}+\|u_{A}\|_{L^{2}}\|\psi_{\rm
tw}\|_{H^{3}(\mathbb{R};\mathbb{R}^{n})}|\gamma_{A}-\gamma_{B}|,$

completing the proof. ∎

###  4.3 Stochastic terms

We here collect the estimates that we will need for the stochastic terms in
our evolution systems. As a preparation, we provide bounds for $g$ in $H^{1}$,
$H^{k},$ and $H^{k+1}$, respectively, noting that the first and the last will
be required in §4.4 below.

######  Lemma 4.12.

Pick $k>d/2$ and suppose that (HSt) and (HPar) are satisfied. Then there
exists a constant $K>0$, which does not depend on the pair $(\Phi,c)$, so that
for any $v\in H^{1}$ we have the bound

$\begin{array}[]{lcl}\|g(\Phi+v)\|_{H^{1}(\mathcal{D};\mathbb{R}^{n\times
m})}&\leq&K(1+\|v\|_{H^{1}}),\end{array}$ (4.62)

while for any pair $v_{A},v_{B}\in H^{k}$ we have the bound

$\begin{array}[]{lcl}\|g(\Phi+v_{A})-g(\Phi+v_{B})\|_{H^{1}(\mathcal{D};\mathbb{R}^{n\times
m})}&\leq&K(1+\|v_{A}\|_{H^{k}})\|v_{A}-v_{B}\|_{H^{1}}\\\\[5.69046pt]
&&\qquad+\,K(1+\|v_{B}\|_{H^{1}})\|v_{A}-v_{B}\|_{H^{k}}.\end{array}$ (4.63)

###### Proof.

These bounds follow directly by noting that $Dg$ is uniformly bounded and by
inspecting the pointwise estimate (4.2). ∎

######  Corollary 4.13.

Pick $k>d/2$ and suppose that (Hq), (HSt) and (HPar) are satisfied. Then for
any $v\in H^{k}$ we have the bound

$\begin{array}[]{lcl}\|g(\Phi+v)\|_{HS(L^{2}_{Q};H^{k})}&\leq&K(1+\|v\|_{H^{k}}^{k+1}),\end{array}$
(4.64)

while for any pair $v_{A},v_{B}\in H^{k}$ we have the estimate

$\begin{array}[]{lcl}\|g(\Phi+v_{A})-g(\Phi+v_{B})\|_{HS(L^{2}_{Q};H^{k})}&\leq&K\big{(}1+\|v_{A}\|_{H^{k}}^{k}+\|v_{B}\|_{H^{k}}^{k}\big{)}\|v_{A}-v_{B}\|_{H^{k}}.\end{array}$
(4.65)

###### Proof.

Using the bound (4.9), these estimates follow by combining Lemmas 4.1 and 4.5.
∎

######  Corollary 4.14.

Pick $k>d/2$ and suppose that (Hq), (HSt) and (HPar) are satisfied. Then there
exists a constant $K>0$, which does not depend on the pair $(\Phi,c)$, so that
for any $v\in H^{k+1}$ we have the bound

$\begin{array}[]{lcl}\|g(\Phi+v)\|_{HS(L^{2}_{Q};H^{k+1})}&\leq&K(1+\|v\|_{H^{k}}^{k+1})(1+\|v\|_{H^{k+1}}),\end{array}$
(4.66)

while for any pair $v_{A},v_{B}\in H^{k+1}$ the expression

$\begin{array}[]{lcl}\Delta_{AB}g&=&g(\Phi+v_{A})-g(\Phi+v_{B})\end{array}$
(4.67)

satisfies the estimate

$\begin{array}[]{lcl}\|\Delta_{AB}g\|_{HS(L^{2}_{Q};H^{k+1})}&\leq&K(1+\|v_{A}\|_{H^{k}}^{k}+\|v_{B}\|_{H^{k}}^{k})\\\\[5.69046pt]
&&\qquad\qquad\times(1+\|v_{A}\|_{H^{k+1}}+\|v_{B}\|_{H^{k+1}})\|v_{A}-v_{B}\|_{H^{k}}\\\\[5.69046pt]
&&\qquad+\,K(1+\|v_{A}\|_{H^{k}}^{k}+\|v_{B}\|_{H^{k}}^{k})\|v_{A}-v_{B}\|_{H^{k+1}}.\end{array}$
(4.68)

###### Proof.

Using the bound (4.10), these estimates follow by combining Lemmas 4.2 and
4.5. ∎

We now turn to the functions $\mathcal{S}$ and $\mathcal{M}_{\sigma}$ that act
on $\xi\in L^{2}_{Q}$ as

$\begin{array}[]{lcl}\mathcal{S}(v;\Phi)[\xi]&=&g(\Phi+v)[\xi]+\partial_{x}(\Phi+v)b(\Phi+v,0)[\xi],\\\\[5.69046pt]
\mathcal{M}_{\sigma}(v)[\xi]&=&\kappa_{\sigma}(\Phi_{\sigma}+v,0)^{-1/2}\mathcal{S}(v;\Phi_{\sigma})[\xi].\end{array}$
(4.69)

######  Corollary 4.15.

Pick $k>d/2$ and suppose that (Hq), (HSt) and (HPar) are satisfied. Then there
exists a constant $K>0$, which does not depend on the pair $(\Phi,c)$, so that
for any $v\in H^{k+1}$ and any sufficiently small $\sigma\geq 0$ we have the
bounds

$\begin{array}[]{lcl}\|\mathcal{S}(v;\Phi)\|_{HS(L^{2}_{Q};H^{k})}&\leq&K(1+\|v\|_{H^{k}}^{k+1}+\|v\|_{H^{k+1}}),\\\\[5.69046pt]
\|\mathcal{M}_{\sigma}(v)\|_{HS(L^{2}_{Q};H^{k})}&\leq&K(1+\|v\|_{H^{k}}^{k+1}+\|v\|_{H^{k+1}}).\end{array}$
(4.70)

###### Proof.

These estimates follow readily from (4.64), the inequality in (4.46) for
$\vartheta=-1/2$, and the bounds for $b$ in Corollary 4.9. ∎

###  4.4 Deterministic terms

Proceeding with the terms in the deterministic part of our evolution
equations, our ultimate goal is to obtain a bound for the nonlinearity
$\mathcal{N}_{\sigma}$ that will feature in our stability arguments. One of
the main tasks is to track the dependence on the $H^{k+1}$-norm of our
perturbation $v(t)$. Indeed, in the sequel we will only have integrated
control over this norm, as opposed to the pointwise control that we will have
over the $H^{k}$-norm. We start by providing basic bounds for the nonlinearity
$f$ and correction term $h$.

######  Corollary 4.16.

Pick $k>d/2$ and suppose that (Hf-Lip), (HCor) and (HPar) are satisfied. Then
there exists a constant $K>0$, which does not depend on the pair $(\Phi,c)$,
so that for any $v\in H^{k}$ we have the bound

$\|f(\Phi+v)\|_{H^{k}}+\|h(\Phi+v)\|_{H^{k}}\leq K(1+\|v\|_{H^{k}}^{k+1}),$
(4.71)

while for any pair $v_{A},v_{B}\in H^{k}$ we have the estimate

$\|f(\Phi+v_{A})-f(\Phi+v_{B})\|_{H^{k}}\leq
K(1+\|v_{A}\|_{H^{k}}^{k}+\|v_{B}\|_{H^{k}}^{k})\|v_{A}-v_{B}\|_{H^{k}},$
(4.72)

which also holds for $h$. Finally, for any $v\in H^{k}$ we have the quadratic
bound

$\|f(\Phi+v)-f(\Phi)-Df(\Phi)[v]\|_{H^{k}}\leq
K(1+\|v\|_{H^{k}}^{k+1})\|v\|^{2}_{H^{k}}.$ (4.73)

###### Proof.

These estimates follow from Lemmas 4.1 and 4.3 together with the bound (4.9).
∎

We proceed by studying the term $\mathcal{K}_{C}$, which we must understand in
$H^{k+1}$. For any multi-index $\beta\in\mathbb{Z}^{d}_{\geq 0}$ with
$|\beta|=1$, we will use the estimate

$\begin{array}[]{lcl}\|\partial^{\beta}\mathcal{K}_{C}(u,\gamma)\|_{H^{k}}&\leq&\|\chi_{h}(u,\gamma)\partial^{\beta}g(U)\|_{HS(L^{2}_{Q};H^{k})}\|\widetilde{\mathcal{K}}_{C}(u,\gamma)\|_{L^{2}_{Q}}\\\\[5.69046pt]
&&\qquad+\,\|\chi_{h}(u,\gamma)g(U)\|_{HS(L^{2}_{Q};H^{k})}\|\partial^{\beta}\widetilde{\mathcal{K}}_{C}(u,\gamma)\|_{L^{2}_{Q}},\end{array}$
(4.74)

together with its natural analogue for
$\partial^{\beta}[\mathcal{K}_{C}(u_{A},\gamma)-\mathcal{K}_{C}(u_{B},\gamma)]$.
We start by considering the term $\partial^{\beta}\widetilde{K}_{C}$, for
which it suffices to understand $g$ in $H^{1}$.

######  Lemma 4.17.

Pick $k>d/2$ together with a multi-index $\beta\in\mathbb{Z}^{d}_{\geq 0}$
with $|\beta|=1$ and suppose that (Hq), (HSt) and (HPar) are satisfied. Then
there exists a constant $K>0$, which does not depend on the pair $(\Phi,c)$,
so that the following holds true. For any $v\in H^{1}$ and
$\gamma\in\mathbb{R}$ we have the bound

$\begin{array}[]{lcl}\|\partial^{\beta}\widetilde{\mathcal{K}}_{C}(\Phi+v,\gamma)\|_{L^{2}_{Q}}&\leq&K(1+\|v\|_{H^{1}}),\end{array}$
(4.75)

while for any pair $v_{A},v_{B}\in H^{k}$ and $\gamma\in\mathbb{R}$, the
expression

$\begin{array}[]{lcl}\Delta_{AB}\widetilde{\mathcal{K}}_{C}&=&\widetilde{\mathcal{K}}_{C}(\Phi+v_{A},\gamma)-\widetilde{\mathcal{K}}_{C}(\Phi+v_{B},\gamma)\end{array}$
(4.76)

satisfies the estimate

$\begin{array}[]{lcl}\|\partial^{\beta}\Delta_{AB}\widetilde{\mathcal{K}}_{C}\|_{L^{2}_{Q}}&\leq&K(1+\|v_{A}\|_{H^{1}})\|v_{A}-v_{B}\|_{L^{2}}\\\\[5.69046pt]
&&\qquad+\,K(1+\|v_{A}\|_{H^{k}})\|v_{A}-v_{B}\|_{H^{1}}\\\\[5.69046pt]
&&\qquad+\,K(1+\|v_{B}\|_{H^{1}})\|v_{A}-v_{B}\|_{H^{k}}\end{array}$ (4.77)

###### Proof.

Note first that for any $z\in H^{1}(\mathcal{D},\mathbb{R}^{m\times n})$ and
any $\gamma\in\mathbb{R}$ we have

$\begin{array}[]{lcl}\|\partial^{\beta}QzT_{\gamma}\psi_{\rm
tw}\|_{L^{2}_{Q}}^{2}&=&\|Q\partial^{\beta}[zT_{\gamma}\psi_{\rm
tw}]\|_{L^{2}_{Q}}^{2}\\\\[5.69046pt]
&\leq&\|q\|_{L^{1}(\mathcal{D};\mathbb{R}^{m\times
m})}\|\partial_{\xi}[zT_{\gamma}\psi_{\rm
tw}]\|_{L^{2}(\mathcal{D};\mathbb{R}^{m})}^{2}\\\\[5.69046pt]
&\leq&\|q\|_{L^{1}(\mathcal{D};\mathbb{R}^{m\times
m})}\|z\|^{2}_{H^{1}(\mathcal{D};\mathbb{R}^{m\times n})}\big{[}\|\psi_{\rm
tw}\|_{\infty}+\|\psi_{\rm tw}^{\prime}\|_{\infty}\big{]}^{2}.\end{array}$
(4.78)

The bound (4.75) hence follows directly from (4.62). Turning to the estimate
(4.77), we recall the identity (4.37) and set $\gamma=\gamma_{A}=\gamma_{B}$.
The stated bound can now be obtained by combining Lemmas 4.8 and 4.12. ∎

######  Lemma 4.18.

Pick $k>d/2$ and suppose that (Hq), (HSt) and (HPar) are satisfied. Then there
exists a constant $K>0$, which does not depend on the pair $(\Phi,c)$, so that
the following holds true. For any $v\in H^{k+1}$ and $\gamma\in\mathbb{R}$ we
have the bound

$\begin{array}[]{lcl}\|\mathcal{K}_{C}(\Phi+v,\gamma)\|_{H^{k+1}}&\leq&K(1+\|v\|^{k+1}_{H^{k}})(1+\|v\|_{H^{k+1}}),\end{array}$
(4.79)

while for any pair $v_{A},v_{B}\in H^{k+1}$ and $\gamma\in\mathbb{R}$, the
expression

$\Delta_{AB}\mathcal{K}_{C}=\mathcal{K}_{C}(\Phi+v_{A},\gamma)-\mathcal{K}_{C}(\Phi+v_{B},\gamma)$
(4.80)

satisfies the estimate

$\begin{array}[]{lcl}\|\Delta_{AB}\mathcal{K}_{C}\|_{H^{k+1}}&\leq&K(1+\|v_{A}\|_{H^{k}}^{k+1}+\|v_{B}\|_{H^{k}}^{k+1})\\\\[5.69046pt]
&&\qquad\qquad\times(1+\|v_{A}\|_{H^{k+1}}+\|v_{B}\|_{H^{k+1}})\|v_{A}-v_{B}\|_{H^{k}}\\\\[5.69046pt]
&&\qquad+\,K(1+\|v_{A}\|_{H^{k}}^{k}+\|v_{B}\|_{H^{k}}^{k})\|v_{A}-v_{B}\|_{H^{k+1}}.\end{array}$
(4.81)

###### Proof.

The bound (4.79) follows from the decomposition (4.74), using (4.29), (4.64),
(4.66), and (4.75). Turning to the Lipschitz estimate in $H^{k}$, we pick a
multi-index $\beta\in\mathbb{Z}^{d-1}_{\geq 0}$ and consider the splitting

