mathx”17 standard global,type,entry[boolean]subentry[true]

# Multidimensional Stability of Planar Travelling Waves for Stochastically
Perturbed Reaction-Diffusion Systems

M. van den Bosch${}^{\,\rm a,}$111Corresponding author.
Email addresses<EMAIL_ADDRESS>and
<EMAIL_ADDRESS>, H. J. Hupkes${}^{\,\rm a}$

###### Abstract

We consider reaction-diffusion systems with multiplicative noise on a spatial
domain of dimension two or higher. The noise process is white in time,
coloured in space, and invariant under translations. Inspired by previous
works on the real line, we establish the multidimensional stability of planar
waves on a cylindrical domain on time scales that are exponentially long with
respect to the noise strength. This is achieved by means of a stochastic phase
tracking mechanism that can be maintained over such long time scales. The
corresponding mild formulation of our problem features stochastic integrals
with respect to anticipating integrands, which hence cannot be understood
within the well-established setting of Itô-integrals. To circumvent this
problem, we exploit and extend recently developed theory concerning forward
integrals.

aMathematical Institute, Leiden University,
P.O. Box 9512, 2300 RA Leiden, The Netherlands

Keywords: propagating fronts and pulses, translation invariant noise,
nonlinear stability, stochastic phase shift, forward integrals

##  1 Introduction

In this paper we set out to study the multidimensional stability of planar
travelling wave solutions to stochastic reaction-diffusion systems of the form

$\mathrm{d}u=[D\Delta u+f(u)]\mathrm{d}t+\sigma g(u)\mathrm{d}W_{t}^{Q},$
(1.1)

posed on the cylindrical spatial domain
$\mathcal{D}=\mathbb{R}\times\mathbb{T}^{d-1}$ for some $d\geq 2$, where
$\mathbb{T}$ denotes a one-dimensional torus of arbitrary fixed size. The
waves of interest will be propagating in the direction of the first
(unbounded) coordinate. In addition, for $\textbf{x}\in\mathcal{D}$ and $t>0$
we have $u(\mathbf{x},t)\in\mathbb{R}^{n}$ for some $n\geq 1$, and the matrix
$D$ is diagonal with strictly positive entries. The deterministic dynamics is
driven by a “generalised Gaussian” noise process which is considered white in
time and coloured in space. The noise satisfies the formal relations

$\displaystyle\mathbb{E}[\mathrm{d}W_{t}^{Q}(\textbf{x})]$ $\displaystyle=0,$
(1.2)
$\displaystyle\mathbb{E}[\mathrm{d}W_{t}^{Q}(\textbf{x})\mathrm{d}W_{t^{\prime}}^{Q}(\textbf{x}^{\prime})]$
$\displaystyle=\delta(t-t^{\prime})q(\textbf{x}-\textbf{x}^{\prime}),$

for some smooth covariance function $q$ that characterises the correlation in
space. In particular, the noise and hence system (1.1) are translationally
invariant. The $Q$ refers to the convolution operator induced by $q.$ As
currently written, the noise in (1.1) should be interpreted in the Itô sense,
but for many physical applications—especially involving external noise—it is
more natural to interpret stochastic terms in the Stratonovich sense [44, 95].
We denote such stochastic reaction-diffusion systems as

$\mathrm{d}u=[D\Delta u+f(u)]\mathrm{d}t+\sigma g(u)\circ\mathrm{d}W_{t}^{Q},$
(1.3)

and note that these can also be incorporated into our framework. This paper is
based on preceding works where $d=1$; see [27, 28, 29, 30]. We emphasise that
in [30] the mathematical expressions (1.1), (1.2), and (1.3) are linked back
to the relevant equations and notation commonly used in the physics
literature.

Our aim is to explore the influence of the multiplicative noise term for small
$\sigma$ on the propagation of planar wave solutions. As is discussed in [19]
and several references therein, multiplicative noise is associated with
external fluctuations. Typical scenarios are whenever a control parameter of
the otherwise deterministic system fluctuates around its intended value. For
example, in chemical systems—especially within experimental setups—the
reaction rates serve as control parameters that are highly sensitive to local
variations in temperature, illumination conditions in photochemical systems
and other environmental variables [36, 66, 89]. As an illustration, let us
consider the effect of fluctuations in the intensity of the light source
driving photosensitive Belousov-Zhabotinskii chemical reactions. The authors
in [84] demonstrate, both numerically and experimentally, that for an
effectively one-dimensional front the velocity decreases when random spatial
fluctuations of the light intensity are present. In the two-dimensional
setting, however, the front becomes distorted and curvature seems to favour an
increase in the wave velocity.

One may mathematically model the latter by stochastically perturbing, in an
appropriate manner, the two-component system

$\displaystyle\partial_{t}u$ $\displaystyle=\Delta
u+\frac{1}{\varepsilon}\left(u-u^{2}-(\alpha
v+\beta)\frac{u-\gamma}{u+\gamma}\right),$ (1.4) $\displaystyle\partial_{t}v$
$\displaystyle=\delta\Delta v+u-v,$

where $u$ and $v$ are dimensionless versions of the concentrations of bromous
acid and the catalyst, respectively, while the parameters satisfy
$\alpha,\beta,\gamma,\delta,\varepsilon>0$. These equations are derived in
[46] from a modified version of the Oregenator model by performing a quasi-
steady state approximation. These (minor) modifications are motivated by the
fact that inhibiting effects were observed after irradiating photosensitive
chemicals with both ultraviolet and visible light [97].

The control parameter $\beta$ in system (1.4) is proportional to the applied
light intensity. Note that existence and temporal stability of travelling
waves, and other spatial patterns such as spirals, have been extensively
studied in the light-insensitive case (i.e., $\beta=0$); see [23, 43, 54, 64]
and references therein. Allowing $\beta$ to fluctuate randomly by performing
the formal substitution
$\beta\mapsto\beta-\varepsilon\sigma\frac{\partial}{\partial t}W_{t}^{Q}$, we
obtain the stochastically perturbed system

$\displaystyle\mathrm{d}u$ $\displaystyle=\left[\Delta
u+\frac{1}{\varepsilon}\left(u-u^{2}-(\alpha
v+\beta)\frac{u-\gamma}{u+\gamma}\right)\chi(u)\right]\mathrm{d}t+\sigma\frac{u-\gamma}{u+\gamma}\pi(u)\circ\mathrm{d}W_{t}^{Q},$
(1.5) $\displaystyle\mathrm{d}v$ $\displaystyle=\left[\delta\Delta
v+u-v\right]\mathrm{d}t,$

after introducing cut-off functions $\chi$ and $\pi$ that satisfy
$\chi(u)=\pi(u)=1$ for all values of $u$ that are chemically relevant. We
exploit such cut-off functions to enforce convenient pointwise Lipschitz
properties on certain nonlinearities and to ensure that the noise does not
affect the homogeneous background states of the front solutions. We refer to
[27, 30] for many other examples of model systems that fit within our
framework.

#### Deterministic setting

From now on, we shall consider the setting where (1.1) with $\sigma=0$ admits
a spectrally stable travelling front or pulse in one spatial dimension. More
specifically, the operator associated to the linearisation about the wave in
the one-dimensional setting is assumed to have only the translational
eigenvalue at zero, with the remainder of the spectrum bounded away from the
imaginary axis in the left half plane. Such spectrally stable waves are known
to exist under quite general hypotheses [20, 27, 40, 82]. A common analytical
approach for establishing this spectral gap condition is to invoke methods
from geometric singular perturbation theory [26, 31], which rely on a strict
separation between the diffusive length scales. In terms of the example system
(1.4) this would mean $\delta\ll 1$, which for instance is a natural
assumption when modelling the propagation of waves in a silica gel where the
catalyst is being immobilised [46, 84].

The spectral gap allowed Kapitula [38, 39] to use semigroup methods to prove
under very mild conditions that the associated planar waves on
$\mathbb{R}^{d}$ are orbitally stable for every $d\geq 2$. Our main purpose
here is to use the spirit of his approach to establish similar conclusions for
stochastically forced systems. Earlier deterministic approaches, such as [56],
depend heavily on the maximum principle and energy methods which are generally
not applicable to most systems, which we therefore also choose to avoid. It is
also worth pointing out that the spectrum of the operator associated to the
linearisation of a planar wave $(d\geq 2)$ is no longer bounded away from the
imaginary axis. In particular, the resulting algebraic decay of perturbations
required the use of a refined decomposition in which the phase of the wave
plays a crucial role.

#### Cylindrical spatial domain

The first main reason to restrict to a cylindrical spatial domain is our
desire to consider noise that is translationally invariant. In more detail,
suppose $(W_{t}^{Q})_{t\geq 0}$ is a cylindrical $Q$-Wiener process in222In
fact, the noise process is rigorously constructed in an extended space
$\mathcal{W}_{\rm ext}\supset\mathcal{W}$, see §3.2 and [30]. some space
$\mathcal{W}$, where $Q:\mathcal{W}\to\mathcal{W}$ is a linear, symmetric,
positive semi-definite operator $Q:\mathcal{W}\to\mathcal{W}$. All these
assumptions on $Q$ are basically essential to make sense of it as a covariance
operator. For us to be able to interpret either system (1.1) or (1.3), it is
important that stochastic integrals of the form

$\int_{0}^{t}g(\Phi_{0})\mathrm{d}W_{s}^{Q}$ (1.6)

are well-defined in a sense suitable for our analysis. More specifically, we
need to be able to interpret $g(\Phi_{0})$ as a Hilbert-Schmidt operator from
$\mathcal{W}_{Q}=Q^{1/2}(\mathcal{W})$ to the Sobolev space
$H^{k}(\mathcal{D};\mathbb{R}^{n})$, for some $k\geq 0$. Consider now
$\mathcal{W}=L^{2}(\mathcal{D};\mathbb{R}^{m})$ to be the space of square-
integrable functions, where $m>0$ denotes the number of noise components; we
have $n=2$ and $m=1$ in system (1.5). This subsequently allows us to interpret
the $n\times m$ matrix $g(\Phi_{0})$ as a Nemytskii operator, i.e., we proceed
via the pointwise multiplication
$g(\Phi_{0})[\xi](\textbf{x})=g(\Phi_{0}(\textbf{x}))\xi(\textbf{x})$, for any
$\xi\in\mathcal{W}_{Q}$.

If $Q$ is of trace class, i.e.,
Tr$(Q)=\|Q^{1/2}\|_{HS(L^{2};H^{k}(\mathcal{D};\mathbb{R}^{n}))}^{2}<\infty,$
then it suffices to demand

$\sup_{\textbf{x}\in\mathcal{D}}|g(\Phi_{0}(\textbf{x}))|<\infty,$ (1.7)

which is likely to hold in many situations, since it is supposed that the non-
linearity $g$ vanishes at the endpoints of the wave. This is also true if we
replace $\mathcal{D}$ by $\mathbb{R}^{d}$. However, for our setting (1.2) in
combination with the fact that our domain is unbounded, we see that the
convolution operator $Q$, which acts as $Qv=q*v$, cannot be of trace class
(even for $d=1$; see [30]). The computations in §4.1 show that the well-
posedness condition for (1.6) becomes

$\|g(\Phi_{0})\|_{H^{k}(\mathcal{D};\mathbb{R}^{n\times m})}<\infty.$ (1.8)

Note that for $\mathcal{D}=\mathbb{R}^{d}$ this condition holds if and only if
$d=1$, since $\Phi_{0}$ depends only on the first spatial coordinate. Hence,
we consider domains of the form
$\mathcal{D}=\mathbb{R}\times\mathbb{T}^{d-1}$, where (1.8) does hold for
dimensions $d\geq 2.$

The second main reason is that on $\mathbb{R}^{d}$, with $d\geq 2,$ one may
only expect algebraic decay of perturbations [38]. In addition, if the
disturbance is not sufficiently “localised” then one need not have any decay
at all [5]. One therefore needs to carefully balance the technical
requirements for the noise term with the machinery necessary to handle the
slowly decaying terms. Scalar noise for instance, also known as spatially
homogeneous noise (take $\mathcal{W}=\mathbb{R}^{m}$ and $Q$ a positive semi-
definite matrix as in the previous works [27, 28]), is of trace class but
simply reduces the problem back to the one-dimensional case and hence
generates no decay in the transverse direction.

We do intend to investigate the impact of coloured noise on travelling waves
evolving on $\mathbb{R}^{d}$ in the future, and the present paper can be seen
as a preparatory study. For example, our work here can be used to extract
detailed information concerning the stochastic behaviour of the phase and the
dependence on the size $|\mathbb{T}|$ of the torus. We envision that the
translational invariance with respect to the transverse direction will need to
be loosened, utilising localised or weighted noise as interesting
alternatives.

#### Main result

We prove that a spectrally stable planar wave on
$\mathcal{D}=\mathbb{R}\times\mathbb{T}^{d-1}$ survives in a suitable sense
under the influence of the small multiplicative noise terms in (1.1) or (1.3).
This is achieved by analysing perturbations of the form

$v(x,y,t)=u(x+\gamma(t),y,t)-\Phi_{\sigma}(x),$ (1.9)

and following the spirit of the procedure developed in [27, 30]. We extend the
pair $(\Phi_{0},c_{0})$ to a branch of so-called instantaneous stochastic
waves $(\Phi_{\sigma},c_{\sigma})$ that satisfy333Throughout this paper, we
shall often use the abbreviations $L^{2}=L^{2}(\mathcal{D};\mathbb{R}^{n})$
and $H^{k}=H^{k}(\mathcal{D};\mathbb{R}^{n})$. At times we also need to
consider other domains and codomains, in which case we will always be explicit
to prevent any confusion.

$\|\Phi_{\sigma}-\Phi_{0}\|_{H^{k}}+|c_{\sigma}-c_{0}|=\mathcal{O}(\sigma^{2}),$
(1.10)

which only feel stochastic forcing at onset. The phase shift $\gamma(t)$ is
intended to stochastically “freeze” the solution $u$ by enforcing the
orthogonality condition

$\langle v(t),\psi_{\rm tw}\rangle_{L^{2}}=0,$ (1.11)

for some function $\psi_{\rm tw}$ related the adjoint of operator associated
to the linearisation about the wave. In particular, the phase shift
$\gamma(t)$ satisfies the stochastic (ordinary) differential equation

$\mathrm{d}\gamma=\big{[}c_{\sigma}+\mathcal{O}(\|u(t)-\Phi_{\sigma}(\cdot-\gamma(t))\|_{H^{k}})\big{]}\mathrm{d}t+\mathcal{O}(\sigma)\mathrm{d}W_{t}^{Q},$
(1.12)

and can, loosely speaking, be seen as the “location” of the wave after
spatially averaging over the transverse coordinates. For a rigorous
construction and more information, we refer to §2.

We are interested in the probabilistic behaviour of the exit-time

$t_{\rm st}(\eta;k)=\inf\\{t\geq
0:\|v(t)\|_{H^{k}}^{2}+\int_{0}^{t}e^{-\varepsilon(t-s)}\|v(s)\|_{H^{k+1}}^{2}\mathrm{d}s>\eta\\},$
(1.13)

which measures at what time the solution $u(t)$ deviates too much from the
expected “location” of the profile $\Phi_{\sigma}$. By taking $k>d/2$ and
using a standard Sobolev embedding, the first term ensures that the pointwise
size of $v(t)$ stays under control until this stopping time. The second term
(with a small regularisation parameter $\varepsilon>0$) provides integrated
control over higher derivatives. This is an important bonus feature of our
method and can be seen as a type of optimal regularity result. For example,
this enables us to provide $H^{1}$-results in dimensions $2\leq d\leq 4$,
where no pointwise control is available. In addition, it could be exploited to
consider nonlinearities $f$ that include a dependence on the first derivatives
$\nabla u$.

Our metastability result below not only improves upon the main findings
presented in [29], but also extends these to higher spatial dimensions. In
particular, we may take $\mathcal{D}=\mathbb{R}$ when $d=1$ (see Table 2.1),
and the probability bound in (1.14) is now in line with those found in the
context of large deviations theory [11, 86, 90, 96]. In words, the theorem
states that with probability exponentially close to one, our exit-time is
exponentially long with respect to the parameter $1/\sigma$.

######  Theorem 1.1 (Interpretation of Theorem 2.6).

Under certain technical assumptions, planar fronts and pulses in
stochastically perturbed reaction-diffusion equations evolving over
$\mathcal{D}=\mathbb{R}\times\mathbb{T}^{d-1}$ persist on exponentially long
times scales. In particular, there exists a constant $0<\mu<1$ so that for all
sufficiently small $\sigma>0$ there is a stochastic scalar process $\gamma(t)$
such that

$\mathbb{P}(t_{\rm st}(\eta;k)<T)\leq
2T\exp\left(-\frac{\mu\eta}{\sigma^{2}}\right)$ (1.14)

holds for any integer $T\geq 2$, any sufficiently small exit value $\eta>0$,
and any initial value $u(0)$ that satisfies
$\|u(0)-\Phi_{\sigma}\|_{H^{k}}^{2}<\mu\eta$.

In one spatial dimension, several distinct approaches have been developed to
study the behaviour of patterns under stochastic perturbations, using various
alternative definitions for the stochastic phase [3, 34, 47, 52, 53, 60, 87,
88]. An alternative technique that also leads to stability results on
exponentially long timescales uses renormalisation to reset the phase tracking
mechanism at suitable time points [14, 22, 47, 63]. In two dimensions, the
latter approach has been used to study rotating spirals [50] on a compact
spatial domain. As we will explain in the sequel, our spatially unbounded
setting introduces complications that are not covered by these results.

#### Local existence and uniqueness

While rigorous results on the impact of stochastic forcing on deterministic
patterns are still relatively scarce [49], general existence and uniqueness
results for SPDEs have been well-developed over the past decades; see [8, 10,
11, 16, 21, 24, 25, 33, 35, 73, 100] and the references therein. Nevertheless,
in order to be able to talk about $H^{k}$-valued (local) variational solutions
of (1.1), we need to exploit developments that are quite recent.

In spatial dimension $d=1$ it has been shown—after substitution of
$u=z+\Phi_{\rm ref}$ into system (1.1) with $\Phi_{\rm ref}=\Phi_{0}$—that

$\mathrm{d}z=[\partial_{x}^{2}z+\Phi_{\rm ref}^{\prime\prime}+f(z+\Phi_{\rm
ref})]\mathrm{d}t+\sigma g(z+\Phi_{\rm ref})\mathrm{d}W_{t}^{Q}$ (1.15)

admits a global unique solution [27, 30], under suitable conditions on the
nonlinearities $f$ and $g$. In particular, it crucially involves the sign
condition

$\langle f(u)-f(v),u-v\rangle\leq K|u-v|^{2}$ (1.16)

to control the super-linear terms in $f$; consider, e.g., the Allen-Cahn
nonlinearity $f(u)=u-u^{3}$. This can be used to derive local monotonicity
properties such as

$\displaystyle\langle
f(\Phi+v_{A})-f(\Phi+v_{B}),v_{A}-v_{B}\rangle_{L^{2}}\leq
K\|v_{A}-v_{B}\|_{L^{2}}^{2},$ (1.17)

which should be contrasted with the estimate444This follows directly from the
proof of [27, Lem. 2.3.1], with the slight modification that instead of
invoking the Sobolev embedding $\|u\|_{\infty}\leq C\|u\|_{H^{1}}$, we use the
Gagliardo-Nirenberg inequality
$\|u\|_{\infty}\leq\|u\|_{L^{2}}^{1/2}\|u\|_{H^{1}}^{1/2}$; see [57, 67].

$\displaystyle\|f(\Phi+v_{A})-f(\Phi+v_{B})\|_{L^{2}}\leq
K\big{(}1+\|v_{A}\|_{L^{2}}\|v_{A}\|_{H^{1}}+\|v_{B}\|_{L^{2}}\|v_{B}\|_{H^{1}}\big{)}\|v_{A}-v_{B}\|_{L^{2}},$
(1.18)

that is available for general cubic nonlinearities. To be more precise, this
global solution fits into the classical variational framework as described in
Liu and Röckner [58], which uses the Gelfand triple $(H^{1},L^{2},H^{-1})$ to
embed the SPDE and its solution and relies heavily on these monotonicity
properties. For additive noise, these conditions have been weakend by the same
authors to allow cross terms as in (1.18) [59, Sec. 5.2]. Further refinements
are also possible in the setting where the Gelfand triple embeddings are
compact [77], yet both these extensions do not apply to our spatially extended
system with multiplicative noise.

