# Pathwise Uniform Convergence of Time Discretisation Schemes for SPDEs

Katharina Klioba  Hamburg University of Technology, Institute of Mathematics,
D-21073 Hamburg, Germany<EMAIL_ADDRESS>and  Mark Veraar  Delft
Institute of Applied Mathematics
Delft University of Technology
P.O. Box 5031
2600 GA Delft
The Netherlands<EMAIL_ADDRESS>
###### Abstract.

In this paper, we prove convergence rates for time discretisation schemes for
semi-linear stochastic evolution equations with additive or multiplicative
Gaussian noise, where the leading operator $A$ is the generator of a strongly
continuous semigroup $S$ on a Hilbert space $X$, and the focus is on non-
parabolic problems. The main results are optimal bounds for the uniform strong
error

${\rm
E}_{k}^{\infty}\coloneqq\Big{(}\mathbb{E}\sup_{j\in\\{0,\ldots,N_{k}\\}}\|U(t_{j})-U^{j}\|^{p}\Big{)}^{1/p},$

where $p\in[2,\infty)$, $U$ is the mild solution, $U^{j}$ is obtained from a
time discretisation scheme, $k$ is the step size, and $N_{k}=T/k$. The usual
schemes such as exponential Euler, implicit Euler, and Crank-Nicolson, etc.
are included as special cases. Under conditions on the nonlinearity and the
noise, we show

  * •

${\rm E}_{k}^{\infty}\lesssim k\log(T/k)$ (linear equation, additive noise,
general $S$);

  * •

${\rm E}_{k}^{\infty}\lesssim\sqrt{k}\log(T/k)$ (nonlinear equation,
multiplicative noise, contractive $S$);

  * •

${\rm E}_{k}^{\infty}\lesssim k\log(T/k)$ (nonlinear wave equation,
multiplicative noise)

for a large class of time discretisation schemes. The logarithmic factor can
be removed if the exponential Euler method is used with a (quasi)-contractive
$S$. The obtained bounds coincide with the optimal bounds for SDEs. Most of
the existing literature is concerned with bounds for the simpler pointwise
strong error

${\rm
E}_{k}\coloneqq\bigg{(}\sup_{j\in\\{0,\ldots,N_{k}\\}}\mathbb{E}\|U(t_{j})-U^{j}\|^{p}\bigg{)}^{1/p}.$

Applications to Maxwell equations, Schrödinger equations, and wave equations
are included. For these equations, our results improve and reprove several
existing results with a unified method and provide the first results known for
implicit Euler and Crank-Nicolson.

###### Key words and phrases:

time discretisation schemes, pathwise uniform convergence, SPDEs, stochastic
convolutions, stochastic wave equation

###### 2020 Mathematics Subject Classification:

Primary: 65C30; Secondary: 47D06, 60H15, 60H35, 65J08, 65M12

The second author is supported by the VICI subsidy VI.C.212.027 of the
Netherlands Organisation for Scientific Research (NWO)

##  1\. Introduction

In this paper, we consider stochastic PDEs driven by an additive or
multiplicative Gaussian noise. The equations we consider can be written as
abstract stochastic evolution equations on a Hilbert space $X$ of the form

(1.1)
$\displaystyle\Bigg{\\{}\begin{split}\mathrm{d}U&=(AU+F(U))\,\mathrm{d}t+G(U)\,\mathrm{d}W_{H}~{}~{}~{}\text{
on }[0,T],\\\ U(0)&=u_{0}\in L^{p}(\Omega;X).\end{split}$

Here, $A$ is the generator of a $C_{0}$-semigroup $(S(t))_{t\geq 0}$, $W_{H}$
is a cylindrical Brownian motion, $F$ and $G$ are globally Lipschitz, $u_{0}$
is the initial data, and $p\in[2,\infty)$.

Our aim is to obtain strong convergence rates for temporal discretisation
schemes that cover the hyperbolic setting. The hyperbolic setting has been
extensively studied in recent years (see [2, 3, 5, 8, 13, 16, 17, 18, 19, 21,
23, 24, 33, 38, 41, 49, 50, 51, 54, 70, 71] and references therein). In the
parabolic setting, (i.e., $(S(t))_{t\geq 0}$ being an analytic semigroup)
regularisation phenomena occur, which make it possible to prove very different
convergence results. In the non-parabolic case, new methods to show
convergence rates are needed and related to a way to obtain regularity. Kato’s
setting for the hyperbolic case from his seminal work [47] creates a way to
obtain this regularity, which has proven to be very useful in the analysis of
quasilinear equations as well as their numerical treatment [30, 35, 36, 48,
62].

The main idea in Kato’s setting is to consider two spaces $X$ and $Y$ with
$Y\hookrightarrow X$ (or sometimes even three spaces) on which the operator
$A$ and the nonlinearities $F$ and $G$ can be analysed. In this way, one can
create regularity of $U$, and obtain better mapping properties of the
nonlinearities. In numerical approximations, the obtained regularity can be
used to obtain convergence rates, as illustrated for the deterministic case in
the references above.

The above setting often also applies to the parabolic case, in which, however,
the required mapping properties of $F$ on $Y$ can often be avoided due to the
regularising effect of the convolution with the analytic semigroup $S$. For
these equations, it does not seem necessary to work with the Kato setting, as
regularisation phenomena can be exploited. For details on the parabolic case,
the reader is referred to [4, 6, 7, 10, 22, 29, 32, 42, 43, 44, 46, 52, 53,
55, 57, 69] and references therein, as well as Remark 6.7. Consequently, our
focus lies on the hyperbolic setting.

###  1.1. Setting

In the above-mentioned literature on the hyperbolic case (and often in the
parabolic case), the error considered is the pointwise strong error

(1.2)
$\displaystyle\sup_{j\in\\{0,\ldots,N_{k}\\}}\mathbb{E}\|U(t_{j})-U^{j}\|^{p},$

where $U$ is the mild solution to (1.1), and $(U^{j})_{j=0}^{N_{k}}$ is an
approximation of the solution given by a temporal discretisation scheme of the
form $U^{0}=u_{0}$,

(1.3) $U^{j}=R_{k}U^{j-1}+kR_{k}F(U^{j-1})+R_{k}G(U^{j-1})\Delta W_{j},\ \
j=1,\ldots,N_{k}.$

Here, $N_{k}=T/k$ is the number of points, $k=t_{j}-t_{j-1}$ is the uniform
step size, $t_{j}=jk$, and $\Delta W_{j}=W_{H}(t_{j})-W_{H}(t_{j-1})$. The
operator $R_{k}$ is an approximation of the semigroup $S$ at time $k$.

When performing numerical simulations to approximate the solution of a
stochastic equation, one naturally wants the simulation to be close to the
solution of (1.1). However, (1.2) being small does not provide enough
information to conclude this, see Example 1.1. Also, from a probabilistic
point of view, (1.2) contains no information on the convergence of the path.
Instead, it is a more meaningful question to find convergence rates for the
uniform strong error

(1.4)
$\displaystyle\mathbb{E}\sup_{j\in\\{0,\ldots,N_{k}\\}}\|U(t_{j})-U^{j}\|^{p},$

where now the supremum over $j$ is inside the expectation. In the
deterministic setting, there is no difference between (1.2) and (1.4). It is a
widely known open problem in the field to find optimal estimates for (1.4).
Such estimates where the supremum is inside the expectation are usually called
maximal estimates, and there is an enormous literature on maximal estimates
for general stochastic processes [64]. However, for processes that do not have
any Gaussian or martingale structure, it can be quite complicated to prove
(sharp) maximal estimates. Even maximal estimates for the mild solution $U$ to
(1.1) with $F=0$ and $G(u)$ replaced by a progressively measurable $g\in
L^{2}(\Omega\times(0,T);X)$, are unknown in general (see the survey [67,
Section 4] for details). The difference between the errors (1.2) and (1.4) is
illustrated in the following simple example.

######  Example 1.1.

Let $\Omega=[0,1]$ and let $\mathbb{P}$ denote the Lebesgue measure. For
$\gamma\in(0,1]$, let $v_{N}:\Omega\times[0,1]\to\mathbb{R}$ be given by
$v_{N}(\omega,t)=1$ if $|t-\omega|<1/(2N^{\gamma})$, and zero otherwise. Then
one can check that the following error estimates hold:

$\sup_{t\in[0,1]}\mathbb{E}|v_{N}(t)|^{p}\leq\frac{1}{N^{\gamma}}\ \ \
\text{and}\ \ \ \mathbb{E}\sup_{t\in[0,1]}|v_{N}(t)|^{p}=1.$

One even has $\sup_{t\in[0,1]}|v_{N}(\omega,t)|=1$ for any $\omega\in\Omega$.
This shows the discrepancy between having the supremum inside the expectation
or not. Continuity of $v_{N}$ plays no role here. Indeed, one can easily
replace the indicator function by a continuous piecewise constant function
without influencing the above error estimates.

In the case where $S$ generates a $C_{0}$-group, it is known how to estimate
the uniform strong error (1.4) for the exponential Euler method (i.e.,
$R_{k}=S(k)$). In this case, one can use the group structure in the following
way

$\int_{0}^{t}S(t-s)g(s)dW_{H}(s)=S(t)\int_{0}^{t}S(-s)g(s)dW_{H}(s),$

and, similarly, for the discrete approximation. This makes it possible to
avoid maximal estimates for stochastic convolutions and use martingale
techniques instead. This technique was first applied in [70] to obtain optimal
convergence rates for the uniform strong error of the exponential Euler method
for abstract wave equations. Later, this technique was extended to other
settings (see [2, 8, 16, 25]), and, in particular, applied to stochastic
Schrödinger and Maxwell equations. However, if $S$ is not a group, this
technique is no longer applicable. Equations in which $S$ is not a group
include transport equations, equations with dissipation (e.g. damped wave
equations), parabolic equations, etc. Of course, there are also many important
systems where groups are unavailable (e.g. if a parabolic equation is coupled
to a wave or transport equation). Even more importantly, for schemes involving
rational approximations (e.g. implicit Euler, Crank-Nicolson), it is unclear
how to use the $C_{0}$-group structure to estimate the uniform strong error,
since the group does not appear in the scheme.

On the other hand, for other discretisation schemes estimates for the simpler
pointwise strong error (1.2) are available (see e.g. the above-mentioned
papers in the hyperbolic case). Moreover, simulations suggest that optimal
rates of convergence for the uniform strong error (1.4) hold as well. The main
goal of our work is to prove such optimal bounds for (1.4) for more general
semigroups and more general schemes. In particular, we prove such bounds under
the condition that $S$ and $R$ are contractive. This solves the open problem
on optimal rates for (1.4) for this class of semigroups and numerical schemes
up to a logarithmic factor.

###  1.2. Some of the main results for multiplicative noise

As in Kato’s setting for the hyperbolic case, let $X$ and $Y$ be Hilbert
spaces with $Y\hookrightarrow X$. For $\alpha\in(0,1]$ we say that $R$
approximates $S$ to order $\alpha$ on $Y$ if there is a constant
$C_{\alpha}\geq 0$ such that for all $x\in Y$, $k>0$, and
$j\in\\{0,\ldots,N_{k}\\}$

$\|(S(t_{j})-R_{k}^{j})x\|_{X}\leq C_{\alpha}k^{\alpha}\|x\|_{Y},$

where $R_{k}^{j}=(R_{k})^{j}$ denotes the $j$-th power of the scheme at time
step $k$. Our main result on convergence rates for (1.4) is as follows.

######  Theorem 1.2.

Let $X$ and $Y$ be Hilbert spaces such that $Y\hookrightarrow X$. Let $A$ be
the generator of a $C_{0}$-contraction semigroup $(S(t))_{t\geq 0}$ on $X$ and
$Y$. Suppose that $(R_{k})_{k>0}$ is a time discretisation scheme which is
contractive on both $X$ and $Y$, that $R$ approximates $S$ to order
$\alpha\in(0,1/2]$ on $Y$, and that $Y\hookrightarrow D((-A)^{\alpha})$.
Suppose that $F:X\to X$ and $G:X\to\mathcal{L}_{2}(H,X)$ are Lipschitz
continuous, and that $F:Y\to Y$ and $G:Y\to\mathcal{L}_{2}(H,Y)$ are of linear
growth. Let $p\in[2,\infty)$, $u_{0}\in L^{p}(\Omega;Y)$, and $U$ be the mild
solution to (1.1). Let $k\in(0,T/2]$ and let $(U^{j})_{j=0}^{N_{k}}$ be given
by (1.3). Then there is a constant $C_{T}>0$ not depending on $u_{0}$ and $k$
such that

(1.5) $\bigg{\lVert}\max_{0\leq j\leq
N_{k}}\|U(t_{j})-U^{j}\|_{X}\bigg{\rVert}_{L^{p}(\Omega)}\leq
C_{T}(1+\|u_{0}\|_{L^{p}(\Omega;Y)})k^{\alpha}\log(T/k).$

In particular, the approximations $(U^{j})_{j}$ converge at rate $\alpha$ as
$k\to 0$ up to a logarithmic factor.

Theorem 1.2 applies to, among others,

  * •

exponential Euler (EE): $R_{k}=S(k)$;

  * •

implicit Euler (IE): $R_{k}=(1-kA)^{-1}$;

  * •

Crank-Nicolson (CN): $R_{k}=(2+kA)(2-kA)^{-1}$.

Higher-order implicit Runge-Kutta methods such as Radau methods, BDF(2),
Lobatto IIA, IIB, and IIC, and some DIRK schemes are covered as well. The
contractivity of the scheme $R$ in the case of (EE) and (IE) follows from the
contractivity of the semigroup $S$. For other rational schemes, the
contractivity of $R_{k}=r(kA)$ follows from the holomorphy of the
corresponding rational function $r:\mathbb{C}_{-}\to\mathbb{C}$ and
$|r(z)|\leq 1$ for all $z\in\mathbb{C}_{-}$, which, in particular, is
satisfied for $A$-acceptable or $A$-stable schemes. These assertions follow
from functional calculus (see Proposition 2.5).

In the above, one usually takes $Y$ to be a suitable intermediate space
between $X$ and $D(A)$. In the special and important case that $Y=D(A)$, one
can take $\alpha=\frac{1}{2}$ for all of the aforementioned schemes. More
general convergence rates can be found in Table 1.

| Exponential Euler | Implicit Euler | Crank-Nicolson
---|---|---|---
$\alpha$ | $\beta\wedge\frac{1}{2}$ | $\frac{\beta}{2}\wedge\frac{1}{2}$ | $\frac{2\beta}{3}\wedge\frac{1}{2}$
Table 1. Convergence rates $\alpha$ in case $Y=D((-A)^{\beta})$ in Theorem 1.2

Up to the logarithmic factor, the estimate (1.5) is optimal in the sense that
the rate is the same as the rate for the initial value term on its own (i.e.
with $F=0$ and $G=0$). Theorem 1.2 follows from Theorem 6.4. In the case of
exponential Euler, we show that the logarithmic factor can be omitted, see
Corollary 6.6. In the case of additive noise, a similar result is obtained in
Theorem 3.1 for the range $\alpha\in(0,1]$ for semigroups and schemes which
are not necessarily contractive.

The error estimate (1.5) can be extended from the grid points to the full time
interval $[0,T]$ assuming higher integrability of the initial values. Provided
that $u_{0}\in L^{p_{0}}(\Omega;Y)$ holds for some $p_{0}\in(2,\infty)$ in
addition to the assumptions of Theorem 1.2, the pathwise uniform error on the
full time interval can be estimated as (see Theorem 6.12 below)

(1.6)
$\bigg{\lVert}\sup_{t\in[0,T]}\|U(t)-\tilde{U}(t)\|_{X}\bigg{\rVert}_{L^{p}(\Omega)}\leq
C_{T}(1+\|u_{0}\|_{L^{p_{0}}(\Omega;Y)})k^{\alpha}\log(T/k)$

for all $p\in[2,p_{0})$ and the piecewise constant extension $\tilde{U}$ of
$(U_{j})_{j=0,\ldots,N_{k}}$ to $[0,T]$. For exponential Euler, the
logarithmic correction factor can be replaced by $\log(T/k)^{1/2}$ (see
Corollary 6.13), which is known to be optimal already for scalar SDEs. In
practice, this implies that the rate of convergence in the grid points is
maintained already for a piecewise constant interpolation to other times. The
error estimate relies on new optimal path regularity estimates of stochastic
convolutions in suitable log-Hölder spaces, which will be presented in
Proposition 6.11.

Applications to Schrödinger and Maxwell equations are included in the main
text (see Subsections 3.3, 6.4, and 6.5). Our results improve several results
from the literature to more general schemes and general rates $\alpha$. In
Section 7, we include a setting for abstract wave equations, which was
considered in [70] only for the exponential Euler method. We prove similar
higher-order convergence rates for more general schemes and, in particular,
recover [70] as a special case.