$\begin{array}[]{lcl}\partial^{\beta}\Delta_{AB}\mathcal{K}_{C}&=&\mathcal{E}_{I}+\mathcal{E}_{II},\end{array}$
(4.82)

in which we have defined

$\begin{array}[]{lcl}\mathcal{E}_{I}&=&(\chi_{h}(\Phi+v_{A},\gamma)-\chi_{h}(\Phi+v_{B},\gamma))[\partial^{\beta}g(\Phi+v_{A})]\widetilde{\mathcal{K}}_{C}(\Phi+v_{A},\gamma)\\\\[5.69046pt]
&&\qquad+\chi_{h}(\Phi+v_{B})[\partial^{\beta}g(\Phi+v_{A})-\partial^{\beta}g(\Phi+v_{B})]\widetilde{\mathcal{K}}_{C}(\Phi+v_{A},\gamma)\\\\[5.69046pt]
&&\qquad+\chi_{h}(\Phi+v_{B})\partial^{\beta}g(\Phi+v_{B})\big{[}\widetilde{\mathcal{K}}_{C}(\Phi+v_{A},\gamma)-\widetilde{\mathcal{K}}_{C}(\Phi+v_{B},\gamma)\big{]},\\\\[5.69046pt]
\end{array}$ (4.83)

together with

$\begin{array}[]{lcl}\mathcal{E}_{II}&=&(\chi_{h}(\Phi+v_{A},\gamma)-\chi_{h}(\Phi+v_{B},\gamma))[g(\Phi+v_{A})]\partial^{\beta}\widetilde{\mathcal{K}}_{C}(\Phi+v_{A},\gamma)\\\\[5.69046pt]
&&\qquad+\chi_{h}(\Phi+v_{B})[g(\Phi+v_{A})-g(\Phi+v_{B})]\partial^{\beta}\widetilde{\mathcal{K}}_{C}(\Phi+v_{A},\gamma)\\\\[5.69046pt]
&&\qquad+\chi_{h}(\Phi+v_{B})g(\Phi+v_{B})[\partial^{\beta}\widetilde{\mathcal{K}}_{C}(\Phi+v_{A},\gamma)-\partial^{\beta}\widetilde{\mathcal{K}}_{C}(\Phi+v_{B},\gamma)].\\\\[5.69046pt]
\end{array}$ (4.84)

Using decompositions analogous to (4.74), the bounds in Lemmas 4.8 and 4.17,
and Corollaries 4.13 and 4.14, lead to expressions that can all be absorbed in
the stated estimate. Indeed, we obtain

$\begin{array}[]{lcl}\|\mathcal{E}_{I}\|_{H^{k}}&\leq&K(1+\|v_{A}\|^{k+1}_{H^{k}}+\|v_{B}\|^{k+1}_{H^{k}})(1+\|v_{A}\|_{H^{k+1}}+\|v_{B}\|_{H^{k+1}})\|v_{A}-v_{B}\|_{L^{2}}\\\\[5.69046pt]
&&\qquad+\,K(1+\|v_{A}\|_{H^{k}}^{k}+\|v_{B}\|_{H^{k}}^{k})(1+\|v_{A}\|_{H^{k+1}}+\|v_{B}\|_{H^{k+1}})\|v_{A}-v_{B}\|_{H^{k}}\\\\[5.69046pt]
&&\qquad+\,K(1+\|v_{A}\|_{H^{k}}^{k}+\|v_{B}\|_{H^{k}}^{k})\|v_{A}-v_{B}\|_{H^{k+1}},\end{array}$
(4.85)

together with

$\begin{array}[]{lcl}\|\mathcal{E}_{II}\|_{H^{k}}&\leq&K(1+\|v_{A}\|^{k+1}_{H^{k}}+\|v_{B}\|^{k+1}_{H^{k}})(1+\|v_{A}\|_{H^{1}})\|v_{A}-v_{B}\|_{L^{2}}\\\\[5.69046pt]
&&\qquad+\,K(1+\|v_{A}\|^{k+1}_{H^{k}}+\|v_{B}\|^{k+1}_{H^{k}}))(1+\|v_{A}\|_{H^{1}}+\|v_{B}\|_{H^{1}})\|v_{A}-v_{B}\|_{H^{k}}\\\\[5.69046pt]
&&\qquad+\,K(1+\|v_{B}\|_{H^{k}}^{k+1})(1+\|v_{A}\|_{H^{k}})\|v_{A}-v_{B}\|_{H^{1}},\end{array}$
(4.86)

which proves the assertion. ∎

Upon introducing the expressions

$\Xi_{I}(u,c)=f(u)+c\partial_{x}u,\qquad\qquad\Xi_{II;\sigma}(u,\gamma)=\sigma^{2}[h(u)+\partial_{x}\mathcal{K}_{C}(u,\gamma)],$
(4.87)

we point out that the function $\mathcal{J}_{\sigma}$ defined in (A.23) can be
written in the form

$\displaystyle\mathcal{J}_{\sigma}(u,\gamma;c)$
$\displaystyle=\kappa_{\sigma}(u,\gamma)^{-1}\big{[}\Xi_{I}(u,c)+\Xi_{II;\sigma}(u,\gamma)\big{]}.$
(4.88)

The estimates above can be used to formulate convenient bounds for $\Xi_{I}$
and $\Xi_{II;\sigma}$.

######  Corollary 4.19.

Pick $k>d/2$ and suppose that (Hf-Lip), (HCor), (Hq), (HSt) and (HPar) are
satisfied. Then there exists a constant $K>0$, which does not depend on the
pair $(\Phi,c)$, so that the following holds true. For any $v\in H^{k+1}$, any
$\gamma\in\mathbb{R}$, and any $\sigma\geq 0$, we have the bounds

$\begin{array}[]{lcl}\langle\Xi_{I}(\Phi+v,c),\psi_{\rm
tw}\rangle_{L^{2}}&\leq&K(1+\|v\|_{L^{2}}),\\\\[5.69046pt]
\langle\Xi_{II;\sigma}(\Phi+v,\gamma),\psi_{\rm
tw}\rangle_{L^{2}}&\leq&K\sigma^{2}(1+\|v\|_{L^{2}}),\end{array}$ (4.89)

together with

$\begin{array}[]{lcl}\|\Xi_{I}(\Phi+v,c)\|_{H^{k}}&\leq&K\big{(}1+\|v\|_{H^{k}}^{k+1}+\|v\|_{H^{k+1}}\big{)},\\\\[5.69046pt]
\|\Xi_{II;\sigma}(\Phi+v,\gamma)\|_{H^{k}}&\leq&K\sigma^{2}(1+\|v\|_{H^{k}}^{k+1})(1+\|v\|_{H^{k+1}}).\end{array}$
(4.90)

In addition, for any pair $v_{A},v_{B}\in H^{k+1}$, any $\gamma\in\mathbb{R}$,
and any $\sigma\geq 0$, the expressions

$\begin{array}[]{lcl}\Delta_{AB}\Xi_{I}&=&\Xi_{I}(\Phi+v_{A},c)-\Xi_{I}(\Phi+v_{B},c),\\\\[5.69046pt]
\Delta_{AB}\Xi_{II;\sigma}&=&\Xi_{II;\sigma}(\Phi+v_{A},\gamma)-\Xi_{II;\sigma}(\Phi+v_{B},\gamma),\end{array}$
(4.91)

satisfy the estimates

$\begin{array}[]{lcl}\langle\Delta_{AB}\Xi_{I},\psi_{\rm
tw}\rangle_{L^{2}}&\leq&K\|v_{A}-v_{B}\|_{L^{2}},\\\\[5.69046pt]
\langle\Delta_{AB}\Xi_{II;\sigma},\psi_{\rm
tw}\rangle_{L^{2}}&\leq&\sigma^{2}K\|v_{A}-v_{B}\|_{L^{2}},\\\\[5.69046pt]
\end{array}$ (4.92)

together with

$\begin{array}[]{lcl}\|\Delta_{AB}\Xi_{I}\|_{H^{k}}&\leq&K(1+\|v_{A}\|^{k}_{H^{k}}+\|v_{B}\|_{H^{k}}^{k})\|v_{A}-v_{B}\|_{H^{k}}+K\|v_{A}-v_{B}\|_{H^{k+1}},\\\\[5.69046pt]
\|\Delta_{AB}\Xi_{II;\sigma}\|_{H^{k}}&\leq&\sigma^{2}K(1+\|v_{A}\|_{H^{k}}^{k+1}+\|v_{B}\|_{H^{k}}^{k+1})\\\\[5.69046pt]
&&\qquad\qquad\times(1+\|v_{A}\|_{H^{k+1}}+\|v_{B}\|_{H^{k+1}})\|v_{A}-v_{B}\|_{H^{k}}\\\\[5.69046pt]
&&\qquad+\,\sigma^{2}K(1+\|v_{A}\|_{H^{k}}^{k}+\|v_{B}\|_{H^{k}}^{k})\|v_{A}-v_{B}\|_{H^{k+1}}.\end{array}$
(4.93)

###### Proof.

Inspecting the definitions (4.87), the $L^{2}$-bounds follow from Lemma 4.6
and Corollary 4.9 after transferring the spatial derivatives onto $\psi_{\rm
tw}$, while the $H^{k}$-bounds follow from Corollary 4.16 and Lemma 4.18. ∎

We are now ready to consider the nonlinearity $\mathcal{N}_{\sigma}$ defined
in (A.28). It is exactly at this point where we need the pair
$(\Phi,c)=(\Phi_{\sigma},c_{\sigma})$ to be the instantaneous stochastic
waves, which requires $\sigma\geq 0$ to be sufficiently small. In particular,
for any $v\in H^{k+1}$ we will use the representation

$\mathcal{N}_{\sigma}(v)=\mathcal{N}_{I;\sigma}(v)-\chi_{l}(\Phi_{\sigma}+v,\gamma)\langle\mathcal{N}_{I;\sigma}(v),\psi_{\rm
tw}\rangle_{L^{2}}[\partial_{x}\Phi_{\sigma}+\partial_{x}v],$ (4.94)

which involves the intermediate function

$\mathcal{N}_{I;\sigma}(v)=\Phi_{\sigma}^{\prime\prime}+\mathcal{J}_{\sigma}(\Phi_{\sigma}+v,0;c_{\sigma})-c_{0}\partial_{x}v-Df(\Phi_{0})v;$
(4.95)

see (A.29) and (A.30). By construction, we have
$\Phi_{\sigma}^{\prime\prime}+\mathcal{J}_{\sigma}(\Phi_{\sigma};0;c_{\sigma})=0$,
which allows us to write

$\begin{array}[]{lcl}\mathcal{N}_{I;\sigma}(v)&=&\mathcal{J}_{\sigma}(\Phi_{\sigma}+v,0;c_{\sigma})-\mathcal{J}_{\sigma}(\Phi_{\sigma},0;c_{\sigma})-c_{0}\partial_{x}v-Df(\Phi_{0})v\\\\[5.69046pt]
&=&f(\Phi_{\sigma}+v)-f(\Phi_{\sigma})-Df(\Phi_{\sigma})v+\big{(}Df(\Phi_{\sigma})-Df(\Phi_{0})\big{)}v+(c_{\sigma}-c_{0})\partial_{x}v\\\\[5.69046pt]
&&\qquad+\,\kappa_{\sigma}(\Phi_{\sigma},0)^{-1}\big{(}\Xi_{II;\sigma}(\Phi_{\sigma}+v,0)-\Xi_{II;\sigma}(\Phi_{\sigma},0)\big{)}\\\\[5.69046pt]
&&\qquad+\,(\nu^{(-1)}_{\sigma}(\Phi_{\sigma}+v,0)-\nu^{(-1)}_{\sigma}(\Phi_{\sigma}))\big{[}\Xi_{I}(\Phi_{\sigma}+v,c_{\sigma})+\Xi_{II;\sigma}(\Phi_{\sigma}+v,0)\big{]}\\\\[5.69046pt]
&&\qquad+\,\nu^{(-1)}_{\sigma}(\Phi_{\sigma},0)\big{(}\Xi_{I}(\Phi_{\sigma}+v,c_{\sigma})-\Xi_{I}(\Phi_{\sigma},c_{\sigma})\big{)},\end{array}$
(4.96)

recalling the function
$\nu^{(-1)}_{\sigma}(u,\gamma)=\kappa_{\sigma}^{-1}(u,\gamma)-1$.

######  Corollary 4.20.

Pick $k>d/2$ and suppose that (Hf-Lip), (HCor), (Hq) and (HSt) are satisfied.
Then there exists a constant $K>0$ so that for all sufficiently small
$\sigma\geq 0$ and all $v\in H^{k+1}$ we have the estimates

$\begin{array}[]{lcl}\langle\mathcal{N}_{I;\sigma}(v),\psi_{\rm
tw}\rangle_{L^{2}}&\leq&K\|v\|_{L^{2}}\|v\|_{H^{k}}+\sigma^{2}K\|v\|_{L^{2}},\\\\[5.69046pt]
\|\mathcal{N}_{I;\sigma}(v)\|_{H^{k}}&\leq&K(1+\|v\|^{k+1}_{H^{k}})\|v\|_{H^{k}}^{2}+\sigma^{2}K\|v\|_{H^{k+1}}\big{(}1+\|v\|_{H^{k}}^{k+2}\big{)}.\end{array}$
(4.97)

###### Proof.

The $L^{2}$-inner product bound follows from Lemma 4.10 and Corollary 4.19,
together with the estimate $\|v^{2}\|_{L^{2}}\leq\|v\|_{L^{2}}\|v\|_{H^{k}}$.
The $H^{k}$-bounds follow in a similar fashion, using (4.73) to obtain the
intermediate estimate

$\begin{array}[]{lcl}\|\mathcal{N}_{I;\sigma}(v)\|_{H^{k}}&\leq&K(1+\|v\|^{k+1}_{H^{k}})\|v\|_{H^{k}}^{2}+\sigma^{2}K\|v\|_{H^{k}}\big{(}1+\|v\|_{H^{k}}^{k+1})\\\\[5.69046pt]
&&\qquad+\,\sigma^{2}K\|v\|_{H^{k+1}}\big{(}1+\|v\|_{H^{k}}^{k+2}\big{)},\end{array}$
(4.98)

which can be absorbed in the stated bound. ∎

######  Corollary 4.21.

Pick $k>d/2$ and suppose that (Hf-Lip), (HCor), (Hq) and (HSt) are satisfied.
Then there exists a constant $K>0$ so that for all sufficiently small
$\sigma\geq 0$ and all $v\in H^{k+1}$ we have the estimate

$\begin{array}[]{lcl}\|\mathcal{N}_{\sigma}(v)\|_{H^{k}}&\leq&K\|v\|_{H^{k}}^{2}(1+\|v\|^{k}_{H^{k}}+\|v\|_{H^{k+1}})+\sigma^{2}K\|v\|_{H^{k+1}}\big{(}1+\|v\|_{H^{k}}^{k+2}\big{)}.\end{array}$
(4.99)

###### Proof.

Inspecting the representation (4.94), this bound follows from (4.29) and
Corollary 4.20. ∎

###  4.5 Low dimensional bounds

In this section, we consider the case $1\leq d\leq 4$ and search for bounds in
$H^{1}$. We will make frequent use of the Sobolev embeddings

$H^{1}\hookrightarrow L^{4},\qquad H^{4/3}\hookrightarrow L^{6},\qquad
H^{3/2}\hookrightarrow L^{8},$ (4.100)

and

$H^{3/2}\hookrightarrow W^{1,8/3},\qquad H^{7/4}\hookrightarrow L^{16},\qquad
H^{2}\hookrightarrow W^{1,4}.$ (4.101)