The key issue is that our stability arguments require pointwise control over
the perturbation $v(t)$, requiring us to work in $H^{k}$ with $k>d/2$. Due to
the fact that derivative terms such as $f(u)^{\prime}=f^{\prime}(u)u^{\prime}$
automatically involve products, the monotonicity property (1.17) fails to hold
in $H^{k}$, even if the sign condition (1.16) is satisfied. Recently, a new
variational framework by Agresti and Veraar [4], the so-called critical
variational framework, has been developed which basically replaces local
monotonicity assumptions by local Lipschitz conditions. Our approach here is
hybrid: we first construct a global solution with respect to
$(H^{1},L^{2},H^{-1})$ and then apply the results in [4] to conclude that
these global solutions exist locally in $(H^{k+1},H^{k},H^{k-1})$. For
convenience, we impose global Lipschitz-smoothness on the nonlinearities in a
pointwise fashion, which can readily be obtained by multiplying any
sufficiently smooth nonlinearity with a smooth cut-off function. However, we
caution the reader that this does not carry over to the nonlinearities viewed
as operators on the relevant function spaces. In §5.3 we also explain how
cubic nonlinearities can be accomodated in dimensions $1\leq d\leq 4.$

#### Mild formulation

After performing some computations, we conclude that the perturbation $v(t)$
is a variational solution to an initial value problem of the form

$\begin{cases}\mathrm{d}v(t)&=[A\big{(}t,v(t)\big{)}v(t)+F(t,v(t))]\mathrm{d}t+B(t,v(t))\mathrm{d}W^{Q}_{t},\\\
v(0)&=v_{0},\end{cases}$ (1.19)

in which the unbounded operator-valued function $A$ now also carries a path-
dependence that cannot be transformed away, in contrast to [27, 28, 29, 30]
where $d=1$. Nevertheless, we want to pass to a mild formulation of the form

$v(t)=E(t,0,\omega)v_{0}+\int_{0}^{t}E(t,s,\omega)F(s,v(s))\mathrm{d}s+\underbrace{\int_{0}^{t}E(t,s,\omega)B(s,v(s))\mathrm{d}W^{Q}_{s}}_{\textnormal{Q:
How to interpret this integral?}},$ (1.20)

to exploit the stability of the linearised flow encoded in the evolution
family $E(t,s)$. Through the parameter $\omega\in\Omega$ this family depends
explicitly on the probability space $(\Omega,\mathcal{F},\mathbb{P})$, which
should be seen as a direct consequence of the fact that $A$ depends on $v(t)$.
Consequently, the integrand $E(t,s)B(s,v(s))$ is anticipating, and hence we
cannot make sense of the stochastic integral in (1.20) with the usual Itô
calculus. Mild formulations where $E(t,s)$ is independent of $\omega\in\Omega$
have been studied thoroughly in the survey [69]. In the previous papers [27,
28, 29, 30] it was possible to write $E(t,s)=S_{\rm tw}(t-s)$, using the
semigroup $S_{\rm tw}(t)$ associated to the linearisation of the travelling
wave, for which general theory has been available for some time [11, 21, 24].

The Skorokhod integral is one of the most well-studied extensions of the Itô
integral to a certain set of anticipating integrands [85]. However, this is
known to be inconvenient when studying mild solutions to parabolic equations
with a random family of operators, primarily due to the introduction of a
drift and the reliance on so-called Mallivian derivatives which results in
technical assumptions that cannot be easily verified; see [55] and [70, Rem.
6.7].

#### Forward integrals

In our setting it turns out to be appropriate to use so-called forward
integrals, introduced by Russo and Vallois [79, 80, 81]. In the operator-
valued setting, Léon and Nualart [55] were the first (and one of the few) to
study forward integrals and apply them to stochastic evolution equations.
Their existence and uniqueness results have been obtained by linking the
forward integral back to the Skorokhod integral, which transfers the
associated inconveniences. Recent progress has been made through the
observation [75, eq. (5.4)] that links these integrals to the theory of
pathwise mild solutions as developed by Pronk and Veraar [74]; see for example
[70, Sec. 6] and the work of Kuehn and Neamţu [51] on stochastically perturbed
quasilinear problems.

In §6.3 we will show that the variational solution $v(t)$ to (1.19) satisfies
the mild formulation

$v(t)=E(t,0,\omega)v_{0}+\int_{0}^{t}E(t,s,\omega)F(s,v(s))\mathrm{d}s+\underbrace{\int_{0}^{t}E(t,s,\omega)B(s,v(s))\mathrm{d}W^{-}_{s}}_{\textnormal{A:
As a forward integral!}}$ (1.21)

involving a forward integral. To appreciate this concept, let us consider a
standard Brownian motion $\beta=(\beta(t))_{t\geq 0}$. Formally, we have

$\displaystyle\int_{0}^{t}g(s)\mathrm{d}\beta(s)$
$\displaystyle=\int_{0}^{t}g(s)\frac{\mathrm{d}\beta(s)}{\mathrm{d}s}\mathrm{d}s=\int_{0}^{t}g(s)\lim_{h\searrow
0}\frac{\beta(s+h)-\beta(s)}{h}\mathrm{d}s$ (1.22)
$\displaystyle=\lim_{h\searrow
0}\frac{1}{h}\int_{0}^{t}g(s)[\beta(s+h)-\beta(s)]\mathrm{d}s,$

which motivates the definition

$\int_{0}^{t}g(s)\mathrm{d}\beta^{-}(s):=\lim_{n\to\infty}n\int_{0}^{t}g(s)[\beta(s+1/n)-\beta(s)]\mathrm{d}s$
(1.23)

for the forward integral, interpreting the limit in the ucp-topology. The
intermediate steps in (1.22) do not make sense, since a Brownian motion is
nowhere differentiable, yet we do have the identity
$\int_{0}^{t}g(s)\mathrm{d}\beta(s)=\int_{0}^{t}g(s)\mathrm{d}\beta^{-}(s)$
for adapted processes $g$ [81, Prop 1.1]. Also, anticipating processes $g$
exist for which the forward integral is well-defined, implying that it is
indeed a proper extension of the Itô integral. In the same spirit, Russo and
Vallois also studied so-called backward and symmetric integrals, which are
extensions of the backward Itô and the Stratonovich integral, respectively.

Maximal inequalities for stochastic convolutions (a type of Burkholder-Davis-
Gundy inequalities [41, 76]) play an important role in the series [27, 28, 29,
30]. These inequalities are also essential in the current paper, now
underlying both the well-posedness of our mild formulation and our stability
analysis. For a general random $C_{0}$-evolution family $E(t,s)=E(t,s,\omega)$
of contractions, one can use [70, Thm. 6.4] to obtain bounds of the form

$\mathbb{E}\sup_{0\leq t\leq
T}\left\|\int_{0}^{t}E(t,s)B(s)\mathrm{d}W^{-}_{s}\right\|_{H^{k}}^{2p}\leq
K^{2p}p^{p}\mathbb{E}\left[\int_{0}^{t}\|B(s)\|_{HS(\mathcal{W}_{Q};H^{k})}^{2}\mathrm{d}s\right]^{p},$
(1.24)

which hold for any $p\geq 1$ and some $K>0$ independent of time $T$. This
recent result sharpens the earlier bounds obtained by Léon and Nualart
employing a factorisation method [55, Thm. 4.4].

One of the contributions of this paper is that we derive estimates for forward
integrals that work for certain $C_{0}$-evolution families that are only
eventually contractive, i.e., satisfy the bound $\|E(t,s)\|_{H^{k}\to
H^{k}}\leq M$ with $M>1$. To achieve this, we exploit the fact that the
evolution families we encounter can be decomposed into a family of
contractions and the semigroup associated to the linearisation about the wave
in one spatial dimension, which admits $H^{\infty}$-calculus after projecting
out the translational eigenfunction. For certain one-component systems, it is
known that the linearised flow is immediately contractive in the direction
orthogonal to the translational eigenfunction. This is exploited by many
authors in order to prove (nonlinear) stabilty results in stochastically
perturbed systems [34, 52, 53, 87, 88], but generally speaking it is not true
or unclear whether the semigroup associated to the linearisation about the
wave is contractive [27, 99]. In addition, we provide novel maximal regularity
estimates for these forward integrals, allowing us to control their
$H^{k+1}$-norm in an integrated sense. This is a subtle task due to the
delicate limiting process underlying their definition. In any case, several of
our results generalise naturally to $\mathbb{R}^{d}$ and are also applicable
to other settings that involve random evolution families.

#### Expansions in $\sigma$

One of the main features of our freezing method is that the fluctuations
around the instantaneous stochastic wave $(\Phi_{\sigma},c_{\sigma})$ can be
readily investigated by expanding our equations for $v(t)$ and $\gamma(t)$ in
powers of $\sigma$. For example, we may follow [30] to find the expansions

$\displaystyle\gamma(t)$
$\displaystyle=c_{0}t+\sigma\gamma_{1}(t)+\mathcal{O}(\sigma^{2}),\qquad
c_{\sigma}=c_{0}+\sigma^{2}c_{0;2}+\mathcal{O}(\sigma^{4}),$ (1.25)

with the explicit expressions

$\gamma_{1}(t)=-\frac{1}{|\mathbb{T}|^{d-1}}\int_{0}^{t}\langle
g(\Phi_{0})\mathrm{d}W_{s}^{Q},\psi_{\rm
tw}\rangle_{L^{2}(\mathcal{D};\mathbb{R}^{n})}$ (1.26)

and

$\displaystyle c_{0;2}$
$\displaystyle=-\,\frac{1}{2|\mathbb{T}|^{d-1}}\langle\Phi_{0}^{\prime\prime},\psi_{\rm
tw}\rangle_{L^{2}(\mathbb{R};\mathbb{R}^{n})}\langle
g(\Phi_{0})Qg(\Phi_{0})^{\top}\psi_{\rm tw},\psi_{\rm
tw}\rangle_{L^{2}(\mathbb{R};\mathbb{R}^{n})}$ (1.27)
$\displaystyle\qquad\qquad-\frac{1}{|\mathbb{T}|^{d-1}}\langle
g(\Phi_{0})Qg(\Phi_{0})^{\top}\psi_{\rm tw},\psi_{\rm
tw}^{\prime}\rangle_{L^{2}(\mathbb{R};\mathbb{R}^{n})}-\langle
h(\Phi_{0}),\psi_{\rm tw}\rangle_{L^{2}(\mathbb{R};\mathbb{R}^{n})}.$

As before, the function $h$ denotes the Itô-Stratonovich correction term. Also
note that

$\displaystyle\textrm{Var}\big{[}\gamma_{1}(t)\big{]}$
$\displaystyle=\frac{1}{(|\mathbb{T}|^{d-1})^{2}}\mathbb{E}\left[\int_{0}^{t}\langle
g(\Phi_{0})\mathrm{d}W^{Q}_{s},\psi_{\rm
tw}\rangle_{L^{2}(\mathcal{D};\mathbb{R}^{n})}\right]^{2}$ (1.28)
$\displaystyle=\frac{1}{|\mathbb{T}|^{d-1}}\langle
g(\Phi_{0})Qg(\Phi_{0})^{\top}\psi_{\rm tw},\psi_{\rm
tw}\rangle_{L^{2}(\mathbb{R};\mathbb{R}^{n})}t.$

To illustrate the effects of the dimensions transverse to the propagation of
the wave, we simply take $m=n=1$ and consider a convolution $Qv=q*v$ whose
kernel $q$ can be factorised as follows: $q(x,y)=q_{\rm wv}(x)q_{\perp}(y)$
for $(x,y)\in\mathcal{D}$. Writing $Q_{\rm wv}v=q_{\rm wv}*v$ for the one-
dimensional convolution operator along the wave direction yields

$Qg(\Phi_{0})^{\top}\psi_{\rm tw}=\mathfrak{q}_{\rm
avg}|\mathbb{T}|^{d-1}Q_{\rm wv}g(\Phi_{0})\psi_{\rm
tw},\qquad\mathfrak{q}_{\rm
avg}=\frac{1}{|\mathbb{T}|^{d-1}}\int_{\mathbb{T}^{d-1}}q_{\perp}(y)\,\mathrm{d}y.$
(1.29)

In addition, we define $h(u)=\frac{\mu}{2}q(0)g^{\prime}(u)g(u)$ for
$\mu\in\\{0,1\\}$ and recall that $\mu=0$ corresponds with the Itô
interpretation (1.1) while $\mu=1$ encodes the Stratonovich interpretation
(1.3). These choices allow us to find

$\displaystyle\textrm{Var}\big{[}\gamma_{1}(t)\big{]}$
$\displaystyle=\mathfrak{q}_{\rm avg}\langle g(\Phi_{0})Q_{\rm
wv}g(\Phi_{0})\psi_{\rm tw},\psi_{\rm
tw}\rangle_{L^{2}(\mathbb{R};\mathbb{R})}t$ (1.30)

together with

$\displaystyle c_{0;2}$ $\displaystyle=-\,\frac{1}{2}\mathfrak{q}_{\rm
avg}\langle\Phi_{0}^{\prime\prime},\psi_{\rm
tw}\rangle_{L^{2}(\mathbb{R};\mathbb{R}^{n})}\langle g(\Phi_{0})Q_{\rm
wv}g(\Phi_{0})\psi_{\rm tw},\psi_{\rm
tw}\rangle_{L^{2}(\mathbb{R};\mathbb{R})}$ (1.31)
$\displaystyle\qquad\qquad-\mathfrak{q}_{\rm avg}\langle g(\Phi_{0})Q_{\rm
wv}g(\Phi_{0})\psi_{\rm tw},\psi_{\rm
tw}^{\prime}\rangle_{L^{2}(\mathbb{R};\mathbb{R})}-\frac{\mu}{2}\langle
g^{\prime}(\Phi_{0})g(\Phi_{0}),\psi_{\rm
tw}\rangle_{L^{2}(\mathbb{R};\mathbb{R})}.$

To recover the one-dimensional results in [30, Sec. 2.3], one simply sets
$q_{\perp}\equiv 1$ which implies $\mathfrak{q}_{\rm avg}=1$, independent of
the size of the torus $\mathbb{T}.$ Indeed, this choice models spatially
homogeneous noise in the transverse direction. The situation changes however
if the noise correlation decays in the $y$-direction; for example, take a
kernel $q_{\perp}$ that is supported on $[-1,1]$ and does not depend on the
value $|\mathbb{T}|$. In this case one has $\mathfrak{q}_{\rm avg}\to 0$ as
$|\mathbb{T}|\to\infty$. Consequently, we see that

$\mathrm{Var}\big{[}\gamma_{1}(t)\big{]}\to 0,\qquad
c_{0;2}\to-\frac{\mu}{2}\langle g^{\prime}(\Phi_{0})g(\Phi_{0}),\psi_{\rm
tw}\rangle_{L^{2}(\mathbb{R};\mathbb{R}^{n})}.$ (1.32)

The vanishing variance of $\gamma_{1}(t)$ can intuitively be appreciated by
noting that the energy inserted by the noise can be dissipated in the
transverse direction rather than causing fluctuations in the phase of the
wave. This seems to suggests that on the whole space
$\mathcal{D}=\mathbb{R}^{d}$ one can proceed in the spirit of the
deterministic approach [39] and replace the global phase function by local
phase functions.

Under appropriate parameter regimes, the examples in [30] displayed a change
of sign for $c_{0;2}$ when switching between $\mu=0$ and $\mu=1$ (i.e.,
$\mathfrak{q}_{\rm avg}=1$). Now, in the Stratonovich setting $\mu=1$, it is
conceivable that $c_{0;2}$ changes sign as the parameter $|\mathbb{T}|$ is
adjusted between 0 and infinity. This leads to a potential explanation for the
dimension-dependent speed changes observed in [84]; recall the discussion
above in the context of (1.4). We intend to examine this further in a
forthcoming paper, which will include higher order expansions that are able to
capture curvature-driven effects. This will guide the development of an
appropriate theory for the full case $\mathcal{D}=\mathbb{R}^{d}$.

#### Organisation

The structure of this paper is outlined as follows. In §2 we formulate our
assumptions, state our main results and provide an overview of the main steps
in the proof. In §3 we study random evolution families, introduce the concept
of the forward integral, and obtain key estimates and regularity results for
the associated stochastic convolutions. In §4 we provide bounds for our
nonlinearities, which we use in §5 to construct global and local solutions to
our problem. In §6 we work towards a mild representation for the perturbation
$v(t)$, for which we then provide stability estimates in §7.

#### Acknowledgements

The corresponding author wishes to thank Mark Veraar for the very fruitful
discussion in 2023.

##  2 Main results

In this section we state our findings for the existence and metastability of
planar wave solutions to stochastic reaction-diffusion systems of the form

$\mathrm{d}u=[D\Delta u+f(u)+\sigma^{2}h(u)]\mathrm{d}t+\sigma
g(u)\mathrm{d}W_{t}^{Q},$ (2.1)

where $u(x,y,t)\in\mathbb{R}^{n}$ evolves in time $t\geq 0$ on a cylindrical
domain $\mathcal{D}=\mathbb{R}\times\mathbb{T}^{d-1}\ni(x,y)$ with dimension
$d\geq 2$, and is driven by a translationally invariant noise process
$(W^{Q}_{t})_{t\geq 0}$. The Laplacian acts in a standard fashion on the
spatial coordinates $(x,y)$ as

$\Delta u=\partial_{x}^{2}u+\Delta_{y}u,$ (2.2)

and the diffusion matrix $D$ is diagonal with strictly positive diagonal
elements. In particular, we will denote a derivative with respect to $x$
interchangeably by $\partial_{x}$ and a prime ′. Note that throughout this
work, we also allow $d=1$ with $\mathcal{D}=\mathbb{R}$.

######  Remark 2.1.

To improve the readability of our arguments, we will assume that $D=I_{n}$
from this point forward, where $I_{n}$ is the $n\times n$ identity matrix.
Indeed, the approach used in [28, 30] to handle varying constants on the
diagonal also works in the present multidimensional setting.

To set the stage, we impose several conditions on the nonlinearity $f$ and
also the deterministic planar wave that travels in the $x$-direction. In §2.2,
we formulate conditions on the noise term and the Itô-Stratonovich correction
term $h$, guaranteeing the existence and uniqueness of solutions in various
variational settings. In §2.3, we couple an extra SDE to the SPDE above that
will serve as a phase-tracking mechanism. This enables us to formulate and
discuss our main stability results.

###  2.1 Deterministic setup

We start by formulating our conditions for the nonlinearity $f$, which involve
an arbitrary integer $k\geq 0$ that varies depending on the context. This
parameter is associated to the degree of smoothness (in the $H^{k}$-sense)
that we can expect our solutions to have. Our most general result that is
valid in arbitrary dimensions $d\geq 1$ will require $k>d/2$ and hence provide
pointwise control. In this case, we need to impose global Lipschitz conditions
on $f$ and its derivatives. We do point out that the pointwise control in our
final stability result means that we can safely modify our nonlinearities
outside the region of interest to enforce these conditions.

  * (Hf-Lip)

We have $f\in C^{k+2}(\mathbb{R}^{n};\mathbb{R}^{n})$ with
$f(u_{-})=f(u_{+})=0$ for some pair $u_{\pm}\in\mathbb{R}^{n}.$ In addition,
there is a constant $K_{f}>0$ such that

$|f(u_{A})-f(u_{B})|+\ldots+|D^{k+1}f(u_{A})-D^{k+1}f(u_{B})|\leq
K_{f}|u_{A}-u_{B}|$ (2.3)

holds for all $u_{A},u_{B}\in\mathbb{R}^{n}$.

In lower dimensions $1\leq d\leq 4$ it is also possible to consider solutions
(and noise) with a lower degree of smoothness. In fact, we will be able to
take $k=1$ and allow our nonlinearity $f$ to have cubic growth. Observe that
for $2\leq d\leq 4$ this can no longer be artifically imposed by modifying $f$
since we no longer have pointwise control over our $H^{1}$-valued solutions.

  * (Hf-Cub)

We have $1\leq d\leq 4$ and $f\in C^{3}(\mathbb{R}^{n},\mathbb{R}^{n})$ with
$f(u_{-})=f(u_{+})=0$ for some pair $u_{\pm}\in\mathbb{R}^{n}$. In addition,
there exists a constant $K_{f}>0$ so that the bound

$|D^{3}f(u)|\leq K_{f}$ (2.4)

holds for all $u\in\mathbb{R}^{n}.$

We assume the existence of a planar wave solution
$u(x,y,t)=\Phi_{0}(x-c_{0}t)$ that approaches its limits
$u_{\pm}\in\mathbb{R}^{n}$ at an exponential rate; a common assumption
connected to asymptotic hyperbolicity [39, 82] that holds in many
applications. Together with (Hf-Lip) or (Hf-Cub), this condition implies that
$\Phi_{0}\in C^{k+4}(\mathbb{R};\mathbb{R}^{n})$ holds together with
$|\Phi_{0}^{(\ell)}(x)|\to 0$ exponentially fast as $|x|\to\infty$, for any
$1\leq\ell\leq k+4$, showing that $\Phi_{0}^{\prime}\in
H^{k+3}(\mathbb{R};\mathbb{R}^{n})$.

  * (HTw)

There exists a waveprofile $\Phi_{0}\in C^{2}(\mathbb{R};\mathbb{R}^{n})$ and
a wavespeed $c_{0}\in\mathbb{R}$ that satisfy the travelling wave ODE

$\Phi_{0}^{\prime\prime}+c_{0}\Phi_{0}^{\prime}+f(\Phi_{0})=0.$ (2.5)

In addition, there is a constant $K>0$ together with exponents $\nu_{\pm}>0$
so that the bound

$|\Phi_{0}(x)-u_{-}|+|\Phi_{0}^{\prime}(\xi)|\leq Ke^{-\nu_{-}|x|}$ (2.6)

holds for all $x\leq 0,$ whereas the bound

$|\Phi_{0}(x)-u_{+}|+|\Phi_{0}^{\prime}(\xi)|\leq Ke^{-\nu_{+}|x|}$ (2.7)

holds for all $x\geq 0.$

Linearising (2.5) around the travelling wave $(\Phi_{0},c_{0})$ leads to the
linear operator

$\mathcal{L}_{\rm tw}:H^{2}(\mathbb{R};\mathbb{R}^{n})\to
L^{2}(\mathbb{R};\mathbb{R}^{n})$ (2.8)

that acts as

$[\mathcal{L}_{\rm
tw}u](x)=u^{\prime\prime}(x)+c_{0}u^{\prime}(x)+Df(\Phi_{0}(x))u(x).$ (2.9)

In particular, observe $\mathcal{L}_{\rm tw}\Phi_{0}^{\prime}=0$. Its
associated adjoint operator, which we will denote by

$\mathcal{L}_{\rm tw}^{\rm adj}:H^{2}(\mathbb{R};\mathbb{R}^{n})\to
L^{2}(\mathbb{R};\mathbb{R}^{n}),$ (2.10)

acts as

$[\mathcal{L}_{\rm tw}^{\rm
adj}w](x)=w^{\prime\prime}(x)-c_{0}w^{\prime}(x)+Df(\Phi_{0}(x))w(x),$ (2.11)

and it is easily verified that $\langle\mathcal{L}_{\rm
tw}v,w\rangle_{L^{2}(\mathbb{R};\mathbb{R}^{n})}=\langle v,\mathcal{L}_{\rm
tw}^{\rm adj}w\rangle_{L^{2}(\mathbb{R};\mathbb{R}^{n})}$ holds for $v,w\in
H^{2}(\mathbb{R};\mathbb{R}^{n}).$

We continue with a standard spectral stability condition for the operator
$\mathcal{L}_{\rm tw}$. In particular, we demand that the translational
eigenvalue at zero is an isolated simple eigenvalue and that the remainder of
the spectrum can be strictly bounded to the left of the imaginary axis, i.e.,
there is a spectral gap.