Let us emphasise that schemes involving rational approximations, such as
implicit Euler or Crank-Nicolson, are in the focus of our work. While we
improve existing results for exponential Euler, the main novelty of our work
lies in the possibility to treat other schemes with a semigroup approach. To
the best of the authors’ knowledge, the present work is the first contribution
to pathwise uniform convergence rates for hyperbolic problems from a
theoretical standpoint, both in the generality and for the concrete examples
listed above. The main innovations are:

  * •

first optimal pathwise uniform convergence rates for implicit Euler, Crank-
Nicolson, and any other contractive time discretisation scheme for hyperbolic
SPDEs

  * •

first use of Kato’s framework for SPDEs to systematically treat hyperbolic
problems

  * •

maximal estimates for the convergence rate rather than pointwise estimates

  * •

path regularity results allowing to consider the error on the full time
interval

  * •

novel pathwise uniform stability estimates

  * •

convergence up to order $1$ for abstract wave equations for any contractive
scheme

To make the above results applicable to implementable numerical schemes for
SPDEs, one would additionally need a space discretisation. Since the main
novelty of our work lies in the treatment of temporal discretisations, we will
only consider the latter. Space discretisation is usually performed by means
of spectral Galerkin methods [41, 44, 46, 71], finite differences [4, 19, 32]
or finite elements [3, 18, 49, 50, 51, 54, 55], sometimes combined with a
discontinuous Galerkin approach [5, 38], or other methods in space or space-
time [6, 24, 25, 29, 33, 56].

A detailed understanding of the global Lipschitz setting is a quintessential
step towards the treatment of local Lipschitz nonlinearities, which occur more
frequently in practice. Our result should be seen as a first step, and we plan
to continue our work on uniform strong errors in a local Lipschitz setting in
the near future.

It was recently shown in [20] that one can transfer (1.2) to (1.4) using some
of the Hölder continuity in the $p$-th moment at the price of decreasing the
convergence rate via the Kolmogorov-Chentsov theorem. The strength of this
lies in the generality of possible applications. However, to get practically
useful bounds in concrete cases, there are limitations. A more detailed
comparison is made in Remark 6.5.

###  1.3. Method of proof

For the proof of the convergence rate, we need several ingredients. First of
all, we need to prove that the mild solution actually is continuous with
values in the subspace $Y$. This can be seen as the replacement of the usual
regularisation one has for parabolic equations in spirit of the Kato setting
explained before. Surprisingly, we do not need any Lipschitz assumptions on
$F$ and $G$ as mappings from $Y$ to $Y$, but linear growth conditions suffice.
This is crucial since Lipschitz estimates typically fail for Nemytskij
mappings on Sobolev spaces of higher order (see [27] and Remark 4.5).

A key estimate in the proof is a new maximal inequality for discrete
convolutions. In particular, this inequality will be used to prove the
stability of schemes such as (1.3), i.e.,

$\mathbb{E}\sup_{j\in\\{0,\ldots,N_{k}\\}}\|U^{j}\|^{p}_{Y}\leq C,$

where $C$ is independent of the step size $k$. But it also plays a role in
further estimates for the convergence.

A second key ingredient is another estimate recently proven in [68], which
allows estimating stochastic integral processes that contain a supremum

(1.7) $\mathbb{E}\sup_{i\in\\{1,\ldots,n\\}}\sup_{t\geq
0}\Big{\|}\int_{0}^{t}\Phi_{i}(s)dW_{H}(s)\Big{\|}_{X}^{p}$

by certain square functions with a logarithmic dependency on $n$ (see
Proposition 2.3 below).

Finally, to prove the desired convergence rate of Theorem 1.2 we need to split
the error obtained in (1.3) into

$1\text{ (initial value part)}+4\text{ (deterministic terms)}+5\text{
(stochastic terms)}=10\text{ terms}.$

To estimate these terms we require precise estimates for
$\|S(t_{j})-R_{k}^{j}\|_{\mathcal{L}(Y,X)}$, $\mathbb{E}\|U(t)-U(s)\|^{p}$,
stability estimates, and maximal estimates for continuous and discrete
convolutions.

In the end, we derive an estimate for the error in terms of itself, and we
apply a standard discrete Gronwall argument to deduce the desired error bound.
In the case of exponential Euler, some terms disappear since
$S(t_{j})=R_{k}^{j}$, which makes it possible to omit the logarithmic terms
originating from terms such as (1.7).

###  1.4. Overview

  * •

Section 2 contains the preliminaries for the rest of the paper.

  * •

Section 3 discusses the case of additive noise and semigroups that are not
necessarily contractive. We prove convergence of rate $\alpha$ up to order
one, in case the noise and data are regular enough. This is proved under the
assumption that the numerical scheme $R_{k}$ approximates the semigroup at
rate $\alpha$. Results are illustrated for the Schrödinger equation in which
case the obtained results improve several bounds from the literature for the
exponential Euler scheme, and provide the first uniform bounds for a large
class of other numerical methods including implicit Euler and Crank-Nicolson.

  * •

In Section 4 we introduce the nonlinear evolution equation with multiplicative
noise that we consider in the rest of the paper. After recalling a standard
well-posedness result, we introduce a special case of the Kato setting and
prove that the solution has regularity in the subspace $Y$ in case of linear
growth in the $Y$-setting (see Theorem 4.4).

  * •

Section 5 is concerned with the stability of the discretisation schemes for
the nonlinear evolution equation introduced in Section 4. The main stability
result can be found in Proposition 5.1 and only requires linear growth. Hence,
it is applicable on both $X$ and $Y$.

  * •

Section 6 is central in the paper, and here we prove Theorem 1.2 for the
nonlinear evolution equation introduced in Section 4 (see Theorem 6.4 for the
extended version). Moreover, we prove the error bound (1.6) on the full time
interval in Theorem 6.12. For this, we first establish a new optimal path
regularity result for the solution in Proposition 6.11, which is of
independent interest. In Subsections 6.4 and 6.5, we present applications to
the Schrödinger equation as well as the Maxwell equation.

  * •

In Section 7, we consider abstract stochastic wave equations, and obtain
convergence rates up to order one (see Theorem 7.6). Although we are not in
the setting of Section 6, an inspection of the proofs given there shows that
certain terms behave better for abstract wave equations due to their second-
order nature. Again, convergence rates are obtained for a large class of
numerical schemes, and versions of (1.6) are obtained. Examples with trace
class, space-time white noise, and smooth noise are included and can be found
in Subsections 7.4, 7.5, and 7.6, respectively. All these results are new for
schemes different from exponential Euler. Most notably, for smooth noise, we
can explain the numerical convergence rates one sees in [70, Figure 6.1] for
implicit Euler and Crank-Nicolson.

#### Acknowledgements

The first author wishes to thank the DAAD for the financial support to visit
TU Delft for one semester in 2022, and the colleagues in Delft for their
hospitality. The authors also thank Jan van Neerven and Christian Seifert for
helpful discussion and comments, and Martin Hutzenthaler for suggesting adding
error estimates on the full time interval. Further, the authors thank the
anonymous referees for their feedback, which has helped to improve the quality
and readability of the paper significantly.

##  2\. Preliminaries

### Notation

Throughout the paper, we fix a probability space
$(\Omega,\mathscr{F},\mathbb{P})$ with filtration
$(\mathscr{F}_{t})_{t\in[0,T]}$ and write $\mathcal{P}$ for the progressive
$\sigma$-algebra. Moreover, $H$, $X$, and $Y$ denote Hilbert spaces, where $H$
is used to define the $(\mathscr{F}_{t})_{t\in[0,T]}$-cylindrical Brownian
motion $W_{H}$. Subsequently, the space of Hilbert-Schmidt operators from $H$
to $X$ is denoted by $\mathcal{L}_{2}(H,X)$ and the Borel $\sigma$-algebra of
$X$ by $\mathcal{B}(X)$. Subsequently, we consider the final time $T>0$ to be
fixed and consider a uniform time grid with $t_{j}=jk$, where $k>0$ is the
time step and $j=0,\ldots,N_{k}$ with $N_{k}=T/k\in\mathbb{N}$, and define
$\lfloor t\rfloor\coloneqq\max\\{t_{j}:\,t_{j}\leq t\\}$ for $t\in[0,T]$. By
$(S(t))_{t\geq 0}$, we denote a $C_{0}$-semigroup and by $(R_{k})_{k>0}$ a
numerical scheme that approximates $S$. For a given evolution equation,
$(U(t))_{t\in[0,T]}$ is the exact solution and $U^{j}$ the numerical solution
approximating $U$ at time $t_{j}$ for $j=0,\ldots,N_{k}$. For $f$ and $g$ in
the respective spaces, let
$\|f\|_{p,q,Z}\coloneqq\|f\|_{L^{p}(\Omega;L^{q}(0,T;Z))}$ and
$|\\!|\\!|g|\\!|\\!|_{p,q,Z}\coloneqq\|g\|_{L^{p}(\Omega;L^{q}(0,T;\mathcal{L}_{2}(H,Z)))}$.
We use the notation $f(x)\lesssim g(x)$ to denote that there is a constant
$C\geq 0$ such that for all $x$ in the respective set, $f(x)\leq Cg(x)$.

###  2.1. Stochastic integration

Throughout the paper, we fix a probability space
$(\Omega,\mathscr{F},\mathbb{P})$ and a filtration
$(\mathscr{F}_{t})_{t\in[0,T]}$ on this probability space. Unless otherwise
stated, all random variables and stochastic processes considered are defined
on $(\Omega,\mathscr{F},\mathbb{P})$. Let $H,X$ be Hilbert spaces and denote
the space of Hilbert-Schmidt operators from $H$ to $X$ by
$\mathcal{L}_{2}(H,X)$. It consists of all bounded operators $R:H\to X$ such
that

$\|R\|_{{\mathcal{L}_{2}(H,X)}}^{2}\coloneqq\sum_{i\in
I}\|Rh_{i}\|_{X}^{2}<\infty,$

where $(h_{i})_{i\in I}$ is an orthonormal basis of $H$. If
$R\in\mathcal{L}_{2}(H,X)$, the sum contains at most countably many non-
vanishing terms. For $R\in{\mathcal{L}_{2}(H,X)}$, $(h_{i})_{i\in I}$ as
before, and $\gamma=(\gamma_{n})_{n\geq 1}$ centered i.i.d. normally
distributed random variables we define

(2.1) $R\gamma=\sum_{n\geq 1}\gamma_{n}Rh_{n},$

where the convergence is in $L^{p}(\Omega;X)$ for $p<\infty$ and almost surely
(see [40, Corollary 6.4.12]).

In the stochastic integrals appearing in expressions such as (1.7), the
integrator is a $H$-cylindrical Brownian motion to take
$\mathcal{L}_{2}(H,X)$-valued integrands into account. An _$H$ -cylindrical
Brownian motion_ is a mapping $W_{H}:L^{2}(0,T;H)\to L^{2}(\Omega)$ such that

  1. (i)

$W_{H}b$ is Gaussian for all $b\in L^{2}(0,T;H)$,

  2. (ii)

$\mathbb{E}(W_{H}b_{1}\cdot W_{H}b_{2})=\langle
b_{1},b_{2}\rangle_{L^{2}(0,T;H)}$ for all $b_{1},b_{2}\in L^{2}(0,T;H)$,

  3. (iii)

$W_{H}b$ is $\mathcal{F}_{t}$-measurable for all $b\in L^{2}(0,T;H)$ with
support in $[0,t]$,

  4. (iv)

$W_{H}b$ is independent of $\mathcal{F}_{s}$ for all $b\in L^{2}(0,T;H)$ with
support in $[s,T]$,

where we include a complex conjugate on $W_{H}b_{2}$ in case we want to use a
complex $H$-cylindrical Brownian motion. For $h\in H$ and $t\in[0,T]$, we use
the shorthand notation $W_{H}(t)h\coloneqq W_{H}(\mathbf{1}_{(0,t)}\otimes
h)$. Consequently, $(W_{H}(t)h)_{t\in[0,T]}$ is a Brownian motion for each
fixed $h\in H$, which is standard if and only if $\|h\|_{H}=1$. In the special
case $H=\mathbb{R}$, this notion coincides with real-valued Brownian motions.
We refer to an $H$-valued stochastic process $(W(t))_{t\geq 0}$ as a _$Q$
-Wiener process_ if $W(0)=0$, $W$ has continuous trajectories and independent
increments, and $W(t)-W(s)$ is normally distributed with parameters $0$ and
$(t-s)Q$ for $t\geq s\geq 0$. The operator $Q$ is in $\mathcal{L}(H)$,
positive self-adjoint, and of trace class. One can show that $W$ is a
$Q$-Wiener process if and only if there exists an $H$-cylindrical Brownian
motion $W_{H}$ such that $Q^{1/2}W_{H}\coloneqq\sum_{n\geq
1}Q^{1/2}h_{n}W_{H}(t)h_{n}=W(t)$ for an orthonormal basis $(h_{n})_{n\geq 1}$
of $H$ (cf. (2.1)). To consider an equation such as (1.1) with a $Q$-Wiener
process $W$ instead of a cylindrical Brownian motion, one can replace $G$ by
$GQ^{1/2}$ and reduce to the cylindrical case. For further properties of
$H$-cylindrical Brownian motions, $Q$-Wiener processes and the Itô integral,
we refer to [26].

To estimate Itô integrals w.r.t. such $H$-cylindrical Brownian motions, the
Burkholder-Davis-Gundy inequalities are particularly helpful. They imply that

(2.2)
$\bigg{(}\mathbb{E}\sup_{t\in[0,T]}\left\|\int_{0}^{t}g(s)\,\mathrm{d}W_{H}(s)\right\|_{X}^{p}\bigg{)}^{1/p}\leq
B_{p}\|g\|_{L^{p}(\Omega;L^{2}(0,T;{\mathcal{L}_{2}(H,X)}))}.$

In particular, one can take $B_{2}=2$ (by Doob’s maximal inequality [39, Thm.
3.2.2] and the Itô isometry) and $B_{p}=2\sqrt{2}\sqrt{p}$ for $p>2$. Indeed,
this follows by combining the scalar result of [15, Theorem A] with the
reduction technique in [45, Theorem 3.1] and the simple estimate
$\|(\xi^{2}+\eta^{2})^{1/2}\|_{p}\leq(\|\xi\|_{p}^{2}+\|\eta\|_{p}^{2})^{1/2}$
valid for real-valued random variables $\xi$ and $\eta$ and $p\in[2,\infty)$.

######  Definition 2.1.

A $C_{0}$-semigroup $(S(t))_{t\geq 0}$ is said to be quasi-contractive with
parameter $\lambda\geq 0$ if $\|S(t)\|\leq e^{\lambda t}$ for all $t\geq 0$.

The following maximal inequality for stochastic convolutions follows from
[34], where the contractive case is treated. The quasi-contractive case
follows from a scaling argument.

######  Theorem 2.2.

Let $X$ be a Hilbert space and let $(S(t))_{t\geq 0}$ be a quasi-contractive
semigroup on $X$ with parameter $\lambda\geq 0$. Then for $p\in[2,\infty)$

$\mathbb{E}\sup_{t\in[0,T]}\left\|\int_{0}^{t}S(t-s)g(s)\,\mathrm{d}W_{H}(s)\right\|_{X}^{p}\leq\mathrm{e}^{p\lambda
T}B_{p}^{p}\|g\|_{L^{p}(\Omega;L^{2}(0,T;{\mathcal{L}_{2}(H,X)}))}^{p},$

where $B_{p}$ is the constant from (2.2). In particular, one can take
$B_{2}=2$ and $B_{p}=2\sqrt{2}\sqrt{p}$ for $2<p<\infty$.

Next, we state a special case of [68, Proposition 2.7], which will be needed
to estimate stochastic integral terms without semigroups. Denote the
progressive $\sigma$-algebra on $(\Omega,\mathscr{F},\mathbb{P})$ by
$\mathcal{P}$ and the progressively measurable subspace of a given space by
the index $\mathcal{P}$.

######  Proposition 2.3.

Let $X$ be a Hilbert space and let $0<p<\infty$. Let
$\Phi\coloneqq(\Phi^{(j)})_{j=1}^{N}$ be a finite sequence in
$L_{\mathcal{P}}^{p}(\Omega;L^{2}(0,T;{\mathcal{L}_{2}(H,X)}))$ and set

$I_{N}^{\Phi}\coloneqq\left(\mathbb{E}\sup_{t\in[0,T],j\in\\{1,\ldots,N\\}}\left\|\int_{0}^{t}\Phi_{s}^{(j)}\,\mathrm{d}W_{H}(s)\right\|_{X}^{p}\right)^{1/p}.$

Then for some $K_{p}\geq 0$

$I_{N}^{\Phi}\leq
K_{p}\log(N)\|\Phi\|_{L^{p}(\Omega;L^{2}(0,T;\ell_{N}^{\infty}({\mathcal{L}_{2}(H,X)})))}\quad\text{
if }N\geq 2.$

If $2\leq p<\infty$, this estimate holds with $K_{p}=10\mathrm{e}\sqrt{p}$.