In addition, we often encounter the interpolation bounds

$\begin{array}[]{lcl}\|v\|_{H^{4/3}}&\leq&K\|v\|_{H^{1}}^{2/3}\|v\|_{H^{2}}^{1/3},\\\\[5.69046pt]
\|v\|_{H^{3/2}}&\leq&K\|v\|_{H^{1}}^{1/2}\|v\|_{H^{2}}^{1/2},\\\\[5.69046pt]
\|v\|_{H^{7/4}}&\leq&K\|v\|_{H^{1}}^{1/4}\|v\|_{H^{2}}^{3/4}.\end{array}$
(4.102)

######  Lemma 4.22.

Pick $1\leq d\leq 4$, assume that $\Phi$ is bounded with $\Phi^{\prime}\in
H^{2}$, and consider a $C^{1}$-smooth function
$\Theta:\mathbb{R}^{n}\to\mathbb{R}^{N}$ for which $D^{\ell}\Theta$ is
globally Lipschitz for all $0\leq\ell\leq 1$. Then there exists a constant
$K>0$ so that for any pair $v_{A},v_{B}\in H^{3/2}$ we have the bound

$\|\Theta(\Phi+v_{A})-\Theta(\Phi+v_{B})\|_{H^{1}(\mathcal{D};\mathbb{R}^{N})}\leq
K(1+\|v_{B}\|_{H^{3/2}})\|v_{A}-v_{B}\|_{H^{3/2}}.$ (4.103)

If in addition $v_{B}\in H^{2}$, then we have

$\|\Theta(\Phi+v_{A})-\Theta(\Phi+v_{B})\|_{H^{1}(\mathcal{D};\mathbb{R}^{N})}\leq
K(1+\|v_{B}\|_{H^{2}})\|v_{A}-v_{B}\|_{H^{1}}.$ (4.104)

###### Proof.

The bound is clear in $L^{2}$. Inspecting the pointwise estimate (4.2), the
bound (4.103) follows from applying Hölder’s inequality which yields

$\|(v_{A}-v_{B})v_{B}^{\prime}\|_{L^{2}}\leq\|(v_{A}-v_{B})\|_{L^{8}}\|v_{B}^{\prime}\|_{L^{8/3}}$
(4.105)

and by appealing to the Sobolev embeddings (4.100)–(4.101). On the other hand,
the estimate (4.104) follows from

$\|(v_{A}-v_{B})v_{B}^{\prime}\|_{L^{2}}\leq\|v_{A}-v_{B}\|_{L^{4}}\|v_{B}^{\prime}\|_{L^{4}}$
(4.106)

and using the embeddings $H^{1}\hookrightarrow L^{4}$ and
$H^{2}\hookrightarrow W^{1,4}$. ∎

Observe, in the next result we are restricting our Lipschitz estimate to the
special case $v_{A}=v$ and $v_{B}=0$. This is on account of the pointwise term
$|v_{A}-v_{B}||\partial^{\gamma_{1}}v_{B}||\partial^{\gamma_{2}}v_{B}|$
arising from the third line of (4.109), which requires higher order norms.

######  Lemma 4.23.

Pick $1\leq d\leq 4$, assume that $\Phi$ is bounded with $\Phi^{\prime}\in
H^{2}$, and consider a $C^{2}$-smooth function
$\Theta:\mathbb{R}^{n}\to\mathbb{R}^{N}$ for which $D^{\ell}\Theta$ is
globally Lipschitz for all $0\leq\ell\leq 2$. Then there exists a constant
$K>0$ so that for any $v\in H^{2}$ we have the bound

$\|\Theta(\Phi+v)-\Theta(\Phi)\|_{H^{2}(\mathcal{D};\mathbb{R}^{N})}\leq
K(1+\|v\|_{H^{2}})\|v\|_{H^{2}}.$ (4.107)

###### Proof.

Pick two multi-indices $\gamma_{1},\gamma_{2}\in\mathbb{Z}^{d}_{\geq 0}$ with
$|\gamma_{1}|=|\gamma_{2}|=1$. Writing

$\Delta_{\gamma_{1},\gamma_{2}}\Theta=\partial^{\gamma_{1}+\gamma_{2}}\big{[}\Theta(\Phi+v_{A})-\Theta(\Phi+v_{B})\big{]},$
(4.108)

with $v_{A},v_{B}\in H^{2}$, we may compute

$\begin{array}[]{lcl}\Delta_{\gamma_{1},\gamma_{2}}\Theta&=&D^{2}\Theta(\Phi+v_{A})[\partial^{\gamma_{1}}\Phi+\partial^{\gamma_{1}}v_{A}][\partial^{\gamma_{2}}v_{A}-\partial^{\gamma_{2}}v_{B}]\\\\[5.69046pt]
&&\qquad+\,D\Theta(\Phi+v_{A})[\partial^{\gamma_{1}+\gamma_{2}}v_{A}-\partial^{\gamma_{1}+\gamma_{2}}v_{B}]\\\\[5.69046pt]
&&\qquad+\,\big{(}D^{2}\Theta(\Phi+v_{A})-D^{2}\Theta(\Phi+v_{B})\big{)}[\partial^{\gamma_{1}}\Phi+\partial^{\gamma_{1}}v_{B},\partial^{\gamma_{2}}\Phi+\partial^{\gamma_{2}}v_{B}]\\\\[5.69046pt]
&&\qquad+\,\big{(}D^{2}\Theta(\Phi+v_{A})\big{)}[\partial^{\gamma_{1}}v_{A}-\partial^{\gamma_{1}}v_{B},\partial^{\gamma_{2}}\Phi+\partial^{\gamma_{2}}v_{B}].\end{array}$
(4.109)

Note that $D\Theta$ and $D^{2}\Theta$ are globally bounded, and that for any
pair $w_{1},w_{2}\in H^{2}$ we have

$\|\partial^{\gamma_{1}}w_{1}\partial^{\gamma_{2}}w_{2}\|_{L^{2}}\leq\|\partial^{\gamma_{1}}w_{1}\|_{L^{4}}\|\partial^{\gamma_{2}}w_{2}\|_{L^{4}}\leq\|w_{1}\|_{H^{2}}\|w_{2}\|_{H^{2}}.$
(4.110)

In particular, for $v_{A}=v$ and $v_{B}=0$ we may proceed term by term to
compute

$\begin{array}[]{lcl}\|\Delta_{\gamma_{1},\gamma_{2}}\Theta\|_{L^{2}(\mathcal{D};\mathbb{R}^{N})}&\leq&K\big{[}|v\|_{H^{1}}+\|v\|_{H^{2}}^{2}+\|v\|_{H^{2}}+\|v\|_{L^{2}}+\|v\|_{H^{1}}\big{]},\end{array}$
(4.111)

which can be absorbed in the stated bound. ∎

######  Lemma 4.24.

Pick $1\leq d\leq 4$, assume that $\Phi$ is bounded with $\Phi^{\prime}\in
H^{2}$, and consider a $C^{3}$-smooth function
$\Theta:\mathbb{R}^{n}\to\mathbb{R}^{N}$ for which $D^{3}\Theta$ is globally
bounded. Then there exists a constant $K>0$ so that for any pair
$v_{A},v_{B}\in H^{4/3}$ we have the bound

$\|\Theta(\Phi+v_{A})-\Theta(\Phi+v_{B})\|_{L^{2}(\mathcal{D};\mathbb{R}^{N})}\leq
K(1+\|v_{A}\|_{H^{4/3}}^{2}+\|v_{B}\|_{H^{4/3}}^{2})\|v_{A}-v_{B}\|_{H^{4/3}}.$
(4.112)

###### Proof.

In view of the pointwise bound

$|\Theta(\Phi+v_{A})-\Theta(\Phi+v_{B})|\leq[1+|v_{A}|^{2}+|v_{B}|^{2}]|v_{A}-v_{B}|,$
(4.113)

we may compute

$\displaystyle\|\Theta(\Phi+v_{A})-\Theta(\Phi+v_{B})\|_{L^{2}(\mathcal{D};\mathbb{R}^{N})}$
$\displaystyle\leq
K[1+\|v_{A}\|^{2}_{L^{6}}+\|v_{B}\|^{2}_{L^{6}}]\|v_{A}-v_{B}\|_{L^{6}},$
(4.114)

which leads to the stated estimate using the embedding $H^{4/3}\hookrightarrow
L^{6}$. ∎

######  Lemma 4.25.

Pick $1\leq d\leq 4$, assume that $\Phi$ is bounded with $\Phi^{\prime}\in
H^{2}$, and consider a $C^{3}$-smooth function
$\Theta:\mathbb{R}^{n}\to\mathbb{R}^{N}$ for which $D^{3}\Theta$ is globally
bounded. Then there exists a constant $K>0$ so that for any pair
$v_{A},v_{B}\in H^{2}$ we have the bound

$\begin{array}[]{lcl}\|\Theta(\Phi+v_{A})-\Theta(\Phi+v_{B})\|_{H^{1}(\mathcal{D};\mathbb{R}^{N})}&\leq&K(1+\|v_{A}\|_{H^{1}}+\|v_{B}\|_{H^{2}})\|v_{A}-v_{B}\|_{H^{1}}\\\\[5.69046pt]
&&\qquad+K\|v_{A}\|_{H^{7/4}}^{2}\|(v_{A}-v_{B})\|_{H^{3/2}}\\\\[5.69046pt]
&&\qquad+K(\|v_{A}\|_{H^{7/4}}+\|v_{B}\|_{H^{7/4}})\\\\[5.69046pt]
&&\qquad\qquad\times\|v_{B}\|_{H^{3/2}}\|v_{A}-v_{B}\|_{H^{7/4}}.\end{array}$
(4.115)

###### Proof.

Pick a multi-index $\gamma\in\mathbb{Z}^{d}_{\geq 0}$ with $|\gamma|=1$.
Writing

$\Delta_{\gamma}\Theta=\partial^{\gamma}[\Theta(\Phi+v_{A})-\Theta(\Phi+v_{B})],$
(4.116)

we may inspect (4.1) to obtain the pointwise bound

$|\Delta_{\gamma}\Theta|\leq
K(1+|v_{A}|^{2})|\partial^{\gamma}v_{A}-\partial^{\gamma}v_{B}|+K(1+|v_{A}|+|v_{B}|)|v_{A}-v_{B}|(|\partial^{\gamma}\Phi|+|\partial^{\gamma}v_{B}|).$
(4.117)

Proceeding term by term, we obtain

$\begin{array}[]{lcl}\|\Delta_{\gamma}\Theta\|_{L^{2}(\mathcal{D};\mathbb{R}^{N})}&\leq&K\|v_{A}-v_{B}\|_{H^{1}}+K\|v_{A}\|_{L^{16}}^{2}\|\partial^{\gamma}(v_{A}-v_{B})\|_{L^{8/3}}\\\\[5.69046pt]
&&\qquad+K(\|v_{A}\|_{L^{4}}+\|v_{B}\|_{L^{4}})\|v_{A}-v_{B}\|_{L^{4}}\\\\[5.69046pt]
&&\qquad+K\|v_{A}-v_{B}\|_{L^{4}}\|\partial^{\gamma}v_{B}\|_{L^{4}}\\\\[5.69046pt]
&&\qquad+K(\|v_{A}\|_{L^{16}}+\|v_{B}\|_{L^{16}})\|v_{A}-v_{B}\|_{L^{16}}\|\partial^{\gamma}v_{B}\|_{L^{8/3}},\end{array}$
(4.118)

which leads to the stated bound upon applying the Sobolev embeddings
(4.100)–(4.101). ∎

######  Corollary 4.26.

Pick $1\leq d\leq 4$, assume that $\Phi$ is bounded with $\Phi^{\prime}\in
H^{2}$, and consider a $C^{3}$-smooth function
$\Theta:\mathbb{R}^{n}\to\mathbb{R}^{N}$ for which $D^{3}\Theta$ is globally
bounded. Then there exists a constant $K>0$ so that for any $v\in H^{2}$ we
have the bound

$\begin{array}[]{lcl}\|\Theta(\Phi+v)-\Theta(\Phi)\|_{L^{2}(\mathcal{D};\mathbb{R}^{N})}&\leq&K\|v\|_{H^{2}}\big{(}1+\|v\|_{H^{1}}^{2}\big{)},\\\\[5.69046pt]
\|\Theta(\Phi+v)-\Theta(\Phi)\|_{H^{1}(\mathcal{D};\mathbb{R}^{N})}&\leq&K\|v\|_{H^{2}}\big{(}1+\|v\|_{H^{1}}+\|v\|_{H^{1}}\|v\|_{H^{2}}\big{)}.\end{array}$
(4.119)

###### Proof.

Applying the interpolation estimates (4.102) to the bound (4.112) with
$v_{A}=v$ and $v_{B}=0$ leads directly to the first estimate. Performing the
same for (4.115), we find

$\begin{array}[]{lcl}\|\Theta(\Phi+v)-\Theta(\Phi)\|_{H^{1}(\mathcal{D};\mathbb{R}^{N})}&\leq&K\|v\|_{H^{1}}+K\|v\|_{H^{1}}\|v\|_{H^{2}}^{2}+K\|v\|_{H^{1}}^{2},\end{array}$
(4.120)

which can be absorbed in the stated bound. ∎

######  Lemma 4.27.

Pick $1\leq d\leq 4$, assume that $\Phi$ is bounded with $\Phi^{\prime}\in
H^{2}$, and consider a $C^{3}$-smooth function
$\Theta:\mathbb{R}^{n}\to\mathbb{R}^{N}$ for which $D^{3}\Theta$ is globally
bounded. Then there exists a constant $K>0$ so that for any $v\in H^{2}$ we
have the bounds

$\begin{array}[]{lcl}\|\Theta(\Phi+v)-\Theta(\Phi)-D\Theta(\Phi)[v]\|_{L^{2}(\mathcal{D};\mathbb{R}^{N})}&\leq&K(1+\|v\|_{H^{2}})\|v\|_{H^{1}}^{2},\\\\[5.69046pt]
\|\Theta(\Phi+v)-\Theta(\Phi)-D\Theta(\Phi)[v]\|_{H^{1}(\mathcal{D};\mathbb{R}^{N})}&\leq&K(1+\|v\|_{H^{1}})\|v\|_{H^{2}}^{2}.\end{array}$
(4.121)

###### Proof.

Recall the identity (4.12). Writing

$\mathcal{Q}=D^{2}(\Phi+stv)[v,v],$ (4.122)

for $0\leq s,t\leq 1,$ and picking a multi-index $\gamma$ with $|\gamma|=1$,
we first compute

$\partial^{\gamma}\mathcal{Q}=2D^{2}\Theta(\Phi+stv)[\partial^{\gamma}v,v]+D^{3}\Theta(\Phi+stv)[\partial^{\gamma}\Phi+st\partial^{\gamma}v,v,v],$
(4.123)

which provides the pointwise bounds

$\begin{array}[]{lcl}|\mathcal{Q}|&\leq&K(1+|v|)|v|^{2},\\\\[5.69046pt]
|\partial^{\gamma}\mathcal{Q}|&\leq&K(1+|v|)|v||\partial^{\gamma}v|+K(1+|\partial^{\gamma}v|)|v|^{2}\leq
K|v|\big{(}|v|+|\partial^{\gamma}v|+|v||\partial^{\gamma}v|\big{)}.\end{array}$
(4.124)