  * (HS)

The operator $\mathcal{L}_{\rm tw}:H^{2}(\mathbb{R};\mathbb{R}^{n})\to
L^{2}(\mathbb{R};\mathbb{R}^{n})$ has a simple eigenvalue at $\lambda=0$ and
there exists a constant $\beta>0$ so that the operator $\mathcal{L}_{\rm
tw}-\lambda:H^{2}(\mathbb{R};\mathbb{R}^{n})\to
L^{2}(\mathbb{R};\mathbb{R}^{n})$ is invertible for all $\lambda\in\mathbb{C}$
satisfying Re $\lambda\geq-2\beta$.

Assuming (Hf-Lip) or (Hf-Cub) together with (HTw) and (HS), we note that the
operators $\mathcal{L}_{\rm tw}$ and $\mathcal{L}_{\rm tw}^{\rm adj}$ map
$H^{\ell+2}(\mathbb{R};\mathbb{R}^{n})$ into
$H^{\ell}(\mathbb{R};\mathbb{R}^{n})$, for any $0\leq\ell\leq k+1$. This
follows either directly from inspection or general (Sobolev tower)
interpolation theory [15, 62, 93]. Furthermore, the resolvent set of
$\mathcal{L}_{\rm tw}$ restricted to the domain
$H^{k+3}(\mathbb{R};\mathbb{R}^{n})$ contains the resolvent set of
$\mathcal{L}_{\rm tw}$ seen as the original operator with domain
$H^{2}(\mathbb{R};\mathbb{R}^{n}).$ Consequently, the operator
$\mathcal{L}_{\rm tw}:H^{k+3}(\mathbb{R};\mathbb{R}^{n})\to
H^{k+1}(\mathbb{R};\mathbb{R}^{n})$ is Fredholm with index zero, we have

$\ker(\mathcal{L}_{\rm tw})=\text{span}\\{\Phi_{0}^{\prime}\\}\subset
H^{k+3}(\mathbb{R};\mathbb{R}^{n}),\quad\ker(\mathcal{L}_{\rm tw}^{\rm
adj})=\text{span}\\{\psi_{\rm tw}\\}\subset
H^{k+3}(\mathbb{R};\mathbb{R}^{n}),$ (2.12)

for some $\psi_{\rm tw}$ that satisfies the normalised identity

$\langle\Phi_{0}^{\prime},\psi_{\rm
tw}\rangle_{L^{2}(\mathbb{R};\mathbb{R}^{n})}=1,$ (2.13)

and $|\psi_{\rm tw}^{(\ell)}(x)|\to 0$ exponentially fast as $|x|\to\infty$
for any $0\leq\ell\leq k+3$.

Since $\mathcal{L}_{\rm tw}$ is a lower order perturbation of the diffusion
operator $\partial_{x}^{2}$, we see that $\mathcal{L}_{\rm tw}$ is
sectorial555Following a definition as in [32, Sec. 10], we have that
$-\mathcal{L}_{\rm tw}$ is sectorial. in $L^{2}(\mathbb{R};\mathbb{R}^{n})$,
$H^{k}(\mathbb{R};\mathbb{R}^{n})$ and $H^{k+1}(\mathbb{R};\mathbb{R}^{n})$
(the latter two after restriction), and hence generates analytic semigroups on
these spaces, which we all denote by $(S_{\rm tw}(t))_{t\geq 0}$ since they
agree where they overlap [15, 62, 93]. The same conclusions hold for the
adjoint operator $\mathcal{L}_{\rm tw}^{\rm adj}$.

Lastly, the orthogonal projection $P_{\rm
tw}:H^{k}(\mathbb{R};\mathbb{R}^{n})\to H^{k}(\mathbb{R};\mathbb{R}^{n})$,
defined by

$P_{\rm tw}v=\langle v,\psi_{\rm
tw}\rangle_{L^{2}(\mathbb{R};\mathbb{R}^{n})}\Phi_{0}^{\prime},$ (2.14)

plays a crucial role for our stability analysis. It is used to project out the
translational eigenfunction in order to circumvent the neutral mode of the
semigroup $S_{\mathrm{tw}}(t)$.

###  2.2 Stochastic setup

In this part we formulate conditions to ensure that the stochastic terms in
(2.1) are well-defined. We start by considering the covariance function $q$
that governs the noise process, writing $\hat{q}$ for the Fourier transform of
$q$ (see also Appendix C). Recall, the integer $m$ corresponds to the
dimension of the space from which the noise will be sampled.

  * (Hq)

We have $q\in H^{\ell}(\mathcal{D};\mathbb{R}^{m\times m})\cap
L^{1}(\mathcal{D};\mathbb{R}^{m\times m})$ for some integer $\ell>2k+d/2,$
with $q(-\mathbf{x})=q(\mathbf{x})$ and $q^{\top}(\mathbf{x})=q(\mathbf{x})$
for all $\mathbf{x}=(x,y)\in\mathcal{D}=\mathbb{R}\times\mathbb{T}^{d-1}$.
Further, for
$\boldsymbol{\xi}=(\omega,\xi)\in\widehat{D}=\mathbb{R}\times\mathbb{Z}^{d-1}$
the $m\times m$ matrix $\hat{q}(\boldsymbol{\xi})$ is non-negative definite.

Thanks to Young’s inequality, the integrability of the kernel $q$ allows us to
introduce the bounded convolution operator $Q$ on
$L^{2}(\mathcal{D};\mathbb{R}^{m})$ that acts as

$[Qv](\mathbf{x})=[q*v](\mathbf{x})=\int_{\mathcal{D}}q(\mathbf{x}-\mathbf{x}^{\prime})v(\mathbf{x}^{\prime})\mathrm{d}\mathbf{x}^{\prime}=\int_{\mathbb{R}}\int_{\mathbb{T}^{d-1}}q(x-x^{\prime},y-y^{\prime})v(x^{\prime},y^{\prime})\mathrm{d}x^{\prime}\mathrm{d}y^{\prime}.$
(2.15)

The properties in (Hq) imply that $Q$ is symmetric and that $\langle
Qv,v\rangle_{L^{2}(\mathcal{D};\mathbb{R}^{m})}\geq 0$ holds for every $v\in
L^{2}(\mathcal{D};\mathbb{R}^{m})$, showing that $Q$ is indeed a covariance
operator. Consequently, we can follow [11, 21, 24, 30, 42, 73] and construct a
cylindrical $Q$-Wiener process $W^{Q}=(W^{Q}_{t})_{t\geq 0}$ that is defined
on a filtered probability space $(\Omega,\mathcal{F},\mathbb{F},\mathbb{P})$
and takes values in (an extended space containing) the Hilbert space
$L^{2}(\mathcal{D};\mathbb{R}^{m})$. For more information, we refer to §3.2.

In order to perform $H^{k}$-valued stochastic integration with respect to
$W^{Q}$, we need to interpret the noise term $g(u)$ in (2.1) as a Hilbert-
Schmidt operator from $L^{2}_{Q}$ into $H^{k}(\mathcal{D};\mathbb{R}^{n})$, in
which we have introduced the Hilbert space

$L_{Q}^{2}=Q^{1/2}\big{(}L^{2}(\mathcal{D};\mathbb{R}^{m})\big{)}.$ (2.16)

We proceed by viewing $g$ as a Nemytskii operator that acts in a pointwise
fashion as

$(g(u)[\xi])(\mathbf{x})=g(u(\mathbf{x}))\xi(\mathbf{x}),$ (2.17)

for any $\xi\in L^{2}_{Q}$. We impose the following pointwise conditions on
$g$, which most notably require $g$ to vanish at the limits $u_{\pm}$ for the
waveprofile.

  * (HSt)

We have $g\in C^{k+1}(\mathbb{R}^{n};\mathbb{R}^{n\times m})$ with
$g(u_{-})=g(u_{+})=0.$ In addition, there is a constant $K_{g}>0$ such that

$|g(u_{A})-g(u_{B})|+\ldots+|D^{k+1}g(u_{A})-D^{k+1}g(u_{B})|\leq
K_{g}|u_{A}-u_{B}|$ (2.18)

holds for all $u_{A},u_{B}\in\mathbb{R}^{n}.$

The results in §4.1 use the $H^{\ell}$-smoothness of $q$ to show that $g(u)$
has the desired properties for all $u$ in the affine spaces

$\mathcal{U}_{H^{k}(S;\mathbb{R}^{n})}=\Phi_{\rm
ref}+H^{k}(S;\mathbb{R}^{n}),\quad S\in\\{\mathbb{R},\mathcal{D}\\},$ (2.19)

where $\Phi_{\rm ref}$ is a sufficiently smooth reference function that has
$\Phi_{\rm ref}(\pm\infty)=u_{\pm}$; for example $\Phi_{\rm ref}=\Phi_{0}$.
Indeed, in general we cannot work directly with $H^{k}$ since the wave
$\Phi_{0}$ is not included in this space. On the other hand, in the special
case $u_{+}=u_{-}=0$ one can simply take $\Phi_{\rm ref}=0$.

The purpose of the function $h$ in (2.1) is to allow Itô-Stratonovich
correction terms to be incorporated into our framework. For example, in the
scalar case $n=m=1$ the choice $h(u)=\frac{1}{2}q(0)g^{\prime}(u)g(u)$ allows
us to interpret (2.1) as the Stratonovich SPDE

$du=[\Delta u+f(u)]\mathrm{d}t+\sigma g(u)\circ\mathrm{d}W_{t}^{Q}.$ (2.20)

We refer to [16, 92, 94] for further information. In view of the example
above, we require one order of smoothness less on $h$ as compared to $g$.

  * (HCor)

We have $h\in C^{k}(\mathbb{R}^{n};\mathbb{R}^{n})$ with
$h(u_{-})=h(u_{+})=0$. In addition, there is a constant $K_{h}>0$ such that

$|h(u_{A})-h(u_{B})|+\ldots+|D^{k}h(u_{A})-D^{k}h(u_{B})|\leq
K_{h}|u_{A}-u_{B}|$ (2.21)

holds for all $u_{A},u_{B}\in\mathbb{R}^{n}$.

###  2.3 Overview

Our goal here is to provide a comprehensive overview of the key steps and
intermediate results that lead up to Theorem 2.6. In particular, it will
become clear in what sense our solutions are defined, how our phase-tracking
mechanism is implemented, and which expectation estimates tend towards the
main (meta)stability estimate. Our results cover different combinations of the
parameters $(k,d)$, as summarised in Table 2.1.

non-linearity $f$ | smoothness $k$ | dimension $d$ | existence/uniqueness | (meta)stability
---|---|---|---|---
(Hf-Cub) | $k$ | $=$ | $0$ | $d$ | $=$ | $1$ |  global∗ |  yes∗∗
$k$ | $=$ | $1$ | $d$ | $\leq$ | $4$ | global | yes
(Hf-Lip) |  $k$ | = | $0$ |  $d$ | $=$ | $1$ | global |  yes∗∗
$k$ | = | $0$ | $d$ | $\geq$ | $2$ | global | no
$k$ | = | $1$ | $d$ | $\leq$ | $4$ | global | yes
$k$ | $>$ | $1$ | $d$ | $=$ | $1$ | local | yes
$k$ | $>$ | $\tfrac{d}{2}$ | $d$ | $\geq$ | $2$ | local | yes
Table 2.1: List of compatible combinations of the smoothness parameter $k$ and
dimension $d$, with (meta)stability on exponentially long time scales. $(*)$:
The case (Hf-Cub) with $k=0$ and $d=1$ is treated in [29, 30] and requires the
one-sided inequality (1.16) as an additional assumption. $(**)$:
(meta)stability for the case $k=0$ and $d=1$ is discussed in Remark 7.2. In
particular, the techniques in this paper can be used to strengthen the bound
obtained in [29].

#### Step 1: Phase-tracking mechanism

Let us continue the discussion from the introduction. We couple the SDE

$\mathrm{d}\gamma=[c+a_{\sigma}(u,\gamma;c)]\mathrm{d}t+\sigma
b(u,\gamma)\mathrm{d}W_{t}^{Q}$ (2.22)

to the SPDE (2.1). The two functions

$a_{\sigma}:\mathcal{U}_{H^{k}(\mathcal{D};\mathbb{R}^{n})}\times\mathbb{R}\to\mathbb{R}\quad\text{and}\quad
b:\mathcal{U}_{H^{k}(\mathcal{D};\mathbb{R}^{n})}\times\mathbb{R}\to
HS(L^{2}_{Q};\mathbb{R})$ (2.23)

are defined in Appendix A, based on technical considerations to ensure that
our perturbation does not feel the neutral translational mode of the linear
flow; see Step 3 for further information. The velocity parameter $c$ is
discussed further in Step 4.

#### Step 2: Existence, uniqueness, and regularity

Our first result will clarify what we mean by a solution to system (2.1). In
particular, under the assumption (Hf-Lip) we will obtain continuous
$L^{2}$-valued solutions that are unique up to indistinguishability [76]. As a
matter of fact, there exists a version of $z(t)$, let us say $\tilde{z}(t),$
which satisfies (i)-(iii) and for which the map
$\tilde{z}:(0,T]\times\Omega\mapsto H^{1}(\mathcal{D};\mathbb{R}^{n})$ is
progressively measurable [4, 59, 73]. However $\tilde{z}(t)$ is not continuous
in $H^{1}(\mathcal{D};\mathbb{R}^{n})$. We refer to §5 for the precise
interpretation of the diffusion operator in the integral equation (2.26),
which should be seen in the context of Gelfand triples [17, Sec. 5.9].

######  Proposition 2.2 (see §5).

Fix $k=0$ and assume that (Hf-Lip), (HSt), (HCor), (HTw) and (Hq) are
satisfied. Pick $T>0$ together with $0\leq\sigma\leq 1$. Then for any initial
condition

$(z_{0},\gamma_{0})\in L^{2}(\mathcal{D};\mathbb{R}^{n})\times\mathbb{R},$
(2.24)

there are progressively measurable maps

$z:[0,T]\times\Omega\to
L^{2}(\mathcal{D};\mathbb{R}^{n}),\quad\gamma:[0,T]\times\Omega\to\mathbb{R},$
(2.25)

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
$C([0,T];L^{2}(\mathcal{D};\mathbb{R}^{n})\times\mathbb{R})$;

  2. (ii)

We have the integrability condition $(z,\gamma)\in
L^{2}([0,T]\times\Omega,\mathrm{d}t\times\mathbb{P};H^{1}(\mathcal{D};\mathbb{R}^{n})\times\mathbb{R})$;

  3. (iii)

The $L^{2}(\mathcal{D};\mathbb{R}^{n})$-valued identity666At first, the
equality in (2.26) should be understood as an equality in the dual space of
$H^{1}$, thus in the analytically weak sense, but by (ii) we can conclude that
we have equality in $L^{2}$. See also [27], [59, Rem. 4.2.2] and §5 for more
information.

$\displaystyle z(t)=z_{0}+\int_{0}^{t}[\Delta z(s)+\Phi_{\rm
ref}^{\prime\prime}]\mathrm{d}s$ $\displaystyle+\int_{0}^{t}f(z(s)+\Phi_{\rm
ref})\mathrm{d}s$ (2.26) $\displaystyle+\sigma^{2}\int_{0}^{t}h(z(s)+\Phi_{\rm
ref})\mathrm{d}s+\sigma\int_{0}^{t}g(z(s)+\Phi_{\rm ref})\mathrm{d}W_{s}^{Q},$

together with the scalar identity

$\gamma(t)=\gamma_{0}+\int_{0}^{t}[c+a_{\sigma}(z(s)+\Phi_{\rm
ref},\gamma(s);c)]\mathrm{d}s+\sigma\int_{0}^{t}b(z(s)+\Phi_{\rm
ref},\gamma(s))\mathrm{d}W_{s}^{Q},$ (2.27)

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

  4. (iv)

Suppose there are another progressively measurable maps $\tilde{z}$ and
$\tilde{\gamma}$ that satisfy (i)–(iii). Then for almost all
$\omega\in\Omega,$ we have

$\tilde{z}(t,\omega)=z(t,\omega)\quad\text{and}\quad\tilde{\gamma}(t,\omega)=\gamma(t,\omega),\quad\text{for
all }0\leq t\leq T.$ (2.28)

Upon fixing $k=1$ and assuming (Hf-Cub) instead of (Hf-Lip), the same results
hold after replacing the pair
$(L^{2}(\mathcal{D};\mathbb{R}^{n}),H^{1}(\mathcal{D};\mathbb{R}^{n}))$ by
$(H^{1}(\mathcal{D};\mathbb{R}^{n}),H^{2}(\mathcal{D};\mathbb{R}^{n}))$.

In §5.2 we investigate the regularity of the solution described in Proposition
2.2. We observe that the solution $z(t)$ with a smooth initial condition
$z_{0}\in H^{k}(\mathcal{D};\mathbb{R}^{n})$ remains $H^{k}$-smooth on (at
least) a short realisation-dependent time interval $[0,\tau(\omega))$, but
could lose smoothness over time. An interesting corollary of our stability
result in Step 5 is that, with very high probability, solutions remain smooth
for a very long time.

#### Step 3: Evolution of the perturbation

For the purposes of this step, we may choose any pair $(\Phi,c)$ that is
sufficiently close to $(\Phi_{0},c_{0})$ as the basis for our perturbation
argument. In particular, we impose the following condition and refer to
Appendix A for more information.

  * (HPar)

The conditions (HTw) and (HS) hold and the pair
$(\Phi,c)\in\mathcal{U}_{H^{k+2}(\mathbb{R};\mathbb{R}^{n})}\times\mathbb{R}$
satisfies

$\|\Phi-\Phi_{0}\|_{H^{k+2}(\mathbb{R};\mathbb{R}^{n})}\leq\min\\{1,|\mathbb{T}|^{\frac{d-1}{2}}[4\|\psi\|_{H^{1}(\mathbb{R};\mathbb{R}^{n})}]^{-1}\\},\quad|c-c_{0}|\leq
1.$ (2.29)

We now decompose $u(t)$ as

$u(t)=T_{\gamma(t)}\Phi+T_{\gamma(t)}v(t),$ (2.30)

or equivalently, define the perturbation $v(t)$ by writing

$v(t)=T_{-\gamma(t)}u(t)-\Phi,$ (2.31)

where $T_{\delta},$ for any $\delta\in\mathbb{R}$, is the right-shift operator
that act on the first coordinate only, i.e.,

$T_{\delta}u(x,y,t)=u(x-\delta,y,t),\quad\delta\in\mathbb{R}.$ (2.32)

The computations in §6.1 exploit the translational invariance of our noise to
show that $v(t)$ satisfies the system

$\mathrm{d}v=\mathcal{R}_{\sigma}(v;c,\Phi)\mathrm{d}t+\sigma\mathcal{S}(v;\Phi)\mathrm{d}W_{t}^{Q},$
(2.33)

with the deterministic part being of the form

$\mathcal{R}_{\sigma}(v;c,\Phi)=\Delta_{y}v+\kappa_{\sigma}(\Phi+v,0)\big{[}\partial_{x}^{2}v+\Phi^{\prime\prime}+\mathcal{J}_{\sigma}(\Phi+v,0;c)\big{]}+a_{\sigma}(\Phi+v,0;c)\partial_{x}(\Phi+v),$
(2.34)

while the stochastic part reads

$\mathcal{S}(v;\Phi)[\xi]=g(\Phi+v)[\xi]+\partial_{x}(\Phi+v)b(\Phi+v,0)[\xi],\quad\xi\in
L^{2}_{Q}.$ (2.35)

The exact definitions of the scalar function $\kappa_{\sigma}$ and the
nonlinearity $\mathcal{J}_{\sigma}$ are provided in Appendix A, yet the main
take away is that these terms are (more or less) the same as in the $d=1$ case
[30] and do not involve second order derivatives. In addition, we have
$\mathcal{R}_{0}(0,c_{0},\Phi_{0})=0$.

