# Stochastic wave equation with Hölder noise coefficient: well-posedness and
small mass limit

Yi HAN  Department of Pure Mathematics and Mathematical Statistics, University
of Cambridge<EMAIL_ADDRESS>
###### Abstract.

We construct unique martingale solutions to the damped stochastic wave
equation

$\mu\frac{\partial^{2}u}{\partial t^{2}}(t,x)=\Delta u(t,x)-\frac{\partial
u}{\partial t}(t,x)+b(t,x,u(t,x))+\sigma(t,x,u(t,x))\frac{dW_{t}}{dt},$

where $\Delta$ is the Laplacian on $[0,1]$ with Dirichlet boundary condition,
$W$ is space-time white noise, $\sigma$ is $\frac{3}{4}+\epsilon$ -Hölder
continuous in $u$ and uniformly non-degenerate, and $b$ has linear growth. The
same construction holds for the stochastic wave equation without damping term.
More generally, the construction holds for SPDEs defined on separable Hilbert
spaces with a densely defined operator $A$, and the assumed Hölder regularity
on the noise coefficient depends on the eigenvalues of $A$ in a quantitative
way. We further show the validity of the Smoluchowski-Kramers approximation:
assume $b$ is Hölder continuous in $u$, then as $\mu$ tends to $0$ the
solution to the damped stochastic wave equation converges in distribution, on
the space of continuous paths, to the solution of the corresponding stochastic
heat equation. The latter result is new even in the case of additive noise.

Supported by EPSRC grant EP/W524141/1.

##  1\. Introduction

In this paper we are interested in the damped stochastic wave equation with
irregularity in both the drift and the diffusion coefficients. On the compact
interval $D=[0,1]\subset\mathbb{R}$ the equation can be formulated as

$\displaystyle\mu\frac{\partial^{2}u^{\mu}}{\partial t^{2}}=\Delta
u^{\mu}(t,x)-\frac{\partial u^{\mu}}{\partial
t}(t,x)+b(t,x,u^{\mu}(t,x))+g(t,x,u^{\mu}(t,x))dW(t,x),$ (1.1) $\displaystyle
u^{\mu}(t,x)=0,\quad x\in\partial D,$ $\displaystyle
u^{\mu}(0,x)=u_{0}(x),\quad\frac{\partial u^{\mu}}{\partial
t}(0,x)=v_{0}(x),\quad x\in D,$

with $\mu>0$. The equation models a material with elasticity, experiencing
friction, as well as deterministic and random forcing. The $-\frac{\partial
u^{\mu}}{\partial t}$ term models elasticity and $\mu>0$ is the mass density.

We will also consider in parallel the stochastic wave equation without
damping, formulated as

$\displaystyle\mu\frac{\partial^{2}u^{\mu}}{\partial t^{2}}=\Delta
u^{\mu}(t,x)+b(t,x,u^{\mu}(t,x))+g(t,x,u^{\mu}(t,x))dW(t,x),$ (1.2)
$\displaystyle u^{\mu}(t,x)=0,\quad x\in\partial D,$ $\displaystyle
u^{\mu}(0,x)=u_{0}(x),\quad\frac{\partial u^{\mu}}{\partial
t}(0,x)=v_{0}(x),\quad x\in D.$

When the coefficients $b$ and $g$ are Lipschitz continuous in $u^{\mu}$, these
SPDEs have been well studied. The well-posedness of a strong solution follows
from a fixed point theorem, and as $\mu\to 0$, the damped stochastic wave
equation (1.1) converges to the corresponding stochastic heat equation.
Motivated by finite dimensional Langevin systems, it is natural to ask whether
we can deduce the same results when the coefficients $b$ and $g$ are not
locally Lipschitz continuous. We will give a positive answer in this paper.

In this paper, we will work with equations (1.1) and (1.2) in their abstract
forms, formulated as evolution equations on an infinite dimensional, separable
Hilbert space $H$. In the literature, (1.1) and (1.2) are usually called
random field solutions of SPDEs, which can be regraded as particular examples
as abstract evolution equations on Hilbert space. Our abstract treatment not
only makes the proof clearer but also shows the result has much wider
applicability. We will rewrite (1.1) and (1.2) as the following evolution
equation defined on $H\times H^{-1}:$ (in the setting of (1.1), (1.2) we take
$H=L^{2}([0,1];\mathbb{R})$, and for a general separable Hilbert space $H$ see
Section 2.1 for the definition of $H^{-1}$)

$\mu\frac{\partial^{2}u_{\mu}(t)}{\partial
t^{2}}=Au_{\mu}(t)-\zeta\frac{\partial u_{\mu}(t)}{\partial
t}+B(t,u_{\mu}(t))+G(t,u_{\mu}(t))\frac{dW_{t}}{dt},$ (1.3)

where $W$ is the space-time white noise on $H$ and $A$ is some positive
operator whose properties will be specified later. The variable constant
$\zeta$ typically takes values 0 or 1. The choice $\zeta=1$ corresponds to the
damped stochastic wave equation (1.1) and the choice $\zeta=0$ corresponds to
the wave equation without damping term (1.2). If we take
$H=L^{2}([0,1];\mathbb{R})$, take $A$ to be the Laplace operator on $[0,1]$
with Dirichlet boundary condition, take $W$ the space-time white noise on
$L^{2}([0,1])$ and set $B(t,u_{\mu}(t))(x)=b(t,x,u^{\mu}(t,x))$ and
$G(t,u_{\mu}(t))(x)=g(t,x,u^{\mu}(t,x))$, we recover random field SPDEs (1.1)
and (1.2).

In this paper we are interested in the well-posedness issue of (1.3) when both
$B$ and $G$ are irregular, and study the $\mu\to 0$ limit for the damped
nonlinear stochastic wave equation (1.1) (more precisely, the abstract
equation (1.3) with $\zeta=1$).

Given that $B$ and $G$ are irregular, in this paper it is more convenient to
work with probabilistic weak solutions. We will mainly consider weak mild
solutions to (1.3). That is, the solution is weak in probabilistic sense and
mild in analytic sense. More precisely: (to simplify notations, we defer the
definition of the Hilbert space $\mathcal{H}_{1}$ and the explicit expression
of the mild formulation of $Z_{\mu}(t)$ to Section 2.1.)

######  Definition 1.1.

A weak mild solution to the abstract stochastic wave equation (1.3) with
initial date $(u_{0},v_{0})\in\mathcal{H}_{1}$ is given by a sequence
$(\Omega,\mathcal{F},(\mathcal{F}_{t}),\mathbb{P},W,Z_{\mu})$, where
$(\Omega,\mathcal{F},(\mathcal{F}_{t}),\mathbb{P})$ is some filtered
probability space defining a cylindrical wiener process $W$ and an
$\mathcal{F}_{t}$-adapted, $H\times H^{-1}$-valued continuous process
$(Z_{\mu}(t))_{t\geq 0}$ (the first coordinate of $Z_{\mu}(t)$ is
$u_{\mu}(t)$,) such that $\mathbb{P}$-a.s. $Z_{\mu}(t)$ is a mild solution to
(1.3) with initial data $(u_{0},v_{0})\in\mathcal{H}_{1}$. The precise
formulation of mild solution that $Z_{\mu}(t)$ should satisfy is given in
(2.5) of Definition 2.1.

The two main theorems of this paper are as follows. The first considers well-
posedness.

######  Theorem 1.2.

Given a separable Hilbert space $H$. Suppose the operator $A$, the nonlinear
mapping $B:[0,\infty)\times H\to H$ and $G:[0,\infty)\times
H\to\mathcal{L}(H_{0},H)$ (with $H_{0}$ specified below)111In this paper if
the subscript of $|\cdot|$ is not specified, then it refers to the norm of
$H$. satisfy the following list of assumptions:

$\mathbf{H1}$ the operator $A:\mathcal{D}(A)\subset H\to H$ is densely
defined, that $A$ has a set of eigenvectors $\\{e_{n}\\}_{n\geq 1}$ forming an
orthogonal basis of $H$, with eigenvalues $Ae_{n}=-\alpha_{n}e_{n}$. Moreover,
we may find some $c>0$ such that $\alpha_{n}\geq c$ for each $n$, and there
exists some $\eta_{0}\in(0,1)$ such that for each $\eta\in(0,\eta_{0})$,

$\sum_{n\geq 1}\frac{1}{\alpha_{n}^{1-\eta}}<\infty.$ (1.4)

Moreover, there exists a Banach space $H_{0}\subset H$ with a continuous
embedding, such that each eigenvalue $e_{n}$ lies in $H_{0}$, and that for
some finite $M<\infty$,

$\sup_{n}|e_{n}|_{H_{0}}\leq M<\infty$ (1.5)

$\mathbf{H2}$ The map $B(t,x):[0,\infty)\times H\to H$ is measurable.

$\mathbf{H3}$ For some $\beta\in(1-\frac{\eta_{0}}{2},1]$, we may find a
constant $M>0$ independent of $t$ such that the function
$G(t,\cdot):H\to\mathcal{L}(H_{0},H)$ satisfies

$|G(t,x)-G(t,y)|_{\mathcal{L}(H_{0},H)}\leq
M|x-y|_{H}^{\beta},\quad|x-y|_{H}\leq 1,\quad x,y\in H.$ (1.6)

Moreover, $G$ is self-adjoint in the sense that for any $t>0,x\in H$,

$\langle G(t,x)e_{j},e_{k}\rangle=\langle e_{j},G(t,x)e_{k}\rangle,\quad
j,k\in\mathbb{N}_{+}.$ (1.7)

$\mathbf{H4}$ We may find an extension of $G(t,x)$ with
$G(t,x)\in\mathcal{L}(\widetilde{H}_{0},H)$ with
$H_{0}\subset\widetilde{H}_{0}\subset H$ such that, for each $x\in H$,
$G(t,x)$ has a right inverse $G^{-1}(t,x)$ on $H$ in the sense that

$G(t,x)G^{-1}(t,x)u=u\text{ for any }u\in H,$

with $G^{-1}(t,x)H\subset\widetilde{H}_{0}$ and satisfies, for some $C>0$
independent of $t>0,$

$\sup_{x\in H}|G^{-1}(t,x)|_{\mathcal{L}(H,H)}\leq C<\infty.$ (1.8)

$\mathbf{H5}$ the nonlinear mappings $B$ and $G$ have linear growth: for some
$M>0,$

$|B(t,x)|_{H}\leq M(1+|x|_{H}),\quad|G(t,x)|_{\mathcal{L}(H_{0},H)}\leq
M(1+|x|_{H}).$ (1.9)

Then there exists a unique weak mild solution to the abstract stochastic wave
equation (1.3) with initial data $(u_{0},v_{0})\in\mathcal{H}_{1}$, in the
sense of Definition 1.1.

######  Remark 1.3.

We give some specific examples where Assumption $\mathbf{H}1$ holds true. The
most important case is the Laplacian on $[0,1]$ with Dirichlet boundary
condition, then $\mathbf{H}1$ is satisfied with $\eta_{0}=\frac{1}{2}$ and
consequently we can take any $\beta\in(\frac{3}{4},1]$ in Assumption
$\mathbf{H}3$. Assumption $\mathbf{H}1$ also covers some random field SPDEs in
dimension two or higher, say one considers the bi-Laplacian $A=-\Delta^{2}$ on
any bounded domain in $\mathbb{R}^{d}$, $d=1,2,3,$ with Dirichlet boundary
conditions.

######  Remark 1.4.

We show how random field solutions (1.1) and (1.2) to the stochastic wave
equation are covered by Assumptions $\mathbf{H}1$ to $\mathbf{H}5$. We take
$H=L^{2}([0,1])$ and $H_{0}=L^{\infty}([0,1])$. Since the eigenfunctions of
Laplacian on $[0,1]$ are given by $\sin(n\pi x)$, they are uniformly bounded
in $L^{\infty}([0,1])$. This justifies $\mathbf{H}1$. Assume $b$ is
$\alpha$-Hölder continuous in $u$, we check $B$ is $\alpha$-Hölder continuous
on $H$, justifying assumption $\mathbf{H}2^{\prime}$ given below: by Cauchy-
Schwartz inequality, for any two $u,v\in H$,

$\begin{aligned}
|B(t,u)-B(t,v)|_{H}=&\sqrt{\int_{0}^{1}|b(t,x,u(x))-b(t,x,v(x))|^{2}dx}\\\
\leq&C\sqrt{\int_{0}^{1}|u(x)-v(x)|^{2\alpha}dx}\leq
C|u-v|_{H}^{\alpha}.\end{aligned}.$

The same estimate applies to $g(t,x,u)$ assuming $g$ is $\beta$-Hölder
continuous in $u$. Then notice that, from the definition: for any $u\in H$ and
$v\in H_{0}$, the map $G:[0,\infty)\times H\to\mathcal{L}(H_{0},H)$ is given
by $(G(t,u)v)(x)=g(t,x,u(t,x))v(x),x\in[0,1]$, we have,

$|G(t,u)-G(t,v)|_{\mathcal{L}(H_{0},H)}\leq
C|g(t,x,u(t,x))-g(t,x,v(t,x))|_{L^{2}([0,1])},$ (1.10)

then assumption $\mathbf{H}3$ is justified given the previous estimate. The
self-adjointness assumption (1.7) follows directly from definition of $G$. For
assumption $\mathbf{H}4$, if we assume that $|g(t,x,u)|\geq\epsilon$ for some
$\epsilon>0$ and all $t,x,u$, then we may define $G^{-1}:[0,\infty)\times
H\to\mathcal{L}(H,H)$ via $G^{-1}(t,u)v(x)=\frac{1}{g(t,x,u(t,x))}v(x),$ where
$u,v\in H$. Then it is clear that we can slightly extend the definition of $G$
such that $GG^{-1}v=v$ for any $v\in H$. We shall assume that $g(t,x,u)$ has
linear growth in $u$, so that $|g(t,x,u)|_{H}\leq C(1+|u|_{H})$. This implies
$G$ is a well-defined map on $\mathcal{L}(H_{o},H)$ and assumption
$\mathbf{H}5$ is satisfied.

We note that we can consider some other choices of $H_{0}$ that lead to new
interesting examples of SPDEs with non-local coefficients. Say for example we
can take $H_{0}=H$ and consider $g(t,x,u(x))=1+h(|u(x)-v(x)|_{L^{2}([0,1])})$
for some $v\in H$ and some $\frac{3}{4}+\epsilon$-Hölder continuous function
h. A lot more different types of SPDEs can be generated under this framework,
whose well-posedness are new when the dependence is only Hölder continuous.

The second result considers the $\mu\to 0$ small mass limit of the damped
abstract stochastic wave equation (1.3) with $\zeta=1$. The limit is the
following $H$-valued stochastic heat equation:

$du(t)=[Au(t)+B(t,u(t))]dt+G(t,u(t))dw(t),\quad u(0)=u_{0}\in H.$ (1.11)

interpreted in terms of mild formulation (see equation (2.7)).

The stochastic heat equation (1.11) under the regularity assumptions stated in
Theorem 1.5 is recently investigated in [23].

######  Theorem 1.5.

Assume all the assumptions $\mathbf{H1}$ to $\mathbf{H5}$ are true. Moreover,
assume the following stronger assumptions:

$\mathbf{H}1^{\prime}$ In addition to $\mathbf{H}_{1}$, assume we may find
some constants $C>0$ such that

$\frac{1}{C}k^{\frac{1}{1-\eta_{0}}}\leq\alpha_{k}\leq
Ck^{\frac{1}{1-\eta_{0}}}\quad\forall k\in\mathbb{N}_{+}$ (1.12)

(this assumption is stronger than and trivially implies estimate (1.4). This
assumption is satisfied by the Laplacian operator on $[0,1]$ once we take
$\eta_{0}=\frac{1}{2}$).

$\mathbf{H}2^{\prime}$: the drift $B$ is Hölder continuous: for some
$\alpha\in(0,1]$, we can find a time independent constant $M>0$ such that

$|B(t,x)-B(t,y)|_{H}\leq M|x-y|_{H}^{\alpha},\quad\forall|x-y|\leq 1,\quad
x,y\in H.$ (1.13)

Consider the damped abstract stochastic wave equation, (1.3) with $\zeta=1$.

Then for any $T>0$, the solution $u_{\mu}(t)$ (the first component of
$Z_{\mu}(t)\in H\times H^{-1}$) converges to $u(t)$, the solution to the
stochastic heat equation (1.11), in terms of convergence in distribution, as
$\mathcal{C}([0,T];H)-$valued processes.

We will actually prove convergence of $u_{\mu}|_{[0,T]}$ towards $u|_{[0,T]}$
with respect to a Wasserstein distance on $\mathcal{P}(\mathcal{C}([0,T];H)),$
the space of Borel probability measures on $\mathcal{C}([0,T];H)$.

This paper provides, to our best knowledge, currently the most general results
on stochastic wave equation with irregular coefficients. We cover irregular
drift and diffusion coefficients, and irregularity in the diffusion
coefficient has never been considered before in this context. We work on a
separable Hilbert space so that random field solutions are contained as
special cases but our theory is much more general. We provide the first proof
of Smoluchowski-Kramers approximation for an infinite dimensional system with
merely Hölder coefficients. Due to the complexity of the wave equation
semigroup (which is not analytic, for example), even the weak existence part
of Theorem 1.2 seems novel and is not covered by standard methods (see Section
3.1 for discussions). Before we discuss the method and plan of this paper, we
first give a brief literature review of related results.

###  1.1. Related literature

####  1.1.1. Works on regularization by noise

In the broadest scope, this work contributes to investigating the
regularization by noise phenomenon for stochastic differential equations with
irregular coefficients. The simplest case one may think of is a finite
dimensional SDE

$dX_{t}=b(X_{t})dt+\sigma(X_{t})dW_{t}$

defined on $\mathbb{R}^{d}$. When $\sigma=I$ is the identity and $b$ is
bounded measurable, one can construct a unique weak solution via Girsanov
transform, and prove strong uniqueness via Zvonkin’s transform [37].
Extensions to unbounded, time dependent coefficients can be found in [26].
Concerning an irregular diffusion coefficient $\sigma$, when $\sigma$ is non-
degenerate and continuous, one can always construct a unique weak solution via
representing the solution as a time changed Brownian motion (see [34], Section
6.5), and under some extra Sobolev regularity assumptions on $\sigma$ one can
show the solution is strong. The two claims concerning $\sigma$ are available
primarily because we work in finite dimensions, so that the solution to the
SDE without drift is a time-changed Brownian motion, and the analytical theory
of parabolic PDEs can be applied to derive quantitative estimates.