In particular, we find

$\begin{array}[]{lcl}\|\mathcal{Q}\|_{L^{2}(\mathcal{D};\mathbb{R}^{N})}&\leq&K\|v\|_{L^{4}}^{2}+K\|v\|_{L^{6}}^{3},\\\\[5.69046pt]
\|\partial^{\gamma}\mathcal{Q}\|_{L^{2}(\mathcal{D};\mathbb{R}^{N})}&\leq&K\|v\|_{L^{4}}^{2}+K\|v\|_{L^{8}}\|\partial^{\gamma}v\|_{L^{8/3}}+K\|v\|_{L^{16}}^{2}\|\partial^{\gamma}v\|_{L^{8/3}},\end{array}$
(4.125)

which in view of the Sobolev embeddings (4.100)–(4.101) leads to the bounds

$\begin{array}[]{lcl}\|\mathcal{Q}\|_{L^{2}(\mathcal{D};\mathbb{R}^{N})}&\leq&K\Big{(}\|v\|_{H^{1}}^{2}+\|v\|_{H^{4/3}}^{3}\Big{)},\\\\[5.69046pt]
\|\partial^{\gamma}Q\|_{L^{2}(\mathcal{D};\mathbb{R}^{N})}&\leq&K\Big{(}\|v\|_{H^{1}}^{2}+\|v\|_{H^{3/2}}\|v\|_{H^{3/2}}+\|v\|_{H^{7/4}}^{2}\|v\|_{H^{3/2}}\Big{)}.\end{array}$
(4.126)

The stated estimates now follow by applying the interpolation bounds (4.102).
∎

######  Corollary 4.28.

Suppose that (Hq), (HSt) and (HPar) are satisfied with $k=1$ and pick a multi-
index $\beta\in\mathbb{Z}^{d}_{\geq 0}$ with $|\beta|=1$. Then there exists a
constant $K>0$, which does not depend on the pair $(\Phi,c)$, so that the
following holds true. For any $v\in H^{2}$ and $\gamma\in\mathbb{R}$ we have
the bounds

$\begin{array}[]{lcl}\|\partial^{\beta}\widetilde{\mathcal{K}}_{C}(\Phi+v,\gamma)-\partial^{\beta}\widetilde{\mathcal{K}}_{C}(\Phi,\gamma)\|_{L^{2}_{Q}}&\leq&K(1+\|v\|_{L^{2}})\|v\|_{H^{1}},\\\\[5.69046pt]
\|\mathcal{K}_{C}(\Phi+v,\gamma)-\mathcal{K}_{C}(\Phi,\gamma)\|_{H^{2}}&\leq&K(1+\|v\|_{H^{1}}^{3})(1+\|v\|_{H^{2}})\|v\|_{H^{2}}.\end{array}$
(4.127)

###### Proof.

The first bound follows from (4.62) and by inspecting (4.37) with
$\gamma=\gamma_{A}=\gamma_{B}$, $v_{A}=v$ and $v_{B}=0$. To obtain the second
bound, we again consider the splitting in (4.82), but now with $v_{A}=v$ and
$v_{B}=0$. Lemma 4.8 together with (4.107) yields

$\|\mathcal{E}_{I}\|_{H^{1}}\leq
K\|v\|_{L^{2}}(1+\|v\|_{H^{2}}^{2})+K(1+\|v\|_{H^{2}})\|v\|_{H^{2}}.$ (4.128)

whereas (4.104) together with the first estimate of (4.127) allows us to find

$\begin{array}[]{lcl}\|\mathcal{E}_{II}\|_{H^{1}}&\leq&K\|v\|_{L^{2}}(1+\|v\|_{H^{1}})(1+\|v\|_{L^{2}})(1+\|v\|_{H^{1}})\\\\[5.69046pt]
&&\qquad+\,K\|v\|_{H^{1}}(1+\|v\|_{L^{2}})(1+\|v\|_{H^{1}})\\\\[5.69046pt]
&&\qquad+\,K(1+\|v\|_{H^{1}})(1+\|v\|_{L^{2}})\|v\|_{H^{1}}.\end{array}$
(4.129)

Note that both bounds can be absorbed by the stated estimate. ∎

Thanks to the preparations above, we are now ready to work towards the final
estimates for $\mathcal{N}_{\sigma}$. To this end, we recall the expressions
$\Xi_{I}$ and $\Xi_{II;\sigma}$ defined in (4.87), together with the
intermediate function $\mathcal{N}_{I;\sigma}$ and the associated
decomposition (4.96).

######  Corollary 4.29.

Suppose that (Hf-Cub), (HCor), (Hq), (HSt) and (HPar) are satisfied with
$k=1$. Then there exists a constant $K>0$, which does not depend on the pair
$(\Phi,c)$, so that the following holds true. For any $v\in H^{2}$, any
$\gamma\in\mathbb{R}$, and any $\sigma\geq 0$, the expressions

$\begin{array}[]{lcl}\Delta_{v}\Xi_{I}&=&\Xi_{I}(\Phi+v,c)-\Xi_{I}(\Phi,c),\\\\[5.69046pt]
\Delta_{v}\Xi_{II;\sigma}&=&\Xi_{II;\sigma}(\Phi+v,\gamma)-\Xi_{II;\sigma}(\Phi,\gamma)\end{array}$
(4.130)

satisfy the bounds

$\begin{array}[]{lcl}\langle\Delta_{v}\Xi_{I},\psi_{\rm
tw}\rangle_{L^{2}}&\leq&K(1+\|v\|_{H^{1}}^{2})\|v\|_{H^{2}},\\\\[5.69046pt]
\langle\Delta_{v}\Xi_{II;\sigma},\psi_{\rm
tw}\rangle_{L^{2}}&\leq&\sigma^{2}K\|v\|_{L^{2}},\\\\[5.69046pt] \end{array}$
(4.131)

together with

$\begin{array}[]{lcl}\|\Delta_{v}\Xi_{I}\|_{H^{1}}&\leq&K(1+\|v\|_{H^{1}}+\|v\|_{H^{1}}\|v\|_{H^{2}})\|v\|_{H^{2}},\\\\[5.69046pt]
\|\Delta_{v}\Xi_{II;\sigma}\|_{H^{1}}&\leq&\sigma^{2}K\big{(}1+\|v\|_{H^{1}}^{3})(1+\|v\|_{H^{2}})\|v\|_{H^{2}}.\end{array}$
(4.132)

###### Proof.

These bounds readily follow by inspecting the definitions (4.87) and applying
(4.25), (4.104), (4.119) and (4.127). ∎

######  Corollary 4.30.

Suppose that (Hf-Cub), (HCor), (Hq) and (HSt) are satisfied with $k=1$. Then
there exists a constant $K>0$ so that for all sufficiently small $\sigma\geq
0$ and all $v\in H^{2}$ we have

$\begin{array}[]{lcl}\langle\mathcal{N}_{I;\sigma}(v),\psi_{\rm
tw}\rangle_{L^{2}}&\leq&K(1+\|v\|_{H^{2}})\|v\|_{H^{1}}^{2}+\sigma^{2}K(1+\|v\|_{H^{1}}^{2})\|v\|_{H^{2}},\\\\[5.69046pt]
\|\mathcal{N}_{I;\sigma}(v)\|_{H^{1}}&\leq&K(1+\|v\|_{H^{1}})\|v\|_{H^{2}}^{2}+\sigma^{2}K(1+\|v\|_{H^{1}}^{3})(1+\|v\|_{H^{2}})\|v\|_{H^{2}}.\end{array}$
(4.133)

###### Proof.

The $L^{2}$-bound follows from Lemma 4.10, Corollary 4.29 and the quadratic
estimate (4.121). Similar computations for the $H^{1}$-bound lead to the
estimate

$\begin{array}[]{lcl}\|\mathcal{N}_{I;\sigma}(v)\|_{H^{1}}&\leq&K(1+\|v\|_{H^{1}})\|v\|_{H^{2}}^{2}\\\\[5.69046pt]
&&\qquad+\,\sigma^{2}K\|v\|_{L^{2}}(1+\|v\|_{H^{2}})(1+\|v\|_{H^{1}}+\|v\|_{H^{1}}\|v\|_{H^{2}})\\\\[5.69046pt]
&&\qquad+\,\sigma^{2}K(1+\|v\|_{H^{1}}^{3})(1+\|v\|_{H^{2}})\|v\|_{H^{2}},\end{array}$
(4.134)

which can be absorbed in the stated bound. ∎

######  Corollary 4.31.

Suppose that (Hf-Cub), (HCor), (Hq) and (HSt) are satisfied with $k=1$. Then
there exists a constant $K>0$ so that for all sufficiently small $\sigma\geq
0$ and all $v\in H^{2}$ we have

$\begin{array}[]{lcl}\|\mathcal{N}_{\sigma}(v)\|_{H^{1}}&\leq&K(1+\|v\|_{H^{1}}^{2})\|v\|_{H^{2}}^{2}+\sigma^{2}K(1+\|v\|_{H^{1}}^{3})(1+\|v\|_{H^{2}})\|v\|_{H^{2}}.\end{array}$
(4.135)

###### Proof.

By inspecting the representation (4.94), we see that Corollary 4.30 together
with (4.29) lead to

$\begin{array}[]{lcl}\|\mathcal{N}_{\sigma}(v)\|_{H^{1}}&\leq&K(1+\|v\|_{H^{1}})\|v\|_{H^{2}}^{2}\\\\[5.69046pt]
&&\qquad+\,\sigma^{2}K(1+\|v\|_{H^{1}}^{3})(1+\|v\|_{H^{2}})\|v\|_{H^{2}}\\\\[5.69046pt]
&&\qquad+\,K(1+\|v\|_{H^{2}})\|v\|_{H^{1}}^{2}(1+\|v\|_{H^{2}})\\\\[5.69046pt]
&&\qquad+\,\sigma^{2}K(1+\|v\|_{H^{1}}^{2})\|v\|_{H^{2}}(1+\|v\|_{H^{2}}),\end{array}$
(4.136)

which can be absorbed in the stated bound. ∎

##  5 Variational solutions

In this section, we establish Proposition 2.2 and study the regularity of the
obtained solutions. In particular, we provide existence and uniqueness results
for the coupled SPDE-SDE system

$\begin{cases}\mathrm{d}u=[\Delta u+f(u)+\sigma^{2}h(u)]\mathrm{d}t+\sigma
g(u)\mathrm{d}W_{t}^{Q},\\\
\mathrm{d}\gamma=[c+a_{\sigma}(u,\gamma;c)]\mathrm{d}t+\sigma
b(u,\gamma)\mathrm{d}W_{t}^{Q},\end{cases}$ (5.1)

where $u$ takes values in the affine space
$\mathcal{U}_{H^{k}(\mathcal{D};\mathbb{R}^{n})}$, i.e.,
$u=\Phi_{\mathrm{ref}}+v$ with $v\in H^{k}(\mathcal{D};\mathbb{R}^{n})$, for
some $k\geq 0$. We start by considering global existence for the case $k=0$ in
§5.1 and move on to study local existence for $k>d/2$ in §5.2, all under our
global Lipschitz assumption (Hf-Lip). We provide an alternative approach for
global existence results in §5.3 that works for $k=1$ and $1\leq d\leq 4$
under the cubic growth condition (Hf-Cub).

Our main findings beyond the $L^{2}$-based statements in Proposition 2.2 are
summarised in the following result. The precise interpretation of the
diffusion operator in the identity (5.4) is discussed throughout this section.
We emphasise that the various types of solutions that we construct agree where
they overlap; see, e.g., Lemma 5.2 below.

######  Proposition 5.1 (Higher regularity).

Suppose that either (Hf-Lip) is satisfied with $k>d/2$ or that $k=1$ and (Hf-
Cub) is satisfied. Assume furthermore that (HCor), (Hq) and (HSt) are
satisfied. Fix $T>0$ and $0\leq\sigma\leq 1$. Then for any initial condition

$(z_{0},\gamma_{0})\in H^{k}(\mathcal{D};\mathbb{R}^{n})\times\mathbb{R},$
(5.2)

there exists an increasing sequence of stopping times $(\tau_{\ell})_{\ell\geq
0}$ and a stopping time $\tau_{\infty}$, with $\tau_{\ell}\to\tau_{\infty}$
and $0<\tau_{\infty}\leq T$ $\mathbb{P}$-a.s., together with progressively
measurable maps

$z:[0,T]\times\Omega\to
H^{k}(\mathcal{D};\mathbb{R}^{n}),\quad\gamma:[0,T]\times\Omega\to\mathbb{R},$
(5.3)

that satisfy the following properties:

  1. (i)

For almost every $\omega\in\Omega$, the map
$t\mapsto(z(t,\omega),\gamma(t,\omega))$ is of class
$C([0,\tau_{\infty}(\omega));H^{k}(\mathcal{D};\mathbb{R}^{n})\times\mathbb{R})$;

  2. (ii)

We have the integrability condition $(z,\gamma)\in
L^{2}(\Omega;L^{2}([0,\tau_{\ell}];H^{k+1}(\mathcal{D};\mathbb{R}^{n})\times\mathbb{R}))$,
for any $\ell\geq 0$;

  3. (iii)

The $H^{k}(\mathcal{D};\mathbb{R}^{n})$-valued identity121212 At first, the
equality in (5.4) should be understood as an equality in $[H^{k+1}]^{*}\equiv
H^{k-1}$, but by (ii) we can conclude that we have equality in $H^{k}$.

$\displaystyle z(t)=z_{0}+\int_{0}^{t}[\Delta z(s)+\Phi_{\rm
ref}^{\prime\prime}]\mathrm{d}s$ $\displaystyle+\int_{0}^{t}f(z(s)+\Phi_{\rm
ref})\mathrm{d}s$ (5.4) $\displaystyle+\sigma^{2}\int_{0}^{t}h(z(s)+\Phi_{\rm
ref})\mathrm{d}s+\sigma\int_{0}^{t}g(z(s)+\Phi_{\rm ref})\mathrm{d}W_{s}^{Q},$

together with the scalar identity

$\gamma(t)=\gamma_{0}+\int_{0}^{t}[c+a_{\sigma}(z(s)+\Phi_{\rm
ref},\gamma(s);c)]\mathrm{d}s+\sigma\int_{0}^{t}b(z(s)+\Phi_{\rm
ref},\gamma(s))\mathrm{d}W_{s}^{Q},$ (5.5)

hold $\mathbb{P}$-a.s. for all $0\leq t<\tau_{\infty}$;

  4. (iv)

Suppose there are other progressively measurable maps $\tilde{z}$ and
$\tilde{\gamma}$ that satisfy (i)–(iii) with another stopping time
$\tilde{\tau}_{\infty}$ and localising sequence
$(\tilde{\tau}_{\ell})_{\ell\geq 0}$. Then for almost all $\omega\in\Omega,$
we have $\tilde{\tau}_{\infty}(\omega)\leq\tau_{\infty}(\omega)$ together with

$\quad\tilde{z}(t,\omega)=z(t,\omega)\quad\text{and}\quad\tilde{\gamma}(t,\omega)=\gamma(t,\omega),\quad\text{for
all}\quad 0\leq t\leq\tilde{\tau}_{\infty}(\omega).$ (5.6)

In the setting where $k=1$ and (Hf-Cub) is satisfied, we may take
$\tau_{\ell}=\tau_{\infty}=T$.