Our goal is to ensure that the orthogonality condition

$\langle v(t),\psi_{\rm tw}\rangle_{L^{2}(\mathcal{D};\mathbb{R}^{n})}=0$
(2.36)

is satisfied as long as possible, because this causes the perturbation to not
feel any of the dangerous neutral modes of our higher spatially dimensional
system (see §3.2). In order to achieve this, we set out to enforce the
conditions

$\langle{\mathcal{R}_{\sigma}(v;c,\Phi)},\psi_{\rm
tw}\rangle_{L^{2}(\mathcal{D};\mathbb{R}^{n})}=0,$ (2.37)

together with

$\langle{\mathcal{S}(v;\Phi)[\xi]},\psi_{\rm
tw}\rangle_{L^{2}(\mathcal{D};\mathbb{R}^{n})}=0,\quad\xi\in L_{Q}^{2},$
(2.38)

which is possible whenever $\|v\|_{L^{2}(\mathcal{D};\mathbb{R}^{n})}$ is
sufficiently small and by choosing the functions $a_{\sigma}$ and $b$
appropriately. Indeed, condition (2.37) is satisfied whenever

$\displaystyle
a_{\sigma}(\Phi+v,0;c)=-\kappa_{\sigma}(\Phi+v,0)\frac{\langle\partial_{xx}v+\Phi^{\prime\prime}+\mathcal{J}_{\sigma}(\Phi+v,0;c),\psi_{\rm
tw}\rangle_{L^{2}(\mathcal{D};\mathbb{R}^{n})}}{\langle\partial_{x}(\Phi+v),\psi_{\rm
tw}\rangle_{L^{2}(\mathcal{D};\mathbb{R}^{n})}},$ (2.39)

which features only derivatives with respect to the coordinate in the
direction of the wave. This is due to the fact that

$\langle\Delta_{y}v,\psi_{\rm
tw}\rangle_{L^{2}(\mathcal{D};\mathbb{R}^{n})}=\int_{\mathbb{R}}\psi_{\rm
tw}\int_{\mathbb{T}^{d-1}}\nabla_{y}\cdot(\nabla_{y}v)\,\mathrm{d}y\,\mathrm{d}x=0,$
(2.40)

as a consequence of Gauss’ divergence theorem, or more simply due to the fact
that $\nabla_{y}v$ is periodic and because the integral of the derivative of a
periodic function vanishes. Likewise, condition (2.38) is satisfied whenever

$b(\Phi+v,0)[\xi]=-\frac{\langle g(\Phi+v)\xi,\psi_{\rm
tw}\rangle_{L^{2}(\mathcal{D};\mathbb{R}^{n})}}{\langle\partial_{x}(\Phi+v),\psi_{\rm
tw}\rangle_{L^{2}(\mathcal{D};\mathbb{R}^{n})}},\quad\xi\in L^{2}_{Q}.$ (2.41)

This briefly motivates the expressions for $a_{\sigma}$ and $b$ found in
Appendix A, which require less regularity on $v$ and also contain cut-off
functions to account for the cases where $v$ is not sufficiently small.

#### Step 4: Instantaneous stochastic waves

In this step we use our remaining degrees of freedom to pick the initial phase
$\gamma_{0}$ from Step 2 and the profile $(\Phi,c)$ appearing in Step 3. To
this end, we construct a branch of profiles and speeds
$(\Phi_{\sigma},c_{\sigma})$ that turns out to be highly convenient for our
computations and Taylor expansions. In particular, we will demand

$\Phi_{\sigma}^{\prime\prime}+\mathcal{J}_{\sigma}(\Phi_{\sigma},0;c_{\sigma})=0,$
(2.42)

which by construction implies that

$\mathcal{R}_{\sigma}(0;c_{\sigma},\Phi_{\sigma})=0\quad\text{and}\quad
a_{\sigma}(\Phi_{\sigma},0;c_{\sigma})=0.$ (2.43)

In particular, when $u(t)=T_{\gamma(t)}\Phi_{\sigma}$ holds for some $t$, the
phase $\gamma(t)-c_{\sigma}t$ and perturbation $v(t)$ will instantaneously
only feel stochastic forcing, motivating their name. Referring to Appendix A,
we note that $\mathcal{J}_{0}(\Phi,0;c)=c\Phi^{\prime}+f(\Phi)$, which in view
of (2.5) implies that the branch $(\Phi_{\sigma},c_{\sigma})$ reduces
naturally to the deterministic pair $(\Phi_{0},c_{0})$ at $\sigma=0$.

######  Proposition 2.3.

Suppose that either (Hf-Lip) is satisfied with $k\geq 0$ or that $k=1$ and
(Hf-Cub) is satisfied. Assume furthermore that (Hq), (HSt), (HCor), (HTw) and
(HS) hold and pick a sufficiently large $K>0$. Then there exists a
$\delta_{\sigma}>0$ so that for every $0\leq\sigma\leq\delta_{\sigma}$ there
is a unique pair

$(\Phi_{\sigma},c_{\sigma})\in\mathcal{U}_{H^{k+2}(\mathbb{R};\mathbb{R}^{n})}\times\mathbb{R}$
(2.44)

that satisfies

$\Phi_{\sigma}^{\prime\prime}+\mathcal{J}_{\sigma}(\Phi_{\sigma},0;c_{\sigma})=0$
(2.45)

and admits the bound

$\|\Phi_{\sigma}-\Phi_{0}\|_{H^{k+2}(\mathbb{R};\mathbb{R}^{n})}+|c_{\sigma}-c_{0}|\leq
K\sigma^{2}.$ (2.46)

###### Proof.

Note that this is a problem in one spatial dimension. In particular, the
result readily follows from a similar fixed point argument as in the proof of
[27, Prop. 2.2.2]; see also [30, Sec. 4.5]. ∎

We are interested in solutions with an initial condition $u(0)=u_{0}$ close to
$\Phi_{\sigma}.$ The result below shows that the initial phase
$\gamma(0)=\gamma_{0}$ can be chosen in such a way that the orthogonality
condition (2.36) holds at $t=0$ (and hence for all later times until we lose
control over the size of $v$).

######  Proposition 2.4.

Suppose that either (Hf-Lip) is satisfied with $k\geq 0$ or that $k=1$ and
(Hf-Cub) is satisfied. Assume furthermore that (Hq), (HSt), (HCor), (HTw) and
(HS) all hold. Then there exist constants $\delta_{0}>0,\delta_{\sigma}>0$ and
$K>0$ so that for every $0\leq\sigma\leq\delta_{\sigma}$ and any initial
condition $u_{0}\in\mathcal{U}_{H^{k}(\mathcal{D};\mathbb{R}^{n})}$ that
satisfies

$\|u_{0}-\Phi_{\sigma}\|_{H^{k}(\mathcal{D};\mathbb{R}^{n})}<\delta_{0},$
(2.47)

there exists a $\gamma_{0}\in\mathbb{R}$ for which the function

$v_{\gamma_{0}}=T_{-\gamma_{0}}u_{0}-\Phi_{\sigma}$ (2.48)

satisfies the orthogonality condition $\langle v_{\gamma_{0}},\psi_{\rm
tw}\rangle_{L^{2}(\mathcal{D};\mathbb{R}^{n})}=0$ and admits the bound

$|\gamma_{0}|+\|v_{\gamma_{0}}\|_{H^{k}(\mathcal{D};\mathbb{R}^{n})}\leq
K\|u_{0}-\Phi_{\sigma}\|_{H^{k}(\mathcal{D};\mathbb{R}^{n})}.$ (2.49)

###### Proof.

The proof is analogous to the proof of [27, Prop. 2.2.3]. ∎

#### Step 5: Mild formulation

In order to utilise the stability properties of the semigroup $S_{\rm tw}(t)$
associated to the one-dimensional travelling wave, we need to establish some
form of mild representation for the perturbation $v(t)$. Due to the
quasilinear structure of our problem, this turns out to be rather subtle. In
§6.2-6.3 we perform a (stochastic) time transformation allowing us to arrive
at the representation777For notational clarity, we continue to use $t$ for the
transformed time in this section.

$v(t)=E(t,0)v(0)+\int_{0}^{t}E(t,s)\mathcal{N}_{\sigma}(v(s))\mathrm{d}s+\sigma\int_{0}^{t}E(t,s)\mathcal{M}_{\sigma}(v(s))\mathrm{d}W_{s}^{-}.$
(2.50)

Observe that $E(t,s)$ denotes the random evolution family generated by the
time-dependent family of linear operators

$t\mapsto\mathcal{L}_{\sigma}(v(t)):=\mathcal{L}_{\rm
tw}+\kappa_{\sigma}(\Phi_{\sigma}+v(t),0)^{-1}\Delta_{y}.$ (2.51)

In addition, the nonlinearities $\mathcal{N}_{\sigma}$ and
$\mathcal{M}_{\sigma}$ are defined in (A.28) and satisfy

$\begin{array}[]{lcl}\mathcal{N}_{\sigma}(v)&=&\kappa_{\sigma}(\Phi_{\sigma}+v,0)^{-1}\mathcal{R}_{\sigma}(v;c_{\sigma},\Phi_{\sigma})-\mathcal{L}_{\sigma}(v)v,\\\\[5.69046pt]
\mathcal{M}_{\sigma}(v)&=&\kappa_{\sigma}(\Phi_{\sigma}+v,0)^{-1/2}\mathcal{S}(v;\Phi_{\sigma}).\end{array}$
(2.52)

We note that the $\kappa_{\sigma}$ factors are generated by the time
transformation. Due to the dependence of the behaviour of the solution on the
interval $[s,t]$, the function $s\mapsto E(t,s)$ is no longer progressively
measurable. In particular, the stochastic integral in (2.50) is a so-called
forward integral, which we discuss at length in §3.2.

As explained in Appendix A, the alternative representation (A.30) can be used
to conclude that also $\langle\mathcal{N}_{\sigma}(v),\psi_{\rm
tw}\rangle_{L^{2}}=0$ holds whenever $\|v\|_{L^{2}}$ is small. Clearly, we
also have $\langle\mathcal{M}_{\sigma}(v),\psi_{\rm tw}\rangle_{L^{2}}=0$
whenever $\|v\|_{L^{2}}$ is small. These orthogonality conditions are crucial
to circumvent the neutral modes of the random evolution family $E(t,s),$
resulting in long-time stability.

#### Step 6: Long-time stability

As explained in the introduction, our strategy to obtain a stability result
for the pair $(\Phi_{\sigma},c_{\sigma})$ is to control the size of the
perturbation $v(t)$ in terms of the expression

$N_{\varepsilon;k}(t)=\|v(t)\|_{H^{k}}^{2}+\int_{0}^{t}e^{-\varepsilon(t-s)}\|v(s)\|_{H^{k+1}}^{2}\mathrm{d}s,$
(2.53)

in which $\varepsilon>0$ is a small fixed parameter. For any $\eta>0$, the
associated stopping time is given by

$t_{\rm st}(\eta;k)=\inf\\{t\geq 0:N_{\varepsilon;k}(t)>\eta\\}.$ (2.54)

The choice of $k$ becomes important here, as we will need to achieve control
over the nonlinear terms in (2.52). In more detail, in §4 we take $k>d/2$ when
(Hf-Lip) is assumed and show that

$\|\mathcal{N}_{\sigma}(v)\|_{H^{k}}\leq
K\|v\|_{H^{k}}^{2}(1+\|v\|_{H^{k+1}})+\sigma^{2}K\|v\|_{H^{k+1}}$ (2.55)

holds for some $K>0$, whenever $\|v\|_{H^{k}}\leq 1$. Alternatively, assuming
(Hf-Cub), we obtain

$\|\mathcal{N}_{\sigma}(v)\|_{H^{1}}\leq
K\|v\|_{H^{2}}^{2}+\sigma^{2}K(1+\|v\|_{H^{2}})\|v\|_{H^{2}},$ (2.56)

whenever $\|v\|_{H^{1}}\leq 1$ holds. In §7 we shall use the fact that both
cases satisfy

$\|\mathcal{N}_{\sigma}(v)\|_{H^{k+1}}\leq
K\|v\|_{H^{k+1}}^{2}+\sigma^{2}K\|v\|_{H^{k+1}},\quad\|v\|_{H^{k}}\leq 1,$
(2.57)

for some appropriate value of $k$.

Our main result here shows how the expected supremum of $N_{\varepsilon;k}(t)$
behaves as we increase $T$.

######  Proposition 2.5 (see §7).

Suppose that either (Hf-Lip) is satisfied with $k>d/2$ or that $k=1$ and (Hf-
Cub) is satisfied. Assume furthermore that (Hq), (HSt), (HCor), (HTw) and (HS)
hold and pick $\varepsilon>0$ sufficiently small. Then there exist constants
$\delta_{\eta}>0$, $\delta_{\sigma}>0$, and $K>0$ 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 moment bound

$\mathbb{E}\left[\sup_{0\leq t\leq t_{\rm st}(\eta;k)\wedge
T}|N_{\varepsilon;k}(t)|^{p}\right]\leq
K^{p}\Bigg{[}\|v(0)\|_{H^{k}}^{2p}+\sigma^{2p}(p^{p}+\log(T)^{p})\Bigg{]}.$
(2.58)

We conclude that we can choose the time $T$ to be exponentially large in
$1/\sigma.$ More specifically, for any $0\leq
T\leq\exp[\delta_{\sigma}^{2}/\sigma^{2}]$, the estimate above shows

$\mathbb{E}\left[\sup_{0\leq t\leq t_{\rm st}(\eta;k)\wedge
T}|N_{\varepsilon,k}(t)|^{p}\right]\lesssim\|v(0)\|_{H^{k}}^{2p}+\mathcal{O}(\delta_{\sigma}^{2p}).$
(2.59)

The control on all powers of $N_{\varepsilon;k}(t)$ enables us to obtain the
probability bound (2.60), which is an improvement upon the result for $k=0$
and $d=1$ in [29] and in line with the estimates found in the literature
regarding large deviations theory [11, 86, 90, 96]. The proof is relatively
direct, using an exponential Markov-type inequality.

######  Theorem 2.6.

Suppose that either (Hf-Lip) is satisfied with $k>d/2$ or that $k=1$ and (Hf-
Cub) is satisfied. Assume furthermore that (Hq), (HSt), (HCor), (HTw) and (HS)
hold and pick $\varepsilon>0$ sufficiently small. Then there exist constants
$0<\mu<1$, $\delta_{\eta}>0$, and $\delta_{\sigma}>0$ such that, for any
integer $T\geq 2,$ any $0<\eta\leq\delta_{\eta}$, any
$0\leq\sigma\leq\delta_{\sigma}$, and any initial value
$u(0)\in\mathcal{U}_{H^{k}}$ that satisfies
$\|u(0)-\Phi_{\sigma}\|_{H^{k}}^{2}<\mu\eta$, we have

$\mathbb{P}(t_{\rm st}(\eta;k)<T)\leq
2T\exp\left(-\frac{\mu\eta}{\sigma^{2}}\right).$ (2.60)

###### Proof.

Introducing the random variable

$Z_{T}=\sup_{0\leq t\leq t_{\rm st(\eta;k)}\wedge T}|N_{\varepsilon;k}(t)|,$
(2.61)

we observe that $\mathbb{P}(t_{\rm st}(\eta;k)<T)=\mathbb{P}(Z_{T}\geq\eta)$
holds. Using the bounds (2.58) for all integers $p$, we may apply the
exponential Markov-type inequality in Lemma B.1 with

$\Theta_{1}=K\sigma^{2},\quad\Theta_{2}=K\|v(0)\|_{H^{k}}^{2}+K\sigma^{2}\log(T),$
(2.62)

to obtain

$\mathbb{P}(Z_{T}\geq\eta)\leq
3T^{1/2e}\exp\left(-\frac{\eta-2eK\|v(0)\|_{H^{k}}^{2}}{2eK\sigma^{2}}\right).$
(2.63)

The bound (2.60) now readily follows upon choosing $\mu\leq(4eK)^{-1}$ and by
noting that $3T^{1/2e}\leq 2T$ holds for $T\geq 2$. ∎

Finally, we remark that the control of the second moment $(p=1)$ in (2.58)
alone already allows us to show that the exit probability increases
logarithmically in time. Indeed, applying the standard Markov inequality to
the random variable (2.61) yields

$\mathbb{P}(t_{\rm
st}(\eta;k)<T)=\mathbb{P}(Z_{T}\geq\eta)\leq\eta^{-1}\mathbb{E}Z_{T}\leq\eta^{-1}K\left[\|u(0)-\Phi_{\sigma}\|_{H^{k}}^{2}+\sigma^{2}\log(T)\right].$
(2.64)

##  3 Random evolution families and the forward integral

In this preparatory section, we consider a type of random evolution family
associated to $\mathcal{L}_{\rm tw}$ that we will encounter throughout this
paper. In particular, we consider the spatial domain
$\mathcal{D}=\mathbb{R}\times\mathbb{T}^{d-1}$ for some fixed dimension $d\geq
1$ and introduce the family of random linear operators
$\mathcal{L}_{\nu}(t):\Omega\to\mathscr{L}(H^{2},L^{2})$ that act as

$[\mathcal{L}_{\nu}(t)(\omega)u](x,y)=\mathcal{[}\mathcal{L}_{\rm
tw}u(\cdot,y)](x)+\nu(t,\omega)[\Delta_{y}u(x,\cdot)](y),$ (3.1)

for $x\in\mathbb{R}$, $y\in\mathbb{T}^{d-1},$ $0\leq t\leq T,$ and
$\omega\in\Omega$. We impose the following conditions on the coefficient
function $\nu$ and the general setting that we consider in this section.

  * (H$\nu$)

The function $\nu:[0,T]\times\Omega\mapsto\mathbb{R}$ is progressively
measurable and continuous with respect to the time variable
$\mathbb{P}$-almost surely. In addition, there exist two positive constants
$k_{\nu},K_{\nu}>0$ such that $k_{\nu}\leq\nu(t)\leq K_{\nu}$ holds for all
$t\in[0,T]$.

  * (HE)

Either (Hf-Lip) is satisfied with $k\geq 0$ or we have $k=1$ with (Hf-Cub).
Furthermore, (HTw), (HS) and (H$\nu$) all hold and we have $T\geq 1$.

It is common practice to suppress the dependency of $\omega\in\Omega$, and we
will do that too unless we want to be explicit. Since both $\mathcal{L}_{\rm
tw}$ and $\Delta_{y}$ can be seen as unbounded operators on
$L^{2}(\mathcal{D};\mathbb{R}^{n})$ that act pointwise on the ‘other’
coordinate, we will typically write (3.1) in the shorter form

$\mathcal{L}_{\nu}(t)=\mathcal{L}_{\rm tw}+\nu(t)\Delta_{y}.$ (3.2)

At times we will restrict the operators $\mathcal{L}_{\nu}(t)$ to spaces of
smoother functions while using the same notation, writing
$\mathcal{L}_{\nu}(t):\Omega\to\mathscr{L}(H^{k+2},H^{k})$, for example.
Observe that $\mathbb{P}$-a.s. the operators $\mathcal{L}_{\nu}(t)$ are
sectorial for all $0\leq t\leq T$, as they are lower order perturbations to
the diffusive operators $\partial_{x}^{2}+\nu(t)\Delta_{y}$ [61, Prop.
3.2.2(iii)].

In §3.1 we show that any family of random linear operators defined in (3.1)
satisfying (H$\nu$) generates an adapted $C_{0}$-evolution family with
convenient analytic properties. For these $\omega$-dependent evolution
families, we introduce in §3.2 a non-standard type of stochastic
integration—referred to as forward integrals—and provide an essential maximal
inequality. We proceed in §3.3 by exploiting the decaying part of the
evolution to formulate a maximal inequality in $H^{k}$ with respect to a
weight that decays exponentially in time. This allows us in §3.4 to formulate
a maximal regularity result that achieves control over the integrated
$H^{k+1}$-norm. Ultimately, in §3.5 we study the time-dependence of our bounds
and provide conditions that gaurantee a logarithmic growth rate for stochastic
convolutions with our evolution family.

###  3.1 Basic properties

In order to understand the random flow $E(t,s)$ generated by the family of
random linear operators in (3.1), we consider the linear initial value problem

$\partial_{t}v=\mathcal{L}_{\nu}(t)v,\quad v(s)=v_{s}.$ (3.3)

Writing $\hat{v}$ to denote the Fourier transform with respect to the
transverse direction, i.e.,

$\hat{v}(x,\xi)=\frac{1}{|\mathbb{T}|^{d-1}}\int_{\mathbb{T}^{d-1}}e^{-\frac{2\pi
i}{|\mathbb{T}|}\langle
y,\xi\rangle}v(x,y)\,\mathrm{d}y,\quad\xi\in\mathbb{Z}^{d-1},$ (3.4)

the initial value problem (3.3) transforms into

$\partial_{t}\hat{v}=(\mathcal{L}_{\rm
tw}-\lambda_{1}\nu(t)|\xi|^{2})\hat{v},\quad\hat{v}(s)=\hat{v}_{s},$ (3.5)

where $\lambda_{1}=4\pi^{2}/|\mathbb{T}|^{2}$ denotes the first non-zero
eigenvalue of the Laplacian $\Delta_{y}$. An explicit solution of the equation
above in terms of the analytic $C_{0}$-semigroup $S_{\rm tw}(t)$ generated by
$\mathcal{L}_{\rm tw}$ is given by

$\hat{v}(t)=S_{\rm
tw}(t)e^{-\lambda_{1}|\xi|^{2}\int_{s}^{t}\nu(r)\mathrm{d}r}\hat{v}_{s}.$
(3.6)

Applying the inversion formula

$v(x,y)=\sum_{\xi\in\mathbb{Z}^{d-1}}e^{\frac{2\pi i}{|\mathbb{T}|}\langle
y,\xi\rangle}\hat{v}(x,\xi),$ (3.7)

we hence obtain the semi-explicit expression

$[E(t,s)v](x,y)=\sum_{\xi\in\mathbb{Z}^{d-1}}e^{\frac{2\pi
i}{|\mathbb{T}|}\langle y,\xi\rangle}S_{\rm
tw}(t-s)e^{-\lambda_{1}|\xi|^{2}\int_{s}^{t}\nu(r)\mathrm{d}r}\hat{v}(x,\xi)$
(3.8)

for the evolution family associated to (3.3). Note that (3.8) reduces to the
expression found in [39] when considering the autonomous case $\nu\equiv 1$.