For an infinite dimensional stochastic system, consider the stochastic heat
equation

$dX_{t}=AX_{t}dt+b(X_{t})dt+\sigma(X_{t})dW_{t}$ (1.14)

where we may as well take $A$ the Laplacian on $[0,1]$ as a motivating
example. There has been much recent investigation of well-posedness with an
irregular drift $b$ in the case of additive noise $\sigma=I$: pathwise
uniqueness is proved for Hölder continuous drifts $b$ in [10] in the context
of evolution equations on Hilbert space, and for bounded measurable drifts
[21] and distributional drifts [1] in the context of random field solutions
(but not for evolution equations on infinite dimensional Hilbert space). The
investigation of an irregular diffusion coefficient in the infinite
dimensional setting turns out to be much more difficult. Notable technical
challenges appear in infinite dimensions, as we may no longer represent the
solution as a time changed Brownian motion and no analytical PDE theory is
available. In [29] the authors made a great achievement in proving pathwise
uniqueness of random field solution to the stochastic heat equation when the
drift $b$ is Lipschitz and diffusion coefficient $\sigma$ is
$\frac{3}{4}+\epsilon$-Hölder continuous in the function argument. The proof
is fairly technical and does not seem to extend to Hilbert space valued
abstract SPDEs. Recently in [23] the author introduced a direct method to
prove uniqueness of martingale solutions of (1.14) with irregular $\sigma$ and
$b$ for (1.14) defined on infinite dimensional separable Hilbert space,
allowing an additional drift $(-A)^{1/2}F(X_{t})$ and studied the long time
behavior of solutions.

For the stochastic wave equation (1.1), there are a few papers considering
irregular drift coefficients and additive white noise, see for example [28],
and papers on a related problem of martingale solutions with reflection, see
[24]. Very recently, in [2] the authors proved pathwise uniqueness and
Malliavin differentiability for the random field solution to the hyperbolic
SPDE $X_{s,t},(s,t)\in[0,T]^{2}:$

$\displaystyle\frac{\partial^{2}X(s,t)}{\partial s\partial
t}=b(s,t,X(s,t))+\frac{\partial^{2}W(s,t)}{\partial s\partial t},\quad
s,t\in[0,T]^{2},$ (1.15) $\displaystyle X(s,0)=X(0,t)=x$

where $W_{s,t}$ is the Wiener sheet and
$b:[0,T]^{2}\times\mathbb{R}^{d}\to\mathbb{R}^{d}$ is bounded measurable.
Their method is based on an adaptation of Davie’s pathwise argument [13] and
does not extend to the case of an irregular diffusion coefficient $\sigma$.
Moreover, the stochastic wave equation with a damping term $\partial_{t}u$ is
not covered by the results in [2], as they consider the SDE (1.15) and then
recover the wave equation via a $\frac{\pi}{4}$ rotation. Thus Theorem 1.2 in
this paper is somewhat orthogonal to [2]: we consider (damped) semilinear
stochastic wave equation with irregular noise coefficients and solve SPDEs on
an infinite dimensional Hilbert space, which has much wider generality and
contains the random field SPDEs as special examples. Due to this infinite
dimensional feature, establishing pathwise uniqueness is particularly
difficult (for the stochastic wave equation with additive noise, pathwise
uniqueness of the Hilbert space valued abstract SPDE is currently only
established when the drift $B$ is $\beta>\frac{2}{3}$-Hölder continuous, see
[28]). Meanwhile, weak well-posedness and convergence in distribution are
already quite useful in many applications, so we do not discuss the proof of
pathwise uniqueness in this paper.

####  1.1.2. Works on stochastic wave equation

This paper is the first, to our best knowledge, to consider stochastic wave
equations with non-Lipschitz noise coefficients. Earlier studies on the
stochastic wave equation focus on Lipschitz continuous $b$ and $\sigma$ in
dimensions 1,2 and 3, working with white noise in dimension 1 and coloured
noise in dimension 2 and 3 so that the solutions are function valued.
Literature in this direction is extensive so we just cite a few and refer to
the references therein: [12], [30], [3], [4]. In recent years, the stochastic
wave equation with polynomial nonlinearity, driven by space-time white noise
in two and three dimensions have been investigated in a number of papers [18],
[19], [17]. The Da Prato-Debussche trick [9] or paracontrolled calculus [16]
are used in these works and the solutions are not function valued. Thus in
these works one has to consider an additive white noise or sufficiently
regular multiplicative noise coefficient.

####  1.1.3. Works on Smoluchowski-Kramers approximation

The second component of this paper, Theorem 1.5, concerns the small noise
limit of stochastic wave equations. This phenomenon was first studied in
finite dimensions for the Langevin equation in [25] and [33] and is then
called the Smoluchowski-Kramers approximation. More results on the convergence
in finite dimensions can be found in [14]. The first proof of Smoluchowski-
Kramers approximation for stochastic wave equation was given in [5], [6], and
later extended in [32] to show that the approximation works for multiplicative
noise in any spatial dimension provided that the covariance of the noise is
suitably chosen. Some more recent investigations of Smoluchowski-Kramers
approximation can be found in [5], [7], [8]. In these works it is assumed that
the drift and diffusion coefficients of the SPDE are Lipschitz continuous in
$u$. In [36] the authors proved Smoluchowski-Kramers approximation for 2D
stochastic wave equation with additive space-time white noise and polynomial
and sine nonlinearities.

In a recent work [35] the authors considered a finite dimensional system with
Hölder continuous drift and non-degenerate noise, and proved Smoluchowski-
Kramers approximation via an application of Zvonkin’s transform. In this paper
Theorem 1.5 we show that Smoluchowski-Kramers approximation is also valid for
irregular coefficients in infinite dimensions, widely extending the scope of
the said topic to cases where analytical techniques are not available.

###  1.2. Plan of this paper

The key technique that we will use to treat irregular coefficients is a
generalized coupling technique introduced in [27] in the setting of stochastic
delay equations. This technique makes use of the non-degeneracy of noise
coefficient $G$ and a combination of both pathwise and probabilistic
techniques. Suppose we want to measure the distance between two solutions with
same initial condition but different coefficients,

$\mu\frac{\partial^{2}u(t)}{\partial t^{2}}=Au(t)-\zeta\frac{\partial
u(t)}{\partial t}+G(t,u(t))\frac{dW_{t}}{dt},$ (1.16)
$\mu\frac{\partial^{2}u^{n}(t)}{\partial t^{2}}=Au^{n}(t)-\zeta\frac{\partial
u^{n}(t)}{\partial t}+G^{n}(t,u^{n}(t))\frac{dW_{t}}{dt},$ (1.17)

we introduce a control parameter $\lambda$, a stopping time $\tau$ and
consider the controlled process

$\mu\frac{\partial^{2}\widetilde{u}^{n}(t)}{\partial
t^{2}}=A\widetilde{u}^{n}(t)-\zeta\frac{\partial\widetilde{u}^{n}(t)}{\partial
t}+\lambda(u(t)-\widetilde{u}^{n}(t))dt1_{t\leq\tau}+G^{n}(t,\widetilde{u}^{n}(t))\frac{dW_{t}}{dt},$
(1.18)

In the probabilistic step we estimate the total variation distance between
$u^{n}|_{[0,T]}$ and $\widetilde{u}^{n}|_{[0,T]}$ via Girsanov transform and
relative entropy computations, which is standard. In the analytic step we
compare the distance between $u(t)$ and $\widetilde{u}^{n}(t)$ via pathwise
estimates. The pathwise estimate is somewhat more involved in infinite
dimensions. In the parabolic literature [23] this estimate was achieved via a
careful use of the stochastic factorization theorem. In the hyperbolic case of
this paper, we further develop the ideas in [23], combining more refined
estimates of second order semigroups in [32]. The proof of Theorem 1.5 goes
through similar lines, yet a further refined semigroup analysis is needed to
make the leading numerical constants independent of the small parameter $\mu$.

This article is organised as follows. In Section 2 we develop the necessary
functional analytic tools and maximal inequality estimates for stochastic
integrals. In section 3 we prove existence and uniqueness of a weak-mild
solutions, Theorem 1.2. In Section 4 we prove the small mass limit, Theorem
1.5.

##  2\. Stochastic estimates of the semigroup

We now summarize the functional analytic background of this paper, and give a
precise definition to the solutions of the (damped) stochastic wave equation.

###  2.1. Functional analytic preliminaries

For any $\delta\in\mathbb{R}$, define the interpolation spaces $H^{\delta}$
(see for example [22], Section 4.4) as the completion of the vector space
$\operatorname{Vect}(e_{1},,e_{2},\cdots)$ of finite linear sums of $e_{k}$,
with respect to the norm

$|f|_{H^{\delta}}^{2}=\sum_{k=1}^{\infty}\alpha_{k}^{\delta}\langle
f,e_{k}\rangle_{H}^{2},$

and define the phase space $\mathcal{H}_{\delta}:=H^{\delta}\times
H^{\delta-1}$. We will mostly consider $\mathcal{H}:=\mathcal{H}_{0}$.

To make the notations more compact, we keep the undetermined constant $\zeta$
that take two values: 0 and 1. The choice $\zeta=1$ corresponds to the damped
stochastic wave equation, and the choice $\zeta=0$ corresponds to the one
without damping. We define a linear operator, for both $\zeta=0$ and
$\zeta=1$:
$\mathcal{A}^{\lambda}_{\mu}:D(\mathcal{A}_{\mu}^{\lambda}):=\mathcal{H}_{\delta-1}\to
H_{\delta}$ via

$\mathcal{A}_{\mu}^{\lambda}(u,v)=(v,(A-\lambda)u/\mu-\zeta v/\mu).$ (2.1)

In the following discussions, whenever $\lambda$ is omitted in the notation,
it means that we take $\lambda=0$.

We give some clarifications for the constants $\lambda,\mu,\zeta$: the
constant $\lambda$ is an intermediate parameter and we sill set it very large
in the proof. It does not appear in the final statements of our theorems. The
constant $\mu$ is the parameter that we will tune to zero in the small mass
limit. The value of $\mu\neq 0$ is irrelevant when we prove the well-posedness
of the wave equation. In the proof of well-posedness, we will consider both
cases $\zeta=0,1$, while in the proof of small mass limit only the case
$\zeta=1$ matters.

Now we denote by $S_{\mu}^{\lambda}(t):H_{\delta}\to H_{\delta}$ the $C_{0}$
semigroup generated by $\mathcal{A}_{\mu}^{\lambda}$. This semigroup is
continuous but not analytic, thus constituting the major analytical difficulty
of this paper.

Now define $\Pi_{1}:\mathcal{H}_{\delta}\to H^{\delta}$ to be the projection
to the first component and $\Pi_{2}:\mathcal{H}_{\delta}\to H^{\delta-1}$
projection to the second component. Now set $\mathcal{I}_{\mu}:H^{\delta}\to
H_{\delta}$ via

$\mathcal{I}_{\mu}u=(0,u/\mu).$ (2.2)

Then for each $\lambda\geq 0,$ the solution to the abstract stochastic wave
equation with $\zeta=0,1$

$\mu\frac{\partial^{2}u_{\mu}^{\lambda}(t)}{\partial
t^{2}}=(A-\lambda)u_{\mu}^{\lambda}(t)-\zeta\frac{\partial
u_{\mu}^{\lambda}(t)}{\partial
t}+B(t,u_{\mu}^{\lambda}(t))+G(t,u_{\mu}^{\lambda}(t))\frac{dW_{t}}{dt},$
(2.3)

can be formulated into the following abstract evolution equation. Denote the
two coordinates by
$Z_{\mu}^{\lambda}(t)=(u_{\mu}^{\lambda}(t),v_{\mu}^{\lambda}(t))$ with
initial value $Z_{\mu}^{\lambda}(0)=Z_{0}=(u_{0},v_{0})\in\mathcal{H}_{1}$, we
can rewrite (2.3) as

$dZ_{\mu}^{\lambda}(t)=\mathcal{A}_{\mu}^{\lambda}Z_{\mu}^{\lambda}(t)dt+\mathcal{I}_{\mu}B(t,\Pi_{1}Z_{\mu}^{\lambda}(t))dt+\mathcal{I}_{\mu}G(t,\Pi_{1}Z_{\mu}^{\lambda}(t))dW_{t}.$
(2.4)

The mild solutions to the stochastic wave equations are defined as follows:

######  Definition 2.1.

The mild solution to the abstract stochastic wave equation (1.3) is defined as
the adapted process $Z_{\mu}$ on $\mathcal{H}_{0}$ that satisfies, for each
$t>0$,

$\displaystyle Z_{\mu}(t)$
$\displaystyle=\mathcal{S}_{\mu}(t)z_{0}+\int_{0}^{t}\Pi_{1}\mathcal{S}_{\mu}(t-s)\mathcal{I}_{\mu}B(s,\Pi_{1}Z_{\mu}(s))ds$
(2.5)
$\displaystyle+\int_{0}^{t}\Pi_{1}\mathcal{S}_{\mu}(t-s)\mathcal{I}_{\mu}G(s,\Pi_{1}Z_{\mu}(s))dW_{s}.$

More generally, the mild solution to the abstract stochastic wave equation
(2.3) is defined as the adapted process $Z_{\mu}^{\lambda}$ on
$\mathcal{H}_{0}$ that satisfies, for each $t>0$,

$\displaystyle Z_{\mu}^{\lambda}(t)$
$\displaystyle=\mathcal{S}_{\mu}^{\lambda}(t)z_{0}+\int_{0}^{t}\Pi_{1}\mathcal{S}_{\mu}^{\lambda}(t-s)\mathcal{I}_{\mu}B(s,\Pi_{1}Z_{\mu}^{\lambda}(s))ds$
(2.6)
$\displaystyle+\int_{0}^{t}\Pi_{1}\mathcal{S}_{\mu}^{\lambda}(t-s)\mathcal{I}_{\mu}G(s,\Pi_{1}Z_{\mu}^{\lambda}(s))dW_{s}.$

The mild solution to the stochastic heat equation (1.11) is defined as the
adapted process $u(t)$ on $H$ that satisfies, for each $t>0$,

$u(t)=\mathcal{S}(t)u_{0}+\int_{0}^{t}\mathcal{S}(t-s)B(s,u(s))ds+\int_{0}^{t}\mathcal{S}(t-s)G(s,u(s))dW_{s}.$
(2.7)

Since $A+\lambda$ has eigenvectors $\\{e_{k}\\}_{k\geq 1}$ for each
$\lambda\geq 0$, the operator $\mathcal{A}_{\mu}^{\lambda}$ leaves invariant
the linear subspace of $\mathcal{H}$ with the form
$\\{(u_{k}e_{k},v_{k}e_{k}):u_{k},v_{k}\in\mathbb{R}\\}$. Given $u\in H,v\in
H^{-1}$, define $u_{k}=\langle u,e_{k}\rangle_{H}$ and $v_{k}=\langle
v,e_{k}\rangle_{H^{-1}}$, and consider

$f_{\mu}^{\lambda}(k)(t,u_{k},v_{k}):=\langle
e_{k},\Pi_{1}\mathcal{S}_{\mu}^{\lambda}(t)(u_{k}e_{k},v_{k}e_{k})\rangle_{H}.$

Then $f_{\mu}^{\lambda}(k)(t,u_{k},v_{k})$ solves, in the case $\zeta=1$,

$\mu{f_{\mu}^{\lambda}(k)}^{\prime\prime}+{f_{\mu}^{\lambda}(k)}^{\prime}+(\alpha_{k}+\lambda)f_{\mu}^{\lambda}(k)=0,\quad
f_{\mu}^{\lambda}(k)(0)=u_{k},f_{\mu}^{\lambda}(k)^{\prime}(0)=v_{k},$ (2.8)

and solves, in the case $\zeta=0,$

$\mu{f_{\mu}^{\lambda}(k)}^{\prime\prime}+(\alpha_{k}+\lambda)f_{\mu}^{\lambda}(k)=0,\quad
f_{\mu}^{\lambda}(k)(0)=u_{k},f_{\mu}^{\lambda}(k)^{\prime}(0)=v_{k}.$ (2.9)

We will use the following estimate from [32]:

######  Lemma 2.2.

Assume $f_{\mu}^{\lambda}(k)(t,u,v)$ is a solution to (2.8). Assume $\zeta=1.$
Then

  * •

assume $u=0$ and $1-4\mu(\alpha_{k}+\lambda)\geq 0$. Then

$|f_{\mu}^{\lambda}(k)(t,0,v)|\leq 4\mu|v|e^{-(\alpha_{k}+\lambda)t},$ (2.10)

and

$|f_{\mu}^{\lambda}(k)^{\prime}(t,0,v)|\leq 2|v|e^{-(\alpha_{k}+\lambda)t}.$
(2.11)

  * •

Assume $u=0$ and $1-4\mu(\alpha_{k}+\lambda)\leq 0$, then

$|f_{\mu}^{\lambda}(k)(t,0,v)|\leq\frac{\sqrt{4\mu}|v|}{\sqrt{\alpha_{k}+\lambda}}e^{-\frac{t}{4\mu}}$
(2.12)

and

$|f_{\mu}^{\lambda}(k)^{\prime}(t,0,v)|\leq 2|v|e^{-\frac{t}{4\mu}},$ (2.13)

  * •

In general, for any $k\in\mathbb{N}$, $\mu>0$ and arbitrary
$u,v\in\mathbb{R}$,

$\mu|f_{\mu}^{\lambda}(k)^{\prime}(t,u,v)|^{2}+(\alpha_{k}+\lambda)|f_{\mu}^{\lambda}(k)(t,u,v)|^{2}\leq\mu|v|^{2}+(\alpha_{k}+\lambda)|u|^{2}.$
(2.14)

We have the following $\zeta=0$ version of the above estimate:

######  Lemma 2.3.

Assume $f_{\mu}^{\lambda}(k)(t,u,v)$ is a solution to (2.9). Assume $\zeta=0.$
Then

$|f_{\mu}^{\lambda}(k)(t,0,v)|\leq\frac{\sqrt{\mu}|v|}{\sqrt{\alpha_{k}+\lambda}}$
(2.15)

and

$|f_{\mu}^{\lambda}(k)^{\prime}(t,0,v)|\leq|v|.$ (2.16)

###### Proof.

The solutions to (2.9) are explicitly given by

$f_{\mu}^{\lambda}(k)(t,u,v)=u\cos(\sqrt{\frac{\alpha_{k}+\lambda}{\mu}}t)+v\sin(\sqrt{\frac{\alpha_{k}+\lambda}{\mu}}t)\sqrt{\frac{\mu}{\alpha_{k}+\lambda}}.$
(2.17)

∎

Now we outline some simple consequences of these estimates.