###  5.1 Global existence in $L^{2}$

Throughout this part, we consider $k=0$ and solutions where $(v,\gamma)$ is
measured with respect to the Gelfand triple
$(\mathcal{V},\mathcal{H},\mathcal{V}^{*})$ given by

$\mathcal{V}=H^{1}(\mathcal{D};\mathbb{R}^{n})\times\mathbb{R},\quad\mathcal{H}=L^{2}(\mathcal{D};\mathbb{R}^{n})\times\mathbb{R},\quad\mathcal{V}^{*}=H^{-1}(\mathcal{D};\mathbb{R}^{n})\times\mathbb{R}.$
(5.7)

The associated inner products are given by

$\displaystyle\langle(v,\gamma_{A}),(w,\gamma_{B})\rangle_{\mathcal{V}}$
$\displaystyle=\langle
v,w\rangle_{H^{1}(\mathcal{D};\mathbb{R}^{n})}+\langle\gamma_{A},\gamma_{B}\rangle_{\mathbb{R}},$
(5.8) $\displaystyle\langle(v,\gamma_{A}),(w,\gamma_{B})\rangle_{\mathcal{H}}$
$\displaystyle=\langle
v,w\rangle_{L^{2}(\mathcal{D};\mathbb{R}^{n})}+\langle\gamma_{A},\gamma_{B}\rangle_{\mathbb{R}},$

while the duality pairing acts as

$\langle(v,\gamma_{A}),(w,\gamma_{B})\rangle_{\mathcal{V}^{*};\mathcal{V}}=\langle
v,w\rangle_{H^{-1}(\mathcal{D};\mathbb{R}^{n});H^{1}(\mathcal{D};\mathbb{R}^{n})}+\langle\gamma_{A},\gamma_{B}\rangle_{\mathbb{R}},$
(5.9)

where we follow [58] to interpret the duality pairing between the spaces
$H^{-1}(\mathcal{D};\mathbb{R}^{n})$ and $H^{1}(\mathcal{D};\mathbb{R}^{n})$.
In particular, for any $v\in L^{2}(\mathcal{D};\mathbb{R}^{n})$ and $w\in
H^{1}(\mathcal{D};\mathbb{R}^{n})$ we have

$\langle
v,w\rangle_{H^{-1}(\mathcal{D};\mathbb{R}^{n});H^{1}(\mathcal{D};\mathbb{R}^{n})}=\langle
v,w\rangle_{L^{2}(\mathcal{D};\mathbb{R}^{n})}.$ (5.10)

In addition, the Laplacian can be interpreted as a bounded linear operator
mapping from $H^{1}(\mathcal{D};\mathbb{R}^{n})$ into
$H^{-1}(\mathcal{D};\mathbb{R}^{n})$ by writing

$\langle\Delta
v,w\rangle_{H^{-1}(\mathcal{D};\mathbb{R}^{n});H^{1}(\mathcal{D};\mathbb{R}^{n})}=-\langle\nabla
v,\nabla w\rangle_{L^{2}(\mathcal{D};\mathbb{R}^{n})}=-\langle
v,w\rangle_{H^{1}(\mathcal{D};\mathbb{R}^{n})}+\langle
v,w\rangle_{L^{2}(\mathcal{D};\mathbb{R}^{n})}$ (5.11)

for any pair $v,w\in H^{1}(\mathcal{D};\mathbb{R}^{n})$. In the parlance of
[4], this is analogous to the so-called weak setting.

We are now ready to prove Proposition 2.2, primarily using the estimates for
$a_{\sigma}$ and $b$ obtained in Corollary 4.9 and Lemma 4.11. As a
consequence of the global Lipschitz assumption on $f$, these estimates allow
us to establish Proposition 2.2 in a more direct manner than the approach in
[27, 30]. Indeed, we are able to directly embed the coupled system (5.1)
within a variational framework, rather than solving first for $u$ and then
interpreting the equation for $\gamma$ as an SDE with random coefficients.

It is worthwhile to point out that we can either use the classical variational
framework [58] of Liu and Röckner or the critical variational framework [4]
developed by Agresti and Veraar. Indeed, the Lipschitz estimates (4.42) and
(4.49) do not depend on both $v_{A}$ and $v_{B}$ simultaneously and only
involve $L^{2}$-norms, enabling us to verify the local monotonicity condition
required by [58] for the full coupled system (5.1).

###### Proof of Proposition 2.2 under (Hf-Lip).

It suffices to check the conditions in [58, Thm. 1.1], which provides our
statements when applied with $\alpha=2$ and $\beta=2$. We shall merely focus
on the parts regarding the local monotonicity, coercivity, and the growth
condition induced by the SDE of $\gamma$. The remaining conditions can be
readily verified by the reader by exploiting the identity (5.11) and appealing
to Lemma 4.6 and Corollary 4.7.

Using the bounds in Lemma 4.11, we see for any $v_{A},v_{B}\in
H^{1}(\mathcal{D};\mathbb{R}^{n})$ and any
$\gamma_{A},\gamma_{B}\in\mathbb{R}$ that

$\displaystyle\langle a_{\sigma}(\Phi+v_{A},\gamma_{A},c)$ $\displaystyle-
a_{\sigma}(\Phi+v_{B},\gamma_{B},c),\gamma_{A}-\gamma_{B}\rangle_{\mathbb{R}}$
(5.12) $\displaystyle\leq
K_{a}\rho(v_{A})\big{[}\|v_{A}-v_{B}\|_{L^{2}(\mathcal{D}:\mathbb{R}^{n})}+|\gamma_{A}-\gamma_{B}|\big{]}|\gamma_{A}-\gamma_{B}|$
$\displaystyle\leq
2K_{a}\rho(v_{A})\big{[}\|v_{A}-v_{B}\|_{L^{2}(\mathcal{D}:\mathbb{R}^{n})}^{2}+|\gamma_{A}-\gamma_{B}|^{2}\big{]},$

in which we have exploited the scalar identity $xy\leq x^{2}+y^{2}$ and
introduced the function

$\rho(v)=\big{[}1+\|v\|_{L^{2}(\mathcal{D};\mathbb{R}^{n})}\big{]}^{2}.$
(5.13)

In addition, for any $v\in H^{1}(\mathcal{D};\mathbb{R}^{n})$ and
$\gamma\in\mathbb{R}$ we may use the scalar identity above again together with
$x\leq 1+x^{2}$ to conclude

$\displaystyle\langle a_{\sigma}(\Phi+v,\gamma,c),\gamma\rangle_{\mathbb{R}}$
$\displaystyle\leq|a_{\sigma}(\Phi+v,\gamma,c)\|\gamma|$ (5.14)
$\displaystyle\leq
K_{a}\big{[}1+\|v\|_{L^{2}(\mathcal{D};\mathbb{R}^{n})}\big{]}|\gamma|$
$\displaystyle\leq
2K_{a}\big{[}1+\|v\|_{L^{2}(\mathcal{D};\mathbb{R}^{n})}^{2}+|\gamma|^{2}\big{]}.$

The second estimate in (4.42) together with the bound in (5.12) now yield the
local monotonicity property [58, (H2)], whereas the coercivity condition [58,
(H3)] and the growth condition [58, (H4)] follow from (4.40) and (5.14). ∎

###  5.2 Local existence in $H^{k}$

In this part we investigate the regularity of the solution found in
Proposition 2.2, which we refer to as $(\tilde{z},\tilde{\gamma})\in
L^{2}(\mathcal{D};\mathbb{R}^{n})\times\mathbb{R}$. We do this by establishing
the (local) existence and uniqueness of solutions in
$H^{k}(\mathcal{D};\mathbb{R}^{n})\times\mathbb{R}$, which we write as
$(z,\gamma).$ If $z$ has an initial value in
$H^{k}(\mathcal{D};\mathbb{R}^{n})$, then it remains in
$H^{k}(\mathcal{D};\mathbb{R}^{n})$ for at least a short time. In particular,
we show that $(z,\gamma)$ coincides with the solution
$(\tilde{z},\tilde{\gamma})$, where they overlap, showing that $z$ persists
globally as a continuous $L^{2}$-valued solution even after the
$H^{k}$-smoothness is lost.

Indeed, one can follow [4, 7, 48] to show that for $k\geq 1$ the bilinear map

$\langle v,w\rangle_{H^{k-1};H^{k+1}}=\langle
v,w\rangle_{H^{k-1}}-\sum_{|\alpha|=k-1}\langle\partial^{\alpha}v,\partial^{\alpha}\Delta
w\rangle_{L^{2}}$ (5.15)

allows $H^{k-1}$ to be interpreted as the dual of $H^{k+1}$. In order to
confirm that this duality pairing is compatible with the inner product of
$H^{k}$, it suffices to compute

$\langle v,w\rangle_{H^{k-1};H^{k+1}}=\langle
v,w\rangle_{H^{k-1}}+\sum_{|\alpha|=k-1}\langle\partial^{\alpha}\nabla
v,\partial^{\alpha}\nabla w\rangle_{L^{2}}=\langle v,w\rangle_{H^{k}},$ (5.16)

whenever $v\in H^{k}$ and $w\in H^{k+1}$. In addition, the diffusion operator
$\Delta$ can—as usual—be seen as an element of $\mathscr{L}(H^{k+1};H^{k-1})$.
The definition (5.15) yields

$\langle\Delta v,w\rangle_{H^{k-1};H^{k+1}}=-\langle\nabla v,\nabla
w\rangle_{H^{k}}=-\langle v,w\rangle_{H^{k+1}}+\langle v,w\rangle_{L^{2}}$
(5.17)

for any pair $v,w\in H^{k+1}$, hence generalising (5.11) and providing an
alternative yet equivalent definition for $\Delta$ from the space $H^{k+1}$
into its dual.

As explained in the introduction, the presence of derivatives generates cross
terms that violate the monotonicity requirements in [58]; see e.g., Lemma
4.16. We therefore appeal to the critical variational framework [4] instead.
We remark that the proof below only requires the $C^{k}$-Lipschitz smoothness
on both $f$ and $g$.

###### Proof of Proposition 5.1 under (Hf-Lip).

For any integer $n\in\mathbb{N},$ the bounds in Corollaries 4.13 and 4.16
allow us to find constants $C_{1,n},C_{2,n}>0$ for which the estimates

$\|f(\Phi+v_{A})-f(\Phi+v_{B})\|_{H^{k-1}}+\|h(\Phi+v_{A})-h(\Phi+v_{B})\|_{H^{k-1}}\leq
C_{1,n}\|v_{A}-v_{B}\|_{H^{k}},$ (5.18)

and

$\|g(\Phi+v_{A})-g(\Phi+v_{B})\|_{HS(L^{2}_{Q};H^{k})}\leq
C_{2,n}\|v_{A}-v_{B}\|_{H^{k}}$ (5.19)

hold whenever $\|v_{A}\|_{H^{k}},\|v_{B}\|_{H^{k}}\leq n$. Together with the
fact that (5.17) implies

$\langle\Delta
v,v\rangle_{H^{k-1};H^{k+1}}\leq-\|v\|_{H^{k+1}}^{2}+\|v\|_{H^{k}}^{2},$
(5.20)

and recalling the estimates related to the $\gamma$-variable obtained in the
proof of Proposition 2.2, we note that the result follows by appealing to [4,
Thm 3.3]. In particular, the latter yields the existence and uniqueness of a
maximal solution $((z,\gamma),\sigma_{\infty})$ with a corresponding
localising sequence $(\sigma_{\ell})_{\ell\geq 0}$. Upon defining the stopping
times

$\tau_{\ell}=\sigma_{\ell}\wedge\inf\\{t\geq 0:\sup_{t\geq
0}\|z(t)\|_{H^{k}}^{2}+\int_{0}^{t}\|z(s)\|_{H^{k+1}}^{2}\mathrm{d}s+\int_{0}^{t}|\gamma(s)|^{2}\mathrm{d}s>\ell\\},$
(5.21)

the blow up criterion of $\sigma_{\infty}$ in [4, Thm 3.3] shows that
$\tau_{\ell}\to\sigma_{\infty}$ holds. We may hence set
$\tau_{\infty}=\sigma_{\infty}$, from which all the claims follow. ∎

######  Lemma 5.2.

Consider the setting of Proposition 5.1 where (Hf-Lip) is satisfied. Writing
$(\tilde{z},\tilde{\gamma})$ for the
$L^{2}(\mathcal{D};\mathbb{R})\times\mathbb{R}$-valued solution constructed in
Proposition 2.2, we have

$\tilde{z}(t,\omega)=z(t,\omega)\quad\text{and}\quad\tilde{\gamma}(t,\omega)=\gamma(t,\omega),$
(5.22)

for all $0\leq t<\tau_{\infty}(\omega)$ and almost every $\omega\in\Omega$.