Our first result establishes several useful properties of this family
$E(t,s)$, which is defined on the set ${\triangle}=\\{(s,t)\in[0,T]^{2}:s\leq
t\\}$. In summary, $E(t,s)$ is an adapted $C_{0}$-evolution family in
$H^{k}=H^{k}(\mathcal{D};\mathbb{R}^{n})$ that inherits several important
features from the analytic semigroup $S_{\rm tw}(t)$.

######  Proposition 3.1.

Suppose that (HE) holds. Then the random family
$(E(t,s,\omega))_{(s,t)\in\triangle,\omega\in\Omega}$ satisfies the following
properties:

  1. (i)

There exists a constant $M>0$ that does not depend on $T$ so that
$\|E(t,s)\|_{\mathscr{L}(H^{k})}\leq M$ holds for all $(s,t)\in\triangle$;

  2. (ii)

$E(s,s)=I$ for all $s\in[0,T]$;

  3. (iii)

$E(t,s)=E(t,r)E(r,s)$ for all $0\leq s\leq r\leq t\leq T$;

  4. (iv)

The mapping $\triangle\ni(t,s)\mapsto E(t,s)$ is strongly continuous, i.e.,
$(t,s)\mapsto E(t,s)v$ is continuous for all $v\in
H^{k}(\mathcal{D};\mathbb{R}^{n})$;

  5. (v)

$E(t,s)v$ is $\mathcal{F}_{t}$-measurable for all $(s,t)\in\triangle$ and
$v\in H^{k}(\mathcal{D};\mathbb{R}^{n})$;

  6. (vi)

For every $s<t,$ one has
$\frac{\mathrm{d}}{\mathrm{d}t}E(t,s)=\mathcal{L}_{\nu}(t)E(t,s)$ and
$\frac{\mathrm{d}}{\mathrm{d}s}E(t,s)=-E(t,s)\mathcal{L}_{\nu}(s)$, and there
exists a constant $C>0$, independent of $\omega\in\Omega$, such that

$\|\mathcal{L}_{\nu}(t)E(t,s)\|_{\mathscr{L}(H^{k})}\leq C(t-s)^{-1}.$ (3.9)

###### Proof.

Observe first that for $k>0$, we have

$\partial_{x}^{\alpha}\partial_{y}^{\beta}E(t,s)v(x,y)=\sum_{\xi\in\mathbb{Z}^{d-1}}e^{\frac{2\pi
i}{|\mathbb{T}|}\langle y,\xi\rangle}\partial_{x}^{\alpha}S_{\rm
tw}(t-s)e^{-\lambda_{1}|\xi|^{2}\int_{s}^{t}\nu(r)\mathrm{d}r}\widehat{\partial_{y}^{\beta}v}(x,\xi),$
(3.10)

where $\alpha\in\mathbb{Z}_{\geq 0}$ and $\beta\in\mathbb{Z}_{\geq 0}^{d-1}$
with $\alpha+|\beta|\leq k.$ Applying Plancherel’s theorem twice yields

$\displaystyle\|\partial^{\alpha}_{x}\partial_{y}^{\beta}E(t,s)v\|_{L^{2}(\mathcal{D};\mathbb{R}^{n})}$
$\displaystyle=\|\partial_{x}^{\alpha}E(t,s)\partial_{y}^{\beta}v\|_{L^{2}(\mathbb{T}^{d-1};L^{2}(\mathbb{R};\mathbb{R}^{n}))}$
(3.11) $\displaystyle\leq
M\|\partial_{y}^{\beta}v\|_{L^{2}(\mathbb{T}^{d-1};H^{\alpha}(\mathbb{R};\mathbb{R}^{n}))}$
$\displaystyle\leq M\|v\|_{H^{k}(\mathcal{D};\mathbb{R}^{n})},$

where we choose a sufficiently large $M\geq 1$ for which $\|S_{\rm
tw}(t)\|_{\mathscr{L}(H^{\alpha}(\mathbb{R};\mathbb{R}^{n}))}\leq M$ holds for
all $\alpha\leq k$. This yields (i). Properties (ii), (iii) and (v) are
immediate from the definition (3.8).

Turning to (iv), the uniform bound in (i) implies that it suffices to
establish the continuity for $v$ in a dense set $W\subset
H^{k}(\mathcal{D};\mathbb{R}^{n})$; see, e.g., the proof of [15, Prop I.5.3].
Without loss, we restrict ourselves to $n=1$ and choose $W$ to be the set of
product functions $w(x,y)=w_{1}(x)w_{2}(y)$. Here $w_{1}(x)$ is any element in
$C^{\infty}_{c}(\mathbb{R};\mathbb{R})$ and $w_{2}(y)$ is either the real or
imaginary part of

$e^{\frac{2\pi i}{|\mathbb{T}|}\langle
y,\zeta\rangle},\quad\zeta\in\mathbb{Z}^{d-1}.$ (3.12)

Exploiting Plancherel’s theorem once more, together with the identity (3.10),
we find

$\displaystyle\|E(\tau,\sigma)w-E(t,s)w\|_{H^{k}(\mathcal{D};\mathbb{R})}$
$\displaystyle=\|S_{\rm tw}(\tau-\sigma)w_{1}h(\tau,\sigma;\zeta)-S_{\rm
tw}(t-s)w_{1}h(t,s;\zeta)\|_{H^{k}(\mathcal{D};\mathbb{R})}$ (3.13)
$\displaystyle\leq\|S_{\rm tw}(\tau-\sigma)w_{1}-S_{\rm
tw}(t-s)w_{1}\|_{H^{k}(\mathbb{R};\mathbb{R})}$
$\displaystyle\quad\quad\quad+M\|w_{1}\|_{H^{k}(\mathbb{R};\mathbb{R})}|h(\tau,\sigma;\zeta)-h(t,s;\zeta)|,$

where $h(t,s;\zeta)=e^{-\lambda_{1}|\zeta|^{2}\int_{s}^{t}\nu(r)\mathrm{d}r}$,
which satisfies $|h(t,s;\zeta)|\leq 1.$ Since $S_{\rm tw}(t)$ is strongly
continuous in $H^{k}(\mathbb{R};\mathbb{R}^{n}),$ it remains to show that
$\Delta\to\mathbb{R},(t,s)\mapsto h(t,s;\zeta)$ is continuous for every
$\zeta\in\mathbb{Z}^{d-1}$ fixed. Since $|1-e^{ax}|\leq 2e^{a}|x|$ for $x\leq
1$, we obtain

$\displaystyle|h(\tau,\sigma;\zeta)-h(t,s;\zeta)|$
$\displaystyle\leq\big{|}1-e^{\lambda_{1}|\zeta|^{2}\int_{\sigma}^{\tau}\nu(r)\mathrm{d}r-\lambda_{1}|\zeta|^{2}\int_{s}^{t}\nu(r)\mathrm{d}r}\big{|}$
(3.14) $\displaystyle\leq
2\lambda_{1}|\zeta|^{2}\left|{\int_{\sigma}^{\tau}\nu(r)\mathrm{d}r-\int_{s}^{t}\nu(r)\mathrm{d}r}\right|$
$\displaystyle\leq
2\lambda_{1}|\zeta|^{2}K_{\nu}\big{[}|\tau-t|+|\sigma-s|\big{]}<\varepsilon,$

assuming $|t-\tau|,|s-\sigma|<\varepsilon/4\lambda_{1}|\zeta|^{2}K_{\nu}$,
where $\varepsilon\ll 1$ needs to be sufficiently small.

Finally, property (vi) follows from the fact that we can swap differentiation
and summation in this situation, that $S_{\rm tw}(t)$ satisfies the properties

$\|\tfrac{\mathrm{d}}{\mathrm{d}t}S_{\rm
tw}(t-s)v\|_{H^{k}(\mathbb{R};\mathbb{R}^{n})}=\|\mathcal{L}_{\rm tw}S_{\rm
tw}(t-s)v\|_{H^{k}(\mathbb{R};\mathbb{R}^{n})}\leq(t-s)^{-1}\|v\|_{H^{k}(\mathbb{R};\mathbb{R}^{n})},$
(3.15)

and that the elementary estimate $ue^{-2u}\leq 1$ for $u\geq 0$ allows us to
deduce the inequality

$\lambda_{1}|\xi|^{2}e^{-2\lambda_{1}|\xi|^{2}k_{\nu}t}\leq t/k_{\nu},\quad
t\geq 0,$ (3.16)

on account of (H$\nu$). ∎

#### Decompositions of $E(t,s)$.

In view of the semi-explicit expression (3.8) for the random evolution family
$E(t,s)$, it is convenient to introduce the bounded linear operators
$F(t,s,\omega):H^{k}(\mathbb{T}^{d-1};\mathbb{R}^{n})\to
H^{k}(\mathbb{T}^{d-1};\mathbb{R}^{n})$ that act as

$F(t,s)v(y)=\sum_{\xi\in\mathbb{Z}^{d-1}}e^{\frac{2\pi
i}{|\mathbb{T}|}y\cdot\xi}e^{-\lambda_{1}|\xi|^{2}\int_{s}^{t}\nu(r)\mathrm{d}r}\hat{v}(\xi),$
(3.17)

for any $(s,t)\in\triangle$ and $\omega\in\Omega$. Recall, $\hat{v}$ now
denotes the usual Fourier transform on
$L^{2}(\mathbb{T}^{d-1};\mathbb{R}^{n}).$ This allows us to obtain the
commuting decomposition

$E(t,s)=F(t,s)S_{\rm tw}(t-s)=S_{\rm tw}(t-s)F(t,s)$ (3.18)

for any $(s,t)\in\triangle$. Here $S_{\rm tw}(t)$ and $F(t,s)$ can be
interpreted as a semigroup and evolution family, respectively, on the full
space $H^{k}(\mathcal{D};\mathbb{R}^{n})$ that act pointwise on the “other”
coordinate, i.e.,

$[S_{\rm tw}(t)v](x,y)=[S_{\rm
tw}(t)v(\cdot,y)](x),\quad[F(t,s)v](x,y)=[F(t,s)v(x,\cdot)](y).$ (3.19)

Indeed, we have

$\displaystyle\|S_{\rm tw}(t)v\|_{H^{k}(\mathcal{D};\mathbb{R}^{n})}$
$\displaystyle\leq C\sum_{\ell=0}^{k}\|S_{\rm
tw}(t)v\|_{H^{\ell}(\mathbb{T}^{d-1};H^{k-\ell}(\mathbb{R};\mathbb{R}^{n}))}$
(3.20) $\displaystyle\leq
CM\sum_{\ell=0}^{k}\|v\|_{H^{\ell}(\mathbb{T}^{d-1};H^{k-\ell}(\mathbb{R};\mathbb{R}^{n}))}$
$\displaystyle\leq CMc^{-1}\|v\|_{H^{k}(\mathcal{D};\mathbb{R}^{n})},$

for some constants $C,c>0$. In fact, $F(t,s)$ represents a $C_{0}$-family of
contractions, since

$\|F(t,s)\|_{\mathscr{L}(H^{k}(\mathcal{D};\mathbb{R}^{n}))}=\|F(t,s)\|_{\mathscr{L}(H^{k}(\mathbb{T}^{d-1};\mathbb{R}^{n}))}\leq
1.$ (3.21)

However in general $E(t,s)$ is not a family of contractions, since $S_{\rm
tw}(t)$ is not. Note that for $\nu\equiv 1$ we simply have
$F(t,s)=e^{(t-s)\Delta_{y}}$.

Importantly, note that we can actually decompose $E$ even further by writing

$E(t,s,\omega)=H(t,s,\omega)G(t-s)S_{\rm tw}(t-s),$ (3.22)

using the contractive random evolution family

$H(t,s,\omega)v(y)=\sum_{\xi\in\mathbb{Z}^{d-1}}e^{\frac{2\pi
i}{|\mathbb{T}|}y\cdot\xi}e^{-\lambda_{1}|\xi|^{2}\int_{s}^{t}[\nu(r)-\frac{1}{2}k_{\nu}]\mathrm{d}r}\hat{v}(\xi)$
(3.23)

and the contractive analytic semigroup

$G(t)=e^{\frac{1}{2}k_{\nu}t\Delta_{y}}.$ (3.24)

The key benefit of this decomposition is that

$E_{\rm tw}(t)=G(t)S_{\rm tw}(t)$ (3.25)

defines a (deterministic) analytic semigroup on
$H^{k}(\mathcal{D};\mathbb{R}^{n})$ that hence also has smoothening properties
in the $y$-direction; see, e.g., (3.138) and (3.149).

#### Spectral projections.

Upon introducing the (one-dimensional) spectral projection

$P_{\rm tw}u=\langle u,\psi_{\rm
tw}\rangle_{L^{2}(\mathbb{R};\mathbb{R}^{n})}\Phi_{0}^{\prime}$ (3.26)

together with its complement $P^{\perp}_{\rm tw}=I-P_{\rm tw}$, we note that
the spectral gap assumption (HS) implies the exponential decay

$\|S_{\rm tw}(t)P^{\perp}_{\rm tw}u\|_{H^{k}(\mathbb{R};\mathbb{R}^{n})}\leq
Me^{-\beta t}\|P^{\perp}_{\rm tw}u\|_{H^{k}(\mathbb{R};\mathbb{R}^{n})};$
(3.27)

see for example [61]. On the other hand, after introducing the averaging
operator

$P_{\rm
avg}u=\frac{1}{|\mathbb{T}|^{d-1}}\int_{\mathbb{T}^{d-1}}u(y)\,\mathrm{d}y$
(3.28)

with respect to the $y$-direction and writing $P_{\rm avg}^{\perp}=I-P_{\rm
avg}$, it is clear from (3.17) that

$\|F(t,s)P_{\rm avg}^{\perp}v\|_{H^{k}(\mathbb{T}^{d-1};\mathbb{R}^{n})}\leq
e^{-\lambda_{1}\kappa_{\nu}(t-s)}\|P_{\rm
avg}^{\perp}v\|_{H^{k}(\mathbb{T}^{d-1};\mathbb{R}^{n})},$ (3.29)

since the $\xi=0$ component of the Fourier transform is averaged out and hence
vanishes.

We extend these projections (which are bounded on
$H^{k}(\mathbb{R};\mathbb{R}^{n})$ and
$H^{k}(\mathbb{T}^{d-1};\mathbb{R}^{n})$, respectively) to bounded operators
on the full function space $H^{k}(\mathcal{D};\mathbb{R}^{n})$ by writing

$[P_{\rm tw}u](x,y)=\langle u(\cdot,y),\psi_{\rm
tw}\rangle_{L^{2}(\mathbb{R};\mathbb{R}^{n})}\Phi_{0}^{\prime}(x)\quad\text{and}\quad[P_{\rm
avg}u](x,y)=\frac{1}{|\mathbb{T}|^{d-1}}\int_{\mathbb{T}^{d-1}}u(x,y^{\prime})\,\mathrm{d}y^{\prime}.$
(3.30)

Observe that the bounds in (3.27) and (3.29) carry over to the full spatial
norm $\|\cdot\|_{H^{k}(\mathcal{D};\mathbb{R}^{n})}$ by estimates such as
(3.20) and direct inspection of the Fourier representation (3.17).

An important role in this paper is reserved for the bounded linear operator
$P$ on $H^{k}(\mathcal{D};\mathbb{R}^{n})$ that acts as

$P=P_{\rm tw}P_{\rm avg}=P_{\rm avg}P_{\rm tw},$ (3.31)

implying that it is also a projection. Observe that $Pv=0$ holds if and only
if $\langle v,\psi_{\rm tw}\rangle_{L^{2}(\mathcal{D};\mathbb{R}^{n})}=0.$
Further, introducing the complement

$P^{\perp}=I-P=P^{\perp}_{\rm tw}+P_{\rm tw}P^{\perp}_{\rm avg},$ (3.32)

we see that $E(t,s)P^{\perp}$ admits exponential decay.

######  Lemma 3.2.

Suppose that (HE) holds. Then there exists a constant $M\geq 1$ for which we
have

$\|E(t,s)P^{\perp}\|_{\mathscr{L}(H^{k},H^{k})}\leq Me^{-\mu(t-s)},$ (3.33)

together with

$\|E(t,s)P^{\perp}\|_{\mathscr{L}(H^{k},H^{k+1})}\leq
M\max((t-s)^{-1/2},1)e^{-\mu(t-s)},$ (3.34)

where $\mu=\min\\{\beta,\lambda_{1}k_{\nu}\\}>0.$

###### Proof.

The bound (3.33) follows directly from the representation (3.32) together with
the estimates

$\begin{array}[]{lcl}\|E(t,s)P^{\perp}_{\rm
tw}v\|_{H^{k}}&\leq&\|F(t,s)\|_{\mathcal{L}(H^{k})}\|S(t-s)P^{\perp}_{\rm
tw}v\|_{H^{k}}\\\\[5.69046pt] &\leq&Me^{-\beta(t-s)}\|P^{\perp}_{\rm
tw}v\|_{H^{k}}\end{array}$ (3.35)

and

$\begin{array}[]{lcl}\|E(t,s)P_{\rm tw}P^{\perp}_{\rm
avg}v\|_{H^{k}}&\leq&\|S(t-s)P_{\rm
tw}\|_{\mathcal{L}(H^{k}(\mathcal{D};\mathbb{R})}\|F(t,s)P^{\perp}_{\rm
avg}v\|_{H^{k}}\\\\[5.69046pt]
&\leq&Me^{-\lambda_{1}k_{\nu}(t-s)}\|P^{\perp}_{\rm
avg}v\|_{H^{k}},\end{array}$ (3.36)

increasing $M$ if necessary. On the other hand, (3.34) follows from the
standard semigroup bound

$\|S_{\rm
tw}(t)\|_{\mathscr{L}(H^{k}(\mathbb{R};\mathbb{R}^{n}),H^{k+1}(\mathbb{R};\mathbb{R}^{n}))}\leq
M\max\\{t^{-1/2},1\\}$ (3.37)

and inspection of the identity (3.10). ∎

######  Remark 3.3.

Note that we do not assume any Hölder continuity on our coefficient $\nu$, as
opposed to the setting of [15, Ch. VI.9] and [1, 2, 74, 98]. Assuming more
regularity would allow us to write the evolution family as

$E(t,s)=e^{(t-s)\mathcal{L}_{\nu}(s)}+\int_{s}^{t}Z(t,r)\,\mathrm{d}r,$ (3.38)

where $Z(t,r)$, see [1, eq. (2.7)] for instance, is expressed in terms of
inductively defined operators that are related to the resolvent of
$\mathcal{L}_{\nu}(s)$.

###  3.2 Forward integrability and maximal inequalities

We will now set out to construct a suitable notion for stochastic convolutions
against the random evolution family $E(t,s)$. The key issue that needs to be
addressed is that $E(t,s)$ is only measurable with respect to
$\mathcal{F}_{t}$ and not $\mathcal{F}_{s}$, precluding the use of the regular
Itô integral. In addition, $E(t,s)$ is not a family of contractions, which
prevents us from appealing directly to existing results.

As customary, we let $\mathcal{W}$ be a real separable Hilbert space with
orthonormal basis $(e_{k})_{k\geq 0}$ and assume
$Q\in\mathscr{L}(\mathcal{W})$ to be a non-negative symmetric operator. We can
then consider the Hilbert space $\mathcal{W}_{Q}=Q^{1/2}(\mathcal{W})$ endowed
with its natural inner product

$\langle v,w\rangle_{\mathcal{W}_{Q}}=\langle
Q^{-1/2}v,Q^{-1/2}w\rangle_{\mathcal{W}},$ (3.39)

which has $(\sqrt{Q}e_{k})_{k\geq 0}$ as an orthonormal basis.888We tacitly
neglect any possible zero element of the set $(\sqrt{Q}e_{k})_{k\geq 0}$
caused by the fact that $Q$ is only a non-negative and not a positive
operator.

Following [11, 21, 24, 30, 42, 73] we consider a filtered probability space
$(\Omega,\mathcal{F},\mathbb{F},\mathbb{P})$ [76] and set to construct a
cylindrical $Q$-Wiener process $W^{Q}=(W^{Q}_{t})_{t\geq 0}$ that is adapted
to the filtration $\mathbb{F}$. We consider a set $(\beta_{k})_{k\geq 0}$ of
independent standard Brownian motions adapted to $\mathbb{F}$ and write

$W_{t}^{Q}=\sum_{k=0}^{\infty}\sqrt{Q}e_{k}\beta_{k}(t),$ (3.40)

which converges in $L^{2}(\Omega;\mathcal{W}_{\rm ext})$ for some larger
(abstract) space $\mathcal{W}\subset\mathcal{W}_{\rm ext}$ that is guaranteed
to exist by the discussion in [24]; see also [30, Sec. 5.1] for additional
background information.

For any Hilbert space $\mathcal{H}$ and $p\geq 2$, we introduce the class of
processes

$\displaystyle\mathcal{N}^{p}([0,T];\mathbb{F};HS(\mathcal{W}_{Q};\mathcal{H}))$
$\displaystyle=\\{B\in
L^{p}(\Omega;L^{2}([0,T];HS(\mathcal{W}_{Q};\mathcal{H}))):$ (3.41)
$\displaystyle\quad\quad\quad\quad\quad B\text{ has a progressively measurable
version}\\},$

for which Itô stochastic integrals with respect to $W^{Q}$ can be defined. In
fact, we have the identity

$\int_{0}^{t}B(s)\mathrm{d}W_{s}^{Q}=\lim_{n\to\infty}\sum_{k=1}^{n}\int_{0}^{t}B(s)[\sqrt{Q}e_{k}]\mathrm{d}\beta_{k}(s),$
(3.42)

where the convergence is in $L^{p}(\Omega;\mathcal{H})$ and hence also in
probability [42]. We remark that for any measurable and adapted process, we
can find a progressive measurable version with an indistinguishable stochastic
integral [65, p. 68].