###### Proof.

We only need to comment on the case $p\in[2,\infty)$ and $2\leq N\leq 7$,
since the result in [68] was stated for $N\geq 8$. In this case, the triangle
inequality and the Burkholder-Davis-Gundy inequalities with $B_{p}\leq
2\sqrt{2}\sqrt{p}$ in $X$ (see (2.2)) give

$\displaystyle I_{N}^{\Phi}$
$\displaystyle\leq\Big{(}\sum_{j=1}^{N}\mathbb{E}\sup_{t\in[0,T]}\Big{\|}\int_{0}^{t}\Phi_{s}^{(j)}\,\mathrm{d}W_{H}(s)\Big{\|}_{X}^{p}\Big{)}^{1/p}$
$\displaystyle\leq
B_{p}\Big{(}\sum_{j=1}^{N}\|\Phi^{(j)}\|_{L^{p}(\Omega;L^{2}(0,T;{\mathcal{L}_{2}(H,X)}))}^{p}\Big{)}^{1/p}$
$\displaystyle\leq
2\sqrt{2}\sqrt{p}N^{1/p}\|\Phi\|_{L^{p}(\Omega;L^{2}(0,T;\ell_{N}^{\infty}({\mathcal{L}_{2}(H,X)})))}$
$\displaystyle\leq
10\mathrm{e}\sqrt{p}\log(N)\|\Phi\|_{L^{p}(\Omega;L^{2}(0,T;\ell_{N}^{\infty}({\mathcal{L}_{2}(H,X)})))},$

where the last estimate follows from $2\sqrt{2}N^{1/2}\leq
10\mathrm{e}\log(N)$ for $2\leq N\leq 7$. ∎

###  2.2. Approximation of semigroups and interpolation

An integral part of approximating solutions of a stochastic evolution equation
concerns the approximation of a semigroup by some scheme. The following
definition allows us to quantify the approximation behaviour.

######  Definition 2.4.

Let $X$ be a Hilbert space. An _$\mathcal{L}(X)$ -valued scheme_ is a function
$R:[0,\infty)\to\mathcal{L}(X)$. We denote $R_{k}\coloneqq R(k)$ for $k\geq
0$. Let $Y$ be a Hilbert space which is continuously and densely embedded in
$X$. If $A$ generates a $C_{0}$-semigroup $(S(t))_{t\geq 0}$ on $X$, an
$\mathcal{L}(X)$-valued scheme $R$ is said to _approximate $S$ to order
$\alpha>0$ on $Y$_ or, equivalently, _$R$ converges of order $\alpha$ on $Y$_
if for all $T>0$ there is a constant $C_{\alpha}\geq 0$ such that

$\|(S(jk)-R_{k}^{j})u\|_{X}\leq C_{\alpha}k^{\alpha}\|u\|_{Y}$

for all $u\in Y$, $k>0$, and $j\in\mathbb{N}$ such that $jk\in[0,T]$. An
$\mathcal{L}(X)$-valued scheme $R$ is said to be _contractive_ if
$\|R_{k}\|_{\mathcal{L}(X)}\leq 1$ for all $k\geq 0$.

Subsequently, we will omit the index for norms in the space $X$. In the
absence of nonlinear and noise terms, the following schemes approximate $S$ to
different orders:

  * •

exponential Euler (EE): $R_{k}=S(k)$, any order $\alpha>0$ on $X$;

  * •

implicit Euler (IE): $R_{k}=(1-kA)^{-1}$, order $\alpha\in(0,1]$ on
$D((-A)^{2\alpha})$;

  * •

Crank-Nicolson (CN): $R_{k}=(2+kA)(2-kA)^{-1}$, order $\alpha\in(0,2]$ on
$D((-A)^{3\alpha/2})$ provided that $(S(t))_{t\geq 0}$ is contractive.

Contractivity of the semigroup and the approximating scheme play a central
role in our theory. While the contractivity of (EE) is immediate from the
contractivity of the semigroup, we state a useful sufficient condition to
verify the contractivity of rational schemes such as (IE) and (CN) below. One
of the standard assumptions in the theory of semigroup approximation is that
the scheme $R$ stems from a rational function $r:\mathbb{C}_{-}\to\mathbb{C}$
with $|r(z)|\leq 1$ for all $z$ in the negative open halfplane
$\mathbb{C}_{-}$. Under an additional consistency condition, this condition is
known as A-acceptability [14], and it certainly holds for $A$-stable schemes
[28].

######  Proposition 2.5.

Let $A$ be the generator of a $C_{0}$-semigroup of contractions on a Hilbert
space $X$. Suppose that $r:\mathbb{C}_{-}\to\mathbb{C}$ is holomorphic,
$|r(z)|\leq 1$ for all $z\in\mathbb{C}_{-}$, and let $R_{k}=r(kA)$ for $k>0$.
Then $R$ is contractive.

###### Proof.

This is a consequence of the properties of the bounded $H^{\infty}$-calculus
of $-A$ as the negative generator of a contraction semigroup, since
$R_{k}=r(kA)=r(-k(-A))$ is defined via $H^{\infty}$-calculus. The underlying
theorem can be found in [40, Thm. 10.2.24]. ∎

As a consequence of this proposition, contractive schemes include (IE), (CN),
and some higher-order implicit Runge-Kutta methods such as Radau methods,
BDF(2), Lobatto IIA, IIB, and IIC as well as some DIRK schemes.

A common choice for the spaces $Y$ on which a given scheme approximates $S$
are domains of fractional powers of $A$. An important property of these spaces
is that they embed into the real interpolation spaces with parameter $\infty$,
i.e., for $\alpha>0$

(2.3) $D(A^{\alpha})\hookrightarrow D_{A}(\alpha,\infty).$

Here, $D_{A}(\alpha,\infty)$ denotes the real interpolation space
$(X,D(A))_{\alpha,\infty}$. On later occasions, also the real interpolation
spaces $(X,D(A))_{\alpha,2}$ will be used. See [58, 66] for details on
interpolation spaces.

Embeddings of the form (2.3) and properties of $D_{A}(\alpha,\infty)$ allow us
to obtain decay rates for semigroup differences as follows. Let $(S(t))_{t\geq
0}$ be a $C_{0}$-semigroup such that $\|S(t)\|\leq Me^{\lambda t}$ for some
$M\geq 1$ and $\lambda\geq 0$ for all $t\geq 0$. Such $M$ and $\lambda$ exist
for every $C_{0}$-semigroup [31, Prop. 5.5]. Then
$\|S(t)-S(s)\|_{\mathcal{L}(X)}\leq 2M\mathrm{e}^{\lambda T}$ for $0\leq s\leq
t\leq T$. Since

$\|[S(t)-S(s)]x\|_{X}=\left\|\int_{s}^{t}S(r)Ax\;\mathrm{d}r\right\|_{X}\leq
M\mathrm{e}^{\lambda T}(t-s)\|x\|_{D(A)}$

for $x\in D(A)$, we have $\|S(t)-S(s)\|_{\mathcal{L}(D(A),X)}\leq
2M\mathrm{e}^{\lambda T}(t-s)$. By interpolation,

$\|S(t)-S(s)\|_{\mathcal{L}(D_{A}(\alpha,\infty),X)}\leq
2^{1-\alpha}M\mathrm{e}^{\lambda T}(t-s)^{\alpha}\leq 2M\mathrm{e}^{\lambda
T}(t-s)^{\alpha}$

for $\alpha\in(0,1)$. Let $Y$ be another Hilbert space such that
$Y\hookrightarrow X$. Under the assumption that $Y\hookrightarrow
D_{A}(\alpha,\infty)$ continuously for some $\alpha\in(0,1)$ or
$Y\hookrightarrow D(A)$ continuously, in which case we set $\alpha=1$, this
implies

(2.4) $\|S(t)-S(s)\|_{\mathcal{L}(Y,X)}\leq 2C_{Y}M\mathrm{e}^{\lambda
T}(t-s)^{\alpha},$

where $C_{Y}$ denotes the embedding constant of $Y$ into
$D_{A}(\alpha,\infty)$ or $D(A)$.

###  2.3. Gronwall type lemmas

We need the following variants of the classical Gronwall inequality.

######  Lemma 2.6.

Let $\phi:[0,T]\to[0,\infty)$ be a continuous function and let
$\alpha,\beta\in[0,\infty)$ be constants. Suppose that

$\phi(t)\leq\alpha+\beta\Big{(}\int_{0}^{t}\phi(s)^{2}ds\Big{)}^{1/2},\ \
t\in[0,T].$

Then

$\phi(t)\leq\alpha(1+\beta^{2}t)^{1/2}\exp\Big{(}\frac{1}{2}+\frac{1}{2}\beta^{2}t\Big{)},\
\ t\in[0,T].$

###### Proof.

Using $(a+b)^{2}\leq(1+\theta)a^{2}+(1+\theta^{-1})b^{2}$ for $a,b\geq 0$ and
$\theta>0$, we can write

$\phi(t)^{2}\leq(1+\theta)\alpha^{2}+\beta^{2}(1+\theta^{-1})\int_{0}^{t}\phi(s)^{2}ds,\
\ t\in[0,T].$

Therefore, applying Gronwall’s inequality we see that

$\phi(t)^{2}\leq(1+\theta)\alpha^{2}\exp(\beta^{2}(1+\theta^{-1})t).$

Taking $\theta=\beta^{2}t$ we obtain

$\phi(t)^{2}\leq(1+\beta^{2}t)\alpha^{2}\exp(\beta^{2}t+1),$

which gives the desired estimate. ∎

In the same way, one can prove the following discrete analogue by using the
discrete version of Gronwall’s lemma instead (see [37, Proposition 5]).

######  Lemma 2.7.

Let $\alpha,\beta\geq 0$ and $(\varphi_{j})_{j\geq 0}$ be a nonnegative
sequence. If

$\varphi_{j}\leq\alpha+\beta\left(\sum_{i=0}^{j-1}\varphi_{i}^{2}\right)^{1/2}~{}\text{
for }j\geq 0,$

then

$\varphi_{j}\leq\alpha(1+\beta^{2}j)^{1/2}\exp\left(\frac{1}{2}+\frac{1}{2}\beta^{2}j\right)~{}\text{
for }j\geq 0.$

##  3\. Convergence rates for additive noise

In this section, we present several results on convergence rates for linear
equations with additive noise. The reason to start with this case is twofold.
Higher convergence rates can be proved in this case. Moreover, it allows us to
explain the new techniques in a simpler setting, which can help understand the
more complicated multiplicative setting of Section 6.

Consider the stochastic evolution equation with additive noise of the form

(3.1) $\mathrm{d}U=AU\,\mathrm{d}t+g(t)\,\mathrm{d}W_{H}(t)\text{ on
}[0,T],~{}~{}U(0)=u_{0}\in L_{\mathcal{F}_{0}}^{p}(\Omega;X),$

where $A$ generates a $C_{0}$-semigroup $(S(t))_{t\geq 0}$ on a Hilbert space
$X$ with norm $\|\cdot\|$, $W_{H}$ is an $H$-cylindrical Brownian motion for
some Hilbert space $H$, and $p\in[2,\infty)$. For Hölder continuous noise
$g\in L_{\mathcal{P}}^{p}(\Omega;C^{\alpha}([0,T];{\mathcal{L}_{2}(H,X)}))$,
$\alpha\in(0,1]$, mapping into a space $Y\hookrightarrow X$, we prove rates of
convergence for time discretisation schemes. An improvement of the rate is
shown for the exponential Euler method for quasi-contractive semigroups.
Results are illustrated for the nonlinear Schrödinger equation in Subsection
3.3.

The mild solution to (3.1) for $t\in[0,T]$ is uniquely given by [26, Chapters
5,6].

(3.2) $U(t)=S(t)u_{0}+\int_{0}^{t}S(t-s)g(s)\,\mathrm{d}W_{H}(s).$

To approximate it, we employ a time discretisation scheme
$R:[0,\infty)\to\mathcal{L}(X)$ with time step $k>0$ on a uniform grid
$\\{t_{j}=jk:~{}j=0,\ldots,N_{k}\\}\subseteq[0,T]$ with final time
$T=t_{N_{k}}>0$ and $N_{k}=\frac{T}{k}\in\mathbb{N}$ being the number of time
steps. The discrete solution is given by $U^{0}\coloneqq u_{0}$ and

(3.3) $\displaystyle U^{j}$ $\displaystyle\coloneqq
R_{k}U^{j-1}+R_{k}g(t_{j-1})\Delta
W_{j}=R_{k}^{j}u_{0}+\sum_{i=0}^{j-1}R_{k}^{j-i}g(t_{i})\Delta
W_{i+1},~{}~{}j=1,\ldots,N_{k},$

with Wiener increments $\Delta W_{j}\coloneqq W_{H}(t_{j})-W_{H}(t_{j-1})$,
where we used (2.1).

###  3.1. General semigroups

Our first result concerns general $C_{0}$-semigroups $S$. A further
improvement under further conditions on $S$ is discussed in Subsection 3.2.
Below, we denote the Hölder seminorm in
$C^{\alpha}([0,T];{\mathcal{L}_{2}(H,X)})$ by $[\cdot]_{\alpha,X}$ for
$\alpha\in(0,1]$ and let

(3.4)
$|\\!|\\!|g|\\!|\\!|_{p,\infty,Y}\coloneqq\|g\|_{L^{p}(\Omega;C([0,T];\mathcal{L}_{2}(H,Y)))},\quad(g\in
L^{p}(\Omega;C([0,T];{\mathcal{L}_{2}(H,Y)}))).$

######  Theorem 3.1.

Let $X$ and $Y$ be Hilbert spaces such that $Y\hookrightarrow X$. Let $A$ be
the generator of a $C_{0}$-semigroup $(S(t))_{t\geq 0}$ on $X$ with
$\|S(t)\|\leq Me^{\lambda t}$ for some $M\geq 1$ and $\lambda\geq 0$. Let
$(R_{k})_{k>0}$ be a time discretisation scheme and assume that $R$
approximates $S$ to order $\alpha\in(0,1]$ on $Y$. Suppose that
$Y\hookrightarrow D_{A}(\alpha,\infty)$ continuously if $\alpha\in(0,1)$ or
$Y\hookrightarrow D(A)$ continuously if $\alpha=1$. Let $p\in[2,\infty)$,
$u_{0}\in L_{\mathcal{F}_{0}}^{p}(\Omega;Y)$, and $g\in
L_{\mathcal{P}}^{p}(\Omega;C([0,T];{\mathcal{L}_{2}(H,Y)}))$ as well as $g\in
L_{\mathcal{P}}^{p}(\Omega;C^{\alpha}([0,T];{\mathcal{L}_{2}(H,X)}))$. Denote
by $U$ the mild solution of (3.1) and by $(U^{j})_{j=0,\ldots,N_{k}}$ the
temporal approximations as defined in (3.3). Then for $N_{k}\geq 2$

$\left\lVert\max_{0\leq j\leq
N_{k}}\|U(t_{j})-U^{j}\|\right\rVert_{p}\leq\left(C_{1}+C_{2}\log\left(\frac{T}{k}\right)\right)k^{\alpha}$

with constants $C_{1}\coloneqq C_{\alpha}\|u_{0}\|_{L^{p}(\Omega;Y)}$ and

$\displaystyle C_{2}$
$\displaystyle\coloneqq\frac{K_{p}\sqrt{T}}{\sqrt{2\alpha+1}}\left(M\mathrm{e}^{\lambda
T}\big{\|}[g]_{\alpha,X}\big{\|}_{p}+\left(2M\mathrm{e}^{\lambda
T}C_{Y}+C_{\alpha}\right)|\\!|\\!|g|\\!|\\!|_{p,\infty,Y}\right),$

where $C_{\alpha}$ is as in Definition 2.4, $K_{p}=10\mathrm{e}\sqrt{p}$, and
$C_{Y}$ denotes the embedding constant of $Y$ into $D_{A}(\alpha,\infty)$ or
$D(A)$.

In particular, the approximations $(U^{j})_{j}$ converge at rate
$\min\\{\alpha,1\\}$ up to a logarithmic correction factor as $k\to 0$.