###### Proof.

Without loss, set $h=0.$ Although we can identify the identity in (5.4) as an
equality in $H^{k}$, it is actually an equality in $[H^{k+1}]^{*}$, which we
will now exploit. Indeed, for all $\zeta\in H^{k+1}$ we have

$\displaystyle\langle z(t),\zeta\rangle_{H^{k}}$ $\displaystyle=\langle
z_{0},\zeta\rangle_{H^{k}}+\int_{0}^{t}\langle\Delta
z(s),\zeta\rangle_{H^{k-1};H^{k+1}}\mathrm{d}s+\int_{0}^{t}\langle
f(z(s)+\Phi_{\rm ref}),\zeta\rangle_{H^{k}}\mathrm{d}s$ (5.23)
$\displaystyle\hskip 85.35826pt+\sigma\int_{0}^{t}\langle g(z(s)+\Phi_{\rm
ref})\mathrm{d}W_{s}^{Q},\zeta\rangle_{H^{k}}.$

Picking an arbitrary $\eta\in C_{c}^{\infty}(\mathcal{D};\mathbb{R}^{n})$, we
write $\zeta=(1-\Delta)^{-k}\eta$ and note that $\zeta\in H^{\ell}$ for any
$\ell\geq 0$. For any $v\in H^{k}$, Parseval’s identity (see Appendix C)
yields

$\begin{array}[]{lcl}\langle
v,\zeta\rangle_{H^{k}}&=&\langle\boldsymbol{\xi}\mapsto(1+|\boldsymbol{\xi}|^{2})^{k/2}\hat{v}(\boldsymbol{\xi}),\boldsymbol{\xi}\mapsto(1+|\boldsymbol{\xi}|^{2})^{k/2}\hat{\zeta}(\boldsymbol{\xi})\rangle_{L^{2}(\widehat{\mathcal{D}};\mathbb{R}^{n})}\\\\[5.69046pt]
&=&\langle\boldsymbol{\xi}\mapsto\hat{v}(\boldsymbol{\xi}),\boldsymbol{\xi}\mapsto(1+|\boldsymbol{\xi}|^{2})^{k}\hat{\zeta}(\boldsymbol{\xi})\rangle_{L^{2}(\widehat{\mathcal{D}};\mathbb{R}^{n})}\\\\[5.69046pt]
&=&\langle v,(1-\Delta)^{k}\zeta\rangle_{L^{2}}\\\\[5.69046pt] &=&\langle
v,\eta\rangle_{L^{2}}.\end{array}$ (5.24)

In addition, we have the pathwise identities

$\langle\Delta z(s),\zeta\rangle_{H^{k-1};H^{k+1}}=\langle
z(s),\Delta\zeta\rangle_{H^{k}}=\langle z(s),\Delta\eta\rangle_{L^{2}},$
(5.25)

for any $0\leq s<\tau_{\infty}$. We find that $z(t)$ is an analytically weak
solution in $L^{2},$ i.e.,

$\displaystyle\langle z(t),\eta\rangle_{L^{2}}$ $\displaystyle=\langle
z_{0},\eta\rangle_{L^{2}}+\int_{0}^{t}\langle
z(s),\Delta\eta\rangle_{L^{2}}\mathrm{d}s+\int_{0}^{t}\langle f(z(s)+\Phi_{\rm
ref}),\eta\rangle_{L^{2}}\mathrm{d}s$ (5.26) $\displaystyle\hskip
85.35826pt+\sigma\int_{0}^{t}\langle g(z(s)+\Phi_{\rm
ref})\mathrm{d}W_{s}^{Q},\eta\rangle_{L^{2}},$

and conclude that $z(t)$ is a solution in the sense of Proposition 2.2 by
invoking a standard density argument. The fact that $\gamma(t)$ and
$\tilde{\gamma}(t)$ coincide directly follows. ∎

###  5.3 Global existence in $H^{1}$ for $1\leq d\leq 4$

We conclude by considering an alternative approach towards global existence
that is valid in low spatial dimensions, namely $1\leq d\leq 4$. The relevant
Gelfand triple is given by

$\mathcal{V}=H^{2}(\mathcal{D};\mathbb{R}^{n})\times\mathbb{R},\quad\mathcal{H}=H^{1}(\mathcal{D};\mathbb{R}^{n})\times\mathbb{R},\quad\mathcal{V}^{*}=L^{2}(\mathcal{D};\mathbb{R}^{n})\times\mathbb{R},$
(5.27)

which is analogous to the strong setting in [4]. The duality pairings are the
same as those in §5.2 upon fixing $k=1$ and we consider the cubic growth
condition (Hf-Cub).

For spatial dimension $d=1$, the classical framework [58] can be used to
construct solutions to (5.1) in the weak setting (5.7), under the additional
variational condition (1.16) [27, 30]. However, the Sobolev embeddings simply
do not work out in $d>1$, while the analogue of (1.16) fails in the strong
setting (5.27). Fortunately, such an inequality is not needed for the approach
developed in the critical variational framework [4].

Let us point out that we cannot solve $(z,\gamma)$ in one go as in the
previous settings. This is because the estimate (5.29) for $f$ below implies

$\displaystyle|a_{\sigma}(\Phi+v_{A},\gamma_{A},c)-a_{\sigma}(\Phi+v_{B},\gamma_{B},c)|\leq$
$\displaystyle\,K\big{[}1+\|v_{A}\|^{2}_{H^{4/3}}+\|v_{B}\|^{2}_{H^{4/3}}\big{]}\|v_{A}-v_{B}\|_{H^{4/3}}$
(5.28)
$\displaystyle\quad\quad+K\big{[}1+\|v_{B}\|_{H^{4/3}}^{3}\big{]}|\gamma_{A}-\gamma_{B}|,$

which does not satisfy Assumption 3.1 in [4] due to the cubic growth term.
Nevertheless, we can solve for $z$ first and follow the random-coefficient
approach developed in [27, 30] to understand $\gamma$.

We remark that within this setting the dimension restriction $1\leq d\leq 4$
is induced by both the nonlinearity $f$ and the noise term $g$. It is related
to the critical exponents $4/3$ and $3/2$ that appear in [4] and the proof
below.

###### Proof of Proposition 5.1 under (Hf-Cub).

As before, the estimates pertaining to the $\gamma$-variable in the proof of
Proposition 2.2 also suffice here, so we only need to focus on the terms
related to the SPDE of $z$. Without loss, we take $h=0$ again. On account of
(Hf-Cub), Lemma 4.24 provides the bound

$\displaystyle\|f(\Phi+v_{A})-f(\Phi+v_{B})\|_{L^{2}}$ $\displaystyle\leq
C_{1}[1+\|v_{A}\|^{2}_{H^{4/3}}+\|v_{B}\|^{2}_{H^{4/3}}]\|v_{A}-v_{B}\|_{H^{4/3}},$
(5.29)

which involves the exponent $4/3$ that is critical in [4]. In addition, Lemmas
4.5 and 4.22 together yield the estimate

$\|g(\Phi+v_{A})-g(\Phi+v_{B})\|_{HS(L^{2}_{Q};H^{1})}\leq
C_{2}(1+\|v_{B}\|_{H^{3/2}})\|v_{A}-v_{B}\|_{H^{3/2}},$ (5.30)

where again the value $3/2$ is critical. The remaining conditions needed to
invoke [4, Thm. 3.4] for the first equation in (5.1) can be readily verified.
As a result, we may write $u=\Phi_{\mathrm{ref}}+z$ for the solution to this
equation, and conclude that $z=(z(t))_{t\in[0,T]}$ lives in the Bochner spaces

$C([0,T];H^{1})\quad\text{and}\quad L^{2}([0,T];H^{2}).$ (5.31)

Using the interpolation bound
$\|v\|_{H^{4/3}}\leq\|v\|_{H^{1}}^{2/3}\|v\|_{H^{2}}^{1/3}$, we obtain

$\displaystyle\int_{0}^{T}\|z(t)\|_{H^{4/3}}^{3}\,\mathrm{d}t$
$\displaystyle\leq\|z\|_{C([0,T];H^{1})}\int_{0}^{T}\|z(t)\|_{H^{2}}\,\mathrm{d}t$
(5.32)
$\displaystyle\leq\sqrt{T}\|z\|_{C([0,T];H^{1})}\|z\|_{L^{2}([0,T];H^{2})},$

which implies $z\in L^{3}([0,T];H^{4/3})$. In particular, the bound (5.28)
yields

$\displaystyle|a_{\sigma}(\Phi+z(t),\gamma_{A},c)-a_{\sigma}(\Phi+z(t),\gamma_{B},c)|\leq
K\rho\big{(}z(t)\big{)}|\gamma_{A}-\gamma_{B}|$ (5.33)

with the weight function

$\rho(v)=1+\|v\|_{H^{4/3}}^{3}.$ (5.34)

Note that $t\mapsto\rho\big{(}z(t)\big{)}$ is integrable. This allows us to
follow the approach in the proof of [30, Prop. 4.5.2]—which refers to [73, Ch.
3]—to establish the global existence and uniqueness of $\gamma(t)$. ∎

We remark that under the weaker condition (Hf-Lip), the same result can be
obtained by simply applying the classical variational framework [58] directly
to the full problem (5.1). In particular, this approach uses the estimates

$\begin{array}[]{lcl}\|f(\Phi+v_{A})-f(\Phi+v_{B})\|_{H^{1}}&\leq&K(1+\|v_{B}\|_{H^{2}})\|v_{A}-v_{B}\|_{H^{1}},\\\\[5.69046pt]
\|g(\Phi+v_{A})-g(\Phi+v_{B})\|_{HS(L^{2}_{Q};H^{1})}&\leq&K(1+\|v_{B}\|_{H^{2}})\|v_{A}-v_{B}\|_{H^{1}},\end{array}$
(5.35)

that follow from Lemma 4.22 and are again valid for $1\leq d\leq 4$ only.
Alternatively, one can invoke the framework in [4], for which it suffices to
use the bound

$\|f(\Phi+v_{A})-f(\Phi+v_{B})\|_{L^{2}}\leq K\|v_{A}-v_{B}\|_{L^{2},}$ (5.36)

in combination with (5.30).

###### Proof of Proposition 2.2 under (Hf-Cub).

This is simply a restatement of the global $H^{1}$-results in Proposition 5.1.
∎

##  6 Evolution equations of the perturbation

In this section, we establish equations for the evolution of the perturbation

$v(t)=T_{-\gamma(t)}u(t)-\Phi=T_{-\gamma(t)}[z(t)+\Phi_{\rm ref}]-\Phi,$ (6.1)

where $z(t)$ and $\gamma(t)$ were constructed in §5. In §6.1 we show that
$v(t)$ satisfies

$\mathrm{d}v=\mathcal{R}_{\sigma}(v;c,\Phi)\mathrm{d}t+\sigma\mathcal{S}(v;\Phi)\mathrm{d}W_{s}^{Q},$
(6.2)

in the variational sense, where the nonlinearities $\mathcal{R}_{\sigma}$ and
$\mathcal{S}$ are given by

$\mathcal{R}_{\sigma}(v;c,\Phi)=\Delta_{y}v+\kappa_{\sigma}(\Phi+v,0)\big{[}\partial_{x}^{2}v+\Phi^{\prime\prime}+\mathcal{J}_{\sigma}(\Phi+v,0;c)\big{]}+a_{\sigma}(\Phi+v,0,c)\partial_{x}(\Phi+v)$
(6.3)

and

$\mathcal{S}(v;\Phi)[\xi]=g(\Phi+v)[\xi]+\partial_{x}(\Phi+v)b(\Phi+v,0)[\xi],\quad\xi\in
L^{2}_{Q};$ (6.4)

see Appendix A for the definitions of $a_{\sigma}$, $b$, $\kappa_{\sigma}$ and
$\mathcal{J}_{\sigma}$, which are analogous to the $d=1$ case and do not
involve second derivatives.

It is worthwhile to point out that $v(t)$ explicitly depends on the phase
$\gamma(t)$, yet $\gamma(t)$ is absent in the equations above. This is a
consequence of the translational invariance of our system, which allows us to
reduce the coupled system (5.1) to a single system of equations; see
(A.25)–(A.26).

Subsequently, in §6.2 we consider a stochastic time transformation to change
the coefficient in front of $\partial_{x}^{2}$ into unity; as in
$\mathcal{L}_{\rm tw}$. In particular, we show that the transformed function
$\bar{v}(t)$ satisfies the system

$\mathrm{d}\bar{v}=[\mathcal{L}_{\rm
tw}\bar{v}+\kappa_{\sigma}(\Phi+\bar{v},0)^{-1}\Delta_{y}\bar{v}+\mathcal{N}_{\sigma}(\bar{v})]\mathrm{d}t+\mathcal{M}_{\sigma}(\bar{v})\mathrm{d}{W}_{t}^{Q},$
(6.5)

where the definitions of $\mathcal{N}_{\sigma}$ and $\mathcal{M}_{\sigma}$ can
be found in (A.28). As before, note that these functions do not contain second
derivatives and are analogous to the $d=1$ case. The main message is that we
have cleanly isolated the linear operator $\mathcal{L}_{\rm tw}$, allowing us
to pass to a mild formulation where we can exploit the semigroup $S_{\rm
tw}(t)$. Indeed, in §6.3 we show that the variational solution to (6.5) is
also a mild solution in some sense, however, the time-dependent coefficient in
front of the Laplacian $\Delta_{y}$ forces us to consider random evolution
families, causing significant complications for our $d>1$ case that were
absent in [27, 30].

###  6.1 An application of Itô’s formula

Our starting point here is the local $H^{k}$-valued solutions constructed in
Proposition 5.1, which we will use to provide a rigorous interpretation for
(6.2). To this end, we note that the second derivatives in
$\mathcal{R}_{\sigma}$ can be interpreted as an element in
$\mathscr{L}(H^{k+1},H^{k-1})$ in the usual sense, which corresponds with the
duality pairing

$\langle[\Delta_{y}+\kappa\partial_{x}^{2}]v,w\rangle_{H^{k-1};H^{k+1}}=-\langle\nabla_{y}v,\nabla_{y}w\rangle_{H^{k}}-\kappa\langle\partial_{x}v,\partial_{x}w\rangle_{H^{k}},$
(6.6)

for any $\kappa\in\mathbb{R}$, $v\in H^{k+1}$, and $w\in H^{k+1}$.

######  Proposition 6.1.

Consider the setting of Proposition 5.1 and suppose that condition (HPar)
holds. Then the map

$v:[0,T]\times\Omega\to H^{k}(\mathcal{D};\mathbb{R}^{n})$ (6.7)

defined in (6.1) is progressively measurable and satisfies the following
properties:

  1. (i)

For almost every $\omega\in\Omega$, the map $t\mapsto v(t,\omega)$ is of class
$C([0,\tau_{\infty}(\omega));H^{k}(\mathcal{D};\mathbb{R}^{n}))$;

  2. (ii)

For any $\ell\geq 0$ we have the integrability condition $v\in
L^{2}(\Omega;L^{2}([0,\tau_{\ell}];H^{k+1}(\mathcal{D};\mathbb{R}^{n}))$;

  3. (iii)

For almost every $\omega\in\Omega$ and any $\ell\geq 0$, we have
$\mathcal{R}_{\sigma}(v(\cdot,\omega);c,\Phi)\in
L^{1}([0,\tau_{\ell}(\omega)];H^{k-1}(\mathcal{D};\mathbb{R}^{n}))$, together
with $\mathcal{S}(v;\Phi)\in
L^{2}(\Omega;L^{2}([0,\tau_{\ell}];HS(L^{2}_{Q};H^{k}(\mathcal{D};\mathbb{R}^{n})))$;

  4. (iv)

The $H^{k}(\mathcal{D};\mathbb{R}^{n})$-valued identity

$v(t)=v(0)+\int_{0}^{t}\mathcal{R}_{\sigma}(v(s);c,\Phi)\mathrm{d}s+\sigma\int_{0}^{t}\mathcal{S}(v(s);\Phi)\mathrm{d}W_{s}^{Q},$
(6.8)

holds $\mathbb{P}$-a.s. for all $0\leq t<\tau_{\infty}$.

###### Proof.

Properties (i)–(iii) follow rather directly from Proposition 5.1. The proof of
(iv) is completely analogous to that of in [30, Sec. 5.4], because an
application of Itô’s formula only results into terms with derivatives with
respect to the direction of the wave, since $\gamma(t)$ only affects the
$x$-coordinate. In particular, the Laplacian with respect to $y$ introduces no
unexpected terms.