Setting out to drop the requirement concerning progressive measurability, we
follow [55] and introduce the notion of a forward integral with respect to the
cylindrical $Q$-Wiener process $W^{Q}$. We restrict ourselves to the Hilbert
space setting, referring the reader to [70] for the analogous definitions for
Banach spaces.

######  Definition 3.4.

Suppose $G:[0,T]\times\Omega\to HS(\mathcal{W}_{Q};\mathcal{H})$ is an
$\mathcal{F}$-measurable process such that for each $\xi\in\mathcal{W}$ we
$\mathbb{P}$-a.s. have $G[\xi]\in L^{1}([0,T];\mathcal{H})$. For any integer
$n\geq 1$ we define

$I^{-}(G,n)=n\sum_{k=1}^{n}\int_{0}^{T}G(s)[\sqrt{Q}{e_{k}}](\beta_{k}(s+1/n)-\beta_{k}(s))\mathrm{d}s.$
(3.43)

If the sequence $\left(I^{-}(G,n)\right)_{n=1}^{\infty}$ converges in
probability, then $G$ is said to be forward integrable (with respect to
$\mathcal{H}$), and we denote the limiting process
$\lim_{n\to\infty}I^{-}(G,n)$ either by $I^{-}(G)$ or

$\int_{0}^{T}G(s)\mathrm{d}W_{s}^{-}.$ (3.44)

In the special case
$G\in\mathcal{N}^{p}([0,T];\mathbb{F};HS(\mathcal{W}_{Q};\mathcal{H}))$, the
infinite sequence $(I^{-}(G,n))_{n=1}^{\infty}$ converges in probability and
the limit coincides with the Itô-integral [75, Prop. 3.2]. In other words, the
forward integral is indeed a proper extension of the Itô-integral. One of the
advantages over other extensions of the Itô-integral, such as the Skorohod
integral, is that one may simply pull any random operator
$A:\Omega\rightarrow\mathscr{L}(\mathcal{H},\mathcal{K})$ out of the integral.
That is, when $G$ is forward integrable with respect to $\mathcal{H}$, then
$AG$ is forward integrable with respect to $\mathcal{K}$, and

$\int_{0}^{t}AG(s)\mathrm{d}W^{-}_{s}=A\int_{0}^{t}G(s)\mathrm{d}W^{-}_{s},$
(3.45)

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

Turning towards stochastic convolutions with $E(t,s)$ in
$\mathcal{H}=H^{k}=H^{k}(\mathcal{D};\mathbb{R}^{n})$, we now follow [70] by
first considering adapted finite-rank step processes $B$ that map into
$HS(\mathcal{W}_{Q};H^{k+2})$, providing extra smoothness as compared to the
target space $H^{k}$ in which we want to understand the convolution. These
processes can be written as

$B(t)[\sqrt{Q}e_{\ell}]=\sum_{i=1}^{I}\sum_{j=1}^{J}\mathbf{1}_{\left(t_{i-1},t_{i}\right]}(t)\mathbf{1}_{A_{ij}}{h_{ij\ell}},$
(3.46)

where $A_{ij}\in\mathcal{F}_{t_{i-1}}$ and $h_{ij\ell}\in H^{k+2}$, for any
$1\leq i\leq I$, $1\leq j\leq J$ and $1\leq\ell\leq L$, for some
$I,J,L<\infty$. For the remaining $\ell>L$ we set $B(s)[\sqrt{Q}e_{\ell}]=0.$

######  Proposition 3.5.

Suppose that (HE) holds. Consider any adapted finite-rank step process $B$
that takes values in $HS(\mathcal{W}_{Q};H^{k+2})$. Then the stochastic
process $(E(t,s)B(s))_{s\in[0,t]}$ is forward integrable with respect to
$H^{k}$ on the interval $[0,t]$, for any $0\leq t\leq T,$ and
$\mathbb{P}$-a.s. we have

$\int_{0}^{t}E(t,s)B(s)\mathrm{d}W_{s}^{-}=E(t,0)\int_{0}^{t}B(s)\mathrm{d}W_{s}^{Q}+\int_{0}^{t}\partial_{s}E(t,s)\int_{s}^{t}B(r)\mathrm{d}W_{r}^{Q}\,\mathrm{d}s.$
(3.47)

Moreover, the process
$\left(\int_{0}^{t}E(t,s)B(s)\mathrm{d}W_{s}^{-}\right)_{t\in[0,T]}$ has a
continuous version in $H^{k}$.

###### Proof.

Since the process $B$ takes values in the the domain of $\mathcal{L}_{\nu}$,
item (vi) of Proposition 3.1 leads to the bound

$\|E(t,s,\omega)v\|_{W^{1,1}([0,t];H^{k})}\leq C\|v\|_{H^{k+2}}$ (3.48)

for some constant $C>0$, independent of $\omega\in\Omega$ and
$(s,t)\in\triangle$. In particular, the right-hand side of (3.47) is well-
defined and can indeed be identified as the forward integral of
$E(t,\cdot)B(\cdot)$ by applying [75, Cor. 5.3]. The pathwise continuity of
the Itô integral $\int_{0}^{t}B(s)\mathrm{d}W_{s}^{Q}$ in $H^{k+2}$ directly
implies that the forward integral (3.47) admits a continuous version in
$H^{k}$. ∎

Combining (3.45) and Proposition 3.5, we see that for any $B$ that satisfies
the conditions of the latter we may split stochastic convolutions, in the
sense that for any $0\leq\sigma\leq\tau\leq t$ we may write

$\displaystyle\int_{\sigma}^{\tau}E(t,s)B(s)\mathrm{d}W_{s}^{-}$
$\displaystyle=\int_{0}^{\tau}E(t,s)B(s)\mathrm{d}W_{s}^{-}-\int_{0}^{\sigma}E(t,s)B(s)\mathrm{d}W_{s}^{-}$
(3.49)
$\displaystyle=E(t,\tau)\int_{0}^{\tau}E(\tau,s)B(s)\mathrm{d}W_{s}^{-}-E(t,\sigma)\int_{0}^{\sigma}E(\sigma,s)B(s)\mathrm{d}W_{s}^{-}.$

More precisely, the integrals on the right hand side are well-defined (when
assuming the conditions in Proposition 3.5) and can hence be seen as a
definition for the left hand side of (3.49).

A major step towards interpreting stochastic convolutions of $E(t,s)$ with
general processes $B$ is the derivation of some maximal inequality, which
provides an estimate for (3.47) that does not rely on the additional
$H^{k+2}$-smoothness. We refer to [69, 99] for discussions on maximal
inequalities for ordinary Itô convolutions, which typically rely on
factorisation or dilation arguments that require the semigroups to be
contractive or to admit an $H^{\infty}$-calculus. In our case, we exploit the
decomposition (3.18) and utilise the fact that $S_{\rm tw}(t)$ admits an
$H^{\infty}$-calculus after projecting out the neutral mode. Note that the
factorisation method, although applicable in a much wider setting, leads to a
bound that is less sharp (see also the discussion after Theorem 3.7).

In particular, we proceed by splitting the process $B$ as

$B(s)=P_{\rm tw}B(s)+P_{\rm tw}^{\perp}B(s),$ (3.50)

noting that the spectral projection acts as

$P_{\rm tw}B(s)[w]=\langle B(s)[w],\psi_{\rm
tw}\rangle_{L^{2}(\mathbb{R};\mathbb{R}^{n})}\Phi_{0}^{\prime}$ (3.51)

for all $w\in\mathcal{W}_{Q}$. Since $S_{\rm
tw}(t)\Phi_{0}^{\prime}=\Phi_{0}^{\prime}$ holds, we have

$E(t,s)P_{\rm tw}B(s)=F(t,s)P_{\rm tw}B(s),$ (3.52)

allowing us to exploit the fact that $F$ is a contractive (random) evolution
family.

Turning to the complementary process $P_{\rm tw}^{\perp}B$, we shall make us
of the subspaces

$H^{k}_{\perp}(\mathbb{R};\mathbb{R}^{n})=\\{v\in
H^{k}(\mathbb{R};\mathbb{R}^{n}):P_{\rm tw}v=0\\},$ (3.53)

for any $k\geq 0$, which are again Hilbert spaces when endowed with the norm
$\|\cdot\|_{H^{k}}$. Considering the restricted operator $\mathcal{L}_{\rm
tw}:H^{k+2}_{\perp}(\mathbb{R};\mathbb{R}^{n})\to
H^{k}_{\perp}(\mathbb{R};\mathbb{R}^{n})$, we note that the perturbation
arguments in [101, Sec. 8] can be combined with the fact that the restriction
of $-\partial_{x}^{2}$ to $H^{k+2}(\mathbb{R};\mathbb{R}^{n})$ admits a
bounded $H^{\infty}$-calculus [32, Prop 10.2.23], to conclude that
$-\mathcal{L}_{\rm tw}$ is sectorial in
$H^{k}_{\perp}(\mathbb{R};\mathbb{R}^{n})$ and admits a bounded
$H^{\infty}$-calculus of angle strictly smaller then $\pi/2$. The details can
be found in [27, Lem. 2.9.7] where the $k=0$ case was considered, relying
crucially on the fact that the simple translational eigenvalue at zero has
been removed from the spectrum of $-\mathcal{L}_{\rm tw}$ .

In view of [99, Prop. 3.1], there exists999 With regards to the notation used
in [37, 71, 99], we point out that
$\gamma(\mathbb{R}_{+};H^{k}(\mathbb{R};\mathbb{R}^{n}))=L^{2}(\mathbb{R}_{+};H^{k}(\mathbb{R};\mathbb{R}^{n}))$,
since $H^{k}(\mathbb{R};\mathbb{R}^{n})$ is a Hilbert space, which is of type
2 as well as cotype 2 [37, Rem. 4.7].  an equivalent norm
$\vvvert\cdot\vvvert_{H^{k}_{\perp}(\mathbb{R};\mathbb{R}^{n})}$ on
$H^{k}_{\perp}(\mathbb{R};\mathbb{R}^{n})$, which is given by

$\vvvert
v\vvvert_{H^{k}_{\perp}(\mathbb{R};\mathbb{R}^{n})}=\Bigg{[}\int_{0}^{\infty}\|\mathcal{L}_{\rm
tw}^{1/2}S_{\rm
tw}(r)v\|^{2}_{H^{k}(\mathbb{R};\mathbb{R}^{n})}\,\mathrm{d}r\Bigg{]}^{1/2},$
(3.54)

ensuring that $S_{\rm tw}(t)$ restricted to
$H^{k}_{\perp}(\mathbb{R};\mathbb{R}^{n})$ is contractive with respect to this
norm. Turning to the full spatial domain, we now introduce the notation

$H^{k}_{\perp}(\mathcal{D};\mathbb{R}^{n})=\\{v\in
H^{k}(\mathcal{D};\mathbb{R}^{n}):P_{\rm tw}v=0\\}$ (3.55)

and provide it with the norm

$\vvvert
v\vvvert_{H^{k}_{\perp}(\mathcal{D};\mathbb{R}^{n})}=\left[\sum_{\beta:|\beta|\leq
k}\int_{\mathbb{T}^{d-1}}\vvvert\partial_{y}^{\beta}v(\cdot,y)\vvvert_{H^{k-|\beta|}_{\perp}(\mathbb{R};\mathbb{R}^{n})}^{2}\,dy\right]^{1/2},$
(3.56)

where the sum is with respect to multi-indices $\beta\in\mathbb{Z}^{d-1}_{\geq
0}$. This norm is again equivalent to the usual $\|\cdot\|_{H^{k}}$ norm,
which means that there exist constants $C,c>0$ for which

$c\|v\|_{H^{k}(\mathcal{D};\mathbb{R}^{n})}\leq\vvvert
v\vvvert_{H^{k}_{\perp}(\mathcal{D};\mathbb{R}^{n})}\leq
C\|v\|_{H^{k}(\mathcal{D};\mathbb{R}^{n})},$ (3.57)

for all $v\in H^{k}_{\perp}(\mathcal{D},\mathbb{R}^{n})$. The full evolution
family $E(t,s)$ can be restricted to this subspace, where it is contractive.

######  Lemma 3.6.

Suppose that (HE) holds. Then for any $(s,t)\in\triangle$ and any $v\in
H^{k}_{\perp}(\mathcal{D};\mathbb{R}^{n})$, we have $E(t,s)v\in
H^{k}_{\perp}(\mathcal{D};\mathbb{R}^{n})$ together with the bound

$\vvvert E(t,s)v\vvvert_{H^{k}_{\perp}(\mathcal{D};\mathbb{R}^{n})}\leq\vvvert
v\vvvert_{H^{k}_{\perp}(\mathcal{D};\mathbb{R}^{n})}.$ (3.58)

###### Proof.

Let us write $H^{k}_{\perp}=H^{k}_{\perp}(\mathcal{D};\mathbb{R}^{n}).$ The
invariance of $H^{k}_{\perp}$ under the evolution $E(t,s)$ follows from the
fact that inner products with respect to $x$ commute with the Fourier
transform with respect to $y$. As in the proof of Proposition 3.1, the
operator $\partial_{y}^{\beta}$ commutes with both $F(t,s)$ and $S_{\rm
tw}(t)$. Hence, for any $v\in H^{k}_{\perp}$ we compute

$\displaystyle\vvvert E(t,s)v\vvvert_{H^{k}_{\perp}}^{2}$
$\displaystyle=\sum_{\beta:|\beta|\leq
k}\int_{\mathbb{T}^{d-1}}\vvvert[\partial_{y}^{\beta}F(t,s)S_{\rm
tw}(t-s)v](\cdot,y)\vvvert_{H^{k-|\beta|}_{\perp}(\mathbb{R};\mathbb{R}^{n})}^{2}\,\mathrm{d}y$
(3.59) $\displaystyle=\sum_{\beta:|\beta|\leq
k}\int_{\mathbb{T}^{d-1}}\int_{0}^{\infty}\|\mathcal{L}_{\rm tw}^{1/2}S_{\rm
tw}(r)[\partial_{y}^{\beta}F(t,s)S_{\rm
tw}(t-s)v](\cdot,y)\|^{2}_{H^{k-|\beta|}(\mathbb{R};\mathbb{R}^{n})}\,\mathrm{d}r\,\mathrm{d}y$
$\displaystyle=\sum_{\beta:|\beta|\leq
k}\int_{0}^{\infty}\int_{\mathbb{T}^{d-1}}\|[F(t,s)\mathcal{L}_{\rm
tw}^{1/2}S_{\rm tw}(r)S_{\rm
tw}(t-s)\partial_{y}^{\beta}v](\cdot,y)\|^{2}_{H^{k-|\beta|}(\mathbb{R};\mathbb{R}^{n})}\,\mathrm{d}y\,\mathrm{d}r$
$\displaystyle\leq\sum_{\beta:|\beta|\leq
k}\int_{0}^{\infty}\int_{\mathbb{T}^{d-1}}\|[\mathcal{L}_{\rm tw}^{1/2}S_{\rm
tw}(r+t-s)\partial_{y}^{\beta}v](\cdot,y)\|^{2}_{H^{k-|\beta|}(\mathbb{R};\mathbb{R}^{n})}\,\mathrm{d}y\,\mathrm{d}r,$

where we used the fact that $F(t,s)$ is a contraction which, in addition, acts
pointwise with respect to $x$. By performing the substitution $r+t-s\mapsto
r$, we find

$\displaystyle\vvvert E(t,s)v\vvvert_{H^{k}_{\perp}}^{2}$
$\displaystyle\leq\sum_{\beta:|\beta|\leq
k}\int_{t-s}^{\infty}\int_{\mathbb{T}^{d-1}}\|[\mathcal{L}_{\rm
tw}^{1/2}S_{\rm
tw}(r)\partial_{y}^{\beta}v](\cdot,y)\|^{2}_{H^{k-|\beta|}(\mathbb{R};\mathbb{R}^{n})}\,\mathrm{d}y\,\mathrm{d}r$
(3.60) $\displaystyle\leq\sum_{\beta:|\beta|\leq
k}\int_{0}^{\infty}\int_{\mathbb{T}^{d-1}}\|[\mathcal{L}_{\rm tw}^{1/2}S_{\rm
tw}(r)\partial_{y}^{\beta}v](\cdot,y)\|^{2}_{H^{k-|\beta|}(\mathbb{R};\mathbb{R}^{n})}\,\mathrm{d}y\,\mathrm{d}r$
$\displaystyle=\sum_{\beta:|\beta|\leq
k}\int_{\mathbb{T}^{d-1}}\int_{0}^{\infty}\|[\mathcal{L}_{\rm tw}^{1/2}S_{\rm
tw}(r)\partial_{y}^{\beta}v](\cdot,y)\|^{2}_{H^{k-|\beta|}(\mathbb{R};\mathbb{R}^{n})}\,\mathrm{d}r\,\mathrm{d}y$
$\displaystyle=\sum_{\beta:|\beta|\leq
k}\int_{\mathbb{T}^{d-1}}\vvvert[\partial_{y}^{\beta}v](\cdot,y)\vvvert^{2}_{H^{k-|\beta|}_{\perp}(\mathbb{R};\mathbb{R}^{n})}\,\mathrm{d}y$
$\displaystyle=\vvvert v\vvvert_{H^{k}_{\perp}}^{2},$

as desired. ∎

We are now ready to state our main result here and provide a maximal
inequality for forward integrals. It can be seen as an extension of [70, Thm.
6.4], in the sense that $E(t,s)$ itself is not a family of contractions on the
full space $H^{k}=H^{k}(\mathcal{D};\mathbb{R}^{n})$. It is worth noting that
for $p=1$ one can use an alternative direct approach utilising the Fourier
transform.

######  Theorem 3.7 (maximal inequality).

Suppose that (HE) holds. Then there exists a constant $K_{\rm cnv}>0$ that
does not depend on $T$ so that for any adapted finite-rank step process $B$
that takes values in $HS(\mathcal{W}_{Q};H^{k+2})$, and for every $p\geq 1$,
we have the bound

$\mathbb{E}\sup_{0\leq t\leq
T}\left\|\int_{0}^{t}E(t,s)B(s)\mathrm{d}W^{-}_{s}\right\|_{H^{k}}^{2p}\leq
p^{p}K_{\rm
cnv}^{2p}\mathbb{E}\left[\int_{0}^{T}\|B(s)\|_{HS(\mathcal{W}_{Q};H^{k})}^{2}\mathrm{d}s\right]^{p}.$
(3.61)

###### Proof.

In light of the decomposition (3.50), we write

$\begin{array}[]{lcl}\mathcal{I}&=&\mathbb{E}\sup_{0\leq t\leq
T}\left\|\int_{0}^{t}E(t,s)P_{\rm
tw}B(s)\mathrm{d}W^{-}_{s}\right\|_{H^{k}}^{2p},\\\\[5.69046pt]
\mathcal{I}_{\perp}&=&\mathbb{E}\sup_{0\leq t\leq
T}\left\|\int_{0}^{t}E(t,s)P_{\rm
tw}^{\perp}B(s)\mathrm{d}W^{-}_{s}\right\|_{H^{k}}^{2p},\end{array}$ (3.62)

and note that

$\mathbb{E}\sup_{0\leq t\leq
T}\left\|\int_{0}^{t}E(t,s)B(s)\mathrm{d}W^{-}_{s}\right\|_{H^{k}}^{2p}\leq
2^{2p-1}\left(\mathcal{I}+\mathcal{I}_{\perp}\right).$ (3.63)

In view of (3.52) and the contractivity of $F(t,s)$, we may apply [70, Thm.
6.4] to find

$\displaystyle\mathcal{I}$
$\displaystyle\leq\mathbb{E}\sup_{t\in[0,T]}\left\|\int_{0}^{t}F(t,s)P_{\rm
tw}B(s)\mathrm{d}W_{s}^{-}\right\|_{H^{k}}^{2p}$ (3.64) $\displaystyle\leq
p^{p}K_{F}^{2p}\mathbb{E}\left[\int_{0}^{T}\|P_{\rm
tw}B(s)\|_{HS(\mathcal{W}_{Q};H^{k})}^{2}\mathrm{d}s\right]^{p}$
$\displaystyle\leq p^{p}K_{F}^{2p}\|P_{\rm
tw}\|_{\mathscr{L}(H^{k}(\mathbb{R};\mathbb{R}^{n}))}^{2p}\mathbb{E}\left[\int_{0}^{T}\|B(s)\|_{HS(\mathcal{W}_{Q};H^{k})}^{2}\mathrm{d}s\right]^{p}$

for some constant $K_{F}$. In a similar fashion, we may exploit Lemma 3.6 and
the equivalence (3.57) to compute

$\displaystyle\mathcal{I}_{\perp}$ $\displaystyle\leq
C^{2p}\mathbb{E}\sup_{0\leq t\leq T}\left\vvvert\int_{0}^{T}E(t,s){P_{\rm
tw}^{\perp}B(s)}\mathrm{d}W_{s}^{-}\right\vvvert_{H^{k}_{\perp}(\mathcal{D};\mathbb{R}^{n})}^{2p}$
(3.65) $\displaystyle\leq
p^{p}C^{2p}K_{E}^{2p}\mathbb{E}\left[\int_{0}^{T}\vvvert P_{\rm
tw}^{\perp}B(s)\vvvert^{2}_{HS(\mathcal{W}_{Q};H^{k}_{\perp}(\mathcal{D};\mathbb{R}^{n}))}\mathrm{d}s\right]^{p}$
$\displaystyle\leq
p^{p}C^{2p}K_{E}^{2p}c^{-2p}\mathbb{E}\left[\int_{0}^{T}\|P_{\rm
tw}^{\perp}B(s)\|^{2}_{HS(\mathcal{W}_{Q};H^{k})}\mathrm{d}s\right]^{p}$
$\displaystyle\leq p^{p}C^{2p}K_{E}^{2p}c^{-2p}\|P_{\rm
tw}^{\perp}\|_{\mathscr{L}(H^{k}(\mathbb{R};\mathbb{R}^{n}))}^{2p}\mathbb{E}\left[\int_{0}^{T}\|B(s)\|^{2}_{HS(\mathcal{W}_{Q};H^{k})}\mathrm{d}s\right]^{p}$

for some constant $K_{E}$. By setting $K_{\rm cnv}=2\max\\{K_{F}\|P_{\rm
tw}\|_{\mathscr{L}(H^{k}(\mathbb{R};\mathbb{R}^{n}))},CK_{E}c^{-1}\|P_{\rm
tw}^{\perp}\|_{\mathscr{L}(H^{k}(\mathbb{R};\mathbb{R}^{n}))}\\},$ the desired
bound follows. ∎

In retrospect, the representation (3.47) allows us (for $T\geq 1$) to derive
the (crude) pathwise bound

$\sup_{0\leq t\leq
T}\left\|\int_{0}^{t}E(t,s)B(s)\mathrm{d}W_{s}^{-}\right\|_{H^{k}}\leq
KT\sup_{0\leq t\leq
T}\left\|\int_{0}^{t}B(s)\mathrm{d}W_{s}^{Q}\right\|_{H^{k+2}},$ (3.66)

for some $K>0.$ Taking $L^{2p}(\Omega)$-norms on both sides, and appealing to
[70, Prop. 2.1 and Rem. 2.2], results into a similar inequality as in (3.61),
but then with $HS(\mathcal{W}_{Q};H^{k+2})$ instead of
$HS(\mathcal{W}_{Q};H^{k})$ and an additional factor of $T^{2p}$ on the right
hand side, which would both be detrimental to our future estimates. Note that
the factorisation method does not lead to a loss of regularity, but the
maximal inequality that one obtains is less sharp. Indeed, the constant
$K_{\rm cnv}$ would depend on time, the power $p$ on the right hand side of
(3.61) would be inside of the integral, and the result would only hold for
$p>1.$

Since the embeddings $H^{k+2}\hookrightarrow H^{k}$ are dense, for any $k\geq
0,$ we can use the maximal inequality above to extend the definition of the
forward integral for progressively measurable processes that attain values in
$H^{k}$. This is achieved in a standard fashion by approximating these
processes by adapted step processes of finite-rank with values in $H^{k+2}$.