###### Proof.

Define $S^{k}(t)\coloneqq R_{k}^{j}$ for $t\in(t_{j-1},t_{j}]$ and let
$\lfloor t\rfloor\coloneqq\max\\{t_{j}:\,t_{j}\leq t\\}$ for $t\in[0,T]$. Then
the discrete solutions are given by the integral representation

$U^{j}=R_{k}^{j}u_{0}+\int_{0}^{t_{j}}S^{k}(t_{j}-s)g(\lfloor
s\rfloor)\,\mathrm{d}W_{H}(s).$

Combining this representation with the mild solution formula (3.2), the error
can be bounded by

$\displaystyle E$ $\displaystyle\coloneqq\Big{\lVert}\max_{0\leq j\leq
N_{k}}\|U(t_{j})-U^{j}\|\Big{\rVert}_{p}\leq\Big{\lVert}\max_{0\leq j\leq
N_{k}}\|[S(t_{j})-R_{k}^{j}]u_{0}\|\Big{\rVert}_{p}$
$\displaystyle\phantom{=}+\Big{\lVert}\max_{0\leq j\leq
N_{k}}\Big{\|}\int_{0}^{t_{j}}S(t_{j}-s)[g(s)-g(\lfloor
s\rfloor)]\,\mathrm{d}W_{H}(s)\Big{\|}\Big{\rVert}_{p}$
$\displaystyle\phantom{=}+\Big{\lVert}\max_{0\leq j\leq
N_{k}}\Big{\|}\int_{0}^{t_{j}}[S(t_{j}-\lfloor s\rfloor)-S(t_{j}-s)]g(\lfloor
s\rfloor)\,\mathrm{d}W_{H}(s)\Big{\|}\Big{\rVert}_{p}$
$\displaystyle\phantom{=}+\Big{\lVert}\max_{0\leq j\leq
N_{k}}\Big{\|}\int_{0}^{t_{j}}[S(t_{j}-\lfloor
s\rfloor)-S^{k}(t_{j}-s)]g(\lfloor
s\rfloor)\,\mathrm{d}W_{H}(s)\Big{\|}\Big{\rVert}_{p}$ (3.5)
$\displaystyle\eqqcolon E_{1}+E_{2}+E_{3}+E_{4}.$

We proceed to estimate all four terms individually. Since $R$ approximates $S$
to order $\alpha$ on $Y$,

(3.6) $E_{1}\leq C_{\alpha}\|u_{0}\|_{L^{p}(\Omega;Y)}k^{\alpha}.$

For the second term, we note that for $s\in[t_{\ell},t_{\ell+1})$ for some
$0\leq\ell\leq N_{k}-1$, the definition of the Hölder seminorm
$[\cdot]_{\alpha}$ implies that $\mathbb{P}$-almost surely

$\displaystyle\Big{\|}\sum_{i=0}^{j-1}\mathbf{1}_{[t_{i},t_{i+1})}(s)S(t_{j}-s)[g(s)-g(t_{i})]\Big{\|}_{\mathcal{L}_{2}(H,X)}$
$\displaystyle\leq\|S(t_{j}-s)\|_{\mathcal{L}(X)}\|g(s)-g(t_{\ell})\|_{\mathcal{L}_{2}(H,X)}$
$\displaystyle\leq M\mathrm{e}^{\lambda
T}[g]_{\alpha,X}(s-t_{\ell})^{\alpha}.$

Proposition 2.3 with
$\Phi_{s}^{(j)}=\sum_{i=0}^{j-1}\mathbf{1}_{[t_{i},t_{i+1})}(s)S(t_{j}-s)[g(s)-g(t_{i})]$
then yields

$\displaystyle E_{2}=\bigg{\lVert}\max_{0\leq j\leq
N_{k}}\bigg{\|}\int_{0}^{t_{j}}\sum_{i=0}^{j-1}\mathbf{1}_{[t_{i},t_{i+1})}(s)S(t_{j}-s)[g(s)-g(t_{i})]\,\mathrm{d}W_{H}(s)\bigg{\|}\bigg{\rVert}_{p}$
$\displaystyle\leq K_{p}\log(N_{k})\Big{\lVert}\Big{(}\int_{0}^{T}\max_{1\leq
j\leq
N_{k}}\|\Phi_{s}^{(j)}\|_{\mathcal{L}_{2}(H,X)}^{2}\,\mathrm{d}s\Big{)}^{1/2}\Big{\rVert}_{p}$
$\displaystyle\leq K_{p}M\mathrm{e}^{\lambda
T}\log(N_{k})\bigg{\lVert}\bigg{(}\sum_{l=0}^{N_{k}-1}\int_{t_{\ell}}^{t_{\ell+1}}[g]_{\alpha,X}^{2}(s-t_{\ell})^{2\alpha}\,\mathrm{d}s\bigg{)}^{1/2}\bigg{\rVert}_{p}$
$\displaystyle\leq K_{p}M\mathrm{e}^{\lambda
T}\frac{1}{\sqrt{2\alpha+1}}\log(N_{k})k^{\alpha+1/2}\bigg{\|}\bigg{(}\sum_{l=0}^{N_{k}-1}[g]_{\alpha,X}^{2}\bigg{)}^{1/2}\bigg{\|}_{p}$
(3.7) $\displaystyle=K_{p}M\mathrm{e}^{\lambda
T}\big{\|}[g]_{\alpha,X}\big{\|}_{p}\frac{\sqrt{T}}{\sqrt{2\alpha+1}}\log(N_{k})k^{\alpha},$

where we have used Hölder continuity of $g$. Analogously, with
$\Phi_{s}^{(j)}=\sum_{i=0}^{j-1}\mathbf{1}_{[t_{i},t_{i+1})}(s)[S(t_{j}-t_{i})-S(t_{j}-s)]g(t_{i})$
for $E_{3}$ we obtain

(3.8) $E_{3}\leq 2K_{p}M\mathrm{e}^{\lambda
T}C_{Y}\frac{\sqrt{T}}{\sqrt{2\alpha+1}}|\\!|\\!|g|\\!|\\!|_{p,\infty,Y}\log(N_{k})k^{\alpha}$

using pathwise boundedness of $g$, i.e.,
$g(\omega,\cdot):[0,T]\to\mathcal{L}_{2}(H,Y)$ being bounded for
$\mathbb{P}$-almost every $\omega\in\Omega$, and noting that by (2.4)

$\displaystyle\big{\|}[S(t_{j}-t_{\ell})-S(t_{j}-s)]g(t_{\ell})\big{\|}_{\mathcal{L}_{2}(H,X)}\leq
2M\mathrm{e}^{\lambda
T}C_{Y}(s-t_{\ell})^{\alpha}\|g(t_{\ell})\|_{\mathcal{L}_{2}(H,Y)}$

holds $\mathbb{P}$-almost surely. Likewise, with
$\Phi_{s}^{(j)}=\sum_{i=0}^{j-1}\mathbf{1}_{[t_{i},t_{i+1})}(s)[S(t_{j}-t_{i})-R_{k}^{j-i}]g(t_{i})$,
we obtain

(3.9) $E_{4}\leq
K_{p}C_{\alpha}\frac{\sqrt{T}}{\sqrt{2\alpha+1}}|\\!|\\!|g|\\!|\\!|_{p,\infty,Y}\log(N_{k})k^{\alpha},$

since $R$ approximates $S$ to order $\alpha$ on $Y$. The error bound follows
from inserting (3.6), (3.1), (3.8), and (3.9) into (3.1). ∎

For the _exponential Euler method_ , less regularity of the initial value
suffices for the same convergence behaviour. The exponential Euler method is
obtained by setting $R_{k}=S(k)$ in (3.3), i.e., we would solve exactly in the
absence of noise $g$.

######  Corollary 3.2 (Exponential Euler).

Let $X$ and $Y$ be Hilbert spaces such that $Y\hookrightarrow X$. Let $A$ be
the generator of a $C_{0}$-semigroup $(S(t))_{t\geq 0}$ on $X$ with
$\|S(t)\|\leq Me^{\lambda t}$ for some $M\geq 1$ and $\lambda\geq 0$. Assume
that $g\in L_{\mathcal{P}}^{p}(\Omega;C([0,T];{\mathcal{L}_{2}(H,Y)}))$ and
$g\in L_{\mathcal{P}}^{p}(\Omega;C^{\alpha}([0,T];{\mathcal{L}_{2}(H,X)}))$
for some $\alpha\in(0,1]$. Suppose that $Y\hookrightarrow
D_{A}(\alpha,\infty)$ continuously if $\alpha\in(0,1)$ or $Y\hookrightarrow
D(A)$ continuously if $\alpha=1$. Let $p\in[2,\infty)$ and $u_{0}\in
L_{\mathcal{F}_{0}}^{p}(\Omega;X)$. Denote by $U$ the mild solution of (3.1)
and by $(U^{j})_{j=0,\ldots,N_{k}}$ the temporal approximations as defined in
(3.3) obtained with the exponential Euler method $R\coloneqq S$. Then for
$N_{k}\geq 2$

$\bigg{\lVert}\max_{0\leq j\leq N_{k}}\|U(t_{j})-U^{j}\|\bigg{\rVert}_{p}\leq
C\log\left(\frac{T}{k}\right)k^{\alpha}$

with constant

$\displaystyle C$ $\displaystyle\coloneqq K_{p}M\mathrm{e}^{\lambda
T}\frac{\sqrt{T}}{\sqrt{2\alpha+1}}\left(\left\|[g]_{\alpha,X}\right\|_{p}+2C_{Y}|\\!|\\!|g|\\!|\\!|_{p,\infty,Y}\right),$

where $K_{p}=10\mathrm{e}\sqrt{p}$ and $C_{Y}$ denotes the embedding constant
of $Y$ into $D_{A}(\alpha,\infty)$ or $D(A)$.

In particular, if $Y\hookrightarrow D(A)$ and $g$ is Lipschitz continuous as a
map to ${\mathcal{L}_{2}(H,X)}$, the approximations $(U^{j})_{j}$ converge at
rate $1$ up to a logarithmic correction factor as $k\to 0$.

###### Proof.

We split the error as in (3.1). For exponential Euler, the terms $E_{1}$ and
$E_{4}$ in (3.1) vanish due to
$S(t_{j})-R_{k}^{j}=S(jk)-S(k)^{j}=S(jk)-S(jk)=0$ and, likewise,
$S(t_{j}-t_{i})-R_{k}^{j-i}=0$. The error bound follows from inserting the
bounds (3.1) and (3.8) of the remaining terms into (3.1). ∎

###  3.2. Quasi-contractive Semigroups

Considering quasi-contractive semigroups, that is, $C_{0}$-semigroups
$(S(t))_{t\geq 0}$ for which $\|S(t)\|\leq e^{\lambda t}$ for some
$\lambda\geq 0$ for all $t\geq 0$, allows us to eliminate the logarithmic
factor for exponential Euler. The principle that lies at the heart of our
proof is the maximal inequality from Theorem 2.2, which is used to estimate
the stochastic convolutions in the error term. Depending on the spatial
regularity of the noise $g$, the convergence rate $\alpha\in(0,1]$ is attained
without a logarithmic correction factor.

######  Theorem 3.3 (exponential Euler, quasi-contractive case).

Adopt the notation and assumptions of Corollary 3.2. In addition, assume that
$\|S(t)\|\leq\mathrm{e}^{\lambda t}$ for some $\lambda\geq 0$ for all
$t\in[0,T]$. Then for $N_{k}\geq 2$

$\left\lVert\max_{0\leq j\leq N_{k}}\|U(t_{j})-U^{j}\|\right\rVert_{p}\leq
Ck^{\alpha}$

with constant

$\displaystyle
C\coloneqq\frac{B_{p}\sqrt{T}}{\sqrt{2\alpha+1}}\left(\mathrm{e}^{\lambda
T}\left\|[g]_{\alpha,X}\right\|_{p}+2C_{Y}\mathrm{e}^{2\lambda
T}|\\!|\\!|g|\\!|\\!|_{p,\infty,Y}\right),$

where $B_{p}$ is the constant from Theorem 2.2.

###### Proof.

We bound the error as in (3.1), where the first and fourth term vanish as
discussed in the proof of Corollary 3.2. We proceed to bound the remaining
terms using the maximal inequality from Theorem 2.2 instead of Proposition 2.3
to obtain

$\displaystyle E_{2}$
$\displaystyle\leq\bigg{\lVert}\sup_{t\in[0,T]}\bigg{\|}\int_{0}^{t}S(t-s)[g(s)-g(\lfloor
s\rfloor)]\,\mathrm{d}W_{H}(s)\bigg{\|}\bigg{\rVert}_{p}$ $\displaystyle\leq
B_{p}\mathrm{e}^{\lambda
T}\bigg{\lVert}\bigg{(}\int_{0}^{T}\left\|g(s)-g(\lfloor
s\rfloor)\right\|_{\mathcal{L}_{2}(H,X)}^{2}\,\mathrm{d}s\bigg{)}^{1/2}\bigg{\rVert}_{p}$
$\displaystyle\leq B_{p}\mathrm{e}^{\lambda
T}\bigg{\lVert}\bigg{(}\sum_{i=0}^{N_{k}-1}\int_{t_{i}}^{t_{i+1}}[g]_{\alpha,X}^{2}(s-t_{i})^{2\alpha}\,\mathrm{d}s\bigg{)}^{1/2}\bigg{\rVert}_{p}$
(3.10) $\displaystyle\leq\frac{B_{p}\mathrm{e}^{\lambda
T}\sqrt{T}}{\sqrt{2\alpha+1}}\big{\|}[g]_{\alpha,X}\big{\|}_{p}k^{\alpha}$

by Hölder continuity of $g$. Analogously, for $E_{3}$ we deduce from the
semigroup bound (2.4) that

$\displaystyle E_{3}$
$\displaystyle\leq\bigg{\lVert}\sup_{t\in[0,T]}\bigg{\|}\int_{0}^{t}S(t-s)[S(s-\lfloor
s\rfloor)-I]g(\lfloor s\rfloor)\,\mathrm{d}W_{H}(s)\bigg{\|}\bigg{\rVert}_{p}$
$\displaystyle\leq B_{p}\mathrm{e}^{\lambda
T}\bigg{\lVert}\bigg{(}\int_{0}^{T}\left\|[S(s-\lfloor s\rfloor)-I]g(\lfloor
s\rfloor)\right\|_{\mathcal{L}_{2}(H,X)}^{2}\,\mathrm{d}s\bigg{)}^{1/2}\bigg{\rVert}_{p}$
$\displaystyle\leq 2B_{p}\mathrm{e}^{2\lambda
T}C_{Y}\bigg{\lVert}\bigg{(}\sum_{i=0}^{N_{k}-1}\int_{t_{i}}^{t_{i+1}}(s-t_{i})^{2\alpha}\left\|g(t_{i})\right\|_{\mathcal{L}_{2}(H,Y)}^{2}\,\mathrm{d}s\bigg{)}^{1/2}\bigg{\rVert}_{p}$
(3.11) $\displaystyle\leq 2B_{p}\mathrm{e}^{2\lambda
T}C_{Y}\frac{\sqrt{T}}{\sqrt{2\alpha+1}}|\\!|\\!|g|\\!|\\!|_{p,\infty,Y}k^{\alpha}.$

The final error bound follows from adding (3.2) and (3.2). ∎

In particular, convergence rate $1$ is attained without logarithmic correction
factor for spatially sufficiently regular noise $g$. General, possibly
irregular initial values $u_{0}\in L_{\mathcal{F}_{0}}^{p}(\Omega;X)$ are
still admissible as the following corollary shows.

######  Corollary 3.4.

Let $X$ be a Hilbert space and let $A$ be the generator of a quasi-contractive
$C_{0}$-semigroup on $X$ with parameter $\lambda>0$. Assume that $g\in
L_{\mathcal{P}}^{p}(\Omega;C([0,T];\mathcal{L}_{2}(H,D(A))))$ and is pathwise
Lipschitz continuous as a map to ${\mathcal{L}_{2}(H,X)}$. Let
$p\in[2,\infty)$ and $u_{0}\in L_{\mathcal{F}_{0}}^{p}(\Omega;X)$. Denote by
$U$ the mild solution of (3.1) and by $(U^{j})_{j=0,\ldots,N_{k}}$ the
temporal approximations as defined in (3.3) obtained with the exponential
Euler method $R\coloneqq S$. Then there is a constant $C\geq 0$ depending on
$(g,T,p,\alpha,\lambda,X,D(A))$ such that for $N_{k}\geq 2$

$\left\lVert\max_{0\leq j\leq N_{k}}\|U(t_{j})-U^{j}\|\right\rVert_{p}\leq
Ck,$

i.e., the approximations $(U^{j})_{j}$ converge at rate $1$ as $k\to 0$.

###  3.3. Application to the linear Schrödinger equation with additive noise

In this subsection, we study convergence rates of time discretisations of the
linear stochastic Schrödinger equation with a potential and additive noise

(3.12)
$\displaystyle\Bigg{\\{}\begin{split}\mathrm{d}u&=-\mathrm{i}(\Delta+V)u\;\mathrm{d}t-\mathrm{i}\;\mathrm{d}W~{}~{}~{}\text{
on }[0,T],\\\ u(0)&=u_{0}\end{split}$

in $\mathbb{R}^{d}$ for $d\in\mathbb{N}$, where $\\{W(t)\\}_{t\geq 0}$ is a
square-integrable $\mathbb{K}$-valued $Q$-Wiener process (see Subsection 2.1),
$\mathbb{K}\in\\{\mathbb{R},\mathbb{C}\\}$, with respect to a normal
filtration $(\mathscr{F}_{t})_{t\geq 0}$, $V$ is a $\mathbb{K}$-valued
potential, $u_{0}$ is an $\mathscr{F}_{0}$-measurable random variable,
$\mathrm{i}$ is the imaginary unit, and $\Delta$ the Laplace operator on
$\mathbb{R}^{d}$. Next, we introduce conditions on the dimension and the
regularity of $V$. With a slight variation of the methods below, one can also
consider (3.12) on $[0,L]^{d}$ with periodic boundary conditions. More general
domains with Dirichlet or Neumann boundary conditions can be treated as well,
but for this, suitable adjustments are needed in the proofs below.