######  Lemma 2.4.

For any $t\geq 0$, $\mu>0$, $\lambda\geq 0$ and $\zeta=\\{0,1\\}$, we have
that

$\|\Pi_{1}\mathcal{S}_{\mu}^{\lambda}(t)\mathcal{I}_{\mu}\|_{\mathcal{L}(H)}\leq
4,\quad\zeta=1,$ (2.18)

and

$\|\Pi_{1}\mathcal{S}_{\mu}^{\lambda}(t)\mathcal{I}_{\mu}\|_{\mathcal{L}(H)}\leq\frac{1}{\sqrt{\alpha_{1}\mu}},\quad\zeta=0,$
(2.19)

###### Proof.

The case $\zeta=1$ and $\lambda=0$ is proved in [32], Lemma 5.3. By
definition,

$\Pi_{1}\mathcal{S}_{\mu}^{\lambda}(t)\mathcal{I}_{\mu}e_{k}=f_{\mu}^{\lambda}(k)(t,0,1/\mu)e_{k}.$

As $e_{k}$ forms an orthogonal basis of $H$,

$\|\Pi_{1}\mathcal{S}_{\mu}^{\lambda}(t)\mathcal{I}_{\mu}\|_{\mathcal{L}(H)}\leq\sup_{k\in\mathbb{N}}|f_{\mu}^{\lambda}(k)(t,0,1/\mu)|.$

Assume $\zeta=1$, for $k$ with $1-4\mu(\alpha_{k}+\lambda)>0$ use (2.10) , and
for the case $1-4\mu(\alpha_{k}+\lambda)>0$ use (2.12). Assume $\zeta=0$, use
the estimate (2.15) and the fact that $\alpha_{k}\geq\alpha_{1}>0$ for each
$k$.

∎

######  Lemma 2.5.

For any $\mu>0$, $t\geq 0$, $\zeta=\\{0,1\\}$ and $\lambda\geq 0$,

$\|\Pi_{1}\mathcal{S}_{\mu}^{\lambda}(t)\begin{pmatrix}I\\\
0\end{pmatrix}\|_{\mathcal{L}(H)}\leq 1.\quad\zeta=0,1.$ (2.20)

###### Proof.

In both cases $\zeta=0,1$, we have

$\|\Pi_{1}\mathcal{S}_{\mu}^{\lambda}(t)\begin{pmatrix}I\\\
0\end{pmatrix}\|_{\mathcal{L}(H)}=\sup_{k\in\mathbb{N}}|f_{\mu}^{\lambda}(k)(t,1,0)|.$
(2.21)

In the case $\zeta=1$ this term is less than or equal to 1 thanks to (2.14).
In the case $\zeta=0$ this is less than or equal to 1 thanks to (2.17). ∎

######  Lemma 2.6.

In the case $\zeta=1$, consider
$N_{\mu}^{\lambda}=\max\\{k\in\mathbb{N}:1-4\mu(\alpha_{k}+\lambda)\geq 0\\}$,
and denote by $P_{N_{\mu}^{\lambda}}$ the orthogonal projection onto the
subspace spanned by $e_{1},\cdots,e_{N_{\mu}^{\lambda}}$. Then for any $t>0$,

$\|\Pi_{1}\mathcal{S}_{\mu}^{\lambda}(t)\begin{pmatrix}0\\\
P_{N_{\mu}^{\lambda}}\end{pmatrix}\|_{\mathcal{L}(H)}\leq 4\mu,\quad\zeta=1$
(2.22)

and

$\|\Pi_{1}\mathcal{S}_{\mu}^{\lambda}(t)\begin{pmatrix}0\\\
I-P_{N_{\mu}^{\lambda}}\end{pmatrix}\|_{\mathcal{L}(H^{-1},H)}\leq\sqrt{4\mu},\quad\zeta=1.$
(2.23)

We also have the following $\zeta=0$ version: for any $t\geq 0,\lambda>0$ and
$\mu>0$,

$\|\Pi_{1}\mathcal{S}_{\mu}^{\lambda}(t)\begin{pmatrix}0\\\
I\end{pmatrix}\|_{\mathcal{L}(H^{-1},H)}\leq\sqrt{\mu},\quad\zeta=0.$ (2.24)

###### Proof.

The first two claims are proved in Lemma 5.5 and 5.6 of [32] when $\lambda=0$.
The first expression is bounded by $\sup_{k\leq
N_{\mu}^{\lambda}}|f_{\mu}^{\lambda}(k)(t,0,1)|\leq 4\mu$, via applying
(2.10). The second expression is bounded by

$\sup_{k>N_{\mu}^{\lambda}}\sqrt{\alpha_{k}}|f_{\mu}^{\lambda}(k)(t,0,1)|\leq\sqrt{4\mu}$

via applying (2.12). The last expression (with $\zeta=0$) is bounded by

$\sup_{k\geq 1}\sqrt{\alpha_{k}}|f_{\mu}^{\lambda}(k)(t,0,1)|\leq\sqrt{\mu}$

via applying (2.15).

∎

######  Lemma 2.7.

For any $\mu\in(0,1)$, we have

$\|\mathcal{S}_{\mu}^{\lambda}(t)\|_{\mathcal{L}(\mathcal{H})}\leq\mu^{-1/2}\sup_{k}\sqrt{\frac{\alpha_{k}+\lambda}{\alpha_{k}}},\quad\zeta=0,1.$
(2.25)

###### Proof.

The result is proved for $\zeta=1$ and $\lambda=0$ in [32], Lemma 5.7. We
provide a complete proof in our more general case.

By definition of $\mathcal{H}$, for $(u,v)\in\mathcal{H}$ and any $t\geq 0$,

$\mu|\mathcal{S}_{\mu}^{\lambda}(t)(u,v)\|_{\mathcal{H}}^{2}\leq\mu\|\Pi_{2}\mathcal{S}_{\mu}^{\lambda}(t)(u,v)|_{H^{-1}}^{2}+|\Pi_{1}\mathcal{S}_{\mu}^{\lambda}(t)(u,v)|H^{2}.$

From the Fourier decomposition it suffices to bound

$\sum_{k=1}^{\infty}\left(\frac{\mu}{\alpha_{k}}|f_{\mu}^{\lambda}(k)^{\prime}(t,u_{k},v_{k})|^{2}+|f_{\lambda}^{\mu}(k)(t,u_{k},v_{k})|^{2}\right).$

First consider the case $\zeta=0$. From (2.17), we see that it is bounded by

$\sup_{k}\frac{\alpha_{k}+\lambda}{\alpha_{k}}(\sum_{k=1}^{\infty}\frac{\mu}{\alpha_{k}}|v_{k}|^{2}+u_{k}|^{2})\leq\sup_{k}\frac{\alpha_{k}+\lambda}{\alpha_{k}}(u,v)_{\mathcal{H}}.$

Then consider the case $\zeta=1$. It suffices to use (2.14) and we get the
same estimate as above. ∎

###  2.2. Estimates in shifted norm

As will be made clear afterwards, for general $\lambda>0$ it is more
convenient to re-define the norm of the interpolation space $H^{\delta}$. Now
we define, for each fixed $\lambda>0$, $H^{\delta}(\lambda)$ as the completion
of the vector space $\operatorname{Vect}(e_{1},\cdots,e_{k},\cdots)$ with
respect to the norm

$|f|_{H^{\delta}(\lambda)}^{2}:=\sum_{k=1}^{\infty}(\alpha_{k}+\lambda)^{\delta}\langle
f,e_{k}\rangle_{H}^{2}.$

This new norm of $H^{\delta}(\lambda)$ is equivalent to the norm of
$H^{\delta}$ for any $\lambda>0$, but it will be more convenient to use.

We now consider the phase space

$\mathcal{H}(\lambda):=H^{0}(\lambda)\times H^{-1}(\lambda)=H\times
H^{-1}(\lambda).$

Note that $H=H^{0}(\lambda)$ so the norm in the first component of
$\mathcal{H}$ is the same as that of $\mathcal{H}(\lambda)$. Now we restate
many of our previous estimates in the new norms $H^{-1}(\lambda)$ and
$\mathcal{H}_{\lambda}$:

######  Lemma 2.8.

With the same notation as Lemma 2.6 and 2.7, we have

$\|\Pi_{1}\mathcal{S}_{\mu}^{\lambda}(t)\begin{pmatrix}0\\\
I-P_{N_{\mu}^{\lambda}}\end{pmatrix}\|_{\mathcal{L}(H^{-1}(\lambda),H)}\leq\sqrt{4\mu},\quad\zeta=1.$
(2.26)

and

$\|\Pi_{1}\mathcal{S}_{\mu}^{\lambda}(t)\begin{pmatrix}0\\\
I\end{pmatrix}\|_{\mathcal{L}(H^{-1}(\lambda),H)}\leq\sqrt{\mu},\quad\zeta=0.$
(2.27)

Moreover, for any $\mu\in(0,1),$ we have

$\|\mathcal{S}_{\mu}^{\lambda}(t)\|_{\mathcal{L}(\mathcal{H}(\lambda))}\leq\mu^{-1/2},\quad\zeta=0,1.$
(2.28)

###### Proof.

The proof is exactly the same. Thanks to and (2.12) and (2.15), we have

$\sup_{k>N_{\mu}^{\lambda}}\sqrt{\alpha_{k}+\lambda}|f_{\mu}^{\lambda}(k)(t,0,1)|\leq\sqrt{4\mu},\quad\zeta=1$

and

$\sup_{k\geq
1}\sqrt{\alpha_{k}+\lambda}|f_{\mu}^{\lambda}(k)(t,0,1)|\leq\sqrt{\mu},\quad\zeta=0.$

The last claim follows again from (2.14) when $\zeta=1$ and (2.17) when
$\zeta=0,$ while the fact that we have $\alpha_{k}+\lambda$ instead of
$\alpha_{k}$ in the definition of the norm of $\mathcal{H}(\lambda)$ allows us
to get rid of the $\sup_{k}\sqrt{\frac{\alpha_{k}+\lambda}{\alpha_{k}}}$
factor in (2.25). ∎

###  2.3. Maximal inequalities of stochastic integrals

In this paper we will extensively use maximal inequalities of stochastic
integrals of the following form

$\Gamma^{\mu,\lambda}(t)=\int_{0}^{t}\mathcal{S}_{\mu}^{\lambda}(t-s)\mathcal{I}_{\mu}\Phi(s)dW_{s}$
(2.29)

for some adapted, self adjoint process $\Phi\in
L^{p}(\Omega;L^{\infty}([0,T];\mathcal{L}(H_{0},H)))$. When we apply the
estimates related to $\Gamma^{\mu,\lambda}$, we will often take
$\Phi(s)=G(s,u(s))$ or $\Phi(s)=G(s,u(s))-G^{n}(s,u^{n}(s))$ for some
approximations $G^{n}$ of $G$. By assumption (1.7) and our approximating
procedure, these choices will make $\Phi(s)$ a self-adjoint adapted processes.
In the rest of this paper we will assume self adjointness of $\Phi$ without
mentioning.

We will use the factorization formula ([11],Section 5.3.1) for stochastic
convolutions which gives, for $0<\alpha<1$ (which we will later choose to be
sufficiently small),

$\Gamma^{\mu,\lambda}(t)=\frac{\sin(\alpha\pi)}{\pi}\int_{0}^{t}(t-s)^{\alpha-1}\mathcal{S}_{\mu}^{\lambda}(t-s)\Gamma_{\alpha}^{\mu,\lambda}(s)ds,$
(2.30)

where

$\Gamma_{\alpha}^{\mu,\lambda}(t)=\int_{0}^{t}(t-s)^{-\alpha}\mathcal{S}_{\mu}^{\lambda}(t-s)\mathcal{I}_{\mu}\Phi(s)dW_{s}.$
(2.31)

We first prove maximal inequality estimates for
$\Gamma_{\alpha}^{\mu,\lambda}$. The proof is similar to [32], Lemma 6.1 but
we make a better use of the summability of
$\sum_{n}\frac{1}{\lambda_{n}^{\sigma}}$, $\sigma<1$, and our result can be
re-interpreted as a maximal inequality of Ornstein-Uhlenbeck process when
$\lambda$ is very large.

######  Notation 2.9.

We abbreviate, for any $T>0$,

$\|\Phi\|_{T}:=\sup_{0\leq t\leq T}|\Phi(t)|_{\mathcal{L}(H_{0},H)}.$

######  Proposition 2.10.

Fix $T>0$. Assume that there is some $\eta_{0}\in(0,1)$ such that

$\sum_{n}\frac{1}{\alpha_{n}^{1-\eta}}<\infty$ (2.32)

for each $\eta\in(0,\eta_{0})$. Then for each $p\geq 2$ and each
$\eta\in(0,\eta_{0}),$ for both $\zeta=0$ and $\zeta=1,$

$\mathbb{E}[|\Pi_{1}\Gamma_{\alpha}^{\mu,\lambda}(t)\|_{H}^{p}]\leq
C_{\eta,p,\mu}\|\Phi\|_{t}^{p}\lambda^{-\frac{\eta p}{2}},\quad t\in[0,T].$
(2.33)

The constant $c_{\eta,p,\mu}$ depends on $\eta$, $p$ and $\mu$.

###### Proof.

Applying the Burkholder-Davis-Gundy inequality to
$\Gamma_{\alpha}^{\mu,\lambda}$:

$\mathbb{E}[\|\Pi_{1}\Gamma_{\alpha}^{\mu,\lambda}(t)\|_{H}^{p}]\leq
C\mathbb{E}\left(\sum_{j=1}^{\infty}\int_{0}^{t}(t-s)^{-2\alpha}|\Pi_{1}\mathcal{S}_{\mu}^{\lambda}(t-s)\mathcal{I}_{\mu}\Phi(s)e_{j}|_{H}^{2}ds\right)^{p/2}$
(2.34)

The quadratic variation

$\displaystyle\Lambda_{\alpha}^{\mu,\lambda}(t):$
$\displaystyle=\sum_{j=1}^{\infty}\int_{0}^{t}(t-s)^{-2\alpha}|\Pi_{1}\mathcal{S}_{\mu}^{\lambda}(t-s)\mathcal{I}_{\mu}\Phi(s)e_{j}|_{H}^{2}ds$
$\displaystyle=\sum_{k=1}^{\infty}\sum_{j=1}^{\infty}\int_{0}^{t}(t-s)^{-2\alpha}\langle\Pi_{1}\mathcal{S}_{\mu}^{\lambda}(t-s)\mathcal{I}_{\mu}\Phi(s)e_{j},e_{k}\rangle_{H}^{2}ds$
$\displaystyle=\sum_{k=1}^{\infty}\sum_{j=1}^{\infty}\int_{0}^{t}(t-s)^{-2\alpha}\langle\Phi(s)e_{j},\mathcal{I}_{\mu}^{*}S_{\mu}^{\lambda*}(t-s)\Pi_{1}^{*}e_{k}\rangle_{H}^{2}ds$

Since

$\displaystyle\langle\mathcal{I}_{\mu}^{*}\mathcal{S}_{\mu}^{\lambda*}(t)\Pi_{1}^{*}e_{k},e_{j}\rangle_{H}$
$\displaystyle=\langle\Pi_{1}\mathcal{S}_{\mu}^{\lambda}(t)\mathcal{I}_{\mu}e_{j},e_{k}\rangle_{H}$
$\displaystyle=\begin{cases}f_{\mu}^{\lambda}(k)(t,0,1/\mu)\quad j=k\\\ 0\quad
j\neq k,\end{cases}$

we further simplify $\Lambda_{\alpha}^{\mu,\lambda}(t)$ as, abbreviating
$f_{\mu}^{\lambda}(k)(t):=f_{\mu}^{\lambda}(k)(t,0,1/\mu)$,

$\Lambda_{\alpha}^{\mu,\lambda}(t)=\sum_{k=1}^{\infty}\sum_{j=1}^{\infty}\int_{0}^{t}(t-s)^{-2\alpha}(f_{\mu}^{\lambda}(k)(t-s))^{2}\langle\Phi(s)e_{j},e_{k}\rangle_{H}^{2}ds.$

By self-adjointness of $\Phi$ we have, using $e_{k}\in H_{0}$ with
$|e_{k}|_{H_{0}}\leq M$,

$\sum_{j=1}^{\infty}\langle\Phi(s)e_{j},e_{k}\rangle_{H}^{2}=\sum_{j=1}^{\infty}\langle\Phi(s)e_{k},e_{j}\rangle_{H}^{2}=|\Phi(s)e_{k}|_{H}^{2}\leq
M|\Phi(s)|^{2}_{\mathcal{L}(H_{0},H)}.$

Thus we bound $\Lambda_{\alpha}^{\mu,\lambda}(t)$ as follows:

$\Lambda_{\alpha}^{\mu,\lambda}(t)\leq\int_{0}^{t}(t-s)^{-2\alpha}\|\Phi(s)\|_{t}^{2}\left(\sum_{k=1}^{\infty}f_{\mu}^{\lambda}(k)^{2}(t-s,0,1/\mu)\right)ds.$
(2.35)

In the following we find effective upper bounds for

$\sum_{k=1}^{\infty}f_{\mu}^{\lambda}(k)^{2}(t,0,1/\mu).$

For the case $\zeta=1$, thanks to (2.10), (2.12), this is bounded by

$\sum_{k=1}^{N_{\mu}^{\lambda}}e^{-2(\alpha_{k}+\lambda)t}+\sum_{k=N_{\mu}^{\lambda}+1}^{\infty}\frac{e^{-\frac{t}{2\mu}}}{\mu(\alpha_{k}+\lambda)}:=J_{1}(t)+J_{2}(t).$

For the case $\zeta=0$, thanks to (2.15), this is bounded by

$\sum_{k=1}^{\infty}\frac{1}{\mu(\alpha_{k}+\lambda)}:=J_{3}.$

For any $\eta\in(0,1)$, by Young’s inequality we may find $c_{\eta}$ such that

$\alpha_{k}+\lambda\geq c_{\eta}{\alpha_{k}}^{1-\eta}\lambda^{\eta}.$ (2.36)

Thus for any $\eta\in(0,\eta_{0})$, by (2.32) we may find $c_{\eta}$ such that

$J_{3}\leq\frac{c_{\eta}}{\mu}\lambda^{-\eta},\quad
J_{2}(t)\leq\frac{c_{\eta}}{\mu}\lambda^{-\eta}e^{-\frac{t}{2\mu}}.$ (2.37)

In the case $\zeta=0$, the proof is already complete by plugging the bound of
$J_{3}$ into (2.35).