######  Corollary 3.8 (unique extension).

Suppose that (HE) holds. Then for all $p\geq 2$, the mapping

$B\mapsto\int_{0}^{t}E(t,s)B(s)\mathrm{d}W_{s}^{-}$ (3.67)

has a unique extension to a continuous linear operator

$J_{p}:\mathcal{N}^{p}([0,t];\mathbb{F};HS(\mathcal{W}_{Q};H^{k}))\rightarrow
L^{p}(\Omega;C([0,t];H^{k})),$ (3.68)

for any $0\leq t\leq T.$

######  Remark 3.9.

We continue to denote the generalised forward integral $J_{p}B$ as
$\int_{0}^{t}E(t,s)B(s)\mathrm{d}W_{s}^{-},$ and emphasise the fact that the
maximal inequality in (3.61) remains valid. However, for general
$B\in\mathcal{N}^{p}$, it is not yet known whether the stochastic process
$J_{p}B$ is given by a forward stochastic convolution again, i.e., whether
$I^{-}\big{(}E(t,\cdot)B(\cdot)\big{)}$ exists in the sense of Definition 3.4.
On the other hand, if the latter does exist, then it is necessarily equal to
$J_{p}B$.

###  3.3 Weighted decay

Although of fundamental importance, the maximal inequality (3.61) in its
current form does not allow us to exploit the decay of the evolution family
$E(t,s)P^{\perp}$. Let us introduce the shorthand notation

$\mathcal{Z}^{\perp}[B](t)=\int_{0}^{t}E(t,s)P^{\perp}B(s)\,\mathrm{d}W_{s}^{-}.$
(3.69)

A first step is taken here by considering the process $B$ in a space that is
exponentially weighted with respect to time. In this case one can also include
the weight in the supremum bound.

######  Proposition 3.10 (weighted decay estimate).

Suppose that (HE) holds and pick $\varepsilon\in(0,2\mu)$. Then there exists a
constant $K_{\rm dc}>0$ that does not depend on $T$ so that for any process

$B\in\mathcal{N}^{2p}\big{(}[0,T];\mathbb{F};HS(\mathcal{W}_{Q};H^{k}\big{)},$
(3.70)

and every $p\geq 1$, we have the bound

$\mathbb{E}\sup_{0\leq t\leq
T}\left\|e^{-\varepsilon/2(T-t)}\mathcal{Z}^{\perp}[B](t)\right\|_{H^{k}}^{2p}\leq
p^{p}K_{\rm
dc}^{2p}\mathbb{E}\left[\int_{0}^{T}e^{-\varepsilon(T-s)}\|B(s)\|_{HS(\mathcal{W}_{Q};H^{k})}^{2}\,\mathrm{d}s\right]^{p}.$
(3.71)

In particular, for any $0\leq t\leq T$, we have

$\mathbb{E}\left\|\mathcal{Z}^{\perp}[B](t)\right\|_{H^{k}}^{2p}\leq
p^{p}K^{2p}_{\rm
dc}\mathbb{E}\left[\int_{0}^{t}e^{-\varepsilon(t-s)}\|B(s)\|_{HS(\mathcal{W}_{Q};H^{k})}^{2}\,\mathrm{d}s\right]^{p}.$
(3.72)

The key ingredient is that representation (3.47) allows us to split up
convolutions in such a way that the decay becomes visible. In particular, we
start again by considering an adapted finite-rank step process $B$ that takes
values in $HS(\mathcal{W}_{Q};H^{k+2})$. This allows us to consider a time
$N\leq t\leq N+1$, for some integer $N$, and split up the convolution as

$\mathcal{Z}^{\perp}[B](t)=I_{N;I}(t)+I_{N;II}(t),$ (3.73)

where

$\begin{array}[]{lcl}I_{N;I}(t)&=&\sum_{j=1}^{N}E(t,j)P^{\perp}\int_{0}^{j}E(j,s)B(s)\mathbf{1}_{j-1\leq
s\leq j}\,\mathrm{d}W_{s}^{-},\\\\[5.69046pt]
I_{N;II}(t)&=&\int_{N}^{t}E(t,s)P^{\perp}B(s)\,\mathrm{d}W_{s}^{-}.\end{array}$
(3.74)

Note that, in the above, we have exploited the commutation relation
$P^{\perp}E(t,s)=E(t,s)P^{\perp}$.

###### Proof of Proposition 3.10.

Without loss we assume $T\in\mathbb{N}$. We first consider an adapted finite-
rank step process $B$ that takes values in $HS(\mathcal{W}_{Q};H^{k+2})$. In
particular, the splitting (3.73) holds. Using the bound (3.33), we hence
obtain the initial estimate

$\displaystyle\left\|I_{N;I}(t)\right\|_{H^{k}}$
$\displaystyle\leq\sum_{j=1}^{N}\|E(t,j)P^{\perp}\|_{\mathscr{L}(H^{k})}\left\|\int_{j-1}^{j}E(j,s)B(s)\mathrm{d}W_{s}^{-}\right\|_{H^{k}}$
(3.75) $\displaystyle\leq
Me^{-\mu(t-N)}\sum_{j=1}^{N}e^{-\mu(N-j)}\left\|\int_{j-1}^{j}E(j,s)B(s)\mathrm{d}W_{s}^{-}\right\|_{H^{k}}.$

An application of Hölder’s inequality yields

$\displaystyle\left\|I_{N;I}(t)\right\|_{H^{k}}^{2p}$ $\displaystyle\leq
M^{2p}e^{-2\mu
p(t-N)}\left[\sum_{j=1}^{N}e^{-\frac{(2\mu-\varepsilon)}{2}(N-j)}\cdot
e^{-\frac{\varepsilon}{2}(N-j)}\left\|\int_{j-1}^{j}E(j,s)B(s)\mathrm{d}W_{s}^{-}\right\|_{H^{k}}\right]^{2p}$
$\displaystyle\leq M^{2p}e^{-2\mu
p(t-N)}\left[\sum_{j=1}^{N}e^{-\frac{p(2\mu-\varepsilon)}{2p-1}(N-j)}\right]^{2p-1}\sum_{j=1}^{N}e^{-p\varepsilon(N-j)}\left\|\int_{j-1}^{j}E(j,s)B(s)\mathrm{d}W_{s}^{-}\right\|_{H^{k}}^{2p}$
$\displaystyle\leq\frac{M^{2p}e^{-2\mu
p(t-N)}}{\big{(}e^{\frac{p(2\mu-\varepsilon)}{2p-1}}-1\big{)}^{2p-1}}\sum_{j=1}^{N}e^{-p\varepsilon(N-j)}\left\|\int_{j-1}^{j}E(j,s)B(s)\mathrm{d}W_{s}^{-}\right\|_{H^{k}}^{2p}.$
(3.76)

Writing $C=Me^{1/(2(2\mu-\varepsilon)^{2})}$ and observing

$\frac{M^{2p}}{(e^{\frac{p(2\mu-\varepsilon)}{2p-1}}-1)^{2p-1}}\leq
M^{2p}e^{p/(2\mu-\varepsilon)^{2}}=C^{2p},$ (3.77)

we hence see that

$\displaystyle\|e^{\frac{1}{2}\varepsilon t}I_{N;I}(t)\|_{H^{k}}^{2p}$
$\displaystyle\leq C^{2p}e^{\varepsilon
pN}\sum_{j=1}^{N}e^{-p\varepsilon(N-j)}\,\left\|\int_{j-1}^{j}E(j,s)B(s)\mathrm{d}W_{s}^{-}\right\|_{H^{k}}^{2p}$
(3.78) $\displaystyle\leq C^{2p}\sum_{j=1}^{N}e^{p\varepsilon
j}\sup_{j^{\prime}\in[j-1,j]}\left\|\int_{j-1}^{j^{\prime}}E(j^{\prime},s)B(s)\mathrm{d}W_{s}^{-}\right\|_{H^{k}}^{2p}.$

In addition, we observe that

$\displaystyle\|e^{\frac{1}{2}\varepsilon t}I_{N;II}(t)\|_{H^{k}}^{2p}$
$\displaystyle\leq e^{\varepsilon
pt}\left\|\int_{N}^{t}E(t,s)B(s)\mathrm{d}W_{s}^{-}\right\|_{H^{k}}^{2p}$
(3.79) $\displaystyle\leq e^{\varepsilon
p(N+1)}\sup_{j^{\prime}\in[N,N+1]}\left\|\int_{N}^{j^{\prime}}E(j^{\prime},s)B(s)\mathrm{d}W_{s}^{-}\right\|_{H^{k}}^{2p}.$

Note that these estimates no longer explicitly depend on $t$. As a
consequence, this yields

$\displaystyle\sup_{0\leq t\leq T}\big{[}e^{\frac{1}{2}\varepsilon
t}\|\mathcal{Z}^{\perp}[B](t)\|_{H^{k}}\big{]}^{2p}\leq
2^{2p}C^{2p}\sum_{j=1}^{T}e^{p\varepsilon
j}\sup_{j^{\prime}\in[j-1,j]}\left\|\int_{j-1}^{j^{\prime}}E(j^{\prime},s)B(s)\mathrm{d}W_{s}^{-}\right\|_{H^{k}}^{2p}.$
(3.80)

Applying the maximal inequality (3.61), we may hence compute

$\displaystyle\mathbb{E}\sup_{0\leq t\leq T}\big{[}e^{\frac{1}{2}\varepsilon
t}\|\mathcal{Z}^{\perp}[B](t)\|_{H^{k}}\big{]}^{2p}$ $\displaystyle\leq
2^{2p}C^{2p}K_{\rm cnv}^{2p}p^{p}\sum_{j=1}^{T}e^{p\varepsilon j}\
\mathbb{E}\left[\int_{j-1}^{j}\|B(s)\|_{HS(\mathcal{W}_{Q};H^{k})}^{2}\,\mathrm{d}s\right]^{p}$
$\displaystyle\leq p^{p}2^{2p}C^{2p}K_{\rm
cnv}^{2p}e^{p\varepsilon}\mathbb{E}\sum_{j=1}^{T}\left[\int_{j-1}^{j}e^{\varepsilon
s}\|B(s)\|_{HS(\mathcal{W}_{Q};H^{k})}^{2}\,\mathrm{d}s\right]^{p}$ (3.81)
$\displaystyle\leq{p^{p}}[2CK_{\rm
cnv}e^{\varepsilon/2}]^{2p}\mathbb{E}\left[\int_{0}^{T}e^{\mu
s}\|B(s)\|_{HS(\mathcal{W}_{Q};H^{k})}^{2}\,\mathrm{d}s\right]^{p},$

using the standard sequence space inequality
$\|\cdot\|_{\ell^{p}}^{p}\leq\|\cdot\|_{\ell^{1}}^{p}$ in the last step. In a
standard fashion, we now use Corollary 3.8 to extend the result to general
processes $B$ that satisfy (3.70). ∎

###  3.4 Maximal regularity

Our goal here is to examine the integrated $H^{k+1}$-norm of stochastic
convolutions with our evolution family $E(t,s)$. In particular, we consider
the integral

$\mathcal{I}^{\perp}_{B}=\int_{0}^{T}e^{-\varepsilon(T-s)}\|\mathcal{Z}^{\perp}[B](s)\|_{H^{k+1}}^{2}\mathrm{d}s,$
(3.82)

with $\mathcal{Z}^{\perp}[B]$ defined as in (3.69). Our result here states
that in a certain sense the $H^{k}$-supremum estimates from Theorem 3.7 and
Proposition 3.10 can be combined to infer integrated control over the
$H^{k+1}$-norm of the convolution.

######  Proposition 3.11 (maximal regularity estimate).

Suppose that (HE) holds and pick $\varepsilon\in(0,2\mu).$ Then there exists a
constant $K_{\rm mr}>0$ that does not depend on $T$ so that for any

$B\in\mathcal{N}^{2p}\big{(}[0,T];\mathbb{F};HS(\mathcal{W}_{Q};H^{k}\big{)},$
(3.83)

and every integer $p\geq 1$, we have the bound

$\displaystyle\mathbb{E}[\mathcal{I}^{\perp}_{B}]^{p}$ $\displaystyle\leq
K_{\rm mr}^{p}\,\mathbb{E}\sup_{0\leq t\leq
T}\|\mathcal{Z}^{\perp}[B](t)\|_{H^{k}}^{2p}$ (3.84)
$\displaystyle\qquad+p^{p/2}K_{\rm
mr}^{p}\mathbb{E}\left[\int_{0}^{T}e^{-\varepsilon(T-r)}\|B(r)\|_{HS(\mathcal{W}_{Q};H^{k})}^{2}\,\mathrm{d}r\right]^{p}.$

Instead of appealing directly to a result for forward integrals, we will need
to dive deeper into the limiting process used in [70] to obtain maximal
inequalities. This procedure crucially relies upon the introduction of a
(small) delay $\delta>0$ into the arguments of the random evolution family. In
particular, we introduce the notation $x_{+}=\max\\{0,x\\}$ and write

$E^{\delta}(t,s)=E\big{(}(t-\delta)_{+},(s-\delta)_{+}\big{)},\qquad\nu^{\delta}(t)=\nu((t-\delta)_{+}),$
(3.85)

for $0\leq s\leq t\leq T$.

We proceed under the assumption that $B$ is an adapted finite-rank step
process that takes values in $HS(\mathcal{W}_{Q};H^{k+3})$, instead of
$HS(\mathcal{W}_{Q};H^{k+2})$. Upon defining the process

$Z_{\delta}(t)=\int_{0}^{t}E^{\delta}(t,s)P^{\perp}B(s)\mathrm{d}W_{s}^{-},$
(3.86)

our main task is to establish bounds for the truncated integral

$I_{\delta}=\int_{\delta}^{T}e^{-\varepsilon(T-s)}\|Z_{\delta}(s)\|_{H^{k+1}}^{2}\mathrm{d}s$
(3.87)

which only involve $H^{k}$-norms. To see that this integral is well-defined,
we follow step 3 in the proof of [70, Thm. 6.4] and remark that the arguments
in the proof of Proposition 3.5 can be used to obtain the alternative
representation

$Z_{\delta}(t)=E((t-\delta)^{+},0)P^{\perp}\int_{0}^{t}B(s)\,\mathrm{d}W_{s}^{Q}+\int_{0}^{(t-\delta)^{+}}\partial_{s}E((t-\delta)^{+},s)P^{\perp}\int_{s+\delta}^{t}B(r)\,\mathrm{d}W_{r}^{Q}\,\mathrm{d}s.$
(3.88)

This implies that $Z_{\delta}$ has continuous paths in $H^{k+1}$ and that we
have the pathwise bounds

$\sup_{0\leq t\leq T}\|Z_{\delta}(t)\|_{H^{\ell}}\leq KT\sup_{0\leq t\leq
T}\left\|\int_{0}^{t}B(r)\,\mathrm{d}W_{r}^{Q}\right\|_{H^{\ell+2}},$ (3.89)

for any $0\leq\ell\leq k+1$, with a constant $K>0$ that is independent of
$\delta$.

The key towards eliminating the dependence on the $H^{k+1}$-norms lies within
the introduction of the alternative equivalent inner products101010In this
definition and the subsequent identities (3.91) and (3.93) it is also allowed
to take $\delta=0$.

$\langle v,w\rangle_{s;\delta;k+1}=\langle v,w\rangle_{H^{k}}+\langle
v_{x},w_{x}\rangle_{H^{k}}+\nu^{\delta}(s)\langle\nabla_{y}v,\nabla_{y}w\rangle_{H^{k}},$
(3.90)

for any $v,w\in H^{k+1}$ and any $0\leq s\leq T$. For now, we readily observe

$\max\\{1,k_{\nu}\\}\langle v,v\rangle_{H^{k+1}}\leq\langle
v,v\rangle_{s;\delta;k+1}\leq d\max\\{1,K_{\nu}\\}\langle
v,v\rangle_{H^{k+1}},$ (3.91)

which implies that

$\displaystyle I_{\delta}$ $\displaystyle\leq
C_{\nu}\int_{\delta}^{T}e^{-\varepsilon(T-s)}\langle
Z_{\delta}(s),Z_{\delta}(s)\rangle_{s;\delta;k+1}\mathrm{d}s,$ (3.92)

where $C_{\nu}=\min\\{1,k_{\nu}^{-1}\\}$. To appreciate the benefit of these
inner products, we note that for $t\geq\delta$ we may compute

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\langle
E^{\delta}(t,s)v,E^{\delta}(t,s)w\rangle_{H^{k}}$
$\displaystyle=\langle[\mathcal{L}_{\rm
tw}+\nu^{\delta}(t)\Delta_{y}]E^{\delta}(t,s)v,E^{\delta}(t,s)w\rangle_{H^{k}}$
(3.93) $\displaystyle\quad\quad\quad+\langle E(t,s)v,[\mathcal{L}_{\rm
tw}+\nu^{\delta}(t)\Delta_{y}]E^{\delta}(t,s)w\rangle_{H^{k}}$
$\displaystyle=\langle E^{\delta}(t,s)v,[\mathcal{L}_{\rm tw}+\mathcal{L}_{\rm
tw}^{\rm adj}-2\partial_{x}^{2}]E^{\delta}(t,s)w\rangle_{H^{k}}$
$\displaystyle\quad\quad\quad+2\langle
E^{\delta}(t,s)v,[\partial_{x}^{2}+\nu^{\delta}(t)\Delta_{y}]E^{\delta}(t,s)w\rangle_{H^{k}}$
$\displaystyle=\langle E^{\delta}(t,s)v,[\mathcal{L}_{\rm tw}+\mathcal{L}_{\rm
tw}^{\rm adj}-2\partial_{x}^{2}]E^{\delta}(t,s)w\rangle_{H^{k}}$
$\displaystyle\quad\quad\quad+2\langle
E^{\delta}(t,s)v,E^{\delta}(t,s)w\rangle_{H^{k}}$
$\displaystyle\quad\quad\quad\quad\quad\quad-2\langle
E^{\delta}(t,s)v,E^{\delta}(t,s)w\rangle_{t;\delta;k+1}.$

In order to extract this derivative, we introduce the bilinear form

$\mathcal{J}^{\delta}(s_{2},s_{1})[v,w]=\int_{s_{1}}^{s_{2}}e^{-\varepsilon(s_{2}-s)}\langle
E^{\delta}(s,s_{1})v,E^{\delta}(s,s_{1})w\rangle_{s;\delta;k+1}\mathrm{d}s,$
(3.94)

for any $\delta\leq s_{1}\leq s_{2}\leq T$ and any pair $v,w\in H^{k+1}$.
Performing an integration by parts, we obtain the decomposition

$\mathcal{J}^{\delta}(s_{2},s_{1})[v,w]=\mathcal{J}^{\delta}_{\mathrm{expl}}(s_{2},s_{1})[v,w]+\mathcal{J}^{\delta}_{\mathrm{res}}(s_{2},s_{1})[v,w]$
(3.95)

in terms of the explicit and residual bilinear forms

$\begin{array}[]{lcl}\mathcal{J}^{\delta}_{\mathrm{expl}}(s_{2},s_{1})[v,w]&=&-\frac{1}{2}\langle
E^{\delta}(s_{2},s_{1})v,E^{\delta}(s_{2},s_{1})w\rangle_{H^{k}}+\frac{1}{2}e^{-\varepsilon(s_{2}-s_{1})}\langle
v,w\rangle_{H^{k}},\\\\[5.69046pt]
\mathcal{J}^{\delta}_{\mathrm{res}}(s_{2},s_{1})[v,w]&=&\frac{1}{2}\int_{s_{1}}^{s_{2}}e^{-\varepsilon(s_{2}-s)}\langle
E^{\delta}(s,s_{1})v,A_{\varepsilon}E^{\delta}(s,s_{1})w\rangle_{H^{k}}\,\mathrm{d}s,\end{array}$
(3.96)

where $A_{\varepsilon}=(2+\varepsilon)+\mathcal{L}_{\rm tw}+\mathcal{L}_{\rm
tw}^{\rm adj}-2\partial_{x}^{2}\in\mathscr{L}(H^{k}).$ The main point is that
these expressions can be bounded using $H^{k}$-norms only.