Let $\sigma\geq 0$ and, for this subsection only, write
$L^{2}=L^{2}(\mathbb{R}^{d})$ and $H^{\sigma}=H^{\sigma}(\mathbb{R}^{d})$. We
will also be using the Bessel potential spaces $H^{\sigma,q}(\mathbb{R}^{d})$,
which coincide with the classical Sobolev spaces
$W^{\sigma,q}(\mathbb{R}^{d})$ if $\sigma\in\mathbb{N}$ and $q\in(1,\infty)$.
For details on these spaces the reader is referred to [9, 66].

To ensure the well-posedness of (3.12), we assume one of the following
mutually exclusive conditions holds.

######  Assumption 3.5.

Let $\sigma\geq 0$, $d\in\mathbb{N}$ and $V\in L^{2}$ such that

  1. (i)

$\sigma>\frac{d}{2}$ and $V\in H^{\sigma}$, or

  2. (ii)

$\sigma=0$ and $V\in H^{\beta}$ for some $\beta>\frac{d}{2}$, or

  3. (iii)

$\sigma\in(0,1)$, $d>2\sigma$, and $V\in H^{\beta}$ for some
$\beta>\frac{d}{2}$, or

  4. (iv)

$\sigma=1$, $d\geq 2$, and $V\in H^{\beta}$ for some $\beta>\frac{d}{2}$.

In particular, this assumption implies that $Vu\in H^{\sigma}$ for any $u\in
H^{\sigma}$ and $\|Vu\|_{H^{\sigma}}\leq C_{V}\|u\|_{H^{\sigma}}$ for some
constant $C_{V}\geq 0$ depending on $V$. This follows from the algebra
property of $H^{\sigma}$ in case (i). Note that while (i) is taken verbatim
from [2, Prop. 4.1], cases (ii) and (iv) assume less regularity in our
assumption and case (iii) is new. In the second case (ii), Hölder’s inequality
and the Sobolev embedding $H^{\beta}\hookrightarrow L^{\infty}$ for
$\beta>\frac{d}{2}$ yield

$\|Vu\|_{L^{2}}\leq\|V\|_{L^{\infty}}\|u\|_{L^{2}}\lesssim\|V\|_{H^{\beta}}\|u\|_{L^{2}}$

in the case (ii), see [2, Prop. 4.1]. The case (iii) is covered by Lemma 3.6
below. Lastly, $\|Vu\|_{H^{1}}\lesssim\|u\|_{H^{1}}$ in the case (iv) follows
from Hölder’s inequality, once with $p=2\beta$ and
$q=\frac{4\beta}{2\beta-2}$, $\beta>1$, and the embeddings
$H^{\beta}\hookrightarrow L^{\infty}$, $H^{1}\hookrightarrow L^{q}$, as well
as $H^{\beta}\hookrightarrow H^{1,2\beta}$ via

$\displaystyle\|Vu\|_{H^{1}}^{2}$
$\displaystyle\lesssim\|Vu\|_{L^{2}}^{2}+\|Vu^{\prime}\|_{L^{2}}^{2}+\|V^{\prime}u\|_{L^{2}}^{2}$
$\displaystyle\leq\|V\|_{L^{\infty}}^{2}(\|u\|_{L^{2}}^{2}+\|u^{\prime}\|_{L^{2}}^{2})+\|V^{\prime}\|_{L^{2\beta}}^{2}\|u\|_{L^{q}}^{2}$
$\displaystyle\lesssim(\|V\|_{H^{\beta}}^{2}+\|V\|_{H^{1,2\beta}}^{2})\|u\|_{H^{1}}^{2}\lesssim\|V\|_{H^{\beta}}^{2}\|u\|_{H^{1}}^{2}.$

Hence, multiplication by $V$ is a bounded operator on $H^{\sigma}$ if
Assumption 3.5 holds.

######  Lemma 3.6.

Let $\sigma\in(0,1)$, $d\in\mathbb{N}$ such that $d>2\sigma$, and $V\in
H^{\beta}(\mathbb{R}^{d})$ for some $\beta>\frac{d}{2}$. Then
$\|Vu\|_{H^{\sigma}}\leq C_{V}\|u\|_{H^{\sigma}}$ for some constant $C_{V}\geq
0$ for all $u\in H^{\sigma}(\mathbb{R}^{d})$.

###### Proof.

Let $q_{1}=\frac{2d}{d-2\sigma}$ and $q_{2}=\frac{d}{\sigma}$. Then
$\frac{1}{q_{1}}+\frac{1}{q_{2}}=\frac{1}{2}$ and $q_{1}<\infty$ because
$d>2\sigma$. By classical Sobolev and Bessel potential space embeddings [9,
Thm. 6.5.1], $H^{d/2}\hookrightarrow H^{\sigma,q_{2}}$,
$H^{\sigma}\hookrightarrow L^{q_{1}}$, and $H^{\beta}\hookrightarrow
C_{b}(\mathbb{R}^{d})\hookrightarrow L^{\infty}$. Thus, an application of the
product estimate [65, Prop. 2.1.1] yields

$\displaystyle\|Vu\|_{H^{\sigma}}$
$\displaystyle\lesssim\|V\|_{H^{\sigma,q_{2}}}\|u\|_{L^{q_{1}}}+\|V\|_{L^{\infty}}\|u\|_{H^{\sigma}}\lesssim(\|V\|_{H^{d/2}}+\|V\|_{H^{\beta}})\|u\|_{H^{\sigma}}\lesssim\|V\|_{H^{\beta}}\|u\|_{H^{\sigma}}.\qed$

Since $-\mathrm{i}\Delta$ generates a contractive semigroup [2, Lemma 2.1],
its bounded perturbation $-\mathrm{i}(\Delta+V)$ generates a quasi-contractive
semigroup [31, Thm. III.1.3]. Thus, we are in the setting of Subsection 3.2.
Global existence and uniqueness of mild solutions $U\in
L^{p}(\Omega;C([0,T];H^{\sigma}))$ to (3.12) in $H^{\sigma}$ are guaranteed
provided that $p\in[2,\infty)$, $u_{0}\in
L_{\mathcal{F}_{0}}^{p}(\Omega;H^{\sigma})$,
$Q^{1/2}\in\mathcal{L}_{2}(L^{2},H^{\sigma})$, and Assumption 3.5 holds.

Therefore, the Schrödinger equation (3.12) can be rewritten in the form of
(3.1) on $X=H^{\sigma}$ with an $H$-cylindrical Brownian motion $W_{H}$ for
$H=L^{2}$.

For exponential Euler, we recover the error bound from [2, Thm. 4.3], showing
convergence of rate $1$ in the case of sufficiently regular $Q^{1/2}$ under
less regularity assumptions on $V$. Moreover, under weaker regularity
assumptions on $Q^{1/2}$ and $V$, we additionally provide an error bound for
fractional convergence rates $\alpha\in(0,1]$.

######  Theorem 3.7.

Let $\sigma\geq 0$, $d\in\mathbb{N}$, and $V\in L^{2}$ satisfy Assumption 3.5,
and let $p\in[2,\infty)$. Assume that $u_{0}\in
L_{\mathcal{F}_{0}}^{p}(\Omega;H^{\sigma})$ and
$Q^{1/2}\in\mathcal{L}_{2}(L^{2},H^{\sigma+2\alpha})$ for some
$\alpha\in(0,1]$. Denote by $U$ the mild solution of the linear stochastic
Schrödinger equation with additive noise (3.12) and by
$(U^{j})_{j=0,\ldots,N_{k}}$ the temporal approximations as defined in (3.3)
obtained with the exponential Euler method $R\coloneqq S$. Then there exists a
constant $C\geq 0$ depending on $(V,u_{0},T,p,\alpha,\sigma,d)$ such that for
$N_{k}\geq 2$

$\left\|\max_{0\leq j\leq
N_{k}}\|U(t_{j})-U^{j}\|_{H^{\sigma}}\right\|_{p}\leq
C\|Q^{1/2}\|_{\mathcal{L}_{2}(L^{2},H^{\sigma+2\alpha})}k^{\alpha}.$

###### Proof.

As discussed above, $A=-\mathrm{i}(\Delta+V)$ generates a quasi-contractive
semigroup on $H^{\sigma}$. Furthermore, setting $g=-\mathrm{i}Q^{1/2}$ allows
us to rewrite (3.12) in the form of a stochastic evolution equation (3.1).
Thus, Theorem 3.3 is applicable with $X=H^{\sigma}$ and $H=L^{2}$. It remains
to check that $g\in
L_{\mathcal{P}}^{p}(\Omega;C([0,T];{\mathcal{L}_{2}(H,Y)}))$ for some
$Y\hookrightarrow D_{A}(\alpha,\infty)$ and that $g\in
L^{p}_{\mathcal{P}}(\Omega;C^{\alpha}([0,T];{\mathcal{L}_{2}(H,X)}))$. The
latter holds for any $\alpha\in(0,1]$ due to $g$ being constant in time.
Taking
$Y=H^{\sigma+2\alpha}=(H^{\sigma},H^{\sigma+2})_{\alpha,2}=(H^{\sigma},D(A))_{\alpha,2}\hookrightarrow(H^{\sigma},D(A))_{\alpha,\infty}$,
the first condition is satisfied as well. Corollary 3.2 yields the desired
error bound. ∎

Furthermore, Theorem 3.1 enables us to extend [2, Thm. 4.3] to general
discretisation schemes $R$ involving rational approximations, at the price of
an additional logarithmic factor. We state it for implicit Euler and Crank-
Nicolson.

######  Theorem 3.8.

Let $\sigma\geq 0$, $d\in\mathbb{N}$, and $V\in L^{2}$ satisfy Assumption 3.5,
and let $p\in[2,\infty)$. Let $(R_{k})_{k>0}$ be the implicit Euler method
(IE) or the Crank-Nicolson method (CN) and set $\ell=4$ or $\ell=3$,
respectively. Assume that $u_{0}\in
L_{\mathcal{F}_{0}}^{p}(\Omega;H^{\sigma+\ell\alpha})$ and
$Q^{1/2}\in\mathcal{L}_{2}(L^{2},H^{\sigma+\ell\alpha})$ for some
$\alpha\in(0,1]$. Denote by $U$ the mild solution of the linear stochastic
Schrödinger equation with additive noise (3.12) and by
$(U^{j})_{j=0,\ldots,N_{k}}$ the temporal approximations as defined in (3.3).
Then there exists a constant $C\geq 0$ depending on
$(V,u_{0},T,p,\alpha,\sigma,d,\ell)$ such that for $N_{k}\geq 2$

$\left\|\max_{0\leq j\leq
N_{k}}\|U(t_{j})-U^{j}\|_{H^{\sigma}}\right\|_{p}\leq
C\|Q^{1/2}\|_{\mathcal{L}_{2}(L^{2},H^{\sigma+\ell\alpha})}\log\left(\frac{T}{k}\right)k^{\alpha}.$

###### Proof.

This follows from Theorem 3.1 noting that (IE) approximates $S$ to order
$\alpha$ on $D((-A)^{2\alpha})$ and this fractional domain is given by
$D((\mathrm{i}\Delta)^{2\alpha})=[H^{\sigma},H^{\sigma+2}]_{2\alpha}=H^{\sigma+4\alpha}$,
which is chosen as the space $Y$. Here, $[\cdot,\cdot]_{\alpha}$ denotes the
complex interpolation space with parameter $\alpha$. Likewise, (CN)
approximates $S$ to order $\alpha$ on
$D((-A)^{3\alpha/2})=[H^{\sigma},H^{\sigma+2}]_{3\alpha/2}=H^{\sigma+3\alpha}$.
∎

Comparing this result to Theorem 3.7 for exponential Euler (EE), it becomes
apparent that lower-order schemes like (IE) need higher regularity of the
noise $Q^{1/2}$ to achieve the same rate of convergence
($\mathcal{L}_{2}(L^{2},H^{\sigma+4\alpha})$ compared to
$\mathcal{L}_{2}(L^{2},H^{\sigma+2\alpha})$). For instance, for
$Q^{1/2}\in\mathcal{L}_{2}(L^{2},H^{\sigma+2})$, the rates for (EE), (CN), and
(IE) are $1$, $\frac{2}{3}$, and $\frac{1}{2}$, respectively. If
$Q^{1/2}\in\mathcal{L}_{2}(L^{2},H^{\sigma+3})$, (EE) and (CN) have the same
convergence rates up to a logarithmic factor, and if
$Q^{1/2}\in\mathcal{L}_{2}(L^{2},H^{\sigma+4})$, so does (IE), all provided
that $V$ and $u_{0}$ are sufficiently smooth.

Note that in the absence of a potential, the same convergence rates are
obtained without any limitation on the dimension $d\in\mathbb{N}$ in terms of
the parameter $\sigma$. An analogue of Theorem 3.8 can be obtained for other
implicit Runge-Kutta methods if the space is known on which the scheme
approximates the semigroup to a given order.

##  4\. Well-posedness

We consider the stochastic evolution equation with multiplicative noise

(4.1)
$\displaystyle\Bigg{\\{}\begin{split}\mathrm{d}U&=(AU+F(t,U))\,\mathrm{d}t+G(t,U)\,\mathrm{d}W_{H}~{}~{}~{}\text{
on }[0,T],\\\ U(0)&=u_{0}\in L_{\mathscr{F}_{0}}^{p}(\Omega;X)\end{split}$

for $1\leq p<\infty$ and $A$ generating a $C_{0}$-semigroup $(S(t))_{t\geq 0}$
of contractions on $X$. In this section, we present progressive measurability,
linear growth and global Lipschitz conditions on $F$ and $G$ ensuring the
well-posedness of the above equation.

######  Assumption 4.1.

Let $X$ be a Hilbert space and let $p\in[2,\infty)$. Let
$F:\Omega\times[0,T]\times X\to
X,F(\omega,t,x)=\tilde{F}(\omega,t,x)+f(\omega,t)$ and
$G:\Omega\times[0,T]\times
X\to{\mathcal{L}_{2}(H,X)},G(\omega,t,x)=\tilde{G}(\omega,t,x)+g(\omega,t)$ be
strongly $\mathcal{P}\otimes\mathcal{B}(X)$-measurable, and such that
$\tilde{F}(\cdot,\cdot,0)=0$ and $\tilde{G}(\cdot,\cdot,0)=0$, and suppose

  1. (a)

_(global Lipschitz continuity on $X$)_ there exist constants
$C_{F,X},C_{G,X}\geq 0$ such that for all $\omega\in\Omega,t\in[0,T]$ and
$x,y\in X$, it holds that

$\displaystyle\|\tilde{F}(\omega,t,x)-\tilde{F}(\omega,t,y)\|$
$\displaystyle\leq C_{F,X}\|x-y\|,$
$\displaystyle\|\tilde{G}(\omega,t,x)-\tilde{G}(\omega,t,y)\|_{\mathcal{L}_{2}(H,X)}$
$\displaystyle\leq C_{G,X}\|x-y\|,$

  2. (b)

_(integrability)_ $f\in L^{p}_{\mathcal{P}}(\Omega;L^{1}(0,T;X))$ and $g\in
L^{p}_{\mathcal{P}}(\Omega;L^{2}(0,T;{\mathcal{L}_{2}(H,X)}))$.

Note that Assumption 4.1 implies linear growth of $F$ and $G$:

(4.2) $\|\tilde{F}(\omega,t,x)\|\leq
C_{F,X}(1+\|x\|),~{}\|\tilde{G}(\omega,t,x)\|_{\mathcal{L}_{2}(H,X)}\leq
C_{G,X}(1+\|x\|),$

where the constant $1$ can be left out, but is included for later use in
Theorem 4.4.

Well-posedness shall be understood in the sense of existence and uniqueness of
mild solutions to (4.1). Denote by $L^{0}(\Omega;V)$ the space of all strongly
measurable $V$-valued random variables for Banach spaces $V$.

######  Definition 4.2.

A $U\in L^{0}_{\mathcal{P}}(\Omega;C([0,T];X))$ is called a _mild solution_ to
(4.1) if a.s. for all $t\in[0,T]$

$U(t)=S(t)u_{0}+\int_{0}^{t}S(t-s)F(s,U(s))\,\mathrm{d}s+\int_{0}^{t}S(t-s)G(s,U(s))\,\mathrm{d}W_{H}(s).$

The following well-posedness result is more or less standard [26, Chapters
6,7].