In the following we focus on the case $\zeta=1$. We first compute the
stochastic integral relating to $J_{1}(t)$. The analysis of $J_{1}(t)$ is
similar to that of the heat semigroup in [23]. For any $r<1$ we can find
constants $c_{r}$ such that, by using (2.36) in the second inequality,

$e^{-2(\alpha_{k}+\lambda)t}\leq
c_{r}\frac{1}{((\alpha_{k}+\lambda)t)^{r}}\leq
c_{r,\eta}\frac{1}{\alpha_{k}^{(1-\eta)r}}\lambda^{-\eta r}\frac{1}{t^{r}}.$
(2.38)

Since $\int_{0}^{t}s^{-2\alpha-r}ds<\infty$ by choosing $\alpha$ sufficiently
small followed by choosing $r$ sufficiently close to 1, one can ensure that
$\sum_{k=1}^{\infty}\frac{1}{\alpha_{k}^{(1-\eta)r}}<\infty$ for $r$
sufficiently close to 1, and then $r\eta$, the exponent of $\lambda^{-1},$ can
achieve any value in $(0,\eta_{0})$ by our choice. This produces the desired
$\lambda^{\eta}$ factor in the upper bound of
$\Lambda_{\alpha}^{\mu,\lambda}$.

Finally we deal with the component involving $J_{2}(t)$. It suffices to plug
in the bound (2.37) into (2.35). This finishes the proof with a constant
$c_{\eta,p,\mu}$ that is $\mu$-dependent. ∎

######  Proposition 2.11.

Fix $T>0$. For each $p\geq 2$ and each $\eta\in(0,\eta_{0}),$ we may choose
$\alpha>0$ sufficiently small such that, for $\zeta=1,$

  1. (1)

for $\mu\in(0,1)$ and $t\in[0,T],$

$\mathbb{E}|P_{N_{\mu}^{\lambda}}\Pi_{2}\Gamma_{\alpha}^{\mu,\lambda}(t)|_{H}^{p}\leq\frac{c_{\eta,p,\mu}}{\mu^{p}}\lambda^{-\frac{\eta
p}{2}}\mathbb{E}\|\Phi\|_{t}^{p},$ (2.39)

  2. (2)

for $\mu\in(0,1)$ and $t\in[0,T],$

$\mathbb{E}|(I-P_{N_{\mu}^{\lambda}})\Pi_{2}\Gamma_{\alpha}^{\mu,\lambda}(t)|_{H^{-1}(\lambda)}^{p}\leq\frac{c_{\eta,p,\mu}}{\mu^{p}}\lambda^{-\frac{\eta
p}{2}}\mathbb{E}\|\Phi\|_{t}^{p}$ (2.40)

and for $\zeta=0$, for any $\mu\in(0,1)$ and $t\in[0,T],$

$\mathbb{E}|(I-P_{N_{\mu}^{\lambda}})\Pi_{2}\Gamma_{\alpha}^{\mu,\lambda}(t)|_{H^{-1}(\lambda)}^{p}\leq\frac{c_{\eta,p,\mu}}{\mu^{p}}\lambda^{-\frac{\eta
p}{2}}\mathbb{E}\|\Phi\|_{t}^{p}.$ (2.41)

Here the constant $c_{\eta,p,\mu}$ depends on $\eta$, $p$ and $\mu>0$, and
also implicitly on $\alpha>0$ sufficiently small, but does not depend on
$\lambda$.

Moreover, all these estimates are true if we evaluate the left hand side of
each estimate at a stopping time $t\wedge\tau$, where $\tau$ is a stopping
time adapted to the filtration $(\mathcal{F}_{t})_{t\geq 0}$, and we write on
the right hand side of each estimate
$\sup_{s\in[0,t\wedge\tau]}\|\Phi(s)\|_{H}^{p}.$

###### Proof.

We first deal with the case $\zeta=1$, where we are required to separately
deal with the higher and lower modes. For the lower modes, let
$\Lambda_{1}(t)$ denote the quadratic variation of
$P_{N_{\mu}^{\lambda}}\Pi_{2}\Gamma_{\alpha}^{\mu,\lambda}.$ Then

$\displaystyle\Lambda_{1}(t)=$
$\displaystyle\sum_{j=1}^{\infty}\int_{0}^{t}(t-s)^{-2\alpha}|P_{N_{\mu}^{\lambda}}\Pi_{2}\mathcal{S}_{\mu}^{\lambda}(t-s)\mathcal{I}_{\mu}\Phi(s)e_{j}\|_{H}^{2}ds$
$\displaystyle=$
$\displaystyle\sum_{k=1}^{N_{\mu}^{\lambda}}\sum_{j=1}^{\infty}\int_{0}^{t}(t-s)^{-2\alpha}\langle\Phi(s)e_{j},\mathcal{I}_{\mu}^{*}\mathcal{S}_{\mu}^{\lambda*}(t-s)\Pi_{2}^{*}e_{k}\rangle_{H}^{2}ds.$

The eigenvalues $e_{j}$ satisfy
$\mathcal{I}_{\mu}^{*}\mathcal{S}_{\mu}^{\lambda*}\Pi_{2}^{*}e_{k}=f_{\mu}^{\lambda}(k)^{\prime}(t-s)e_{k}$,
where $f_{\mu}^{\lambda}(k)$ solves (2.8) with $u_{k}=0$ and $v_{k}=1/\mu$. By
(2.11) with $v=\frac{1}{\mu},$ for $1\leq k\leq N_{\mu}^{\lambda},$ we have

$(f_{\mu}^{\lambda}(k)^{\prime}(t)\leq\frac{2e^{-(\alpha_{k}+\lambda)t}}{\mu},$

so that, with the same reasoning as the previous proof,

$\Lambda_{1}(t)\leq\frac{C}{\mu^{2}}\int_{0}^{t}(t-s)^{-2\alpha}\|\Phi\|_{s}^{2}(\sum_{k=1}^{N_{\mu}^{\lambda}}e^{-2(\alpha_{k}+\lambda)(t-s)})ds.$

By the same reasoning as in the proof of Proposition 2.10, for any
$\eta\in(0,\eta_{0})$ we may find $C_{\eta}>0$ such that

$\Lambda_{1}(t)\leq\frac{C_{\eta}}{\mu^{2}}\|\Phi\|_{t}^{2}\lambda^{-\eta}.$

By the BDG inequality

$\mathbb{E}|P_{N_{\mu}}\Pi_{2}\Gamma_{\alpha}^{\mu,\lambda}(t)|_{H}^{p}\leq
C\mathbb{E}(\Lambda_{1}(t))^{p/2}$

which implies (2.39) .

For the higher modes, compute the quadratic variation

$\displaystyle\Lambda_{2}(t)$
$\displaystyle=\sum_{j=1}^{\infty}\int_{0}^{t}(t-s)^{-2\alpha}|(I-P_{N_{\mu}^{\lambda}})\Pi_{2}\mathcal{S}_{\mu}^{\lambda}(t-s)\mathcal{I}_{\mu}\Phi(s)e_{j}\|_{H^{-1}(\lambda)}^{2}ds$
$\displaystyle=\sum_{j=1}^{\infty}\int_{0}^{t}(t-s)^{-2\alpha}|(-A-\lambda)^{-1/2}(I-P_{N_{\mu}^{\lambda}})\Pi_{2}\mathcal{S}_{\mu}^{\lambda}(t-s)\mathcal{I}_{\mu}\Phi(s)e_{j}|_{H}^{2}ds,$

Expanding the squares, the right hand side is

$\Lambda_{2}(t)\leq\sum_{k=N_{\mu}^{\lambda}+1}^{\infty}\sum_{j=1}^{\infty}\int_{0}^{t}(t-s)^{-2\alpha}\langle\Phi(s)e_{j},\mathcal{I}_{\mu}^{*}\mathcal{S}_{\mu}^{\lambda*}(t-s)\Pi_{2}^{*}(I-P_{N_{\mu}^{\lambda}}^{*})(-A-\lambda)^{-1/2}e_{k}\rangle_{H}^{2}ds.$

Since

$\displaystyle\langle\mathcal{I}_{\mu}^{*}\mathcal{S}_{\mu}^{\lambda*}(t-s)\Pi_{2}^{*}(I-P_{N_{\mu}^{\lambda}}^{*})(-A-\lambda)^{-1/2}e_{k},e_{j}\rangle_{H}^{2}$
$\displaystyle=\begin{cases}(\alpha_{k}+\lambda)^{-1/2}f_{\mu}^{\lambda}(k)^{\prime}(t-s),\quad\text{
if }k=j>N_{\mu},\\\ 0\quad\text{ otherwise },\end{cases}$

and by (2.13),

$(\alpha_{k}+\lambda)^{-1/2}|f_{\mu}^{\lambda}(k)^{\prime}(t-s)|\leq
C(\alpha_{k}+\lambda)^{-1/2}\mu^{-1}.$

By (2.37) and the assumptions on $\alpha_{k}$, for any $\eta\in(0,\eta_{0})$
we may find $c_{\eta}>0$ such that

$\sum_{k=N_{\mu}^{\lambda}+1}^{\infty}(\alpha_{k}+\lambda)^{-1}|f_{\mu}^{\lambda}(k)^{\prime}(t-s)|^{2}\leq\frac{1}{\mu^{2}}\sum_{k=1}^{\infty}\frac{1}{(\alpha_{k}+\lambda)}\leq\frac{c_{\eta}}{\mu^{2}}\lambda^{-\eta}.$

Thus

$\Lambda_{2}(t)\leq\frac{c_{\eta}}{\mu^{2}}\lambda^{-\eta}\|\Phi\|_{t}^{2}.$

By the BDG inequality,

$\mathbb{E}|(I-P_{N_{\mu}^{\lambda}})\Pi_{2}\Gamma_{\alpha}^{\mu,\lambda}(t)|_{H^{-1}(\lambda)}^{p}\leq\frac{c_{\eta}}{\mu^{p}}\lambda^{-\eta
p/2}\mathbb{E}\|\Phi\|_{t}^{p}.$

This completes the proof for the $\zeta=1$ case. The case $\zeta=0$ is
analogous, and essentially repeats the computations in the high-modes part of
$\zeta=1$, using (2.16) in place of (2.13)

The proof regarding evaluation at a stopping time $t\wedge\tau$ is easy: one
may just set $\Phi=0$ on $[t\wedge\tau,t]$ and use the estimates we already
obtained.

∎

######  Theorem 2.12.

Fix $T>0$. Recall $\Gamma^{\mu,\lambda}$ defined in (2.29). Then for any
$p\geq 2$ and any $\eta\in(0,\eta_{0})$, we may find $c_{\eta,p,\mu}>0$ such
that for all $\lambda\geq 0,$ for both $\zeta=0$ and $\zeta=1,$

$\mathbb{E}\sup_{t\in[0,T]}|\Pi_{1}\Gamma^{\mu,\lambda}(t)|_{H}^{p}\leq
c_{\eta,p,\mu}\lambda^{-\frac{\eta p}{2}}\mathbb{E}\|\Phi\|_{T}^{p}.$ (2.42)

The constant $c_{\eta,p,\mu}$ is independent of $\lambda>0$.

Moreover, for any stopping time $\tau$ adapted to the filtration
$(\mathcal{F}_{t})_{t\geq 0}$, we have

$\mathbb{E}\sup_{t\in[0,T\wedge\tau]}|\Pi_{1}\Gamma^{\mu,\lambda}(t)|_{H}^{p}\leq
c_{\eta,p,\mu}\lambda^{-\frac{\eta
p}{2}}\mathbb{E}\|\Phi\|_{T\wedge\tau}^{p}.$ (2.43)

######  Remark 2.13.

A crucial observation shall be made here: the estimate (2.42) involves the
norm of the first component $H\times\\{0\\}\subset\mathcal{H}$, so that
changing the norm of $H^{-1}$ into $H^{-1}(\lambda)$ has no quantitative
impact on this estimate. When we solve the stochastic wave equation, as the
drift and diffusion coefficients depend only on $u$ but not on
$\partial_{t}u$, we only need to establish uniqueness in law of $u$, then
uniqueness in law of $\partial_{t}u$ follows from integration. Therefore we
only need to derive estimates in the first component $H$ of $\mathcal{H}$. The
reason why we insist on changing the norm of $H^{-1}$ to $H^{-1}(\lambda)$ is
that, when using the factorization formula of stochastic convolutions, the
latter choice can generate the desired $\lambda^{-\eta/2}$ coefficient, but
the original choice does not give us this coefficient. This will be clear from
the proof below.

###### Proof.

In the case $\zeta=1,$ by the stochastic factorization formula ([11],Section
5.3.1),

$\displaystyle\Gamma^{\mu,\lambda}(t)$
$\displaystyle=\frac{\sin(\alpha\pi)}{\pi}\int_{0}^{t}(t-s)^{\alpha-1}\mathcal{S}_{\mu}^{\lambda}(t-s)[\begin{pmatrix}I\\\
0\end{pmatrix}\Pi_{1}\Gamma_{\alpha}^{\mu,\lambda}(s)$ (2.44)
$\displaystyle+\begin{pmatrix}0\\\
P_{N_{\mu}^{\lambda}}\end{pmatrix}P_{N_{\mu}^{\lambda}}\Pi_{2}\Gamma_{\alpha}^{\mu,\lambda}(s)+\begin{pmatrix}0\\\
(1-P_{N_{\mu}^{\lambda}})\end{pmatrix}(1-P_{N_{\mu}^{\lambda}})\Pi_{2}\Gamma_{\alpha}^{\mu,\lambda}(s)]ds.$

Now we choose $\alpha>0$ sufficiently small, and bound the $L^{p}$-th norm of
$\Gamma^{\mu,\lambda}(t)$ term by term. We start with the first term, applying
Jensen’s inequality222One may also apply Hölder’s inequality here as in [32],
but this will lead to an additional requirement $p>\frac{1}{\alpha}$. Applying
Jensen’s inequality does not lead to this requirement.:

$\begin{aligned}
&\mathbb{E}\sup_{t\in[0,T]}\left|\int_{0}^{t}(t-s)^{\alpha-1}\Pi_{1}\mathcal{S}_{\mu}^{\lambda}(t-s)\begin{pmatrix}I\\\
0\end{pmatrix}\Pi_{1}\Gamma_{\alpha}^{\mu,\lambda}(s)ds\right|_{H}^{p}\\\
&\leq
c_{\eta,p}\sup_{s\in[0,T]}\left|\Pi_{1}\mathcal{S}_{\mu}^{\lambda}(s)\begin{pmatrix}I\\\
0\end{pmatrix}\right|_{\mathcal{L}(H)}^{p}\mathbb{E}\int_{0}^{T}(t-s)^{\alpha-1}|\Pi_{1}\Gamma_{\alpha}^{\mu,\lambda}(s)|_{H}^{p}ds\\\
&\leq c_{\eta,p}\lambda^{-\frac{\eta
p}{2}}\mathbb{E}\|\Phi\|_{T}^{p}.\end{aligned},$ (2.45)

where in the second inequality we use Lemma 2.5 and Proposition 2.10.

For the second term

$\mathbb{E}\sup_{t\in[0,T]}\left|\int_{0}^{t}(t-s)^{\alpha-1}\Pi_{1}\mathcal{S}_{\mu}^{\lambda}(t-s)\begin{pmatrix}0\\\
P_{N_{\mu}^{\lambda}}\end{pmatrix}P_{N_{\mu}^{\lambda}}\Pi_{2}\Gamma_{\alpha}^{\mu,\lambda}(s)ds\right|_{H}^{p},$

we use Lemma 2.6 and use Proposition 2.11 (1).

For the third term

$\mathbb{E}\sup_{t\in[0,T]}\left|\int_{0}^{t}(t-s)^{\alpha-1}\Pi_{1}\mathcal{S}_{\mu}^{\lambda}(t-s)\begin{pmatrix}0\\\
I-P_{N_{\mu}^{\lambda}}\end{pmatrix}(I-P_{N_{\mu}^{\lambda}})\Pi_{2}\Gamma_{\alpha}^{\mu,\lambda}(s)ds\right|_{H}^{p},$

we use Lemma 2.8 and use Proposition 2.11 (2). Notice that we are working with
the Hilbert space $H^{-1}(\lambda)$ rather than $H^{-1}$. The proof in the
case $\zeta=0$ is very similar and indeed easier, so we omit the details.

The proof for the stopping time version $\mathbb{E}\sup_{t\in[0,T\wedge\tau]}$
is exactly the same, where we just integrate over $[0,T\wedge\tau]$, and use
the stopping time version of Proposition 2.11. ∎

##  3\. Existence and uniqueness of solutions

The proof of well-posedness in this section is a bit technical due to
difficulties of approximating a continuous map by Lipschitz maps in an
infinite dimension setting. The reader may first read Section 3.5 where we
work under a slightly stronger assumption, and consequently we have a much
shorter proof with the main ideas well illustrated. The general case follows
similar lines, but a lot more care has to be taken.

While we prove existence and uniqueness to the stochastic wave equation (1.3),
the value of $\mu>0$ is irrelevant. Thus we suppress $\mu$ from the notation
and simply write the solution as $Z(t)=(u(t),v(t))$.

###  3.1. Proof of weak existence: step 1

For weak existence of solutions to hyperbolic SPDEs on Hilbert space, we
cannot use the standard approach in [15], [11], [20] that are designed for the
parabolic case, because the semigroup $\mathcal{S}_{\mu}(t)$ of the stochastic
wave equation is not a compact operator. Thus we choose to prove weak
existence directly.

First note that since $G$ is uniformly non-degenerate and $B$ has linear
growth, we may use Girsanov transform to construct the drift $B$, so we may
first without loss of generality assume $B=0$. See [28], Remark 2.1 and
references therein for a detailed discussion on applying Girsanov transform
for SPDEs in infinite dimensions.