In more detail, we pick an arbitrary test function $\zeta\in
C_{c}^{\infty}(\mathcal{D};\mathbb{R}^{n})$ and consider the maps

$\psi_{1;\zeta}:H^{k-1}\times\mathbb{R}\to\mathbb{R},\quad\psi_{2;\zeta}:\mathbb{R}\to\mathbb{R},$
(6.9)

defined by

$\psi_{1;\zeta}(z,\gamma)=\langle z,T_{\gamma}\zeta\rangle_{H^{k-1};H^{k+1}}$
(6.10)

and

$\psi_{2;\zeta}(z,\gamma)=\langle\Phi_{\rm
ref}-T_{\gamma}\Phi_{\sigma},T_{\gamma}\zeta\rangle_{H^{k-1};H^{k+1}}=\langle\Phi_{\rm
ref}-T_{\gamma}\Phi_{\sigma},T_{\gamma}\zeta\rangle_{H^{k}}.$ (6.11)

By construction, we have the identity

$\langle
v(t),\zeta\rangle_{H^{k}}=\psi_{1;\zeta}(z(t),\gamma(t))+\psi_{2;\zeta}(\gamma(t)).$
(6.12)

Performing computations as in [30, Lem. 5.5] and [30, Cor. 5.6] leads to the
expression

$\langle v(t),\zeta\rangle_{H^{k}}=\langle
v(0),\zeta\rangle_{H^{k}}+\int_{0}^{t}\langle\mathcal{R}_{\sigma}(v(s);c,\Phi),\zeta\rangle_{H^{k-1};H^{k+1}}\mathrm{d}s+\sigma\int_{0}^{t}\langle\mathcal{S}(v(s);\Phi)\mathrm{d}\widetilde{W}_{s}^{Q},\zeta\rangle_{H^{k}},$
(6.13)

where

$\widetilde{W}_{t}^{Q}=\sum_{k=0}^{\infty}\int_{0}^{t}T_{-\gamma(s)}\sqrt{Q}e_{k}\mathrm{d}\beta_{k}(s)$
(6.14)

is a stochastic process that is, in fact, indistinguishable from the
cylindrical $Q$-Wiener process $W^{Q}_{t}$ on account of the translational
invariance of $Q$; see the proof of [30, Prop 5.4] for more information. By
convention, we may replace the stochastic process $\widetilde{W}^{Q}_{t}$ by
$W^{Q}_{t}$, which proves the assertion. ∎

######  Remark 6.2.

As a matter of fact, throughout the proof of Proposition 6.1, one only needs
to exploit the translation invariant property of our noise and our system in
the $x$-coordinate. Therefore, it is also possible to consider other types of
noise with a more general $y$-dependence.

###  6.2 Stochastic time transformation

Our goal here is to introduce the time transformation that will lead to the
system (6.5). As in the one-dimensional setting [27, 30], we will rescale time
homogeneously over space in order to divide out the troublesome
$\kappa_{\sigma}$ coefficient in front of the $\partial_{x}^{2}$ term in
(6.2). However, it will reappear in front of the $\Delta_{y}$ term, which
means that the resulting problem is still quasi-linear instead of semi-linear
as in the one-dimensional setting. Nevertheless, this repositioning is an
essential part of our analysis.

Consider $v=(v(t))_{t\in[0,T]}$ to be the map defined in (6.1), where we let
$(z,\gamma)$ be the global solution constructed in Proposition 2.2, and where
we take $(\Phi,c)=(\Phi_{\sigma},c_{\sigma})$ from this point forward. Let us
introduce the notation

$\tau_{\sigma;v}(t,\omega)=\int_{0}^{t}\kappa_{\sigma}(\Phi_{\sigma}+v(s,\omega),0)\,\mathrm{d}s.$
(6.15)

Lemma 4.10 shows that $t\mapsto\tau_{\sigma;v}(t)$ is a continuous strictly
increasing $\mathbb{F}$-adapted process. This implies that it admits an
inverse map $t_{\sigma;v}:[0,T]\times\Omega\to[0,T]$ that satisfies

$\tau_{\sigma;v}(t_{\sigma;v}(\tau,\omega),\omega)=\tau,\qquad
t_{\sigma;v}(\tau_{\sigma;v}(t,\omega),\omega)=t,$ (6.16)

for (almost) every $\omega\in\Omega.$ Specifically, we may use (4.46) to find
a constant $K_{\kappa}\geq 1$ such that

$t\leq\tau_{\sigma;v}(t)\leq K_{\kappa}t,\qquad K_{\kappa}^{-1}\tau\leq
t_{\sigma;v}(\tau)\leq\tau,$ (6.17)

for all $0\leq t,\tau\leq T$.

We now define the stochastic time transformed perturbation as

$\bar{v}(\tau,\omega)=v(t_{\sigma;v}(\tau,\omega),\omega).$ (6.18)

Applying standard time transformation rules [27, Lem. 6.2] to (6.2) formally
leads to the system

$\mathrm{d}\bar{v}=\kappa_{\sigma}^{-1}(\Phi_{\sigma}+\bar{v},0)\mathcal{R}_{\sigma}(\bar{v};c_{\sigma},\Phi_{\sigma})\mathrm{d}\tau+\sigma\kappa_{\sigma}^{-1/2}(\Phi_{\sigma}+\bar{v},0)\mathcal{S}(\bar{v};\Phi_{\sigma}){\mathrm{d}\overline{W}}_{\tau}^{Q},$
(6.19)

in which $\overline{W}_{\tau}^{Q}$ is again a $Q$-cylindrical Wiener process,
but now adapted to the filtration
$\overline{\mathbb{F}}=(\overline{\mathcal{F}}_{\tau})_{\tau\geq 0}$ given by

$\overline{\mathcal{F}}_{\tau}=\\{A\in\mathcal{F}:A\cap\\{t_{\sigma;v}(\tau)\leq
t\\}\in\mathcal{F}_{t},\,\text{for all }t\geq 0\\}.$ (6.20)

In particular, we have

$\overline{W}^{Q}_{\tau}=\sum_{k=0}^{\infty}\sqrt{Q}e_{k}\bar{\beta}_{k}(\tau),\qquad\bar{\beta}_{k}(\tau)=\int_{0}^{\tau}\frac{1}{\sqrt{\partial_{\tau}t_{\sigma;v}(\tau^{\prime})}}\mathrm{d}\beta_{k}(t_{\sigma;v}(\tau^{\prime})).$
(6.21)

For our purposes here, it suffices to note that $\overline{W}^{Q}_{\tau}$ has
the same statistical properties as $W^{Q}_{t}$. Upon recalling the definitions
(A.28), we see that (6.19) can be written in the form

$\mathrm{d}\bar{v}=[\mathcal{L}_{\rm
tw}\bar{v}+\kappa_{\sigma}(\Phi_{\sigma}+\bar{v},0)^{-1}\Delta_{y}\bar{v}+\mathcal{N}_{\sigma}(\bar{v})]\mathrm{d}t+\mathcal{M}_{\sigma}(\bar{v})\mathrm{d}{\overline{W}}_{t}^{Q}.$
(6.22)

These computations are made rigorous in the following result, in which we have
introduced the transformed stopping times

$\bar{\tau}_{\ell}=\tau_{\sigma;v}(\tau_{\ell})\wedge
T,\qquad\bar{\tau}_{\infty}=\tau_{\sigma;v}(\tau_{\infty})\wedge T.$ (6.23)

######  Proposition 6.3 (stochastic time transform).

Consider the setting of Proposition 5.1 and suppose that condition (HPar)
holds. Then the map

$\bar{v}:[0,T]\times\Omega\to H^{k}(\mathcal{D};\mathbb{R}^{n})$ (6.24)

defined in (6.18) is progressively measurable with respect to the filtration
$\overline{\mathbb{F}}=(\overline{\mathcal{F}}_{t})_{t\geq 0}$ in (6.20) and
satisfies the following properties:

  1. (i)

For almost every $\omega\in\Omega$, the map $t\mapsto\bar{v}(t,\omega)$ is of
class $C([0,\bar{\tau}_{\infty}(\omega));H^{k}(\mathcal{D};\mathbb{R}^{n}))$;

  2. (ii)

For any $\ell\geq 0$ we have the integrability condition $\bar{v}\in
L^{2}(\Omega;L^{2}([0,\bar{\tau}_{\ell}];H^{k+1}(\mathcal{D};\mathbb{R}^{n})))$;

  3. (iii)

For almost every $\omega\in\Omega$ and any $\ell\geq 0$, we have
$\mathcal{N}_{\sigma}(\bar{v}(\cdot,\omega))\in
L^{1}([0,\bar{\tau}_{\ell}(\omega)];H^{k}(\mathcal{D};\mathbb{R}^{n}))$,
together with $\mathcal{M}_{\sigma}(\bar{v})\in
L^{2}(\Omega;L^{2}([0,\bar{\tau}_{\ell}];HS(L^{2}_{Q},H^{k}(\mathcal{D};\mathbb{R}^{n})))$;

  4. (iv)

The $H^{k}(\mathcal{D};\mathbb{R}^{n})$-valued identity

$\displaystyle\bar{v}(t)$
$\displaystyle=\bar{v}(0)+\int_{0}^{t}\big{[}\mathcal{L}_{\rm
tw}\bar{v}(s)+\kappa_{\sigma}(\Phi_{\sigma}+\bar{v}(s),0)^{-1}\Delta_{y}\bar{v}(s)\big{]}\mathrm{d}s$
(6.25)
$\displaystyle\qquad\qquad\qquad\qquad\qquad\quad+\int_{0}^{t}\mathcal{N}_{\sigma}(\bar{v}(s))\mathrm{d}s+\sigma\int_{0}^{t}\mathcal{M}_{\sigma}(\bar{v}(s))\mathrm{d}\overline{W}_{s}^{Q}$

holds $\mathbb{P}$-a.s. for all $0\leq t<\bar{\tau}_{\infty}$.

###### Proof.

The result can be obtained by following the proof of [27, Prop. 6.3] and [30,
Lem. 6.3]. ∎

###  6.3 Mild formulation

We are now ready to recast the system (6.22) into an appropriate mild
formulation. The quasi-linear nature of the problem causes several
complications that need to be addressed, which we achieve by utilising the
theory of forward integration discussed in §3.2.

Let $\bar{v}=(\bar{v}(t))_{t\in[0,T]}$ denote the stochastic process defined
by (6.18), seen as the globally continuous $L^{2}$-valued process on account
of Proposition 2.2. Observe that this process is indistinguishable from the
one in Proposition 6.3 on the interval $[0,\bar{\tau}_{\infty})$ as a result
of Lemma 5.2. Having a globally defined $\bar{v}$ allows us to introduce the
random function

$\nu_{\sigma}(t,\omega):=\kappa_{\sigma}(\Phi_{\sigma}+\bar{v}(t,\omega),0)^{-1}$
(6.26)

for all time $0\leq t\leq T.$ Notice that this function is progressively
measurable and continuous with respect to $t$ for almost every
$\omega\in\Omega$ due to Lemma 4.10 and the fact

$\displaystyle|\nu_{\sigma}(t)-\nu_{\sigma}(s)|=|\kappa_{\sigma}(\Phi+\bar{v}(t),0)^{-1}-\kappa_{\sigma}(\Phi+\bar{v}(s),0)^{-1}|\leq
K\sigma^{2}\|\bar{v}(t)-\bar{v}(s)\|_{L^{2}},\quad t,s\geq 0.$ (6.27)

Indeed, recall $\bar{v}\in C([0,T],L^{2}).$ In particular, property (H$\nu$)
is satisfied, with constants $K_{\nu}=1$ and
$k_{\nu}=\frac{1}{1+\sigma^{2}K}>0$ for some $K>0.$ Note that for $\sigma=0$
we simply have $\nu(t)\equiv 1$.

By construction, we can now use the family of random linear operators
$\mathcal{L}_{\nu_{\sigma}}(t)$ defined as in (3.1) with $\nu=\nu_{\sigma}$ to
recast (6.22) into the form

$\mathrm{d}\bar{v}=[\mathcal{L}_{\nu_{\sigma}}(t)\bar{v}+\mathcal{N}_{\sigma}(\bar{v})]\mathrm{d}t+\sigma\mathcal{M}_{\sigma}(\bar{v})\mathrm{d}\overline{W}_{t}^{Q}.$
(6.28)

Furthermore, we denote by $E(t,s)$ the associated evolution family (3.8) that
features in our mild formulation below. We emphasise that the stochastic
integral in (6.30) is referred to as a generalised forward integral and needs
to be understood as $J_{2}\big{(}\mathcal{M}_{\sigma}(\bar{v})\big{)}$; see
Corollary 3.8. We point out that it is unknown whether this integral is a
(strict) forward integral in the sense of Definition 3.4, i.e., whether we
have
$I^{-}\big{(}E(t,\cdot)\mathcal{M}_{\sigma}(\bar{v}(\cdot))\big{)}=J_{2}\big{(}\mathcal{M}_{\sigma}(\bar{v})\big{)}$.

######  Proposition 6.4 (mild solution).

Consider either the setting of Proposition 5.1 and suppose that condition
(HPar) holds. Then the map

$\bar{v}:[0,T]\times\Omega\to H^{k}(\mathcal{D};\mathbb{R}^{n})$ (6.29)

defined in (6.18) satisfies the $H^{k}(\mathcal{D};\mathbb{R}^{n})$-valued
identity

$\bar{v}(t)=E(t,0)\bar{v}(t)+\int_{0}^{t}E(t,s)\mathcal{N}_{\sigma}(\bar{v}(s))\mathrm{d}s+\sigma\int_{0}^{t}E(t,s)\mathcal{M}_{\sigma}(\bar{v}(s))\mathrm{d}\overline{W}_{s}^{-},$
(6.30)

which holds $\mathbb{P}$-a.s. for all $0\leq t<\bar{\tau}_{\infty}$.

###### Proof.

Recall that $\bar{v}$ is in $C([0,T];L^{2})$. For convenience, we introduce
the shorthand notations

$\textbf{f}(t)=\mathcal{N}_{\sigma}(\bar{v}(t))\quad\text{and}\quad\textbf{g}(t)=\mathcal{M}_{\sigma}(\bar{v}(t)),$
(6.31)

and define the stochastic process $\bar{w}=(\bar{w}(t))_{t\in[0,T]}$ by

$\bar{w}(t)=E(t,0)\bar{v}(0)+\int_{0}^{t}E(t,s)\textbf{f}(s)\mathrm{d}s+\sigma\int_{0}^{t}E(t,s)\textbf{g}(s)\mathrm{d}\overline{W}_{s}^{-},\quad
0\leq t\leq T.$ (6.32)

By inspecting the definitions (A.28) for $\mathcal{N}_{\sigma}$ and
$\mathcal{M}_{\sigma}$ and appealing to Corollary 3.8, we can conclude that
$\bar{w}$ is well-defined and a continuous $L^{2}$-valued process on $[0,T]$
as well.