######  Theorem 4.3.

Suppose that Assumption 4.1 holds for some $p\in[2,\infty)$. Let $A$ be the
generator of a $C_{0}$-contraction semigroup $(S(t))_{t\geq 0}$ on $X$. Let
$u_{0}\in L_{\mathcal{F}_{0}}^{p}(\Omega;X)$. Then (4.1) has a unique mild
solution $U\in L^{p}(\Omega;C([0,T];X))$. Moreover,

$\displaystyle\|U\|_{L^{p}(\Omega;C([0,T];X))}\leq C_{\text{bdd}}^{X}\Big{(}$
$\displaystyle
1+\|u_{0}\|_{L^{p}(\Omega;X)}+\|f\|_{L^{p}(\Omega;L^{1}(0,T;X))}+B_{p}\|g\|_{L^{p}(\Omega;L^{2}(0,T;{\mathcal{L}_{2}(H,X)}))}\Big{)},$

where $C_{\text{bdd}}^{X}\coloneqq(1+C^{2}T)^{1/2}\mathrm{e}^{(1+C^{2}T)/2}$
with $C\coloneqq C_{F,X}T^{1/2}+B_{p}C_{G,X}$, and $B_{p}$ is the constant
from Theorem 2.2.

###### Proof.

First, the local existence and uniqueness of solutions are to be proven.
Second, local solutions are concatenated to obtain global existence and
uniqueness. We only sketch the steps. Let $\delta\in(0,T]$. Define the spaces
$Z_{\delta}\coloneqq L^{p}(\Omega;C([0,\delta];X))$, $Z\coloneqq Z_{T}$,
$Z_{\delta}^{\mathcal{P}}$ as the subset of all adapted $v\in Z_{\delta}$ and
$Z^{\mathcal{P}}\coloneqq Z_{T}^{\mathcal{P}}$. For $v\in
Z_{\delta}^{\mathcal{P}}$, we define the fixed point functional

(4.3) $\Gamma v(t)\coloneqq
S(t)u_{0}+\int_{0}^{t}S(t-s)F(s,v(s))\,\mathrm{d}s+\int_{0}^{t}S(t-s)G(s,v(s))\,\mathrm{d}W_{H}(s).$

The problem of finding local mild solutions of (4.1) then reduces to finding
fixed points $v\in Z_{\delta}^{\mathcal{P}}$ of $\Gamma$. The contraction
mapping theorem yields such unique fixed points provided that $\Gamma$ is a
contraction which maps $Z^{\mathcal{P}}$ and thus $Z_{\delta}^{\mathcal{P}}$
into itself. That is, (i) continuity of paths of $\Gamma v$ and maximal
estimates for $v\in Z_{\delta}^{\mathcal{P}}$ (see Theorem 2.2) as well as
(ii) adaptedness of $\Gamma v$, and that (iii) $\Gamma$ is a (strict)
contraction on $Z_{\delta}^{\mathcal{P}}$. Lastly, we consider the evolution
equation on $[\delta,2\delta]$ with initial value $U(\delta)$ to extend the
solution to larger time intervals.

It remains to prove the a priori estimate for the mild solution $U$. Let
$r\in[0,T]$. Let $\psi(r)=1+\left\|\sup_{t\in[0,r]}\|U(t)\|\right\|_{p}$. From
the triangle inequality, Theorem 2.2 and (4.2) we see that

$\displaystyle\psi(r)$ $\displaystyle\leq
1+\|u_{0}\|_{L^{p}(\Omega;X)}+C_{F,X}\left\|\int_{0}^{r}1+\|U(s)\|\,\mathrm{d}s\right\|_{p}+\left\|f\right\|_{L^{p}(\Omega;L^{1}(0,r;X))}$
$\displaystyle\
+B_{p}\left[C_{G,X}\left\|\left(\int_{0}^{r}(1+\|U(s)\|)^{2}ds\right)^{1/2}\right\|_{p}\,\mathrm{d}s+\|g\|_{L^{p}(\Omega;L^{2}(0,r;{\mathcal{L}_{2}(H,X)}))}\right]$
$\displaystyle\leq
c_{u_{0},f,g}+C_{F,X}\int_{0}^{r}\psi(s)\,\mathrm{d}s+B_{p}C_{G,X}\left(\int_{0}^{r}\psi(s)^{2}\,\mathrm{d}s\right)^{1/2}$
$\displaystyle\leq
c_{u_{0},f,g}+C\left(\int_{0}^{r}\psi(s)^{2}\,\mathrm{d}s\right)^{1/2},$

where
$c_{u_{0},f,g}=1+\|u_{0}\|_{L^{p}(\Omega;X)}+\|f\|_{L^{p}(\Omega;L^{1}(0,T;X))}+B_{p}\|g\|_{L^{p}(\Omega;L^{2}(0,T;{\mathcal{L}_{2}(H,X)}))}$
and $C=C_{F,X}T^{1/2}+B_{p}C_{G,X}$. Here we used Minkowski’s inequality to
pull in the $L^{p}(\Omega)$ and $L^{p/2}(\Omega)$ norms. Lastly, the version
of Gronwall’s inequality from Lemma 2.6 yields the desired result

$\psi(T)\leq c_{u_{0},f,g}(1+C^{2}T)^{1/2}e^{(1+C^{2}T)/2}.\quad\qed$

Lastly, we present a well-posedness result on subspaces $Y\hookrightarrow X$
which does not require Lipschitz continuity of $\tilde{F},\tilde{G}$ on $Y$
but merely linear growth. The reader is referred to Remark 4.5 below for a
discussion where we explain why Lipschitz continuity on $Y$ should be avoided.

######  Theorem 4.4.

Suppose that Assumption 4.1 holds. Let $Y\hookrightarrow X$ be a Hilbert space
and $A$ the generator of a $C_{0}$-contraction semigroup $(S(t))_{t\geq 0}$ on
both $X$ and $Y$. Let $p\in[2,\infty)$ and $u_{0}\in
L_{\mathcal{F}_{0}}^{p}(\Omega;Y)$. Additionally, suppose that $f\in
L^{p}_{\mathcal{P}}(\Omega;L^{1}(0,T;Y))$, $g\in
L_{\mathcal{P}}^{p}(\Omega;L^{2}(0,T;{\mathcal{L}_{2}(H,Y)}))$,
$F:\Omega\times[0,T]\times Y\to Y$, $G:\Omega\times[0,T]\times
Y\to{\mathcal{L}_{2}(H,Y)}$ are strongly
$\mathcal{P}\otimes\mathcal{B}(Y)$-measurable, and there are
$L_{F,Y},L_{G,Y}\geq 0$ such that for all $\omega\in\Omega$, $t\in[0,T]$, and
$x\in Y$,

$\|\tilde{F}(\omega,t,x)\|_{Y}\leq
L_{F,Y}(1+\|x\|_{Y}),~{}\|\tilde{G}(\omega,t,x)\|_{\mathcal{L}_{2}(H,Y)}\leq
L_{G,Y}(1+\|x\|_{Y}).$

Under these conditions the mild solution $U\in L^{p}(\Omega;C([0,T];X))$ to
(4.1) is in $L^{p}(\Omega;C([0,T];Y))$ and

$\displaystyle\|U\|_{L^{p}(\Omega;C([0,T];Y))}\leq C_{\text{bdd}}^{Y}\Big{(}$
$\displaystyle
1+\|u_{0}\|_{L^{p}(\Omega;Y)}+\|f\|_{L^{p}(\Omega;L^{1}(0,T;Y))}+B_{p}\|g\|_{L^{p}(\Omega;L^{2}(0,T;{\mathcal{L}_{2}(H,Y)}))}\Big{)},$

where $C_{\text{bdd}}^{Y}\coloneqq(1+C^{2}T)^{1/2}\mathrm{e}^{(1+C^{2}T)/2}$
with $C\coloneqq L_{F,Y}T^{1/2}+B_{p}L_{G,Y}$, and $B_{p}$ is the constant
from Theorem 2.2.

The constant $C$ appears exponentially in the above. In the special case
$p=2$, $L_{F,Y}=L_{G,Y}=T=1$, this leads to
$C_{\text{bdd}}^{Y}\leq\sqrt{10}e^{5}\leq 470$.

###### Proof.

Recall that by Banach’s fixed point theorem for $\delta\leq T_{0}$, where
$T_{0}\in(0,1]$ only depends on $p$, $C_{F,X}$, $C_{G,X}$ and $X$, one has
$U=\lim_{n\to\infty}U_{n}$ in $L^{p}(\Omega;C([0,\delta];X))$, where
$U_{0}=u_{0}$ and $U_{n+1}=\Gamma(U_{n})$ with $\Gamma$ as defined in (4.3).
Since $F$ and $G$ map $Y$ into $Y$, we can also consider $\Gamma$ as a mapping
on $Z^{2}\coloneqq L_{\mathcal{P}}^{p}(\Omega;L^{2}(0,\delta;Y))$ to
eventually show that $U$ is in
$L^{p}_{\mathcal{P}}(\Omega;C([0,\delta];Y))\subseteq Z^{2}$. Note that for
$U\in Z^{2}$, $F(\cdot,U)$ and $G(\cdot,U)$ are progressively measurable as
$Y$ and ${\mathcal{L}_{2}(H,Y)}$-valued mappings by [39, Theorem 1.1.6].
Moreover, we claim that for all $v\in Z^{2}$,

$\displaystyle\|\Gamma(v)\|_{L^{p}(\Omega;C([0,\delta];Y))}$
$\displaystyle\leq\|u_{0}\|_{L^{p}(\Omega;Y)}+\|f\|_{L^{p}(\Omega;L^{1}(0,\delta;Y))}$
(4.4)
$\displaystyle\phantom{\leq}+B_{p}\|g\|_{L^{p}(\Omega;L^{2}(0,\delta;{\mathcal{L}_{2}(H,Y)}))}+\big{(}L_{F,Y}+B_{p}L_{G,Y}\big{)}(1+\|v\|_{Z^{2}}).$

Indeed, since $S$ is contractive, the maximal inequality, linear growth of
$\tilde{F}$ and $\tilde{G}$ on $Y$, and $\delta\leq 1$ imply

$\displaystyle\|\Gamma(v)-S(\cdot)u_{0}\|_{L^{p}(\Omega;C([0,\delta];Y))}$
$\displaystyle\leq\|F(\cdot,v)\|_{L^{p}(\Omega;L^{1}(0,\delta;Y))}+B_{p}\|G(\cdot,v)\|_{L^{p}(\Omega;L^{2}(0,\delta;{\mathcal{L}_{2}(H,Y)}))}$
$\displaystyle\leq\|f\|_{L^{p}(\Omega;L^{1}(0,\delta;Y))}+L_{F,Y}\left(\delta+\|v\|_{L^{p}(\Omega;L^{1}(0,\delta;Y))}\right)$
$\displaystyle\phantom{\leq}+B_{p}\left(\|g\|_{L^{p}(\Omega;L^{2}(0,\delta;{\mathcal{L}_{2}(H,Y)}))}+L_{G,Y}\left(\sqrt{\delta}+\|v\|_{L^{p}(\Omega;L^{2}(0,\delta;Y))}\right)\right)$
$\displaystyle\leq\|f\|_{L^{p}(\Omega;L^{1}(0,\delta;Y))}+B_{p}\|g\|_{L^{p}(\Omega;L^{2}(0,\delta;{\mathcal{L}_{2}(H,Y)}))}$
$\displaystyle\phantom{\leq}+\big{(}L_{F,Y}+B_{p}L_{G,Y}\big{)}\left(1+\|v\|_{Z^{2}}\right).$

Therefore, (4) follows. Now (4) implies

$\displaystyle\|\Gamma(v)\|_{Z^{2}}$
$\displaystyle\leq\delta^{1/2}\|\Gamma(v)\|_{L^{p}(\Omega;C([0,\delta];Y))}$
$\displaystyle\leq\theta(1+\|u_{0}\|_{L^{p}(\Omega;Y)}+\|f\|_{L^{p}(\Omega;L^{1}(0,\delta;Y))}+\|g\|_{L^{p}(\Omega;L^{2}(0,\delta;{\mathcal{L}_{2}(H,Y)}))}+\|v\|_{Z^{2}}),$

where $\theta=\delta^{1/2}\max\\{1,B_{p},L_{F,Y}+B_{p}L_{G,Y}\\}$. Choosing
$\delta\in(0,T_{0}]$ such that $\theta\leq\frac{1}{2}$, iteratively we obtain
that for $n\geq 1$,

$\displaystyle\|U_{n}\|_{Z^{2}}$
$\displaystyle\leq\theta(1+\|u_{0}\|_{L^{p}(\Omega;Y)}+\|f\|_{L^{p}(\Omega;L^{1}(0,\delta;Y))}+\|g\|_{L^{p}(\Omega;L^{2}(0,\delta;{\mathcal{L}_{2}(H,Y)}))})+\theta\|U_{n-1}\|_{Z^{2}}$
$\displaystyle\leq\theta(1+\|u_{0}\|_{L^{p}(\Omega;Y)}+\|f\|_{L^{p}(\Omega;L^{1}(0,\delta;Y))}+\|g\|_{L^{p}(\Omega;L^{2}(0,\delta;{\mathcal{L}_{2}(H,Y)}))})$
$\displaystyle\phantom{\leq}+\theta^{2}(1+\|u_{0}\|_{L^{p}(\Omega;Y)}+\|f\|_{L^{p}(\Omega;L^{1}(0,\delta;Y))}+\|g\|_{L^{p}(\Omega;L^{2}(0,\delta;{\mathcal{L}_{2}(H,Y)}))}+\|U_{n-2}\|_{Z^{2}})$
$\displaystyle\leq\ldots\leq\sum_{j=1}^{n}\theta^{j}(1+\|u_{0}\|_{L^{p}(\Omega;Y)}+\|f\|_{L^{p}(\Omega;L^{1}(0,\delta;Y))}+\|g\|_{L^{p}(\Omega;L^{2}(0,\delta;{\mathcal{L}_{2}(H,Y)}))})+\theta^{n}\|U_{0}\|_{Z^{2}}$
$\displaystyle\leq
1+\|f\|_{L^{p}(\Omega;L^{1}(0,\delta;Y))}+\|g\|_{L^{p}(\Omega;L^{2}(0,\delta;{\mathcal{L}_{2}(H,Y)}))}+2\|u_{0}\|_{L^{p}(\Omega;Y)}.$

In conclusion, $(U_{n})_{n\in\mathbb{N}}$ is bounded in $Z^{2}$. By
reflexivity of $Y$, and thus of $Z^{2}$ (see [39, Corollary 1.3.22]), there is
a subsequence $(U_{n_{j}})_{j\in\mathbb{N}}$ and $V\in Z^{2}$ such that
$U_{n_{j}}\to V$ weakly in $Z^{2}$ and

(4.5) $\displaystyle\|V\|_{Z^{2}}\leq
1+\|f\|_{L^{p}(\Omega;L^{1}(0,\delta;Y))}+\|g\|_{L^{p}(\Omega;L^{2}(0,\delta;{\mathcal{L}_{2}(H,Y)}))}+2\|u_{0}\|_{L^{p}(\Omega;Y)}.$

Since $U_{n}\to U$ in $L^{p}(\Omega;C([0,\delta];X))$, it follows that $V=U$.
Since $U=\Gamma(U)$, (4) and (4.5) give that $U$ is in
$L^{p}(\Omega;C([0,\delta];Y))$. The same argument can be applied on
$[j\delta,(j+1)\delta]$ using the initial value $U(j\delta)\in
L^{p}(\Omega;Y)$ for $j=1,2,\ldots$ to obtain the statement on $[0,T]$.

The final a priori estimate follows as in Theorem 4.3, where we note that the
Lipschitz conditions on $F$ and $G$ were not used in the estimate. ∎

######  Remark 4.5.

In applications, one often takes $X=L^{2}(O)$ and $Y=H^{1}(O)$ with
$O\subseteq\mathbb{R}^{d}$, and $F$ is a Nemtskij operator for a given
nonlinearity $\phi:\mathbb{R}\to\mathbb{R}$, i.e. $F(x)(\xi)=\phi(x(\xi))$ for
$x\in L^{2}(O)$ and $\xi\in O$. Lipschitz continuity of such mappings holds
for $F$ seen as a mapping from $X$ to $X$ if $\phi$ is Lipschitz. Also, linear
growth holds for $F$ as a mapping from $Y$ into $Y$ if $\phi$ is Lipschitz. A
less trivial fact is that $F$ is continuous from $Y$ into $Y$ (see [65,
Proposition 2.6.4]), but nothing more can be expected. For instance, Lipschitz
continuity of $F:Y\to Y$ would require the estimate

$\|\phi^{\prime}(x)x^{\prime}-\phi^{\prime}(y)y^{\prime}\|_{L^{2}(O)}\leq
C\|x-y\|_{H^{1}(O)}.$

The latter is true if and only if
$\|(\phi^{\prime}(x)-\phi^{\prime}(y))x^{\prime}\|_{L^{2}(O)}\leq\tilde{C}\|x-y\|_{H^{1}(O)}$.
This cannot be expected even if $\phi\in C^{\infty}(\mathbb{R}^{d})$ with
bounded derivatives. Indeed, a product of $x-y$ and $x^{\prime}$ needs to be
estimated, but this cannot be done in terms of $\|x-y\|_{H^{1}(O)}$.
Similarly, problems would occur for $Y=H^{\alpha}(O)$ for other values of
$\alpha>0$. For a detailed exposition which estimates can be expected for
$\phi(x)-\phi(y)$, the reader is referred to [65, Section 2.7].