We begin with a standard approximation procedure333This approximation is used
as an illustration but will not be used in the formal proof to be given..
Consider $P_{n}$, the projection of $H$ onto the eigenspace spanned by the
first $n$ eigenvectors $\\{e_{1},\cdots,e_{n}\\}$, and consider truncated
coefficients

$G^{n}(t,x):=P_{n}G(t,P_{n}x).$

Now we consider the following SPDEs

$\mu\frac{\partial^{2}u^{n}(t)}{\partial t^{2}}=Au^{n}(t)-\zeta\frac{\partial
u^{n}(t)}{\partial t}+G^{n}(t,u^{n}(t))\frac{dW_{t}}{dt},$ (3.1)

with initial value $(P_{n}u_{0},P_{n}v_{0})\in\mathcal{H}_{1}$. The initial
value will converge to $(u_{0},v_{0})$ and thus does not matter now. We focus
on the stochastic integration, which is given in mild formulation as

$\int_{0}^{t}\Pi_{1}\mathcal{S}_{\mu}(t-s)\mathcal{I}_{\mu}G^{n}(s,u^{n}(s))dW(s).$

First using the mild formulation and Gronwall’s inequality one may get an
estimate

$\mathbb{E}[\sup_{t\in[0,T]}|u^{n}(t)|^{p}]\leq M_{p}$

for some $n$-independent constant $M_{p}$. We now wish to establish
compactness. We need the following two results from [6]: abbreviating, for any
$z\in L^{p}(\Omega,L^{p}([0,T];\mathcal{H}_{0})),$

$\Gamma_{\mu}(z)(t)=\int_{0}^{t}\mathcal{S}_{\mu}(t-s)\mathcal{I}_{\mu}G(s,\Pi_{1}z(s))dW(s),$

######  Proposition 3.1.

Fix $T>0$, given any $p>4$ consider $z\in
L^{p}(\Omega,L^{p}([0,T];\mathcal{H}_{0}))$ and any $\delta<\eta_{0}-2/p$, we
have, for the case of damped stochastic wave equation $\zeta=1$:

$\sup_{\mu>0}\mathbb{E}[\Pi_{1}|\Gamma_{\mu}(z)(t)|_{\mathcal{C}([0,T];H^{\delta})}]\leq
C_{p,T}(1+\mathbb{E}\int_{0}^{T}|\Pi_{1}z(t)|_{H}^{p}dt),$ (3.2)

where $C_{p,T}$ does not depend on $\mu$. The parameter $\eta_{0}$ is the one
given in Assumption $\mathbf{H}1$.

For the case $\zeta=0$, i.e. with no damping term $\partial_{t}u$, we have the
same estimate but with a constant $C_{p,T,\mu}$ that is $\mu$-dependent.

The proof of this result in the case $\zeta=1$ is given in [6], Proposition
3.1. The Laplacian on $[0,1]$ was considered there, but generalization to the
operator $A$ satisfying $\mathbf{H}1$ is straightforward. This Proposition can
also be proved via modifying the proof of Theorem 4.4. The proof for the
$\zeta=0$ version is even easier as the constant can be $\mu$-dependent. We
omit the details but remark that the proof can be written via slightly
modifying the proof of Theorem 2.12. We only need to replace all the $H$-norms
by $H^{\delta}$ norms and use the summability of eigenvalues
$\sum_{k}\frac{1}{(\alpha_{k})^{\omega}}$ for appropriate powers of $\omega$.

The following Proposition is also a useful tool in establishing tightness:

######  Proposition 3.2.

For any $\epsilon$ sufficiently small, $\rho\in(0,\epsilon)$ and $p\geq 1$, we
can find a constant $c=c(\epsilon,\rho,p)$ such that

$\sup_{\mu\geq
0}\mathbb{E}|\Pi_{1}\Gamma_{\mu}(z)(t)-\Pi_{1}\Gamma_{\mu}(z)(s)|^{p}\leq
c|t-s|^{\rho
p/2}(1+\mathbb{E}[\Pi_{1}z]^{p}_{\mathcal{C}([0,T];H^{\epsilon})}),$

given $z\in L^{p}(\Omega;\mathcal{C}([0,T];\mathcal{H}_{\epsilon})$ and
$t,s\in[0,T]$.

The proof of this result is given in [6] as a lemma after the proof of
Proposition 3.1. However, one should be careful that this result is only
proved for random fiend valued SPDEs defined on $[0,1]$, and Sobolev embedding
is used in its proof as an essential procedure. Thus we can use this result
for random field SPDEs, and the proof of existence will follow. But in our
setting of abstract Hilbert space mappings, we cannot use this result so we
choose to give an independent proof.

To deduce compactness in $\mathcal{C}([0,T];H)$, the standard procedure is to
use the infinite dimensional Arzela-Ascoli theorem. In this direction, one
need to show (1) tightness at each time $t$, and (2) uniform continuity of
sample paths. Now (1) can be justified with high probability thanks to
Proposition 3.1 since $\Pi_{1}\Gamma_{\mu}(z)(t)$ is uniformly bounded in
$H^{\delta}$ for each $t$, and $H^{\delta}$ embeds compactly into $H$ (also
note that we assume the initial value $(u_{0},v_{0})\in\mathcal{H}_{1}$). If
one assumes Proposition 3.2, then condition (2) can be justified and we are
done. However, this is not the case for hyperbolic SPDEs on Hilbert space
(though for parabolic SPDEs this is true.) The reason is that if we consider
the undamped wave equation, the solution will be sines and cosines. Under the
evolution of wave semigroup, we have a perfect control of the norms of these
solutions, but our control on the time derivative of the solution is rather
poor– they will have very fast periodic oscillation! For random field
stochastic wave equation, we can still get Hölder regularity in time
increments thanks to the fact that the diffusion coefficient $g(t,u(t,x))$ has
higher spatial regularity than merely being an $L^{2}([0,1])$ function.
Unfortunately, this is not assumed anywhere in our paper. Therefore, the
Arzela-Ascoli argument does not apply and we will turn back in Section 3.4 to
finally establish the proof of weak existence, using estimates derived in
Section 3.3.

###  3.2. Lipschitz approximation of continuous maps

In the later proofs we will frequently consider Lipschitz approximations of
Hölder continuous mappings. Such approximations are well-understood for real-
valued functions, but our target space $H$ has infinite dimensions.
Fortunately, we can still find an approximation procedure, such that the
approximating sequences have good properties inherited from the original
function. We begin with the following Proposition, whose proof is taken from
Alex Ravsky [31].

######  Proposition 3.3.

Given $V$ a separable Hilbert space, $V^{\prime}$ be a normed space and
$f:V\to V^{\prime}$ a continuous map. Then we can find a sequence
$\\{f_{n}\\}$ of bounded Lipschitz maps: $V\to V^{\prime}$ that converge
uniformly to $f$ on any compact subset of $V$.

###### Proof.

We only need to consider $V=\ell^{2}$. Write
$\mathbb{Z}/n=\\{x\in\mathbb{R}:nx\in\mathbb{Z}\\}$. First define
$g_{n}^{*}:(\mathbb{Z}/n)^{n}\to V^{\prime}$ via $g_{n}^{*}(x)=f(x)$ for
$|f(x)|\leq n$ and $g_{n}^{*}(x)=nf(x)/|f(x)|$ if $|f(x)|\geq n$. Then extend
$g_{n}^{*}$ to a map $f_{n}^{*}:\mathbb{R}^{n}\to V^{\prime}$ via affine
interpolation. More precisely, consider a map
$\lfloor\cdot\rfloor_{n}:\mathbb{R}\to\mathbb{Z}/n$ via $\lfloor
x\rfloor_{n}=\lfloor nx\rfloor/n$ for $x\in\mathbb{R}$. Given any
$x=(x_{i})\in\mathbb{R}^{n}$, $x$ is contained in
$\mathcal{C}_{x}=\prod_{i=1}^{n}[\lfloor x_{i}\rfloor_{n};\lfloor
x_{i}\rfloor_{n}+1/n]$. Now define
$f_{n}^{*}(x)=\sum\\{\lambda(d,x)g_{n}^{*}(d):d\text{ is a vertex of
$C_{x}$}\\},$ with $\lambda(d,x)=\prod_{i=1}^{n}\mu_{i}(d_{i},x_{i})$ and
$\mu_{i}(d_{i},x_{i})=n(x_{i}-\lfloor x_{i}\rfloor_{n})$ when $d_{i}=\lfloor
x_{i}\rfloor_{n}+1/n$ and $\mu_{i}(d_{i},x_{i})=1-n(x_{i}-\lfloor
x_{i}\rfloor_{n})$ when $d_{i}=\lfloor x_{i}\rfloor_{n}$. Then we have
$f_{n}^{*}(x)$ lies in $\operatorname{Conv}\\{g_{n}^{*}(d):d\text{ is a vertex
of the cube }C_{x}\\}$.

Finally put $f_{n}=f_{n}^{*}p_{n}$ where $p_{n}:\ell_{2}\to\mathbb{R}^{n}$ is
the orthogonal projection.

The proof that $f_{n}$ is Lipschitz continuous, and that $f_{n}$ converges to
$f$ on compacts, are given in [31]. ∎

Now for any $t>0$, we can apply the above Proposition to the continuous map
$G(t,x):H\to\mathcal{L}(H_{0},H)$ where $H$ is a separable Hilbert space and
$\mathcal{L}(H_{0},H)$ is a normed space. When the drift $B$ is also
continuous, we can apply this proposition to $B(t,x):H\to H$.

When $f$ has some more properties, $f^{*}_{n}$ will inherit the same property
when $n$ is large, and so does $f_{n}$. Say if we consider $f=G(t,x)$ for the
diffusion coefficient $G$, then as $G$ is uniformly non-degenerate,
$f_{n}^{*}$ is also uniformly non-degenerate for $n$ large, and so does
$f_{n}$. This is because $f_{n}^{*}(x)$ is a weighted sum of $g_{n}^{*}(d)$
for $d$ vertices of $C_{x}$, and when $n$ is large, by Hölder continuity of
$G$, the difference of the values of $G$ at different sites of $C_{x}$ becomes
negligible. Use also the fact that given an operator $A$ on $H$ with very
small operator norm, then $I+A$ has a right inverse on $H$. This proves the
claim.

Moreover, assume $G(t,x)$ is $\beta>\frac{3}{4}$-Hölder continuous in $x$,
then $f^{n}$ will also be $\beta$-Hölder with Hölder constant uniform in $n$,
at least for any $x,y\in H$ with $|x-y|>d_{n}$, Here $d_{n}$ is some constant
that tends to $0$ as $n\to\infty$. This is because, when $x$ and $y$ lie in
the same cube $C_{x}$, then $|f_{n}(x)-f_{n}(y)|$ is at most
$C(\frac{1}{n})^{\beta}$, while if they belong to different cubes $C_{x}$ and
$C_{y}$, the difference $|f_{n}(x)-f_{n}(y)|$ can be well approximated by the
difference of $g_{n}^{*}(d)$ when $d$ ranges over vertex of $C_{x}$ and
$C_{y}$ respectively, and the latter is bounded by $|x-y|^{\beta}$ up to some
constant. Thus whenever $|x-y|$ is much larger than $\frac{1}{n}$, which is
the typical size of the cube $C_{x}$, we can regard $f^{*}_{n}(x)$, and hence
$f_{n}(x)$, as $\beta$-Hölder functions with uniform in $n$ Hölder constants.

###  3.3. Proof of weak uniqueness

Now we prove the uniqueness of the weak mild solution under assumptions
$\mathbf{H}1$ through $\mathbf{H}5$. We also assume that444In Section 3.4 we
will show that Assumption $\mathbf{H}6$ is implied by Assumptions
$\mathbf{H}1$ to $\mathbf{H}5$.

$\mathbf{H}6$: for the given initial value $(u_{0},v_{0})\in\mathcal{H}_{1}$,
there exists a weak mild solution to (1.3). Moreover, for any $\epsilon>0$
there exists a compact subset $K\subset H$ such that with probability at least
$1-\epsilon$, $u(t)$ never leaves $K$ for $t\in[0,T]$. The proof in this
section is inspired by [27].

We first make several important observations:

  1. (1)

From the mild formulation (2.5) of the SPDE (1.3), and that
$\Pi_{1}Z(s)=u(s),$ it is clear that we only need to prove weak uniqueness for
the first component $u(s)$ of $Z(s)=(u(s),v(s))$. Once that is proved, weak
uniqueness for the whole process $Z(s)$ follows from the mild formulation
(2.5).

  2. (2)

By the non-degeneracy assumption $\mathbf{H}4$ of the diffusion coefficient
$G$, and the linear growth assumption of $B$ in $\mathbf{H}5$, we see that
once weak uniqueness is proved for the SPDE with $B=0$, weak uniqueness for
general $B$ follows from an application of Girsanov theorem. Therefore in the
following proof we assume $B=0$.

As shown in the previous section, we can find a sequence of operators
$G^{n}(t,x)$, which are Lipschitz in $x$, and moreover they are uniformly non-
degenerate and $\beta$-Hölder in $x$ with $n$-independent coefficients.

Now consider the following three SPDEs, with the same initial value
$(u_{0},v_{0})\in\mathcal{H}_{1}$:

$\mu\frac{\partial^{2}u(t)}{\partial t^{2}}=Au(t)-\zeta\frac{\partial
u(t)}{\partial t}+G(t,u(t))\frac{dW_{t}}{dt},$ (3.3)
$\mu\frac{\partial^{2}u^{n}(t)}{\partial t^{2}}=Au^{n}(t)-\zeta\frac{\partial
u^{n}(t)}{\partial t}+G^{n}(t,u^{n}(t))\frac{dW_{t}}{dt},$ (3.4)
$\mu\frac{\partial^{2}\widetilde{u}^{n}(t)}{\partial
t^{2}}=A\widetilde{u}^{n}(t)-\zeta\frac{\partial\widetilde{u}^{n}(t)}{\partial
t}+\lambda(u(t)-\widetilde{u}^{n}(t))dt1_{t\leq\tau}+G^{n}(t,\widetilde{u}^{n}(t))\frac{dW_{t}}{dt},$
(3.5)

where $\tau$ is some stopping time adapted to the filtration
$(\mathcal{F}_{t})_{t\geq 0}$ and $\lambda>0$ is a constant to be determined
later in the proof.

In this proof we aim at proving weak uniqueness of (3.3). By Assumption
$\mathbf{H}6$, we have constructed a weak mild solution to (3.3) on some
filtered probability space. Since $G^{n}$ is Lipschitz continuous, (3.4) and
(3.5) have strong uniqueness, so we can solve (3.4) and (3.5) on the same
probability space as (3.3) and assume they are driven by the same cylindrical
white noise $W$.

Given any compact subset $K\subset H$ containing the initial value $u_{0}$,
denote by

$\Delta_{K}^{n}:=\sup_{t\geq 0}\sup_{y\in
K}|G^{n}(t,y)-G(t,y)|_{\mathcal{L}(H_{0},H)}.$

Since $G^{n}(t,x)$ converges to $G(t,x)$ for each $x\in H$, that $K$ is
compact and $G(t,x)$ is uniformly continuous in $x$, we deduce that
$\lim_{n\to\infty}\Delta_{K}^{n}=0$ since all the estimates are time
independent. Now define two stopping times

$\tau_{K}^{n}:=\inf\\{t\geq 0:|u(t)-\widetilde{u}^{n}(t)|\geq
2\Delta_{K}^{n}\\},$

and

$\theta_{K}:=\inf\\{t\geq 0:u(t)\notin K\\}.$

In the reset of the proof we consider

$\tau=\tau_{K}^{n}\wedge\theta_{K},\quad\lambda=(\Delta_{K}^{n})^{\gamma-1},$

where $\gamma$ will be specified later. We may slightly enlarge
$\Delta_{k}^{n}$, but still assume $\lim_{n\to\infty}\Delta_{K}^{n}=0,$ such
that $\Delta_{K}^{n}$ is much larger than the length scale for $G^{n}$ to be
uniformly $\beta$-Hölder continuous in $x$, as discussed at the end of Section
3.2.

Now we can apply Girsanov theorem to estimate $u^{n}(t)-\widetilde{u}_{n}(t)$.
Denote by $d_{TV}$ the total variation distance on $\mathcal{C}([0,T];H)$ and
$H_{rel}(\cdot\mid\cdot)$ the relative entropy between two probability
measures on $\mathcal{C}([0,T];H)$. By Pinsker’s inequality,

$d_{TV}(\operatorname{Law}(u^{n}|_{[0,T]}),\operatorname{Law}(\widetilde{u}^{n}|_{[0,T]}))\leq\sqrt{2H_{rel}(\operatorname{Law}(u^{n}|_{[0,T]})\mid\operatorname{Law}(\widetilde{u}^{n}|_{[0,T]}))}.$

To compute the relative entropy, we use Girsanov transform, noting that the
SPDE solved by $\widetilde{u}^{n}$ (3.5) can be obtained by the SPDE solved by
$u^{n}$ (3.4) via adding a drift to the Brownian sample path

$(dW_{t})_{t\geq
o}\mapsto(dW_{t}+(G^{n})^{-1}\lambda(u(t)-\widetilde{u}^{n}(t))1_{t\leq\tau}dt)_{t\geq
0}.$ (3.6)

Since $(G^{n})^{-1}$ is a bounded operator, recalling the definition of the
stopping time $\tau$ and $\tau_{K}^{n}$, and the choice of $\lambda$, we
deduce that

$d_{TV}(\operatorname{Law}(u^{n}|_{[0,T]}),\operatorname{Law}(\widetilde{u}^{n}|_{[0,T]}))\leq
CT^{1/2}(\Delta_{K}^{n})^{\gamma}.$ (3.7)

Now we estimate $u(t)-\widetilde{u}^{n}(t)$ via pathwise arguments. For
$0\leq\tau\leq t$ we have the almost sure bound, whenever $\Delta_{K}^{n}\leq
1:$ in the following we use $|\cdot|_{\mathcal{L}(H_{0},H)}$ for the norm of
$G(t,x)$:

$\displaystyle|G(t,u(t))-G^{n}(t,\widetilde{u}^{n}(t))|$
$\displaystyle\leq|G(t,u(t))-G^{n}(t,u(t))|+|G^{n}(t,u(t))-G^{n}(t,\widetilde{u}^{n}(t))|$
(3.8) $\displaystyle\leq\Delta_{K}^{n}+M(2\Delta_{K}^{n})^{\beta}\leq
C(\Delta_{K}^{n})^{\beta},$

where in the second inequality, for the first term we use that before time
$\tau$, $u(t)$ stays in $K$ and $|G-G^{n}|\leq\Delta_{K}^{n}$ on $K$; and for
the second term we use the Hölder continuity of $G^{n}$ and that
$|u-\widetilde{u}^{n}|\leq\Delta_{K}^{n}$ before $\tau$. Here we also use that
$\Delta_{K}^{n}$ is much larger than the length scale for $G^{n}$ to be
$\beta$-Hölder with uniform in $n$ constants.