######  Lemma 3.12.

Consider the setting of Proposition 3.11. Then there exists a constant $K>0$
that does not depend on $T$ so that for all sufficiently small $\delta\geq 0$
we have the bounds

$\begin{array}[]{lcl}|\mathcal{J}^{\delta}_{\mathrm{res}}(s_{2},s_{1})[v,w]|&\leq&K\|v\|_{H^{k}}\|w\|_{H^{k}}\int_{s_{1}}^{s_{2}}e^{-\varepsilon(s_{2}-s)}\mathrm{d}s,\\\\[5.69046pt]
|\mathcal{J}^{\delta}(s_{2},s_{1})[v,w]|&\leq&K\|v\|_{H^{k}}\|w\|_{H^{k}},\end{array}$
(3.97)

for any $\delta\leq s_{1}\leq s_{2}\leq T$ and any $v,w\in H^{k+1}$.

###### Proof.

These bounds follow directly from inspection. ∎

Setting out to obtain a bound for the integral $I_{\delta}$ for $\delta>0$
fixed, we choose an arbitrary partition $\pi=\\{r_{0},r_{1},\ldots,r_{N}\\}$,
where $0=r_{0}<r_{1}<\ldots<r_{N}=T$ and $|r_{j}-r_{j-1}|<\delta$ for
$j=1,\ldots,N$. We ensure that $\delta\in\pi$ holds, i.e.,
$r_{j_{\delta}-1}=\delta$ for some integer $j_{\delta}$. Furthermore, we
introduce the shorthand notation $B^{j}(r)=B(r)\textbf{1}_{r_{j-1}\leq r\leq
r_{j}}$ together with the processes

$Y_{\delta}^{j}(t)=\int_{r_{j-1}}^{t}E^{\delta}(t,r)B^{j}(r)\mathrm{d}W_{r}^{Q},$
(3.98)

defined on $r_{j-1}\leq t\leq r_{j}$. On account of the delay and the mesh
spacing constraint, we see that $E^{\delta}(t,r)v$ is
$\mathcal{F}_{r_{j-1}}$-measurable for all $v\in H^{k}$ and $r_{j-1}\leq r\leq
t\leq r_{j}$. In particular, the stochastic integrals in (3.98) are indeed
well-defined in the Itô sense. In parallel to step 2 of the proof of [70, Thm.
6.4], we note that we have the recursive relations

$Z_{\delta}(t)=E^{\delta}(t,r_{j-1})Z_{\delta}(r_{j-1})+Y_{\delta}^{j}(t),\qquad
r_{j-1}\leq t\leq r_{j}.$ (3.99)

Exploiting the bound (3.92) and breaking up the integral using the partition
$\pi$ leads to the bound

$\displaystyle\mathcal{I}_{\delta}$ $\displaystyle\leq
C_{\nu}\sum_{j=j_{\delta}}^{N}\int_{r_{j-1}}^{r_{j}}e^{-\varepsilon(T-s)}\langle
E^{\delta}(s,r_{j-1})Z_{\delta}(r_{j-1})+Y_{\delta}^{j}(t),E^{\delta}(s,r_{j-1})Z_{\delta}(r_{j-1})+Y_{\delta}^{j}(s)\rangle_{s;\delta;k+1}\mathrm{d}s$
$\displaystyle=C_{\nu}\big{[}\mathcal{H}_{I}+2\mathcal{H}_{II}+\mathcal{H}_{III}\big{]},$
(3.100)

in which we have introduced the expressions

$\displaystyle\mathcal{H}_{I}$
$\displaystyle=\sum_{j=j_{\delta}}^{N}\int_{r_{j-1}}^{r_{j}}e^{-\varepsilon(T-s)}\langle
E^{\delta}(s,r_{j-1})Z_{\delta}(r_{j-1}),E^{\delta}(s,r_{j-1})Z_{\delta}(r_{j-1})\rangle_{s;\delta;k+1}\mathrm{d}s,$
(3.101) $\displaystyle\mathcal{H}_{II}$
$\displaystyle=\sum_{j=j_{\delta}}^{N}\int_{r_{j-1}}^{r_{j}}e^{-\varepsilon(T-s)}\langle
E^{\delta}(s,r_{j-1})Z_{\delta}(r_{j-1}),Y_{\delta}^{j}(s)\rangle_{s;\delta;k+1}\mathrm{d}s,$
(3.102) $\displaystyle\mathcal{H}_{III}$
$\displaystyle=\sum_{j=j_{\delta}}^{N}\int_{r_{j-1}}^{r_{j}}e^{-\varepsilon(T-s)}\langle
Y_{\delta}^{j}(s),Y_{\delta}^{j}(s)\rangle_{s;\delta;k+1}\mathrm{d}s.$ (3.103)

Recalling the definition (3.94) and introducing the expressions

$\begin{array}[]{lcl}\mathcal{H}_{I;A}&=&\sum_{j=j_{\delta}}^{N}e^{-\varepsilon(T-r_{j})}\mathcal{J}^{\delta}_{\mathrm{res}}(r_{j},r_{j-1})[Z_{\delta}(r_{j-1}),Z_{\delta}(r_{j-1})],\\\\[5.69046pt]
\mathcal{H}_{I;B}&=&\sum_{j=j_{\delta}}^{N}e^{-\varepsilon(T-r_{j})}\mathcal{J}^{\delta}_{\mathrm{expl}}(r_{j},r_{j-1})[Z_{\delta}(r_{j-1}),Z_{\delta}(r_{j-1})],\end{array}$
(3.104)

we see that $\mathcal{H}_{I}=\mathcal{H}_{I;A}+\mathcal{H}_{I;B}$.

We will treat the first term $\mathcal{H}_{I;A}$ in a different—more
direct—fashion than the others. This will be especially convenient in §3.5,
since the maximum of this term over a set of different values for $T$ reduces
to the value for the largest $T$. Note that this term would disappear if one
considers the simple grid $\pi=\\{0,\delta,T\\}$ and sends $\delta\downarrow
0$, which would be possible when considering deterministic evolution families.
Intuitively, this term collects the long-range cross talk between different
elements of the set $\\{B^{j}\\}_{j=1}^{N}$. In the regular setting these
contributions are uncorrelated, but this is no longer the case here due to the
probabilistic forward-looking nature of $\nu(t)$. More concretely, quadratic
terms involving $Y_{\delta}$ can be analysed with a mild Itô formula; this is
not possible for quadratic terms involving $Z_{\delta}$.

######  Lemma 3.13.

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 we have
the pathwise estimate

$\textstyle\mathcal{H}_{I;A}\leq K\sup_{0\leq t\leq
T}\|Z_{\delta}(t)\|_{H^{k}}^{2}.$ (3.105)

###### Proof.

Recalling Lemma 3.12, a direct computation yields

$\begin{array}[]{lcl}\mathcal{H}_{I;A}&\leq&K\sum_{j=j_{\delta}}^{N}\|Z_{\delta}(r_{j-1})\|_{H^{k}}^{2}e^{-\varepsilon(T-r_{j})}\int_{r_{j-1}}^{r_{j}}e^{-\varepsilon(r_{j}-s)}\,\mathrm{d}s\\\\[5.69046pt]
&\leq&K\sup_{0\leq t\leq
T}\|Z_{\delta}(t)\|_{H^{k}}^{2}\sum_{j=j_{\delta}}^{N}\int_{r_{j-1}}^{r_{j}}e^{-\varepsilon(T-s)}\,\mathrm{d}s\\\\[5.69046pt]
&=&K\sup_{0\leq t\leq
T}\|Z_{\delta}(t)\|_{H^{k}}^{2}\varepsilon^{-1}(1-e^{-\varepsilon(T-\delta)}),\end{array}$
(3.106)

which provides the bound. ∎

After appropriate transformations, the remaining terms will all have a similar
structure featuring various bilinear forms $\Gamma^{j}$ on each mesh interval.
All these satisfy the following structural constraint, allowing for a
streamlined estimation procedure.

  * (H$\Gamma$)

There exists $K_{\Gamma}>0$ so that for every $1\leq j\leq N$, any
$r\in[r_{j-1},r_{j}]$, and any $\omega\in\Omega$, the map
$\Gamma^{j}(r,\omega):H^{k}\times H^{k}\to\mathbb{R}$ is a bilinear form that
satisfies the bound

$\Gamma^{j}(r)[v,w]\leq K_{\Gamma}\|v\|_{H^{k}}\|w\|_{H^{k}}.$ (3.107)

In addition, the map $\omega\mapsto\Gamma^{j}(r,\omega)[v,w]$ is
$\mathcal{F}_{r_{j-1}}$-measurable for any integer $1\leq j\leq N$, any
$r\in[r_{j-1},r_{j}]$, and any $v,w\in H^{k}$.

######  Lemma 3.14.

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 $p\geq 1$, any $\Gamma$ satisfying (H$\Gamma$), and any progressively
measurable $G\in L^{2p}(\Omega;C([0,T];H^{k}))$, the integral expressions

$\begin{array}[]{lcl}\mathcal{I}_{s}[G,\Gamma]&=&\sum_{j=j_{\delta}}^{N}e^{-\varepsilon(T-r_{j})}\int_{r_{j-1}}^{r_{j}}\Gamma^{j}(r)\big{[}G(r),B^{j}(r)[\,\cdot\,]\big{]}\,\mathrm{d}W_{r}^{Q},\\\\[5.69046pt]
\mathcal{I}_{d}[G,\Gamma]&=&\sum_{k=0}^{\infty}\sum_{j=j_{\delta}}^{N}e^{-\varepsilon(T-r_{j})}\int_{r_{j-1}}^{r_{j}}\Gamma^{j}(r)[B^{j}(r)\sqrt{Q}e_{k},B^{j}(r)\sqrt{Q}e_{k}]\,\mathrm{d}r\end{array}$
(3.108)

satisfy the estimates

$\begin{array}[]{lcl}\mathbb{E}\,\mathcal{I}_{s}[G,\Gamma]^{p}&\leq&\displaystyle
p^{p/2}(KK_{\Gamma})^{p}\mathbb{E}\sup_{0\leq t\leq
T}\Big{[}e^{-\varepsilon(T-t)}\|G(t)\|_{H^{k}}^{2}\Big{]}^{p}\\\\[5.69046pt]
&&\qquad\,\,\displaystyle+\,p^{p/2}(KK_{\Gamma})^{p}\mathbb{E}\left[\int_{0}^{T}e^{-\varepsilon(T-r)}\|B(r)\|_{HS(\mathcal{W}_{Q};H^{k})}^{2}\,\mathrm{d}r\right]^{p},\\\\[5.69046pt]
\mathcal{I}_{d}[G,\Gamma]&\leq&\displaystyle
KK_{\Gamma}\int_{0}^{T}e^{-\varepsilon(T-r)}\|B(r)\|_{HS(\mathcal{W}_{Q};H^{k})}^{2}\,\mathrm{d}r.\end{array}$
(3.109)

###### Proof.

Upon introducing the functions

$\chi^{j}(r)=e^{-\varepsilon(T-r_{j})}\mathbf{1}_{r_{j-1}\leq r\leq
r_{j}}\Gamma^{j}(r)[G(r),B^{j}(r)]$ (3.110)

and writing $\chi(r)=\sum_{j=1}^{N}\chi^{j}(r)$, we see that

$\mathcal{I}_{s}[G,\Gamma]=\sum_{j=1}^{N}\int_{0}^{t}\chi^{j}(r)\mathrm{d}W_{r}=\int_{0}^{t}\chi(r)\mathrm{d}W_{r}^{Q}.$
(3.111)

Applying the Burkholder-Davis-Gundy inequality [99, Prop. 2.1] yields111111The
stated $p$-dependence holds for all $p\geq 2$ in view of [99, Rem. 2.1]. We
absorb the (single) extra case $p=1$ into the constant $K$, but remark that
one cannot extend the bound uniformly to $p\in(1,2)$. This is why we restrict
our analysis to integer values of $p$.

$\begin{array}[]{lcl}\mathbb{E}\,\mathcal{I}_{s}[G,\Gamma]^{p}&\leq&p^{p/2}K^{p}\mathbb{E}\left[\int_{0}^{t}\|\chi(r)\|_{HS(\mathcal{W}_{Q};\mathbb{R})}^{2}\mathrm{d}r\right]^{p/2}\\\\[5.69046pt]
&=&p^{p/2}K^{p}\mathbb{E}\left[\sum_{j=1}^{N}\int_{r_{j-1}}^{r_{j}}\|\chi^{j}(r)\|_{HS(\mathcal{W}_{Q};\mathbb{R})}^{2}\mathrm{d}r\right]^{p/2},\end{array}$
(3.112)

where we implicitly have used the identity
$\mathbf{1}_{[r_{j-1},r_{j})}\mathbf{1}_{[r_{k-1},r_{k})}=\delta_{jk}$. In
view of the estimate

$\|\chi^{j}(r)\|_{HS(\mathcal{W}_{Q};\mathbb{R})}^{2}\leq
K_{\Gamma}^{2}e^{-2\varepsilon(T-r_{j})}\|G(r)\|_{H^{k}}^{2}\|B^{j}(r)\|_{HS(\mathcal{W}_{Q};H^{k})}^{2},$
(3.113)

we may write $M_{\varepsilon}[G]=\sup_{0\leq t\leq
T}e^{-\varepsilon(T-t)}\|G(t)\|_{H^{k}}^{2}$ and compute

$\displaystyle\sum_{j=1}^{N}\int_{r_{j-1}}^{r_{j}}\|\chi^{j}(r)\|_{HS(\mathcal{W}_{Q};\mathbb{R})}^{2}\mathrm{d}r$
$\displaystyle\leq
K_{\Gamma}^{2}e^{\varepsilon\delta}M_{\varepsilon}[G]\sum_{j=1}^{N}e^{-\varepsilon(T-r_{j})}\int_{r_{j-1}}^{r_{j}}\|B^{j}(r)\|_{HS(\mathcal{W}_{Q};H^{k})}^{2}\mathrm{d}r$
(3.114) $\displaystyle\leq
K_{\Gamma}^{2}e^{2\varepsilon\delta}M_{\varepsilon}[G]\sum_{j=1}^{N}\int_{r_{j-1}}^{r_{j}}e^{-\varepsilon(T-r)}\|B^{j}(r)\|_{HS(\mathcal{W}_{Q};H^{k})}^{2}\mathrm{d}r$
$\displaystyle=K_{\Gamma}^{2}e^{2\varepsilon\delta}M_{\varepsilon}[G]\int_{0}^{T}e^{-\varepsilon(T-r)}\|B(r)\|_{HS(\mathcal{W}_{Q};H^{k})}^{2}\mathrm{d}r,$

exploiting the disjoint supports of $B^{j}$. The first estimate in (3.109) now
follows from the elementary bound $ab\leq\frac{1}{2}(a^{2}+b^{2})$. The
deterministic bound follows in a similar but more straightforward fashion. ∎

As a final preparation, we shall expand the inner products involving pairs of
$Y^{j}_{\delta}$. In particular, treating the inner product parameter $s$ as
fixed and by noting that $\nu^{\delta}(s)$ is
$\mathcal{F}_{r_{j-1}}$-measurable for $r_{j-1}\leq s\leq r_{j}$, we may apply
a mild Itô formula [9] and find

$\displaystyle\langle
Y_{\delta}^{j}(s),Y_{\delta}^{j}(s)\rangle_{s;\delta;k+1}$
$\displaystyle=\sum_{k=0}^{\infty}\int_{r_{j-1}}^{s}\langle
E^{\delta}(s,r)B^{j}(r)\sqrt{Q}e_{k},E^{\delta}(t,r)B^{j}(r)\sqrt{Q}e_{k}\rangle_{s;\delta;k+1}\,\mathrm{d}r$
(3.115) $\displaystyle\qquad\qquad\qquad+2\int_{r_{j-1}}^{t}\langle
E^{\delta}(s,r)Y_{\delta}^{j}(r),E^{\delta}(s,r)B^{j}(r)[\,\cdot\,]\rangle_{s;\delta;k+1}\mathrm{d}W_{r}^{Q}.$

We are now ready to estimate the remaining expressions. We note that
$\mathcal{H}_{I;B}$ involves a telescoping argument that only generates cross-
terms between neighbouring mesh intervals. Due to the delay $\delta>0$, these
can still intuitively be interpreted as uncorrelated.

######  Lemma 3.15.

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

$\displaystyle\mathbb{E}[\mathcal{H}_{I;B}]^{p}$ $\displaystyle\leq
p^{p/2}K^{p}\mathbb{E}\sup_{0\leq t\leq
T}\left[e^{-\varepsilon(T-t)}\|Z_{\delta}(t)\|_{H^{k}}^{2}\right]^{p}$ (3.116)
$\displaystyle\qquad+p^{p/2}K^{p}\mathbb{E}\left[\int_{0}^{T}e^{-\varepsilon(T-r)}\|B(r)\|_{HS(\mathcal{W}_{Q};H^{k})}^{2}\,\mathrm{d}r\right]^{p}.$

###### Proof.

For convenience, we define $\widetilde{\mathcal{H}}_{I;B}=-2\mathcal{H}_{I;B}$
and use the recurrence relation (3.99) to compute

$\displaystyle\widetilde{\mathcal{H}}_{I;B}$
$\displaystyle=\sum_{j=j_{\delta}}^{N}\Big{[}e^{-\varepsilon(T-r_{j})}\langle
Z_{\delta}(r_{j})-Y_{\delta}^{j}(r_{j}),Z_{\delta}(r_{j})-Y_{\delta}^{j}(r_{j})\rangle_{H^{k}}$
(3.117)
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\quad-e^{-\varepsilon(T-r_{j-1})}\langle
Z_{\delta}(r_{j-1}),Z_{\delta}(r_{j-1})\rangle_{H^{k}}\Big{]}.$

Upon introducing the shorthand notations

$\begin{array}[]{lcl}\mathcal{H}_{I;Ba}&=&\sum_{j=j_{\delta}}^{N}e^{-\varepsilon(T-r_{j})}\langle
E^{\delta}(r_{j},r_{j-1})Z_{\delta}(r_{j-1}),Y_{\delta}^{j}(r_{j})\rangle_{H^{k}},\\\\[5.69046pt]
\mathcal{H}_{I;Bb}&=&\sum_{j=j_{\delta}}^{N}e^{-\varepsilon(T-r_{j})}\langle
Y_{\delta}^{j}(r_{j}),Y_{\delta}^{j}(r_{j})\rangle_{H^{k}},\end{array}$
(3.118)

we notice that

$\displaystyle\widetilde{\mathcal{H}}_{I;B}$
$\displaystyle=\displaystyle\sum_{j=j_{\delta}}^{N}\left[e^{-\varepsilon(T-r_{j})}\langle
Z_{\delta}(r_{j}),Z_{\delta}(r_{j})\rangle_{H^{k}}-e^{-\varepsilon(T-r_{j-1})}\langle
Z_{\delta}(r_{j-1}),Z_{\delta}(r_{j-1})\rangle_{H^{k}}\right]-2\mathcal{H}_{I;Ba}-\mathcal{H}_{I;Bb}$
$\displaystyle=\langle
Z_{\delta}(T),Z_{\delta}(T)\rangle_{H^{k}}-e^{-\varepsilon(T-\delta)}\langle
Z_{\delta}(\delta),Z_{\delta}(\delta)\rangle_{H^{k}}-2\mathcal{H}_{I;Ba}-\mathcal{H}_{I;Bb}.$
(3.119)

It hence remains for us to establish bounds for
$\mathbb{E}[\mathcal{H}_{I;Ba}]^{p}$ and $\mathbb{E}[\mathcal{H}_{I;Bb}]^{p}$
that can be absorbed into (3.116). Firstly, observe that we have

$\mathcal{H}_{I;Ba}=\sum_{j=j_{\delta}}^{N}e^{-\varepsilon(T-r_{j})}\int_{r_{j-1}}^{r_{j}}\langle
E^{\delta}(r_{j},r)E^{\delta}(r,r_{j-1})Z_{\delta}(r_{j-1}),E^{\delta}(r_{j},r)B^{j}(r)\,\mathrm{d}W_{r}^{Q}\rangle_{H^{k}},$
(3.120)

which can be estimated accordingly by applying Lemma 3.14 with
$G(r)=G_{I;Ba}(r),$ where

$G_{I;Ba}(r)=E^{\delta}(r,r_{j-1})Z_{\delta}(r_{j-1}),\quad r_{j-1}\leq r\leq
r_{j}.$ (3.121)

Indeed, we see that

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