##  5\. Stability

Before analysing the convergence of temporal approximations to solutions of
the stochastic evolution equation (4.1) with multiplicative noise, the
question of stability of time discretisation schemes arises. We aim to prove
the stability of contractive time discretisation schemes under linear growth
assumptions on $F$ and $G$, and contractivity conditions on the scheme $R$. We
formulate the result for mappings on $X$, but they will also be applied on $Y$
later on.

Let $R_{k}:X\to X$ be a contractive time discretisation scheme with time step
$k>0$ on a uniform grid $\\{t_{j}=jk:~{}j=0,\ldots,N_{k}\\}\subseteq[0,T]$
with final time $T=t_{N_{k}}>0$ and $N_{k}=\frac{T}{k}\in\mathbb{N}$ being the
number of time steps. We consider the temporal approximations of the mild
solution to (4.1) given by $U^{0}\coloneqq u_{0}$ and

(5.1) $\displaystyle U^{j}$ $\displaystyle\coloneqq
R_{k}U^{j-1}+kR_{k}F(t_{j-1},U^{j-1})+R_{k}G(t_{j-1},U^{j-1})\Delta W_{j}$

with Wiener increments $\Delta W_{j}\coloneqq W_{H}(t_{j})-W_{H}(t_{j-1})$
(see (2.1)) for $1\leq j\leq N_{k}$. The above definition of $U^{j}$ can be
reformulated as the discrete variation-of-constants formula

(5.2)
$U^{j}=R_{k}^{j}u_{0}+k\sum_{i=0}^{j-1}R_{k}^{j-i}F(t_{i},U^{i})+\sum_{i=0}^{j-1}R_{k}^{j-i}G(t_{i},U^{i})\Delta
W_{i+1}$

for $j=0,\ldots,N_{k}$.

######  Proposition 5.1 (Stability).

Let $X$ be a Hilbert space, $p\in[2,\infty)$ and $u_{0}\in
L_{\mathcal{F}_{0}}^{p}(\Omega;X)$. Suppose that $F:\Omega\times[0,T]\times
X\to X$, $G:\Omega\times[0,T]\times X\to{\mathcal{L}_{2}(H,X)}$ are strongly
$\mathcal{P}\otimes\mathcal{B}(X)$-measurable, where $F=\tilde{F}+f$ and
$G=\tilde{G}+g$, $f\in L^{p}_{\mathcal{P}}(\Omega;C([0,T];X))$, $g\in
L_{\mathcal{P}}^{p}(\Omega;C([0,T];{\mathcal{L}_{2}(H,X)}))$, and there are
$L_{F,X},L_{G,X}\geq 0$ such that for all $\omega\in\Omega$, $t\in[0,T]$ and
$x\in X$,

$\|\tilde{F}(\omega,t,x)\|_{X}\leq
L_{F,X}(1+\|x\|_{X}),~{}\|\tilde{G}(\omega,t,x)\|_{\mathcal{L}_{2}(H,X)}\leq
L_{G,X}(1+\|x\|_{X}).$

Let $(R_{k})_{k>0}$ be a contractive time discretisation scheme and $N_{k}\geq
2$. Then the temporal approximations $(U^{j})_{j=0,\ldots,N_{k}}$ obtained via
(5.1) are stable in the sense of

$1+\left\|\max_{0\leq j\leq N_{k}}\|U^{j}\|\right\|_{p}\leq
C_{\text{stab}}c_{u_{0},f,g,T},$

where $C_{\text{stab}}\coloneqq(1+C^{2}T)^{1/2}e^{(1+C^{2}T)/2}$ with
$C\coloneqq L_{F,X}T^{1/2}+B_{p}L_{G,X}$,

$c_{u_{0},f,g,T}\coloneqq
1+\|u_{0}\|_{L^{p}(\Omega;X)}+\|f\|_{L^{p}(\Omega;C([0,T];X))}T+\|g\|_{L^{p}(\Omega;C([0,T];{\mathcal{L}_{2}(H,X)}))}B_{p}T^{1/2},$

and $B_{p}$ is the constant from Theorem 2.2.

Examples for contractive schemes include exponential Euler, implicit Euler,
Crank-Nicolson, and $A$-stable higher-order implicit Runge-Kutta methods such
as Radau methods, BDF(2), Lobatto IIA, IIB, and IIC (see Proposition 2.5).

The exponential dependence in Proposition 5.1 comes from an application of
Gronwall’s inequality. Therefore, to make the result suitable for numerical
applications, some optimization of the constants was necessary. In the special
case that $L_{F,X}=L_{G,X}=T=1$, and $p=2$ one can check that
$C_{\text{stab}}=\sqrt{10}e^{5}\leq 470$ which seems a reasonable constant for
error estimates in applications. Later on, we will also apply Proposition 5.1
in case the space $X$ is replaced by $Y$ in the setting of Section 4.

###### Proof.

Let $\varphi_{N}\coloneqq 1+\|\max_{0\leq j\leq N}\|U^{j}\|\|_{p}$ and
$N\in\\{0,\ldots,N_{k}\\}$. Then the variation-of-constants formula (5.2) and
contractivity of $R_{k}$ allow us to bound

$\displaystyle\varphi_{N}$ $\displaystyle\leq
1+\|u_{0}\|_{L^{p}(\Omega;X)}+k\sum_{i=0}^{N-1}\left\|\max_{0\leq j\leq
i}\|F(t_{j},U^{j})\|\right\|_{p}$ (5.3)
$\displaystyle\phantom{\leq}+\left\|\max_{0\leq j\leq
N}\left\|\sum_{i=0}^{j-1}R_{k}^{j-i}G(t_{i},U^{i})\Delta
W_{i+1}\right\|\right\|_{p}.$

Invoking linear growth of $\tilde{F}$ and boundedness of $f$ for the third
term, we obtain the bound

$\displaystyle k\sum_{i=0}^{N-1}\left\|\max_{0\leq j\leq
i}\|F(t_{j},U^{j})\|\right\|_{p}\leq k\sum_{i=0}^{N-1}\left\|\max_{0\leq j\leq
i}\left(L_{F,X}\left(1+\|U^{j}\|\right)+\|f(t_{j})\|\right)\right\|_{p}$
$\displaystyle\leq k\sum_{i=0}^{N-1}\left(L_{F,X}\left(1+\left\|\max_{0\leq
j\leq i}\|U^{j}\|\right\|_{p}\right)+\|f\|_{L^{p}(\Omega;C([0,T];X))}\right)$
(5.4) $\displaystyle=C_{1,f}t_{N}+L_{F,X}k\sum_{i=0}^{N-1}\varphi_{i}\leq
C_{1,f}t_{N}+L_{F,X}t_{N}^{1/2}\left(k\sum_{i=0}^{N-1}\varphi_{i}^{2}\right)^{1/2},$

where we have set $C_{1,f}\coloneqq\|f\|_{L^{p}(\Omega;C([0,T];X))}$, and used
the Cauchy-Schwarz inequality and $Nk=t_{N}$ in the last line. It remains to
bound the last term in (5).

Since $R_{k}$ is a contraction, by the Sz.-Nagy dilation theorem [63, Theorem
I.4.2] we can find a Hilbert space $\widetilde{X}$, a contractive injection
$Q:X\to\widetilde{X}$, a contractive projection $P:\widetilde{X}\to X$, and a
unitary $\widetilde{R}_{k}$ on $\widetilde{X}$ such that

$R_{k}^{i}=P\widetilde{R}_{k}^{i}Q\ \ \text{for all $i\geq 0$}.$

Let $G^{k}(s)\coloneqq G(t_{i},U^{i})$ and
$S^{k}(s)\coloneqq\widetilde{R}_{k}^{-i}$ for $s\in[t_{i},t_{i+1}),0\leq i\leq
N_{k}-1$. Then it follows from Theorem 2.2 that

$\displaystyle\left\|\max_{0\leq j\leq
N}\left\|\sum_{i=0}^{j-1}R_{k}^{j-i}G(t_{i},U^{i})\Delta
W_{i+1}\right\|\right\|_{p}$ $\displaystyle=\left\|\max_{0\leq j\leq
N}\left\|\sum_{i=0}^{j-1}\widetilde{R}_{k}^{j-i}QG(t_{i},U^{i})\Delta
W_{i+1}\right\|\right\|_{p}$ $\displaystyle=\left\|\max_{0\leq j\leq
N}\left\|\sum_{i=0}^{j-1}\widetilde{R}_{k}^{-i}QG(t_{i},U^{i})\Delta
W_{i+1}\right\|\right\|_{p}$
$\displaystyle\leq\left\|\sup_{t\in[0,t_{N}]}\Big{\|}\int_{0}^{t}S^{k}(s)QG^{k}(s)\,\mathrm{d}W_{H}(s)\Big{\|}\right\|_{p}$
$\displaystyle\leq
B_{p}\left\|\left(\int_{0}^{t_{N}}\|G^{k}(s)\|_{{\mathcal{L}_{2}(H,X)}}^{2}ds\right)^{1/2}\right\|_{p}$
$\displaystyle\leq
B_{p}\left(k\sum_{i=0}^{N-1}\left\|\|G(t_{i},U^{i})\|_{{\mathcal{L}_{2}(H,X)}}\right\|_{p}^{2}\right)^{1/2}$
(5.5) $\displaystyle\leq
B_{p}L_{G,X}\left(k\sum_{i=0}^{N-1}\varphi_{i}^{2}\right)^{1/2}+C_{2,g}t_{N}^{1/2},$

where we have set $C_{2,g}\coloneqq
B_{p}\|g\|_{L^{p}(\Omega;C([0,T];{\mathcal{L}_{2}(H,X)}))}$.

Inserting (5) and (5) in (5) gives the bound

$\displaystyle\varphi_{N}\leq
1+\|u_{0}\|_{L^{p}(\Omega;X)}+C_{1,f}t_{N}+C_{2,g}t_{N}^{1/2}+(L_{F,X}t_{N}^{1/2}+B_{p}L_{G,X})\left(k\sum_{i=0}^{N-1}\varphi_{i}^{2}\right)^{1/2}.$

Setting $C\coloneqq L_{F,X}t_{N}^{1/2}+B_{p}L_{G,X}$ and
$c_{u_{0},f,g,t_{N}}\coloneqq
1+\|u_{0}\|_{L^{p}(\Omega;X)}+C_{1,f}t_{N}+C_{2,g}t_{N}^{1/2}$, we obtain from
the discrete version of Gronwall’s Lemma 2.7 that

$\displaystyle\varphi_{N}\leq
c_{u_{0},f,g}(1+C^{2}kN)^{1/2}\mathrm{e}^{(1+C^{2}kN)/2}.$

This implies the desired statement for $N=N_{k}$ noting that
$t_{N_{k}}=kN_{k}=T$. ∎

##  6\. Convergence rates for multiplicative noise

Our aim is to prove rates of convergence of contractive time discretisation
schemes for nonlinear stochastic evolution equations of the form

(6.1)
$\mathrm{d}U=(AU+F(t,U))\,\mathrm{d}t+G(t,U)\,\mathrm{d}W_{H}(t),~{}~{}U(0)=u_{0}\in
L^{p}(\Omega;X)$

with $t\in[0,T]$ on a Hilbert space $X$ with norm $\|\cdot\|$, where $W_{H}$
is an $H$-cylindrical Brownian motion for some Hilbert space $H$ and
$p\in[2,\infty)$. The operator $A$ is assumed to generate a contractive
$C_{0}$-semigroup $(S(t))_{t\geq 0}$ on $X$ and $F,G$ are assumed to be
progressively measurable, of linear growth and globally Lipschitz as detailed
in Assumption 4.1. Hence, we have the unique mild solution given by a fixed
point of

(6.2)
$U(t)=S(t)u_{0}+\int_{0}^{t}S(t-s)F(s,U(s))\,\mathrm{d}s+\int_{0}^{t}S(t-s)G(s,U(s))\,\mathrm{d}W_{H}(s)$

for $t\in[0,T]$, see Section 4.

To obtain convergence rates for temporal discretisations of the mild solution,
we assume additional structure of the nonlinearity $F$ and the noise $G$. Let
$Y$ be another Hilbert space such that $Y\hookrightarrow X$ and the semigroup
$(S(t))_{t\geq 0}$ is also contractive on $Y$. We will assume $F$ and $G$ map
$Y$ into $Y$ and enjoy linear growth conditions as on $X$ also on $Y$. Note
that Lipschitz continuity is not assumed on $Y$ contrary to $X$. This
additional structure resembling the famous Kato setting [47], which was
briefly mentioned in the introduction, allows for convergence rates of
temporal discretisations for a large class of schemes introduced in Subsection
6.1. The quantitative error estimate in Theorem 6.4 is the main result of this
paper, stating that the additional structure suffices to obtain the order of
the scheme as the convergence rate of the temporal approximations up to a
logarithmic correction factor for sufficiently regular initial data. For the
_exponential Euler method_ , the logarithmic correction factor can be omitted,
as illustrated in Subsection 6.2. The main error estimate of Theorem 6.4 is
extended to the full time interval $[0,T]$ in Subsection 6.3 As an
application, we revisit the Schrödinger equation, now with a multiplicative
potential, in Subsection 6.4 and consider the stochastic Maxwell’s equations
in Subsection 6.5.

###  6.1. General contractive time discretisation schemes

We now detail the assumptions on the structure of $F$ and $G$ on $Y$. Note
that the assumption also implies that the conditions of Theorems 4.3 and 4.4
hold.

######  Assumption 6.1.

Let $X,Y$ be Hilbert spaces such that $Y\hookrightarrow X$ continuously, and
let $p\in[2,\infty)$. Let $F:\Omega\times[0,T]\times X\to
X,F(\omega,t,x)=\tilde{F}(\omega,t,x)+f(\omega,t)$ and
$G:\Omega\times[0,T]\times
X\to{\mathcal{L}_{2}(H,X)},G(\omega,t,x)=\tilde{G}(\omega,t,x)+g(\omega,t)$ be
strongly $\mathcal{P}\otimes\mathcal{B}(X)$-measurable, and such that
$\tilde{F}(\cdot,\cdot,0)=0$ and $\tilde{G}(\cdot,\cdot,0)=0$, and suppose

  1. (a)

_(global Lipschitz continuity on $X$)_ there exist constants
$C_{F,X},C_{G,X}\geq 0$ such that for all $\omega\in\Omega,t\in[0,T]$, and
$x,y\in X$, it holds that

$\|\tilde{F}(\omega,t,x)-\tilde{F}(\omega,t,y)\|\leq
C_{F,X}\|x-y\|,~{}\|\tilde{G}(\omega,t,x)-\tilde{G}(\omega,t,y)\|_{\mathcal{L}_{2}(H,X)}\leq
C_{G,X}\|x-y\|,$

  2. (b)

_(Hölder continuity with values in $X$)_ for some $\alpha\in(0,1]$,

$\displaystyle C_{\alpha,F}\coloneqq\sup_{\omega\in\Omega,x\in
X}[F(\omega,\cdot,x)]_{\alpha}<\infty,~{}C_{\alpha,G}\coloneqq\sup_{\omega\in\Omega,x\in
X}[G(\omega,\cdot,x)]_{\alpha}<\infty,$

  3. (c)

_( $Y$-invariance)_ $F:\Omega\times[0,T]\times Y\to Y$ and
$G:\Omega\times[0,T]\times Y\to{\mathcal{L}_{2}(H,Y)}$ are strongly
$\mathcal{P}\otimes\mathcal{B}(Y)$-measurable, $f\in
L^{p}_{\mathcal{P}}(\Omega;C([0,T];Y))$, and $g\in
L^{p}_{\mathcal{P}}(\Omega;C([0,T];{\mathcal{L}_{2}(H,Y)}))$,

  4. (d)

_(linear growth on $Y$)_ there exist constants $L_{F,Y},L_{G,Y}\geq 0$ such
that for all $\omega\in\Omega,t\in[0,T]$, and $x\in Y$, it holds that

$\|\tilde{F}(\omega,t,x)\|_{Y}\leq
L_{F,Y}(1+\|x\|_{Y}),~{}\|\tilde{G}(\omega,t,x)\|_{\mathcal{L}_{2}(H,Y)}\leq
L_{G,Y}(1+\|x\|_{Y}).$

Condition (b) can be weakened to the existence of some $\alpha\in(0,1]$ such
that

$\sup_{x\in X}\sup_{0\leq s\leq t\leq
T}\frac{F(\cdot,t,x)-F(\cdot,s,x)}{(t-s)^{\alpha}}\in L^{p}(\Omega)$

and likewise for $G$, i.e., pathwise Hölder continuity uniformly in $x\in X$
is sufficient together with existence of $p$-th moments of the Hölder
seminorms. Assumption 6.1 implies that (6.1) has a unique mild solution.