By the mild formulation (2.6) and the definition of
$\mathcal{S}_{\mu}^{\lambda}$,

$u(t)-\widetilde{u}^{n}(t)=\int_{0}^{t}\Pi_{1}\mathcal{S}_{\mu}^{\lambda}(t-s)\mathcal{I}_{\mu}[G(s,u(s))-G^{n}(s,\widetilde{u}^{n}(s))]dW_{s},\quad
0\leq\tau\leq t.$

Applying the maximal inequality (2.43), Markov’s inequality and estimate
(3.8), we deduce that for each $\eta\in(0,\eta_{0})$ and $m\in\mathbb{N}_{+}$
we can find some $C=C_{\eta,m,\mu}>0$ such that, for each $R>0$, [Recall the
choice $\lambda=(\Delta_{K}^{n})^{\gamma-1}$]

$\mathbb{P}(\sup_{t\in[0,T\wedge\tau]}|u(t)-\widetilde{u}^{n}(t)|\geq
C(\Delta_{K}^{n})^{\frac{1-\gamma}{2}\eta+\beta}R)\leq\frac{C}{R^{m}}.$ (3.9)

By our assumption $\beta>1-\frac{\eta_{0}}{2}$, we may choose $\gamma>0$
sufficiently small and $\eta<\eta_{0}$ sufficiently close to $\eta_{0}$ such
that $\frac{1-\gamma}{2}\eta+\beta>2\chi+1>1$ for some $\chi>0$. Now we take
$R=(\Delta_{K}^{n})^{-\chi}$ so that

$\mathbb{P}(\sup_{t\in[0,T\wedge\tau]}|u(t)-\widetilde{u}^{n}(t)|\geq
C(\Delta_{K}^{n})^{1+\chi})\leq C(\Delta_{K}^{n})^{m\chi}.$ (3.10)

Consider the event

$\Omega_{\nu}:=\\{\sup_{t\in[0,\tau]}|u(t)-\widetilde{u}^{n}(t)|>C(\Delta_{K}^{n})^{1+\chi}\\},$

then on $\Omega\setminus\Omega_{\nu}$, one must have
$\|u(t)-\widetilde{u}^{n}(t)|\leq\Delta_{K}^{n}$ for $t\in[0,\tau]$, then by
continuity of trajectories this must hold all the way up to
$T\wedge\theta_{K}$. In other words, on $\Omega\setminus\Omega_{\nu}$,

$\theta_{K}\wedge T\leq\tau_{K}^{n}.$

This implies, for any $\kappa>0$, as $n$ gets large $\Delta_{K}^{n}\to 0$, we
have

$\mathbb{P}(\sup_{t\in[0,T\wedge\theta_{K}]}|u(t)-\widetilde{u}^{n}(t)|>\kappa)\to
0,\quad n\to\infty.$ (3.11)

We can now finish the proof of uniqueness in law. Consider $F$ a real-valued,
bounded continuous function defined on $\mathcal{C}([0,T];H)$. Once we can
show that

$\mathbb{E}[F(u|_{[0,T]})-F(u^{n}|_{[0,T]})]\to 0,\quad n\to\infty,$ (3.12)

uniqueness in law for $u$ holds. This is because the law of each $u^{n}$ is
well-defined (note that $G^{n}$ is Lipschitz). If the law of $u|_{[0,T]}$ were
not unique, any weak solution of $u$ should satisfy all the above estimates,
so any weak solution $u$ should satisfy (3.12), but this forces
$\mathbb{E}[F(u|_{[0,T]})]$ to be the same for each weak solution $u$ given
any test function $F$, and hence the law of $u$ is uniquely determined.

To check (3.12), note first that (3.7) implies

$\mathbb{E}[F(\widetilde{u}^{n}|_{[0,T]})-F(u^{n}|_{[0,T]})]\to 0,\quad
n\to\infty,$ (3.13)

and (3.11) implies that on $\\{\theta_{K}\geq
T\\}\cap(\Omega\setminus\Omega_{\nu}),$ we have that
$\widetilde{u}^{n}|_{[0,T]}$ converges in probability to $u_{[0,T]}$ as
$n\to\infty$. Since $\lim_{n\to\infty}\mathbb{P}(\Omega_{\nu})=0,$ we conclude
that

$\lim\sup_{n\to\infty}\left|\mathbb{E}[F(u|_{[0,T]})-F(u^{n}|_{[0,T]})]\right|\leq
2\sup_{x}|F(x)|\mathbb{P}(\theta_{K}\leq T).$ (3.14)

By Assumption $\mathbf{H}6$, we can find compact sets $K\subset H$ such that
$\mathbb{P}(\theta_{K}\leq T)$ is arbitrarily small. As the left hand side of
(3.14) does not depend on $K$, we have established (3.12), and finally
finished the proof of weak uniqueness for $u(t)$.

###  3.4. Proof of weak existence: step 2

In this section we prove that Assumption $\mathbf{H}6$ holds assuming
Assumptions $\mathbf{H}1$ to $\mathbf{H}5$, thus establishing weak existence
of (1.3).

In this proof it will be convenient to work with the following Wasserstein
distance on $\mathcal{C}([0,T];H)$: define a distance $d$ on
$\mathcal{C}([0,T];H)$ as: for any
$\varphi_{1},\varphi_{2}\in\mathcal{C}([0,T];H),$

$d(\varphi_{1},\varphi_{2})=\sup_{0\leq t\leq
T}|\varphi_{1}(t)-\varphi_{2}(t)|_{H}\wedge 1,$

then for any probability measure $\mu$ and $\nu$ on $\mathcal{C}([0,T];H)$, we
define the Wasserstein distance of $\mu$ and $\nu$ as:

$W(\mu,\nu)=\inf_{(X,Y)}\mathbb{E}[d(X,Y)],$

where $(X,Y)$ ranges over all couplings of $\mu$ and $\nu$, with
$\operatorname{Law}(X)=\mu$ and $\operatorname{Law}(Y)=\nu$. Note that the
metric $d$ we defined is equivalent to the standard metric on
$\mathcal{C}([0,T];H)$.

Recall that we have constructed Lipschitz approximations $G^{n}$, and we set
out to compare the Wasserstein distance between

$\mu\frac{\partial^{2}u^{n}(t)}{\partial t^{2}}=Au^{n}(t)-\zeta\frac{\partial
u^{n}(t)}{\partial t}+G^{n}(t,u^{n}(t))\frac{dW_{t}}{dt},$ (3.15)

and

$\mu\frac{\partial^{2}u^{m}(t)}{\partial t^{2}}=Au^{m}(t)-\zeta\frac{\partial
u^{m}(t)}{\partial t}+G^{m}(t,u^{m}(t))\frac{dW_{t}}{dt},$ (3.16)

with the same initial data $(u_{0},v_{0})$.

First note that, for every $\epsilon>0$, we can find a compact set
$K_{\epsilon}\subset H$ such that for each $n$, with probability at least
$1-\epsilon$, $u^{n}(t)$ will stay in $K_{\epsilon}$ throughout $t\in[0,T]$.
This is guaranteed by Proposition 3.1, the compact embedding of $H^{\delta}$
into $H$, and Markov’s inequality.

Now we consider the auxiliary process

$\mu\frac{\partial^{2}\widetilde{u}^{m}(t)}{\partial
t^{2}}=A\widetilde{u}^{m}(t)-\zeta\frac{\partial\widetilde{u}^{m}(t)}{\partial
t}+\lambda(u^{n}(t)-\widetilde{u}^{m}(t))dt1_{t\leq\tau}+G^{m}(t,\widetilde{u}^{m}(t))\frac{dW_{t}}{dt},$
(3.17)

Assume that $m,n\geq N$ and consider, for any compact subset $K\subset H$:

$c_{N,K}:=\sup_{t\geq 0}\sup_{n\geq N}\sup_{x\in
K}|G^{n}(t,x)-G(t,x)|_{\mathcal{L}(H_{0},H)}.$

By definition $\lim_{N\to\infty}c_{N,K}=0.$ Assume $N$ is already chosen large
enough sot that $c_{N,K_{\epsilon}}\leq\frac{\epsilon}{4}$. Now we set
(abbreviating the subscript $K_{\epsilon}$ and simply write $K$):
$\lambda=c_{N,k}^{\gamma-1}$ and $\tau=T\wedge\\{t\geq
0:|u^{n}(t)-\widetilde{u}^{m}(t)|\geq c_{N,K}\\}$, where $\gamma$ takes the
same value as in Section 3.3.

Then a total variation estimate shows that

$d_{TV}(\operatorname{Law}(u^{m}|_{[0,T]},\operatorname{Law}(\widetilde{u}^{m}|_{[0,T]})))\leq
C(c_{N,K})^{\gamma}\leq\epsilon^{\gamma}.$

A pathwise argument as in the previous section shows, for some $\chi>0$,

$\mathbb{P}(\sup_{t\in[0,T]}|u^{n}(t)-\widetilde{u}^{m}(t)|\geq
c_{N,K})\leq(c_{N,K})^{\chi}\leq\epsilon^{\chi}.$ (3.18)

By definition of total variation distance, we can find a coupling of
$u^{m}|_{[0,T]}$ and $\widetilde{u}^{m}|_{[0,T]}$ such that they are identical
on an event of probability at least $1-\epsilon^{\gamma}$, and arbitrary
otherwise. From this we can deduce a coupling of $u^{n}|_{[0,T]}$ and
$u^{m}|_{[0,T]}$: we use the same Wiener process to drive $u^{n}$ and
$\widetilde{u}^{m}$, and then combine with the coupling of $u^{m}$ and
$\widetilde{u}^{m}$ to obtain a coupling of $u^{n}$ and $u^{m}$.

Note that the distance $d$ is upper bounded by $1$, thus we have, for some
$\omega>0$,

$W(u^{n}|_{[0,T]},u^{m}|_{[0,T]})\leq
c_{N,K}+\epsilon^{\chi}+\epsilon^{\gamma}+\epsilon\leq C\epsilon^{\omega}.$

We can restate our finding as follows: for any $\epsilon>0$, we can find $N>0$
such that whenever $n,m\geq N$, we have
$W(u^{n}|_{[0,T]},u^{m}|_{[0,T]})\leq\epsilon.$ This implies that
$u^{n}|_{[0,T]}$ forms a weakly converging subsequence. Applying Skorokhod
embedding theorem, the limit should solve the stochastic wave equation (1.3)
by continuity of the coefficient $G$. This establishes weak existence, and
hence finishes the proof of Theorem 1.2. That the solution will stay in a
compact subset with high probability again follows from Proposition 3.1.

###  3.5. Alternative proof of existence and uniqueness assuming uniform
approximation

In this section we give an alternative, shorter proof of weak existence and
uniqueness under the following stronger assumption

$\mathbf{H}7$: There exists a sequence of mappings
$G^{n}(t,x):[0,\infty)\times H\to\mathcal{L}(H_{0},H)$ that satisfy
assumptions $\mathbf{H}3,\mathbf{H}4,\mathbf{H}5$ with uniform in $n$
constants, that each $G^{n}$ is Lipschitz in $x$ with Lipschitz constant
uniform in $t$, and moreover we have the uniform approximation

$\lim_{n\to\infty}\sup_{t>0}\sup_{x\in
H}|G^{n}(t,x)-G(t,x)|_{\mathcal{L}(H_{0},H)}=0.$ (3.19)

Though assumption $\mathbf{H}7$ looks much stronger, it indeed covers many
interesting examples of random field solutions to the stochastic wave
equation. Say we consider $H=L^{2}([0,1];\mathbb{R})$ and
$G(t,u)(x)=g(t,u(t,x))$ for any $u\in H$. Assume that for some
$\beta\in(\frac{3}{4},1]$ we have

$g(t,u(t,x))=1+|u(t,x)|^{\beta},$ (3.20)

so the only place the noise coefficient $g$ is non-Lipschitz is at $u=0$. We
can find a sequence of $g_{n}$ approximating $g$, each $g_{n}$ being Lipschitz
continuous in $u$, and

$\sup_{t\geq 0}\sup_{u\in\mathbb{R}}|g_{n}(t,u)-g(t,u)|\to 0,\quad
n\to\infty.$ (3.21)

Defining $G^{n}$ accordingly via $G^{n}(t,u)v=g_{n}(t,u(t,x))v(x),x\in[0,1]$
for any $v\in L^{\infty}([0,1])$, so that the approximating sequence $G^{n}$
satisfies $\mathbf{H}7$. More generally, one could consider diffusion
coefficients $g$ that are Lipschitz continuous away from finitely many
irregular points, or consider those coefficients $g$ such that we can find a
sequence $g_{n}$ uniformly approximating $g$ in the sense of (3.21).

We follow roughly the same steps as in the previous section: we may assume
$B=0$ and consider the following SPDEs

$\mu\frac{\partial^{2}u(t)}{\partial t^{2}}=Au(t)-\zeta\frac{\partial
u(t)}{\partial t}+G(t,u(t))\frac{dW_{t}}{dt},$ (3.22)
$\mu\frac{\partial^{2}u^{n}(t)}{\partial t^{2}}=Au^{n}(t)-\zeta\frac{\partial
u^{n}(t)}{\partial t}+G^{n}(t,u^{n}(t))\frac{dW_{t}}{dt},$ (3.23)
$\mu\frac{\partial^{2}\widetilde{u}^{n}(t)}{\partial
t^{2}}=A\widetilde{u}^{n}(t)-\zeta\frac{\partial\widetilde{u}^{n}(t)}{\partial
t}+\lambda(u(t)-\widetilde{u}^{n}(t))dt1_{t\leq\tau}+G^{n}(t,\widetilde{u}^{n}(t))\frac{dW_{t}}{dt},$
(3.24)

Under assumption $\mathbf{H}7$ we no longer need to consider the compact
subset $K$, and we can use the (globally uniformly) approximating sequence
$G^{n}$ given in $\mathbf{H}7$ .

Denote by

$c_{n}:=\sup_{t\geq 0}\sup_{x\in H}|G^{n}(t,x)-G(t,x)|_{\mathcal{L}(H_{0},H)}$

and consider $\tau:=T\wedge\inf\\{t\geq 0:|u(t)-\widetilde{u}^{n}(t)|\geq
2c_{n}\\}$. Then we set $\lambda=(c_{n})^{\gamma-1}$ and find some
$\eta<\eta_{0}$ close to $\eta_{0}$ such that
$\frac{1-\gamma}{2}\eta+\beta>2\chi+1>1$ for some $\chi>0$. Then with exactly
the same proof as in the previous section, we have

$d_{TV}(P_{n}u^{n}|_{[0,T]},P_{n}\widetilde{u}^{n}|_{[0,T]})\leq
CT^{1/2}(c_{n})^{\gamma}.$ (3.25)

and

$\mathbb{P}(\sup_{t\in[0,T\wedge\tau]}|u(t)-\widetilde{u}^{n}(t)|\geq
C(c_{n})^{1+\chi})\leq C(c_{n})^{m\chi}.$ (3.26)

so that for any $\kappa>0$,

$\mathbb{P}(\sup_{t\in[0,T]}|u(t)-\widetilde{u}^{n}(t)|>\kappa)\to 0,\quad
n\to\infty.$ (3.27)

Combining these arguments, we have established uniqueness of weak mild
solution to (1.3) in exactly the same way as in the previous section.

For weak existence, following (a simplified version of) the argument in
Section 3.4, we deduce that $u^{n}|_{[0,T]}$ forms a Cauchy sequence with
respect to the Wasserstein distance, hence has a converging subsequence and
the limit is a weak mild solution to (1.3).

##  4\. The Smoluchowski-Kramers approximation

###  4.1. Mass-independent estimates of stochastic integrals

In this section we only focus on the $\zeta=1$ case, i.e. the stochastic wave
equation with damping term. In proving the small-mass limit of the damped
stochastic wave equation, the maximal inequality estimate in Proposition
(2.10) is not enough because the numerical constant will diverge in the limit
$\mu\to\infty$. We derive some new estimates:

######  Proposition 4.1.

Assume that there is some $\eta_{0}\in(0,1)$ such that for some constant
$C>0$,

$\sum_{n}\frac{1}{\alpha_{n}^{1-\eta}}<\infty,\quad\frac{1}{C}k^{\frac{1}{1-\eta_{0}}}\leq\alpha_{k}\leq
Ck^{\frac{1}{1-\eta_{0}}}\quad\forall k\in\mathbb{N}_{+}$ (4.1)

for each $\eta\in(0,\eta_{0})$. Then for each fixed $\lambda>0$ we can find
some $\mu_{0}=\mu_{0}(\lambda)>0$ as a function of $\lambda$ such that for any
$\mu\in(0,\mu_{0}(\lambda)]$, for each $p\geq 2$ and each
$\eta\in(0,\eta_{0}),$

$\mathbb{E}[|\Pi_{1}\Gamma_{\alpha}^{\mu,\lambda}(t)|_{H}^{p}]\leq
C_{\eta,p}\mathbb{E}\|\Phi\|_{t}^{p}(\lambda^{-\frac{\eta
p}{2}}+\mu^{(\eta_{0}-2\alpha)p/2}).$ (4.2)

The constant $c_{\eta,p}$ depends on $\eta$ and $p$ but does not depend on
$\lambda$ and $\mu$. By choosing $\mu_{0}(\lambda)$ smaller, we may discard
the $\mu^{(\eta_{0}-2\alpha)p/2}$ factor and the estimate does not depend on
$\mu$.

###### Proof.

Recall from the proof of Proposition 2.10 that

$\displaystyle\Lambda_{\alpha}^{\mu,\lambda}(t):$
$\displaystyle=\sum_{j=1}^{\infty}\int_{0}^{t}(t-s)^{-2\alpha}|\Pi_{1}\mathcal{S}_{\mu}^{\lambda}(t-s)\mathcal{I}_{\mu}\Phi(s)e_{j}|_{H}^{2}ds$
$\displaystyle\leq\int_{0}^{t}(t-s)^{-2\alpha}\|\Phi\|_{s}^{2}\left(\sum_{k=1}^{\infty}f_{\mu}^{\lambda}(k)^{2}(t-s,0,1/\mu)\right)ds.$

and

$\sum_{k=1}^{\infty}f_{\mu}^{\lambda}(k)^{2}(t,0,1/\,u)\leq\sum_{k=1}^{N_{\mu}^{\lambda}}e^{-2(\alpha_{k}+\lambda)t}+\sum_{k=N_{\mu}^{\lambda}+1}^{\infty}\frac{e^{-\frac{t}{2\mu}}}{\mu(\alpha_{k}+\lambda)}:=J_{1}(t)+J_{2}(t).$

and that

$N_{\mu}^{\lambda}:=\max\\{k\in\mathbb{N}:1-4\mu(\alpha_{k}+\lambda)\geq
0\\},$

we assume that for the fixed $\lambda>0$, $\mu>0$ is chosen sufficiently small
such that $\mu\lambda\leq\frac{1}{40}$, that is, we set
$\mu_{0}(\lambda)=\frac{1}{40\lambda}$. Under this choice for
$k>N_{\mu}^{\lambda}$ one must have $\mu\alpha_{k}\geq\frac{9}{40}$ and hence
$\alpha_{k}\geq 9\lambda$ for all $k>N_{\mu}^{\lambda}$. Thus we have

$J_{2}(t)\leq
e^{-\frac{t}{2\mu}}\frac{9}{10}\sum_{k=N_{\mu}^{\lambda}+1}^{\infty}\frac{1}{\mu\alpha_{k}}.$

To bound this summation, we need to make use of the asymptotic growth of
$\alpha_{k}$. By the second assumption in (4.1), we have, for some universal
constant $C$ and constant $C_{\eta_{0}}$ depending only on $\eta_{0},$

$\displaystyle\sum_{k=N_{\mu}^{\lambda}+1}^{\infty}\frac{1}{\mu\alpha_{k}}$
$\displaystyle\leq C\sum_{k=C^{-1}\mu^{-(1-\eta_{0})}}^{\infty}\frac{1}{\mu
k^{\frac{1}{1-\eta_{0}}}}$ (4.3) $\displaystyle\leq
C_{\eta_{0}}\mu^{-1}\int_{C^{-1}\mu^{-(1-\eta_{0})}}^{\infty}x^{-\frac{1}{1-\eta_{0}}}dx\leq
C_{\eta_{0}}\mu^{\eta_{0}-1}.$

Note also the elementary calculation

$\int_{0}^{\infty}s^{-2\alpha}e^{-\frac{s}{2\mu}}ds=(2\mu)^{1-2\alpha}\int_{0}^{\infty}t^{-2\alpha}e^{-t}dt=C\mu^{1-2\alpha},$
(4.4)

so that

$\int_{0}^{t}(t-s)^{-2\alpha}J_{2}(t-s)ds\leq\int_{0}^{t}s^{-2\alpha}e^{-\frac{s}{2\mu}}\mu^{\eta_{0}-1}ds\leq\mu^{\eta_{0}-2\alpha}.$

For the term involving $J_{1}(t)$, we follow exactly the proof of Proposition
2.10. Combining everything, we now have

$\Lambda_{\alpha}^{\mu,\lambda}(t)\leq
C\|\Phi\|_{t}^{2}(\lambda^{-\eta}+\mu^{\eta_{0}-2\alpha})$

and then the Proposition follows from application of the BDG inequality. ∎

Before we proceed, we quote the following result on the $\mu\to 0$
convergence.