By a slight adaptation of the proof in [70, Thm. 6.6], using the proof of [21,
Thm. 3.2] to take into account the deterministic term f, one can show that
$\bar{w}$ is an analytically weak solution in $L^{2}$ to the linear problem

$\mathrm{d}z=[\mathcal{L}_{\nu}(t)z+\textbf{f}(t)]\mathrm{d}t+\textbf{g}(t)\mathrm{d}\overline{W}_{t}^{Q},$
(6.33)

which means pathwise that

$\langle\bar{w}(t),\zeta\rangle_{L^{2}}=\langle\bar{w}(0),\zeta\rangle_{L^{2}}+\int_{0}^{t}\langle\bar{w}(s),\mathcal{L}^{\mathrm{adj}}_{\nu}(s)\zeta\rangle_{L^{2}}\,\mathrm{d}s+\int_{0}^{t}\langle\textbf{f}(s),\zeta\rangle_{L^{2}}\,\mathrm{d}s+\int_{0}^{t}\langle\textbf{g}(s)\mathrm{d}\overline{W}_{s}^{Q},\zeta\rangle_{L^{2}}$
(6.34)

holds for all $\zeta\in C_{c}^{\infty}(\mathcal{D};\mathbb{R}^{n})$ and $0\leq
t<\bar{\tau}_{\infty}$. On the other hand, Proposition 6.3 implies that
$\bar{v}$ is also an analytically weak solution to (6.33) on
$[0,\bar{\tau}_{\infty})$. Upon defining $\bar{\phi}=\bar{v}-\bar{w}$, we
observe that $\bar{\phi}$ solves the linear initial value problem

$\partial_{t}\phi=\mathcal{L}_{\nu}(t)\phi,\quad\phi(0)=0,$ (6.35)

on $[0,\bar{\tau}_{\infty})$, which clearly only has the zero solution
$\phi=0$. We therefore see that $\bar{v}=\bar{w}$ holds with equality in
$L^{2}$ on $[0,\bar{\tau}_{\infty})$, which means that they are equal almost
everywhere on this interval. Since $\bar{v}$ is known to take values in
$H^{k}$ on this interval, we conclude that $\bar{w}$ does the same and that we
have equality in this space, completing the proof. ∎

##  7 Nonlinear stability

In this section we prove the estimate in Proposition 2.5, which leads to the
stochastic metastability of planar waves over exponentially long timescales as
described in Theorem 2.6. From this point onward, we replace the mild form in
(6.30) for the time transformed perturbation with the generic equation

$w(t)=E(t,0)w(0)+\int_{0}^{t}E(t,s)[\sigma^{2}F_{\rm lin}(w(s))+F_{\rm
nl}(w(s))]\mathrm{d}s+\sigma\int_{0}^{t}E(t,s)B(w(s))\mathrm{d}W_{s}^{-},$
(7.1)

in which $E(t,s)$ is any random evolution family that is covered by the
results in §3.2. Throughout this section we assume there exists a continuous
process $w:[0,\infty)\times\Omega\to H^{k}$ for some $k\geq 0$ and a stopping
time $\tau_{\infty}$ so that equation (7.1) is satisfied for $0\leq
t<\tau_{\infty}$.

The maps

$F_{\rm lin}:H^{k+1}\to H^{k},\quad F_{\rm nl}:H^{k+1}\to H^{k},\quad
B:H^{k+1}\to HS(L^{2}_{Q};H^{k}),$ (7.2)

are assumed to satisfy the bounds

$\displaystyle\|F_{\rm lin}(w)\|_{H^{k}}$ $\displaystyle\leq K_{\rm
lin}\|w\|_{H^{k+1}},$ (7.3) $\displaystyle\|F_{\rm nl}(w)\|_{H^{k}}$
$\displaystyle\leq K_{\rm nl}\|w\|_{H^{k+1}}^{2},$ (7.4)
$\displaystyle\|B(w)\|_{HS(L^{2}_{Q},H^{k})}$ $\displaystyle\leq
K_{B}(1+\|w\|_{H^{k+1}}),$ (7.5)

whenever $\|w\|_{H^{k}}\leq 1$. In addition, if $\|w\|_{L^{2}}\leq\eta_{0}$
holds for some sufficiently small $\eta_{0}>0$, then we have the orthogonal
identities

$\langle\sigma F_{\rm lin}(w)+F_{\rm nl}(w),\psi_{\rm
tw}\rangle_{L^{2}}=0\quad\text{and}\quad\langle B(w)[\xi],\psi_{\rm
tw}\rangle_{L^{2}}=0.$ (7.6)

These choices reflect the bounds that arise from (Hf-Cub); see Corollary 4.31.
Of course, they also cover the case where (Hf-Lip) holds, but the computations
in Lemma 7.4 could be simplified slightly by replacing (7.4) with $\|F_{\rm
nl}(w)\|_{H^{k}}\leq K_{\rm nl}\|w\|_{H^{k}}\|w\|_{H^{k+1}}$; see Corollaries
4.15 and 4.21. Lastly, we refer to the remarks on the orthogonality conditions
in Appendix A.

Fix $\varepsilon\in(0,2\mu)$ with $\mu$ as in Lemma 3.2. Recall that our main
objective is to control the size of

$N_{\varepsilon;k}(t)=\|w(t)\|_{H^{k}}^{2}+\int_{0}^{t}e^{-\varepsilon(t-s)}\|w(s)\|_{H^{k+1}}^{2}\mathrm{d}s.$
(7.7)

In particular, for any $0<\eta<\eta_{0}$, we write

$t_{\rm st}(\eta;k)=\inf\\{t\geq
0:\|w(t)\|_{H^{k}}^{2}+\int_{0}^{t}e^{-\varepsilon(t-s)}\|w(s)\|_{H^{k+1}}^{2}\mathrm{d}s>\eta\\}$
(7.8)

for the associated stopping time and observe that the conditions in (7.6) are
automatically satisfied for $w=w(t)$ with $t\leq t_{\rm st}(\eta;k)$. In
addition, the definition of the localising sequence (5.21) implies
$t_{\mathrm{st}}(\eta;k)<\tau_{\infty}$. Finally, note that $\langle
w(t),\psi_{\rm tw}\rangle_{L^{2}}=0$ for $t\leq t_{\rm st}(\eta;k)$ if and
only if $\langle w(0),\psi_{\rm tw}\rangle_{L^{2}}=0$.

Our main result here provides logarithmic growth bounds for the expectation of
the maximal value that $N_{\varepsilon;k}$ attains as we increase $T$.

######  Proposition 7.1.

Consider the generic setting above. Pick two sufficiently small constants
$\delta_{\eta}>0$ and $\delta_{\sigma}>0$. Then there exists a constant $K>0$
so that for any integer $T\geq 2,$ any $0<\eta<\delta_{\eta},$ any
$0\leq\sigma\leq\delta_{\sigma},$ and any integer $p\geq 1$, we have the bound

$\mathbb{E}\left[\sup_{0\leq t\leq t_{\rm st}(\eta;k)\wedge
T}|N_{\varepsilon;k}(t)|^{p}\right]\leq
K^{p}\left[\|v(0)\|_{H^{k}}^{2p}+\sigma^{2p}\big{(}p^{p}+\log(T)^{p}\big{)}\right].$
(7.9)

######  Remark 7.2.

The estimates obtained in [27, 28, 30] for $d=1$ also fit into the generic
framework of this section upon choosing $k=0$. In particular, Proposition 7.1
sharpens [29, Prop. 5.1], which means that the bound (2.60) extends to this
setting and improves [29, Thm. 1.1].

Following the earlier work [27], we proceed by providing separate estimates
for the integrals in (7.1). To this end, we introduce the notation

$\mathcal{E}_{0}(t)=E(t,0)P^{\perp}w(0),$ (7.10)

together with the integrals

$\displaystyle\mathcal{E}_{F;\text{lin}}(t)$
$\displaystyle=\int_{0}^{t}E(t,s)P^{\perp}F_{\operatorname{lin}}(w(s))1_{s<t_{\rm
st}(\eta;k)}\mathrm{d}s,$ (7.11) $\displaystyle\mathcal{E}_{F;\mathrm{nl}}(t)$
$\displaystyle=\int_{0}^{t}E(t,s)P^{\perp}F_{\mathrm{nl}}(w(s))1_{s<t_{\rm
st}(\eta;k)}\mathrm{d}s,$ $\displaystyle\mathcal{E}_{B}(t)$
$\displaystyle=\int_{0}^{t}E(t,s)P^{\perp}B(w(s))1_{s<t_{\rm
st}(\eta;k)}\mathrm{d}W_{s}^{-}.$

The presence of the projection $P^{\perp}$ in the above is simply to emphasise
condition (7.6). Using these expressions, we obtain the estimate

$\displaystyle\mathbb{E}\sup_{0\leq t\leq t_{\rm st}(\eta;k)\wedge
T}\|w(t)\|_{H^{k}}^{2p}$ (7.12) $\displaystyle\qquad\quad\leq
4^{2p}\mathbb{E}\sup_{0\leq t\leq
T}\left[\|\mathcal{E}_{0}(t)\|_{H^{k}}^{2p}+\sigma^{4p}\|\mathcal{E}_{F;\rm
lin}(t)\|_{H^{k}}^{2p}+\|\mathcal{E}_{F;\rm
nl}(t)\|_{H^{k}}^{2p}+\sigma^{2p}\|\mathcal{E}_{B}(t)\|_{H^{k}}^{2p}\right].$

Turning to the integrated $H^{k+1}$-bound, we introduce the integrals

$\displaystyle\mathcal{I}_{0}(t)$
$\displaystyle=\int_{0}^{t}e^{-\varepsilon(t-s)}\|\mathcal{E}_{0}(s)\|_{H^{k+1}}^{2}\mathrm{d}s,$
(7.13) $\displaystyle\mathcal{I}_{F;\mathrm{lin}}(t)$
$\displaystyle=\int_{0}^{t}e^{-\varepsilon(t-s)}\|\mathcal{E}_{F;\mathrm{lin}}(s)\|_{H^{k+1}}^{2}\mathrm{d}s,$
(7.14) $\displaystyle\mathcal{I}_{F;\mathrm{nl}}(t)$
$\displaystyle=\int_{0}^{t}e^{-\varepsilon(t-s)}\|\mathcal{E}_{F;\mathrm{nl}}(s)\|_{H^{k+1}}^{2}\mathrm{d}s,$
(7.15) $\displaystyle\mathcal{I}_{B}(t)$
$\displaystyle=\int_{0}^{t}e^{-\varepsilon(t-s)}\|\mathcal{E}_{B}(s)\|_{H^{k+1}}^{2}\mathrm{d}s.$
(7.16)

This leads directly to the estimate

$\displaystyle\mathbb{E}\sup_{0\leq t\leq t_{\rm st}(\eta;k)\wedge
T}\left[\int_{0}^{t}e^{-\varepsilon(t-s)}\|w(s)\|_{H^{k+1}}^{2}\mathrm{d}s\right]^{p}$
(7.17) $\displaystyle\qquad\quad\leq 4^{2p}\mathbb{E}\sup_{0\leq t\leq
T}\Big{[}\mathcal{I}_{0}(t)^{p}+\sigma^{4p}\mathcal{I}_{F;\rm
lin}(t)^{p}+\mathcal{I}_{F;\rm
nl}(t)^{p}+\sigma^{2p}\mathcal{I}_{B}(t)^{p}\Big{]}.$

######  Lemma 7.3.

For any $0<\eta<\eta_{0}$, any $T>0$, and any $p\geq 1$, we have the pathwise
bounds

$\displaystyle\sup_{0\leq t\leq T}\|\mathcal{E}_{0}(t)\|_{H^{k}}^{2p}$
$\displaystyle\leq M^{2p}\|w(0)\|_{H^{k}}^{2p},$ (7.18)
$\displaystyle\sup_{0\leq t\leq T}\|\mathcal{E}_{F;\rm lin}(t)\|_{H^{k}}^{2p}$
$\displaystyle\leq M^{2p}K_{\rm lin}^{2p}\sup_{0\leq t\leq t_{\rm
st}(\eta;k)\wedge T}N_{\varepsilon;k}(t)^{p},$ $\displaystyle\sup_{0\leq t\leq
T}\|\mathcal{E}_{F;\rm nl}(t)\|_{H^{k}}^{2p}$
$\displaystyle\leq\eta^{p}M^{2p}K_{\rm nl}^{2p}\sup_{0\leq t\leq t_{\rm
st}(\eta;k)\wedge T}N_{\varepsilon;k}(t)^{p}.$

###### Proof.

These results follow directly from straightforward norm estimates; see also
[29, Lem. 5.3]. ∎

######  Lemma 7.4.

There exist a constant $K>0$ so that for any $0<\eta<\eta_{0}$, any $T>0$, and
any $p\geq 1$, we have the pathwise bounds

$\displaystyle\sup_{0\leq t\leq T}\mathcal{I}_{0}(t)^{p}$ $\displaystyle\leq
K^{2p}\|w(0)\|_{H^{k}}^{2p},$ (7.19) $\displaystyle\sup_{0\leq t\leq
T}\mathcal{I}_{F;\rm lin}(t)^{p}$ $\displaystyle\leq K^{2p}\sup_{0\leq t\leq
t_{\rm st}(\eta;k)\wedge T}N_{\varepsilon;k}(t)^{p},$
$\displaystyle\sup_{0\leq t\leq T}\mathcal{I}_{F;\rm nl}(t)^{p}$
$\displaystyle\leq\eta^{p}K^{2p}\sup_{0\leq t\leq t_{\rm st}(\eta;k)\wedge
T}N_{\varepsilon;k}(t)^{p}.$

###### Proof.

The key to establish these estimates is to use the time-dependent inner
products $\langle\cdot,\cdot\rangle_{s;0;k+1}$, as defined in (3.90), and use
the equivalence in (3.91). Indeed, the bound for $\mathcal{I}_{0}$ follows by
applying Lemma 3.12 with $\delta=0$. The remaining estimates can be obtained
by following the computations in [27, Lem. 9.8–9.12], using the identity
(3.93) with $\delta=0$. ∎

######  Lemma 7.5.

For any $0<\eta<\eta_{0}$, any integer $T\geq 2$, and any integer $p\geq 1$,
we have the bound

$\mathbb{E}\sup_{0\leq t\leq
T}\|\mathcal{E}_{B}(t)\|_{H^{k}}^{2p}+\mathbb{E}\sup_{0\leq t\leq
T}\mathcal{I}_{B}(t)^{p}\leq 2^{p}K_{\rm
gr}^{2p}M^{2p}K_{B}^{2p}[\varepsilon^{-1}+\eta]^{p}(p^{p}+\log(T)^{p}).$
(7.20)

###### Proof.

The moment estimate will follow from Proposition 3.18, once we have verified
that condition (HB) holds. To see this, we compute

$\int_{0}^{t}e^{-\varepsilon(t-s)}\|B(w(s))\textbf{1}_{s\leq t_{\rm
st}(\eta;k)}\|_{HS(L^{2}_{Q};H^{k})}^{2}\leq
2K_{B}^{2}(\varepsilon^{-1}+\eta),$ (7.21)

for any $0\leq t\leq T.$ In addition, we note that (7.5) implies

$\displaystyle\|E_{\rm tw}(1)B(w)\|_{HS(L^{2}_{Q};H^{k})}$ $\displaystyle\leq
K_{B}M(1+\|w\|_{H^{k}}),$ (7.22)

which yields the bound

$\|E_{\rm tw}(1)B(w(t))\textbf{1}_{t\leq t_{\rm