######  Lemma 4.2.

Let $f_{\mu}^{\lambda}(k)(t,u,v)$ be defined in (2.8) with $\lambda>0$ fixed
in the sequel. Then

  * •

For each $k\in\mathbb{N}$, $T>0$ and $u\in\mathbb{R}$,

$\lim_{\mu\to
0}\sup_{t\in[0,T]}|f_{\mu}^{\lambda}(k)(t,u,o)-ue^{-(\alpha_{k}+\lambda)t}|=0.$
(4.5)

  * •

For each $k\in\mathbb{N},$ $T>0$, $t_{0}\in(0,T]$ and $v\in\mathbb{R}$,

$\lim_{\mu\to
0}\sup_{t\in[t_{0},T]}|f_{\mu}^{\lambda}(k)(t,0,v/\mu)-ve^{-(\alpha_{k}+\lambda)t}|=0.$
(4.6)

  * •

For each $k\in\mathbb{N}$, $t>0$ and $v\in\mathbb{R}$,

$\lim_{\mu\to 0}f_{\mu}^{\lambda}(k)^{\prime}(t,0,v)=0.$ (4.7)

###### Proof.

Let $f_{\mu}^{\lambda}(k)(t):=f_{\mu}^{\lambda}(k)(t,u,v)$, then it solves the
ODE
$\mu(f_{\mu}^{\lambda}(k))^{\prime\prime}+(f_{\mu}^{\lambda}(k))^{\prime}+(\alpha_{k}+\lambda)f_{\mu}^{\lambda}=0$.
The proof for the case $\lambda=0$ can be found in [32],Theorem 5.8, and the
general case for $\lambda>0$ is exactly the same, so we omit the details. ∎

Then we have to rework through the proof of Proposition 2.11 to make sure the
constants are $\mu$-independent for all $\mu\in(0,\mu_{0}(\lambda)]$. Recall
the definition $N_{\mu}^{\lambda}=\max\\{k:1-4\mu(\alpha_{k}+\lambda)\geq
0\\}$, and consider $P_{N_{\mu}^{\lambda}}$ the projection onto the subspace
spanned by $\\{e_{1},\cdots,e_{N_{\mu}^{\lambda}}\\}$.

######  Proposition 4.3.

Fix $T>0$. For each $p\geq 2$ and $\eta\in(0,\eta_{0}),$ we may choose
$\alpha>0$ sufficiently small such that, for the damped stochastic wave
equation $\zeta=1,$

  1. (1)

for $\mu\in(0,1)$ and $t\in[0,T],$

$\mathbb{E}|P_{N_{\mu}^{\lambda}}\Pi_{2}\Gamma_{\alpha}^{\mu,\lambda}(t)|_{H}^{p}\leq\frac{c_{\eta,p}}{\mu^{p}}\lambda^{-\frac{\eta
p}{2}}\mathbb{E}\|\Phi\|_{t}^{p},$ (4.8)

  2. (2)

For any fixed $t\in[0,T]$ and any fixed $\lambda>0$,

$\lim_{\mu\to
0}\mu^{p}\mathbb{E}|P_{N_{\mu}^{\lambda}}\Pi_{2}\Gamma_{\alpha}^{\mu,\lambda}(t)|_{H}^{p}=0.$
(4.9)

  3. (3)

for any fixed $\lambda>0$, we may find a $\mu_{0}(\lambda)>0$ and a constant
$\bar{\zeta}>0$ such that for all $t\in[0,T]$ and all
$\mu\in(0,\mu_{0}(\lambda)]$ we have,

$\mathbb{E}|(I-P_{N_{\mu}^{\lambda}})\Pi_{2}\Gamma_{\alpha}^{\mu,\lambda}(t)|_{H^{-1}(\lambda)}^{p}\leq\frac{c_{\eta,p}}{\mu^{p(1-\bar{\zeta})/2}}\mathbb{E}\|\Phi\|_{t}^{p}.$
(4.10)

In all the above claims, the constant $C_{\eta,p}$ does not depend on the
choice of $\lambda>0$ and the choice of $\mu\in(0,\mu_{0}(\lambda)]$.

###### Proof.

Denote by $\Lambda_{1}(t)$ the quadratic variation of
$P_{N_{\mu}^{\lambda}}\Pi_{2}\Gamma_{\alpha}^{\mu,\lambda}$. We have

$\displaystyle\Lambda_{1}(t)$
$\displaystyle=\sum_{j=1}^{\infty}\int_{0}^{t}(t-s)^{-2\alpha}|P_{N_{\mu}^{\lambda}}\Pi_{2}\mathcal{S}_{\mu}^{\lambda}(t-s)\mathcal{I}_{\mu}\Phi(s)e_{j}|_{H}^{2}ds$
$\displaystyle=\sum_{k=1}^{N_{\mu}}\sum_{j=1}^{\infty}\int_{0}^{t}(t-s)^{-2\alpha}\langle\Phi(s)e_{j},\mathcal{I}_{\mu}^{*}\mathcal{S}_{\mu}^{\lambda*}(t-s)\Pi_{2}^{*}e_{k}\rangle_{H}^{2}ds$

Since
$\mathcal{I}_{\mu}^{*}\mathcal{S}_{\mu}^{\lambda*}(t-s)\Pi_{2}^{*}e_{k}=f_{\mu}^{\lambda}(k)^{\prime}(t-s,0,1/\mu)e_{k}$,
we can bound

$\Lambda_{1}(t)\leq\sum_{k=1}^{N_{\mu}^{\lambda}}\sum_{j=1}^{\infty}\int_{0}^{t}(t-s)^{-2\alpha}|(f_{\mu}^{\lambda}(k)^{\prime}(t-s,0,1/\mu)|^{2}\langle\Phi(s)e_{j},e_{k}\rangle_{H}^{2}ds.$
(4.11)

Using the estimate

$|f_{\mu}^{\lambda}(k)^{\prime}(t,0,1/\mu)|\leq\frac{2e^{-(\alpha_{k}+\lambda)t}}{\mu},$

we deduce that, following (2.38) and the treatment of $J_{1}(t)$ in the proof
of Proposition 2.10,

$\displaystyle\Lambda_{1}(t)$
$\displaystyle\leq\frac{C}{\mu^{2}}\sum_{k=1}^{N_{\mu}^{\lambda}}\int_{0}^{t}(t-s)^{-2\alpha}e^{-2(\alpha_{k}+\lambda)(t-s)}\|\Phi\|_{s}^{2}ds$
$\displaystyle\leq\frac{C}{\mu^{2}}\lambda^{-\eta}\|\Phi\|_{t}^{2}.$

The proof of (4.8) follows from application of BDG inequality:

$\mathbb{E}|P_{N_{\mu}^{\lambda}}\Pi_{1}\Gamma_{\alpha}^{\mu,\lambda}(t)|_{H}^{p}\leq\mathbb{E}(\Lambda_{1}(t))^{p/2}.$

For the second claim, since $\mu
f_{\mu}^{\lambda}(k)^{\prime}(t,0,1/\mu)=f_{\mu}^{\lambda}(k)^{\prime}(t,0,1)$,
using the estimate (4.7), we have for each $s>0$ and $k\leq
N_{\mu}^{\lambda},$ $j\in\mathbb{N}$,

$\lim_{\mu\to
0}(t-s)^{-2\alpha}\mu^{2}|f_{\mu}^{\lambda}(k)^{\prime}(t,0,1/\mu)|^{2}\langle\Phi(s)e_{j},e_{k}\rangle_{H}^{2}=0$

and also the bound thanks to (2.11): as $k\leq N_{\mu}^{\lambda}$,

$\mu^{2}(f_{\mu}^{\lambda}(k)^{\prime}(t,0,1/\mu))^{2}\langle\Phi(s)e_{j},e_{k}\rangle_{H}^{2}\leq
Ce^{-2(\alpha_{k}+\lambda)t}\langle\Phi(s)e_{j},e_{k}\rangle_{H}^{2},$

then by the dominated convergence theorem, $\Lambda_{1}(t)\to 0$ with
probability one as $\mu\to 0$, and then (4.9) follows from BDG inequality.

For the third claim,

$\displaystyle\Lambda_{2}(t)=$
$\displaystyle\sum_{j=1}^{\infty}\int_{0}^{t}(t-s)^{-2\alpha}|(I-P_{N_{\mu}^{\lambda}})\Pi_{2}\mathcal{S}_{\mu}^{\lambda}(t-s)\mathcal{I}_{\mu}\Phi(s)e_{j}|_{H^{-1}(\lambda)}^{2}ds$
$\displaystyle=\sum_{j=1}^{\infty}\int_{0}^{t}(t-s)^{-2\alpha}|(-A-\lambda)^{-1/2}(I-P_{N_{\mu}^{\lambda}})\Pi_{2}\mathcal{S}_{\mu}^{\lambda}(t-s)\mathcal{I}_{\mu}\Phi(s)e_{j}|_{H}^{2}ds$
$\displaystyle\leq\sum_{k=N_{\mu}^{\lambda}+1}^{\infty}\sum_{j=1}^{\infty}\int_{0}^{t}(t-s)^{-2\alpha}\langle\Phi(s)e_{j},\mathcal{I}_{\mu}^{*}\mathcal{S}_{\mu}^{\lambda*}(t-s)\Pi_{2}^{*}(I-P_{N_{\mu}^{\lambda}})^{*}(-A-\lambda)^{-1/2}e_{k}\rangle_{H}^{2}ds$

Since for $k,j\in\mathbb{N}$,

$\displaystyle\langle\mathcal{I}_{\mu}^{*}\mathcal{S}_{\mu}^{\lambda*}(t-s)\Pi_{2}^{*}(I-P_{N_{\mu}^{\lambda}})^{*}(-A-\lambda)^{-1/2}e_{k},e_{j}\rangle_{H}$
$\displaystyle=\begin{cases}(\alpha_{k}+\lambda)^{-1/2}f_{\mu}^{\lambda}(k)^{\prime}(t-s,0,1/\mu)\quad\text{if
}k=j>N_{\mu}^{\lambda},\\\ 0\quad\text{otherwise}.\end{cases}$

By estimate (2.13),

$(\alpha_{k}+\lambda)^{-1/2}|f_{\mu}^{\lambda}(k)^{\prime}(t-s,0,1/\mu)|\leq
C(\alpha_{k}+\lambda)^{-1/2}\mu^{-1}e^{-\frac{t-s}{4\mu}},$

we have, for all $\mu\in(0,\mu_{0}(\lambda)]$ with $\mu_{0}(\lambda)$ as
defined in Proposition 4.1,

$\sum_{k=N_{\mu}^{\lambda}+1}^{\infty}(\alpha_{k}+\lambda)^{-1}|(f_{\mu}^{\lambda}(k)^{\prime}(t-s,0,1/\mu))|^{2}\leq\sum_{k=N_{\mu}^{\lambda}+1}^{\infty}\frac{Ce^{-\frac{t-s}{2\mu}}}{\mu^{2}(\alpha_{k}+\lambda)}\leq
Ce^{-\frac{t-s}{2\mu}}\mu^{\eta_{0}-2},$

where the last inequality is derived in exactly the same way as in (4.3). Thus
we have the bound for $\Lambda_{2}(t)$:

$\Lambda_{2}(t)\leq\frac{C}{\mu^{2-\eta_{0}}}\int_{0}^{t}(t-s)^{-2\alpha}e^{-\frac{t-s}{2\mu}}\|\Phi\|_{s}^{2}ds.$

Using (4.4), we have

$\Lambda_{2}(t)\leq\frac{C}{\mu^{2\alpha+1-\eta_{0}}}\|\Phi\|_{t}^{2}.$

Now it suffices to take $\alpha>0$ small enough so that
$2\alpha+1-\eta_{0}<1$. Then we can find $\bar{\zeta}>0$ such that

$\Lambda_{2}(t)\leq\frac{C}{\mu^{1-\bar{\zeta}}}\|\Phi\|_{t}^{2}.$

The last claim then follows from BDG inequality. ∎

Now we adapt the proof of Theorem 2.12 to get $\mu$-independent estimates for
small $\mu$.

######  Theorem 4.4.

Fix $T>0$. Recall $\Gamma^{\mu,\lambda}$ defined in (2.29). Then for any
$p\geq 2$, any fixed $\lambda>0$ and any $\eta\in(0,\eta_{0})$, we may find
some $c_{\eta,p}>0$, some $\bar{\zeta}>0$ and some $\mu_{0}(\lambda)>0$ such
that for all $\mu\in(0,\mu_{0}(\lambda)],$ we have the estimate

$\mathbb{E}\sup_{t\in[0,T]}|\Pi_{1}\Gamma^{\mu,\lambda}(t)|_{H}^{p}\leq
c_{\eta,p}(\lambda^{-\frac{\eta
p}{2}}+\mu^{\bar{\zeta}p/2})\mathbb{E}\|\Phi\|_{T}^{p}.$ (4.12)

The constant $c_{\eta,p}$ is independent of the choice of $\lambda>0$ and
$\mu\in(0,\mu_{0}(\lambda)]$. We again note that for fixed $\lambda>0$, as
long as $\mu_{0}(\lambda)$ is chosen small enough, the term
$\mu^{\bar{\xi}p/2}$ can be discarded and the right hand side is indeed
$\mu$-independent.

We also recall the following $\lambda=0$ version, which will be used in the
subsequent proof:

$\mathbb{E}\sup_{t\in[0,T]}|\Pi_{1}\Gamma^{\mu}(t)|_{H}^{p}\leq
C\mathbb{E}\int_{0}^{T}\sup_{s\in[0,t]}\|\Phi(s)\|_{\mathcal{L}(H_{0},H)}^{p}dt,$
(4.13)

where $C$ is $\mu$-independent.

###### Proof.

Proof of claim (4.13) can be found in [32], Theorem 6.3. We will focus on the
$\lambda>0$ case to get decay in $\lambda$:

We apply the stochastic factorization as in (2.44) and treat each term
separately. For the first term

$\mathbb{E}\sup_{t\in[0,T]}\left|\int_{0}^{t}(t-s)^{\alpha-1}\Pi_{1}\mathcal{S}_{\mu}^{\lambda}(t-s)\begin{pmatrix}I\\\
0\end{pmatrix}\Pi_{1}\Gamma_{\alpha}^{\mu,\lambda}(s)ds\right|_{H}^{p}$

we use Lemma 2.5 and Proposition 4.1 instead of Proposition 2.10 so that the
numerical constant $c_{\eta,p}$ is $\mu$-independent. The difference compared
with (2.45) is that we have $\lambda^{-\frac{\eta
p}{2}}+\mu^{(\eta_{0}-2\alpha)p/2}$ instead of $\lambda^{-\frac{\eta p}{2}}$
on the right hand side of the estimate.

For the second term

$\mathbb{E}\sup_{t\in[0,T]}\left|\int_{0}^{t}(t-s)^{\alpha-1}\Pi_{1}\mathcal{S}_{\mu}^{\lambda}(t-s)\begin{pmatrix}0\\\
P_{N_{\mu}^{\lambda}}\end{pmatrix}P_{N_{\mu}^{\lambda}}\Pi_{2}\Gamma_{\alpha}^{\mu,\lambda}(s)ds\right|_{H}^{p},$

we use estimate (2.22) and estimate (4.8). We have $\lambda^{-\frac{\eta
p}{2}}$ on the right hand side and the numerical constant $c_{\eta,p}$ is
$\mu$-independent.

For the third term

$\mathbb{E}\sup_{t\in[0,T]}\left|\int_{0}^{t}(t-s)^{\alpha-1}\Pi_{1}\mathcal{S}_{\mu}^{\lambda}(t-s)\begin{pmatrix}0\\\
I-P_{N_{\mu}^{\lambda}}\end{pmatrix}(I-P_{N_{\mu}^{\lambda}})\Pi_{2}\Gamma_{\alpha}^{\mu,\lambda}(s)ds\right|_{H}^{p},$

we use estimate (2.23) and estimate (4.10). We have $\mu^{\bar{\zeta}p/2}$ on
the right hand side of the estimate and the numerical constant $c_{\eta,p}$ is
$\mu$-independent.

Summing up the three estimates, the proof of the theorem is finished. ∎

###  4.2. Small mass limit of stochastic integrals

We will need two more convergence results before we start the proof of small
mass limit.

For the first result, recall the definition (for the stochastic heat equation)

$\Gamma(t)=\int_{0}^{t}\mathcal{S}(t-s)\Phi(s)dW_{s}$ (4.14)

where $\mathcal{S}$ is the analytic semigroup generated by $A$. Recall also
the definition (2.30)

$\Gamma^{\mu}(t)=\int_{0}^{t}\mathcal{S}_{\mu}(t-s)\Phi(s)dW_{s}.$ (4.15)

We need the following result on the convergence of $\Pi_{1}\Gamma^{\mu}$ to
$\Gamma$:

######  Theorem 4.5.

Given $T>0$ and $\alpha>0$, for any self-adjoint, progressively measurable

$\Phi\in L^{p}(\Omega;L^{\infty}([0,T];\mathcal{L}(H_{0},H))),$

any fixed $\lambda>0$ and any $p\geq 2$, let $\Gamma^{\mu}$ and $\Gamma$ be
defined as before. Then

$\lim_{\mu\to
0}\mathbb{E}|\Pi_{1}\Gamma^{\mu}-\Gamma|^{p}_{\mathcal{C}([0,T];H)}=0.$ (4.16)

This theorem is proved in [32], Theorem 8.1. There was a
$p\geq\frac{1}{\alpha}$ assumption in that result, but we can apply Cauchy-
Schwartz to (4.16) to reduce the value of $p$.

For Lebesgue integrals, we have

######  Theorem 4.6.

For any $T>0$ and $\varphi\in L^{\infty}([0,T];H),$

$\lim_{\mu\to
0}\sup_{t\in[0,T]}\left|\int_{0}^{t}\left(\mathcal{S}(t-s)-\Pi_{1}\mathcal{S}_{\mu}(t-s)\mathcal{I}_{\mu}\right)\varphi(s)ds\right|_{H}=0.$
(4.17)

The proof can be found in [32], Theorem 8.2.

We also need the following more quantitative estimate in the exponent of
$\lambda$ as $\lambda\to+\infty$.

######  Theorem 4.7.

For any $T>0,$ $\lambda>0$ fixed there exists some $c_{1}(\lambda)>0$ such
that for any $\mu\in(0,c_{1}(\lambda)],$ we have

$\sup_{t\in[0,T]}\left|\int_{0}^{t}\Pi_{1}\mathcal{S}_{\mu}^{\lambda}(t-s)\mathcal{I}_{\mu}\varphi(s)ds\right|_{H}\leq
C\lambda^{-1}\sup_{t\in[0,T]}|\varphi(t)|_{H},$ (4.18)

where $C$ is some universal constant.

The same estimate is true if we replace
$\Pi_{1}\mathcal{S}_{\mu}^{\lambda}(t-s)\mathcal{I}_{\mu}$ by
$\mathcal{S}^{\lambda}$. Proof in the case of $\mathcal{S}^{\lambda}$ is very
easy: note that $\|\mathcal{S}^{\lambda}(t-s)\|_{H}\leq e^{-\lambda(t-s)}$ and
integrate over $[0,T]$.

###### Proof.

Recall from the proof of Lemma 2.4 that

$\|\Pi_{1}\mathcal{S}_{\mu}^{\lambda}(t)\mathcal{I}_{\mu}\|_{\mathcal{L}(H)}\leq\sup_{k\in\mathbb{N}}|f_{\mu}^{\lambda}(k)(t,0,1/\mu)|.$

For $1\leq k\leq N_{\mu}^{\lambda}$, we have from (2.10) that

$|f_{\mu}^{\lambda}(k)(t,0,1/\mu)|\leq 4e^{-(\alpha_{k}+\lambda)t}.$

We choose $c_{1}(\lambda)=\frac{1}{5\lambda^{2}}$, so that for any
$\mu\in(0,c_{1}(\lambda)]$ and $k\geq N_{\mu}^{\lambda}$, one must have
$\alpha_{k}\geq\lambda^{2}$.

Then for such $k$, by (2.12) we have

$|f_{\mu}^{\lambda}(k)(t,0,1/\mu)|\leq\frac{2}{\sqrt{\alpha_{k}+\lambda}}\leq\frac{2}{\lambda}.$

Then the claim of the theorem follows from noting that

$\int_{0}^{T}4e^{-(\alpha_{k}+\lambda)t}dt\leq
C\lambda^{-1},\quad\int_{0}^{T}\frac{2}{\lambda}dt\leq C\lambda^{-1},$

and the fact that $\Pi_{1}\mathcal{S}_{\mu}^{\lambda}(t)\mathcal{I}_{\mu}$
preserves the one dimensional subspace spanned by each $e_{k}$. ∎

We also need the following result on convergence of initial values as $\mu\to
0$.

######  Proposition 4.8.

Fix any initial value $(u_{0},v_{0})\in H\times H^{-1}$, and any $T>0$. Then
we have

$\lim_{\mu\to
0}\sup_{t\in[0,T]}|\mathcal{S}(t)u_{0}-\Pi_{1}\mathcal{S}_{\mu}(t)(u_{0},v_{0})|_{H}^{2}=0.$
(4.19)

This result is standard and only involves computations of ODEs. A detailed
proof of this result can be found in [32], Section 8.1, Proof of Theorem 4.2.
Our assumptions on eigenvalues of $A$ are slightly more general but the proof
is the same.

###  4.3. Proof of Smoluchowski-Kramers approximation

In this section we impose the slightly stronger semigroup assumption
$\mathbf{H}1^{\prime}$ and the $\alpha$\- Hölder regularity assumption
$\mathbf{H}2^{\prime}$.

For simplicity, we first assume the following uniform approximation assumption
$\mathbf{H}7^{\prime}$:

$\mathbf{H}7^{\prime}$. There exists a sequence of maps
$B^{n}(t,x):[0,\infty)\times H\to H$ and a sequence of maps
$G^{n}(t,x):[0,\infty)\times H\to\mathcal{L}(H_{0},H)$ such that they satisfy
assumptions $\mathbf{H}2^{\prime}$, $\mathbf{H}3,$ $\mathbf{H}4$ and
$\mathbf{H}5$ with uniform in $n$ constants, and each $G^{n}$, $B^{n}$ are
Lipschitz in $x$ with time-independent Lipschitz constants. Moreover, in the
$n\to\infty$ limit,

$g_{n}:=\sup_{t\geq 0}\sup_{x\in
H}|G^{n}(t,x)-G(t,x)|_{\mathcal{L}(H_{0},H)}\to 0,\quad b_{n}:=\sup_{t\geq
0}\sup_{x\in H}|B^{n}(t,x)-B(t,x)|_{H}\to 0.$ (4.20)

As discussed in Section 3.5, Assumption $\mathbf{H}7^{\prime}$ is already
sufficiently general to cover many random field stochastic wave equations with
irregular coefficients. Say if the drift coefficient $B(t,u)=b(t,u(t,x))$ for
any $u\in L^{2}([0,1];\mathbb{R})$, and we can find a sequence of Lipschitz
maps $b_{n}(t,x)$ converging uniformly to $b(t,x)$ on $\mathbb{R}$, then we
have constructed an approximating sequence $B^{n}$ of $B$.

The proof in the general case, i.e. without the assumption
$\mathbf{H}7^{\prime}$, only needs the following slight modifications: By
(4.13), we obtain $\mu$-independent $L^{p}$ norm estimates for the process
$u_{\mu}(t)$ on $\mathcal{C}([0,T];H)$ for any sufficiently small $\mu$. Now
we can apply Proposition 3.1, combined with the compact embedding of
$H^{\delta}$ into $H$ and Markov’s inequality (we should formulate a version
of Proposition 3.1 for the Lebesgue integration of the drift $B(t,u(t))$,
which is easy to work out, and also note that
$(u_{0},v_{0})\in\mathcal{H}_{1}$) , we deduce that, for any $\epsilon>0$, we
can find a compact set $K_{\epsilon}\subset H$ such that with probability at
least $1-\epsilon$ uniform in $\mu>0$, $u_{\mu}(t)\in K_{\epsilon}$ for all
$t\in[0,T]$; and with probability at least $1-\epsilon$, $u(t)\in
K_{\epsilon}$ for all $t\in[0,T]$ . We set out to find Lipschitz
approximations $B^{n},G^{n}$ of $B$ and $G$ that converge uniformly on
$K_{\epsilon}$, which is already outlined in Section 3.2, and we run the
approximating sequence with $B^{n}$ and $G^{n}$. Then the proof proceeds the
same way as in Section 3.3.

We consider the following four SPDEs: the first is the stochastic heat
equation on $H$, with initial value $u_{0}\in H$. The weak well-posedness of
(4.21) was established in [23] (with time independent coefficients, but the
proof is exactly the same for time dependent coefficients.)

$\frac{\partial u(t)}{\partial t}=Au(t)+B(t,u(t))+G(t,u(t))\frac{dW_{t}}{dt},$
(4.21)

The second is the stochastic heat equation with Lipschitz coefficients and the
same initial condition

$\frac{\partial u^{n}(t)}{\partial
t}=Au^{n}(t)+B^{n}(t,u^{n}(t))+G^{n}(t,u^{n}(t))\frac{dW_{t}}{dt},$ (4.22)

The third is the damped stochastic wave equation with initial value
$(u_{0},v_{0})\in\mathcal{H}_{1}$, whose first coordinate we denote by
$u_{\mu}(t).$

$\mu\frac{\partial^{2}u_{\mu}(t)}{\partial t^{2}}=Au_{\mu}(t)-\frac{\partial
u_{\mu}(t)}{\partial t}+B(t,u_{\mu}(t))+G(t,u_{\mu}(t))\frac{dW_{t}}{dt},$
(4.23)

The fourth is the damped stochastic wave equation with Lipschitz coefficients
and the same initial value $(u_{0},v_{0})$, whose first coordinate we denote
by $u_{\mu}^{n}(t)$.

$\mu\frac{\partial^{2}u_{\mu}^{n}(t)}{\partial
t^{2}}=Au_{\mu}^{n}(t)-\frac{\partial u_{\mu}^{n}(t)}{\partial
t}+B^{n}(t,u_{\mu}^{n}(t))+G^{n}(t,u_{\mu}^{n}(t))\frac{dW_{t}}{dt},$ (4.24)

Recall the Wasserstein distance introduced in Section 3.4. We first prove the
following proposition, inspired by [27], Proposition 5.1:

######  Proposition 4.9.

Denote by $c_{n}=\max(g_{n},b_{n})$ with $g_{n},b_{n}$ defined in (4.20). Then
we can find some constant $C>0$ and $\omega>0$ independent of $\mu$ such that

$W(u^{n}_{\mu}|_{[0,T]},u_{\mu}|_{[0,T]})\leq C(c_{n})^{\omega}.$ (4.25)

###### Proof.

Consider an auxiliary SPDE

$\mu\frac{\partial^{2}\widetilde{u}^{n}_{\mu}(t)}{\partial
t^{2}}=A\widetilde{u}^{n}_{\mu}(t)-\frac{\partial\widetilde{u}^{n}_{\mu}(t)}{\partial
t}+B^{n}(t,\widetilde{u}^{n}_{\mu}(t))+\lambda(u(t)-\widetilde{u}^{n}_{\mu}(t))dt1_{t\leq\tau}+G^{n}(t,\widetilde{u}^{n}_{\mu}(t))\frac{dW_{t}}{dt},$
(4.26)

with the same initial value $(u_{0},v_{0})$, whose first coordinate we denote
by $\widetilde{u}_{\mu}^{n}(t)$.

We choose $\lambda=(c_{n})^{\gamma-1}$ (in this proof we choose $\gamma$ and
the forthcoming $\eta$ the same way as in Section 3.3) and consider the
stopping time $\tau:=\inf\\{t\geq
0:|u_{\mu}(t)-\widetilde{u}_{\mu}^{n}(t)|\geq c_{n}\\}$. Then via the same
Girsanov transform argument, we have

$d_{TV}(u_{\mu}^{n}|_{[0,T]},\widetilde{u}_{\mu}^{n}|_{[0,T]})\leq
C(c_{n})^{\gamma}.$

Now we do the pathwise estimate. Applying the $\mu$-independent estimate in
Theorem 4.4, we deduce that, in exactly the same was as in the proof of
Section 3.3, using also Theorem 4.7 for the Lebesgue integral, the following
$\mu$-independent estimate for all $\mu>0$ sufficiently small: (more
precisely, for all $\mu\in(0,\mu_{0}(\lambda)]$ for $\mu_{0}(\lambda)$
decreasing in $\lambda$)

$\mathbb{P}(\sup_{t\in[0,T]}|u_{\mu}(t)-\widetilde{u}_{\mu}^{n}(t)|\geq
Cc_{n})\leq c_{n}^{\chi}$ (4.27)

where $\chi>0$ is some fixed constant.

By definition of total variation distance, we can find a coupling of
$\widetilde{u}_{\mu}^{n}|_{[0,T]}$ and $u_{\mu}^{n}|_{[0,T]}$ such that they
are identical on an event of probability at least $1-C(c_{n})^{\gamma}$. Since
we upper bound the metric by 1 in our definition of Wasserstein distance $W$,
the expectation of transport mass over the complement set is at most
$C(c_{n})^{\gamma}$. This coupling of $u_{\mu}^{n}$ and
$\widetilde{u}_{\mu}^{n}$ leads to a coupling of $u_{\mu}$ and $u_{\mu}^{n}$,
where we simply couple $u_{\mu}$ and $\widetilde{u}_{\mu}^{n}$ via the same
driving Wiener process. Combining all these estimates, and the fact that
distance is upper bounded by 1, we deduce that for some $\omega>0$,

$W(u^{n}_{\mu}|_{[0,T]},u_{\mu}|_{[0,T]})\leq C(c_{n})^{\omega}.$ (4.28)

independent of the choice of $\mu>0$ sufficiently small. ∎

Exactly the same estimate applies to the stochastic heat equation (4.21),
(4.22), yielding

$W(u^{n}|_{[0,T]},u|_{[0,T]})\leq C(c_{n})^{\omega}.$ (4.29)

We now give the proof of Theorem 1.5, the small mass limit.

###### Proof.

The target of our proof is to show that

$\lim_{\mu\to 0}W(u|_{[0,T]},u_{\mu}|_{[0,T]})=0.$

By triangle inequality applied to the Wasserstein distance, we have

$W(u|_{[0,T]},u_{\mu}|_{[0,T]})\leq
W(u_{\mu}|_{[0,T]},u^{n}_{\mu}|_{[0,T]})+W(u^{n}|_{[0,T]},u^{n}_{\mu}|_{[0,T]})+W(u^{n}|_{[0,T]},u|_{[0,T]}).$

Then for each $n$ we have, applying Proposition 4.9 and (4.29), the following
estimate that does not depend on the value
$\mu\in(0,\mu_{0}({c_{n}}^{\gamma-1}))$:

$W(u|_{[0,T]},u_{\mu}|_{[0,T]})\leq
W(u^{n}|_{[0,T]},u^{n}_{\mu}|_{[0,T]})+C(c_{n})^{\omega}.$

Thus we only have to show that, for each $n>0$, we have

$\lim_{\mu\to 0}W(u^{n}|_{[0,T]},u^{n}_{\mu}|_{[0,T]})=0.$ (4.30)

Once this is done, the proof of the whole theorem is complete since
$\lim_{n\to\infty}c_{n}=0$.

The claim (4.30) is nothing but the Smoluchowski-Kramers approximation
approximation of $u_{\mu}^{n}$ to $u^{n}$. Since the coefficients $G^{n}$ and
$B^{n}$ are Lipschitz continuous in $x$, the said convergence follows from the
existing literature [5], [6], [32]. We give a sketch of proof for the claim
(4.30), following the steps of [32], for sake of completeness:

From the mild formulation (2.5) and (2.7), we have the following
decomposition:

$\displaystyle
u^{n}(t)-u_{\mu}^{n}(t)=(\mathcal{S}(t)u_{0}-\Pi_{1}\mathcal{S}_{\mu}(t)(u_{0},v_{0}))$
(4.31)
$\displaystyle+\int_{0}^{t}(\mathcal{S}(t-s)-\Pi_{1}\mathcal{S}_{\mu}(t-s)\mathcal{I}_{\mu})B^{n}(s,u^{n}(s))ds$
$\displaystyle+\int_{0}^{t}\Pi_{1}\mathcal{S}_{\mu}(t-s)\mathcal{I}_{\mu}(B^{n}(s,u^{n}(s))-B^{n}(s,u_{\mu}^{n}(s)))ds$
$\displaystyle+[\int_{0}^{t}\mathcal{S}(t-s)G^{n}(s,u^{n}(s))dW(s)-\int_{0}^{t}\Pi_{1}\mathcal{S}_{\mu}(t-s)\mathcal{I}_{\mu}G^{n}(s,u^{n}(s))dW(s)]$
$\displaystyle+\int_{0}^{t}\Pi_{1}\mathcal{S}_{\mu}(t-s)\mathcal{I}_{\mu}(G^{n}(s,u^{n}(s))-G^{n}(s,u_{\mu}^{n}(s))dW(s)$
$\displaystyle=:\sum_{k=1}^{5}J_{k}^{\mu}(t).$

Applying Proposition 4.8, Theorem 4.5 and 4.6, we deduce that

$\lim_{\mu\to 0}\mathbb{E}\sup_{t\in[0,T]}|J_{i}^{\mu}(t)|_{H}^{p}=0,\quad
i=1,2,4.$ (4.32)

Denote by $C_{n}$ the larger one of the Lipschitz constants of $G^{n}$ and
$B^{n}$. Then,

$\mathbb{E}\sup_{t\in[0,T]}|J_{3}(t)|_{H}^{p}\leq(C_{n})^{p}T^{p-1}\mathbb{E}\int_{0}^{T}\sup_{s\in[0,t]}|u^{n}(s)-u_{\mu}^{n}(s)|_{H}^{p}dt.$
(4.33)

By estimate (4.13) and the Lipschitz continuity of $G$,

$\mathbb{E}\sup_{t\in[0,T]}|J_{5}(t)|_{H}^{p}\leq(C_{n})^{p}C(T)\mathbb{E}\int_{0}^{T}\sup_{s\in[0,t]}|u^{n}(s)-u_{\mu}^{n}(s)|_{H}^{p}dt.$
(4.34)

Taking the norm on both sides of (4.31) and apply Gronwall’s lemma, we deduce
that

$\displaystyle\mathbb{E}$
$\displaystyle\sup_{t\in[0,T]}|u^{n}(t)-u_{\mu}^{n}(t)|_{H}^{p}$
$\displaystyle\leq
C(n,T)e^{TC(n,T)}(\sup_{t\in[0,T]}|J_{1}(t)|_{H}^{p}+\mathbb{E}\sup_{t\in[0,T]}|J_{2}(t)|_{H}^{p}+\mathbb{E}\sup_{t\in[0,T]}|J_{4}(t)|_{H}^{p}).$

Taking $\mu\to 0$, we deduce that $\lim_{\mu\to
0}\mathbb{E}\sup_{t\in[0,T]}|u^{n}(t)-u_{\mu}^{n}(t)|_{H}=0$ and hence
completes the proof of (4.30). ∎

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