To bound the error arising from time discretisation of the mild solution,
moment bounds of differences of the mild solution at different time points as
in the following lemma are required. As a shorthand notation in accordance
with (3.4), let

(6.3)
$\|f\|_{p,q,Z}\coloneqq\|f\|_{L^{p}(\Omega;L^{q}(0,T;Z))},~{}|\\!|\\!|g|\\!|\\!|_{p,q,Z}\coloneqq\|g\|_{L^{p}(\Omega;L^{q}(0,T;\mathcal{L}_{2}(H,Z)))}$

for Hilbert spaces $Z$, $p\in[2,\infty)$, and $q\in[1,\infty]$. We further
introduce the constants

(6.4) $\displaystyle C_{u_{0},f,g,Z}$ $\displaystyle\coloneqq
1+C_{\text{bdd}}^{Z}(1+\|u_{0}\|_{L^{p}(\Omega;Z)}+\|f\|_{p,1,Z}+|\\!|\\!|g|\\!|\\!|_{p,2,Z})$

for $Z\in\\{X,Y\\}$ with $C_{\text{bdd}}^{X}$ and $C_{\text{bdd}}^{Y}$ as in
Theorems 4.3 and 4.4, respectively. Then the estimate

(6.5) $1+\bigg{\|}\sup_{r\in[0,T]}\|U(r)\|_{Z}\bigg{\|}_{p}\leq
C_{u_{0},f,g,Z}<\infty$

holds for $Z\in\\{X,Y\\}$.

######  Lemma 6.2.

Suppose that Assumption 6.1 holds for some $\alpha\in(0,1]$ and
$p\in[2,\infty)$. Let $A$ be the generator of a $C_{0}$-contraction semigroup
$(S(t))_{t\geq 0}$ on both $X$ and $Y$. Suppose that $Y\hookrightarrow
D_{A}(\alpha,\infty)$ continuously if $\alpha\in(0,1)$ or $Y\hookrightarrow
D(A)$ continuously if $\alpha=1$. Let $u_{0}\in
L_{\mathcal{F}_{0}}^{p}(\Omega;Y)$. Then for all $0\leq s\leq t\leq T$ the
mild solution $U$ of (6.1) satisfies

$\displaystyle\left(\mathbb{E}\|U(t)-U(s)\|^{p}\right)^{1/p}\leq
L_{1}(t-s)+L_{2}(t-s)^{1/2}+L_{3}(t-s)^{\alpha}$

with constants $L_{1}\coloneqq C_{F,X}C_{u_{0},f,g,X}+\|f\|_{p,\infty,X}$,
$L_{2}\coloneqq
B_{p}(C_{G,X}C_{u_{0},f,g,X}+|\\!|\\!|g|\\!|\\!|_{p,\infty,X})$, and

$L_{3}\coloneqq
2C_{Y}\Big{[}\|u_{0}\|_{L^{p}(\Omega;Y)}+TL_{F,Y}C_{u_{0},f,g,Y}+\|f\|_{p,1,Y}+B_{p}\big{(}T^{1/2}L_{G,Y}C_{u_{0},f,g,Y}+|\\!|\\!|g|\\!|\\!|_{p,2,Y}\big{)}\Big{]},$

where $C_{u_{0},f,g,X}$ and $C_{u_{0},f,g,Y}$ are as defined in (6.4), $C_{Y}$
denotes the embedding constant of $Y$ into $D_{A}(\alpha,\infty)$ or $D(A)$,
and $B_{p}$ is the constant from Theorem 2.2.

###### Proof.

Since the conditions of Theorems 4.3 and 4.4 are met, $U$ is pathwise
continuous on $X$. By Theorem 4.4, the pathwise continuity of $U$ follows on
$Y$ as well. Moreover, the bound (6.5) holds.

Fix $t,s\in[0,T]$ with $s\leq t$. From the mild solution formula (6.2), we
deduce that

$\displaystyle\left(\mathbb{E}\|U(t)-U(s)\|^{p}\right)^{1/p}\leq\left\|[S(t)-S(s)]u_{0}\right\|_{L^{p}(\Omega;X)}$
$\displaystyle+\left\|\int_{0}^{s}\|[S(t-r)-S(s-r)]F(r,U(r))\|\;\mathrm{d}r\right\|_{p}+\left\|\int_{s}^{t}\|S(t-r)F(r,U(r))\|\;\mathrm{d}r\right\|_{p}$
$\displaystyle+\left\|\int_{0}^{s}[S(t-r)-S(s-r)]G(r,U(r))\;\mathrm{d}W_{H}(r)\right\|_{L^{p}(\Omega;X)}$
$\displaystyle+\left\|\int_{s}^{t}S(t-r)G(r,U(r))\;\mathrm{d}W_{H}(r)\right\|_{L^{p}(\Omega;X)}\eqqcolon
E_{1}+E_{2}+E_{3}+E_{4}+E_{5},$

where $E_{\ell}=E_{\ell}(t,s)$ for $1\leq\ell\leq 5$. We proceed to bound
these five expressions individually. By the semigroup bound (2.4),

$\displaystyle E_{1}$
$\displaystyle\leq\|S(t)-S(s)\|_{\mathcal{L}(Y,X)}\|u_{0}\|_{L^{p}(\Omega;Y)}\leq
2C_{Y}(t-s)^{\alpha}\|u_{0}\|_{L^{p}(\Omega;Y)}.$

Using (6.5) and (2.4) as well as linear growth of $\tilde{F}$ on $Y$ and $f\in
L^{p}(\Omega;L^{1}(0,T;Y))$, we obtain

$\displaystyle E_{2}$ $\displaystyle\leq
2C_{Y}\Big{\|}\int_{0}^{s}[(t-r)-(s-r)]^{\alpha}\|F(r,U(r))\|_{Y}\,\mathrm{d}r\Big{\|}_{p}$
$\displaystyle\leq
2C_{Y}(t-s)^{\alpha}\bigg{(}sL_{F,Y}\bigg{\|}\sup_{r\in[0,T]}(1+\|U(r)\|_{Y})\bigg{\|}_{p}+\bigg{\|}\int_{0}^{s}\|f(r)\|_{Y}\,\mathrm{d}r\bigg{\|}_{p}\bigg{)}$
$\displaystyle\leq
2C_{Y}\big{(}TL_{F,Y}C_{u_{0},f,g,Y}+\|f\|_{p,1,Y}\big{)}(t-s)^{\alpha}.$

Analogously,

$\displaystyle E_{3}$
$\displaystyle\leq(C_{F,X}C_{u_{0},f,g,X}+\|f\|_{p,\infty,X})(t-s)$

is obtained by contractivity of the semigroup, linear growth of $F$ on $X$ and
boundedness of the solution. For the terms involving a stochastic integral, we
apply Theorem 2.2. Additionally making use of the bound (2.4) for semigroup
differences, splitting the integral as in $E_{2}$, and using linear growth of
$\tilde{G}$, (6.5), as well as $g\in
L^{p}(\Omega;L^{2}(0,T;{\mathcal{L}_{2}(H,Y)})))$ results in

$\displaystyle E_{4}$ $\displaystyle\leq
B_{p}\left(\mathbb{E}\left(\int_{0}^{s}\|[S(t-r)-S(s-r)]G(r,U(r))\|_{\mathcal{L}_{2}(H,X)}^{2}\;\mathrm{d}r\right)^{p/2}\right)^{1/p}$
$\displaystyle\leq
2B_{p}C_{Y}\big{(}T^{1/2}L_{G,Y}C_{u_{0},f,g,Y}+|\\!|\\!|g|\\!|\\!|_{p,2,Y}\big{)}(t-s)^{\alpha}.$

For the last term, the contractivity of the semigroup and linear growth of $G$
yield

$\displaystyle E_{5}$ $\displaystyle\leq
B_{p}\left(\mathbb{E}\left(\int_{s}^{t}\|S(t-r)G(r,U(r))\|_{\mathcal{L}_{2}(H,X)}^{2}\;\mathrm{d}r\right)^{p/2}\right)^{1/p}$
$\displaystyle\leq
B_{p}(C_{G,X}C_{u_{0},f,g,X}+|\\!|\\!|g|\\!|\\!|_{p,\infty,X})(t-s)^{1/2}.$

In conclusion from the five individual bounds, we obtain the statement of the
lemma

$\displaystyle\left(\mathbb{E}\|U(t)-U(s)\|^{p}\right)^{1/p}$
$\displaystyle\leq(C_{F,X}C_{u_{0},f,g,X}+\|f\|_{p,\infty,X})(t-s)$
$\displaystyle\phantom{\leq}+B_{p}\big{(}C_{G,X}C_{u_{0},f,g,X}+|\\!|\\!|g|\\!|\\!|_{p,\infty,X}\big{)}(t-s)^{1/2}$
$\displaystyle\phantom{\leq}+2C_{Y}\Big{[}\|u_{0}\|_{L^{p}(\Omega;Y)}+TL_{F,Y}C_{u_{0},f,g,Y}+\|f\|_{p,1,Y}$
$\displaystyle\phantom{\leq}+B_{p}\big{(}T^{1/2}L_{G,Y}C_{u_{0},f,g,Y}+|\\!|\\!|g|\\!|\\!|_{p,2,Y}\big{)}\Big{]}(t-s)^{\alpha}.\qed$

######  Remark 6.3.

Suppose that $\alpha\in(0,\frac{1}{2}]$. Lemma 6.2 implies $\alpha$-Hölder
continuity of $U$ in $p$-th moment. The latter remains true if the pathwise
continuity of $f$ and $g$ with values in $Y$ from Assumption 6.1(c) are
relaxed to $\|f\|_{p,1,Y}<\infty$ and $|\\!|\\!|g|\\!|\\!|_{p,2,Y}<\infty$.
Performing an additional Hölder argument for $E_{3}$ and $E_{5}$, the pathwise
continuity assumption with values in $X$ can be relaxed to
$\|f\|_{p,\frac{1}{1-\alpha},X}<\infty$ and
$|\\!|\\!|g|\\!|\\!|_{p,\frac{2}{1-2\alpha},X}<\infty$, where we use the
convention $\frac{1}{0}=\infty$. Although the lemma could be improved, for our
purposes the above version is enough since even pathwise continuity with
values in $Y$ is required in Theorem 6.4.

For time discretisation, we employ a contractive time discretisation scheme
$R:[0,\infty)\to\mathcal{L}(X)$ with time step $k>0$ on a uniform grid
$\\{t_{j}=jk:~{}j=0,\ldots,N_{k}\\}\subseteq[0,T]$ with final time
$T=t_{N_{k}}>0$ and $N_{k}=\frac{T}{k}\in\mathbb{N}$ being the number of time
steps. As in the previous section, the discrete solution is given by
$U^{0}\coloneqq u_{0}$ and

(6.6) $\displaystyle U^{j}$ $\displaystyle\coloneqq
R_{k}U^{j-1}+kR_{k}F(t_{j-1},U^{j-1})+R_{k}G(t_{j-1},U^{j-1})\Delta W_{j}$
(6.7)
$\displaystyle=R_{k}^{j}u_{0}+k\sum_{i=0}^{j-1}R_{k}^{j-i}F(t_{i},U^{i})+\sum_{i=0}^{j-1}R_{k}^{j-i}G(t_{i},U^{i})\Delta
W_{i+1}$

for $j=1,\ldots,N_{k}$ with Wiener increments $\Delta W_{j}\coloneqq
W_{H}(t_{j})-W_{H}(t_{j-1})$.

We recall from Definition 2.4 that $R$ approximates $S$ to order $\alpha>0$ on
$Y$ or, equivalently, $R$ converges of order $\alpha$ on $Y$ if there is a
constant $C_{\alpha}\geq 0$ such that for all $u\in Y$

$\|(S(t_{j})-R_{k}^{j})u\|\leq C_{\alpha}k^{\alpha}\|u\|_{Y}.$

Under the conditions of Assumption 6.1 we conclude from Proposition 5.1 and
the remark thereafter that $R$ is stable not only on $X$ but also on $Y$
provided that $u_{0}\in L_{\mathcal{F}_{0}}^{p}(\Omega;Y)$ and both $S$ and
$R$ are contractive on both $X$ and $Y$. Thus,

(6.8) $1+\left\lVert\max\limits_{0\leq j\leq
N_{k}}\|U^{j}\|_{Y}\right\rVert_{p}\leq K_{u_{0},f,g,Y},$

where $K_{u_{0},f,g,Y}\coloneqq C_{\text{stab}}c_{u_{0},f,g,T}$ with constants
$C_{\text{stab}},c_{u_{0},f,g,T}$ as in Proposition 5.1 applied on $Y$ instead
of $X$. Furthermore, we recall the shorthand notation $\|f\|_{p,\infty,Y}$ and
$|\\!|\\!|g|\\!|\\!|_{p,\infty,Y}$ from (6.3).

We can now state and prove the main result of this paper.

######  Theorem 6.4.

Suppose that Assumption 6.1 holds for some $\alpha\in(0,1]$ and
$p\in[2,\infty)$. Let $A$ be the generator of a $C_{0}$-contraction semigroup
$(S(t))_{t\geq 0}$ on both $X$ and $Y$. Let $(R_{k})_{k>0}$ be a time
discretisation scheme which is contractive on $X$ and $Y$. Assume $R$
approximates $S$ to order $\alpha$ on $Y$. Suppose that $Y\hookrightarrow
D_{A}(\alpha,\infty)$ continuously if $\alpha\in(0,1)$ or $Y\hookrightarrow
D(A)$ continuously if $\alpha=1$. Let $u_{0}\in
L_{\mathcal{F}_{0}}^{p}(\Omega;Y)$. Denote by $U$ the mild solution of (6.1)
and by $(U^{j})_{j=0,\ldots,N_{k}}$ the temporal approximations as defined in
(6.6). Then for $N_{k}\geq 2$

$\left\lVert\max_{0\leq j\leq N_{k}}\|U(t_{j})-U^{j}\|\right\rVert_{p}\leq
C_{\mathrm{e}}\left(C_{1}k+C_{2}k^{1/2}+\left(C_{3}+C_{4}\log\left(\frac{T}{k}\right)\right)k^{\alpha}\right)$

with constants $C_{\mathrm{e}}\coloneqq(1+C^{2}T)^{1/2}\exp((1+C^{2}T)/2)$,
$C\coloneqq C_{F,X}\sqrt{T}+B_{p}C_{G,X}$, $C_{1}\coloneqq
L_{1}(\frac{C_{F,X}}{2}T^{2}+B_{p}C_{G,X}\sqrt{T})$, $C_{2}\coloneqq
L_{2}(\frac{2}{3}C_{F,X}T+(\frac{3}{2})^{1/2}B_{p}C_{G,X}\sqrt{T})$,
$C_{4}\coloneqq C_{3,\log}\sqrt{T}$, and

$\displaystyle C_{3}$ $\displaystyle\coloneqq
C_{\alpha}\|u_{0}\|_{L^{p}(\Omega;Y)}+C_{2,\alpha}T+C_{3,\alpha}\sqrt{T},$
$\displaystyle C_{2,\alpha}$
$\displaystyle\coloneqq\frac{C_{F,X}L_{3}+C_{\alpha,F}}{\alpha+1}+\big{(}L_{F,Y}K_{u_{0},f,g,Y}+\|f\|_{p,\infty,Y}\big{)}\left(\frac{2C_{Y}}{\alpha+1}+C_{\alpha}\right),$
$\displaystyle C_{3,\alpha}$
$\displaystyle\coloneqq\frac{B_{p}}{\sqrt{2\alpha+1}}\Big{(}\sqrt{3}C_{G,X}L_{3}+C_{\alpha,G}+2C_{Y}\big{(}L_{G,Y}K_{u_{0},f,g,Y}+|\\!|\\!|g|\\!|\\!|_{p,\infty,Y}\big{)}\Big{)},$
$\displaystyle C_{3,\log}$ $\displaystyle\coloneqq
K_{p}C_{\alpha}\big{(}L_{G,Y}K_{u_{0},f,g,Y}+|\\!|\\!|g|\\!|\\!|_{p,\infty,Y}\big{)},$

where $L_{1},L_{2},L_{3}$ are as defined in Lemma 6.2, $K_{u_{0},f,g,Y}$ as in
(6.8), $K_{p}=10\mathrm{e}\sqrt{p}$, $C_{Y}$ denotes the embedding constant of
$Y$ into $D_{A}(\alpha,\infty)$ or $D(A)$, and $B_{p}$ is the constant from
Theorem 2.2.

In particular, the approximations $(U^{j})_{j}$ converge at rate
$\min\\{\alpha,\frac{1}{2}\\}$ up to a logarithmic correction factor as $k\to
0$.

This convergence result applies to schemes such as exponential Euler, implicit
Euler, Crank-Nicolson, and other $A$-acceptable implicit Runge-Kutta methods
such as Radau methods, BDF(2), Lobatto IIA, IIB, and IIC by virtue of
Proposition 2.5. If $R$ commutes with the resolvent of $A$, contractivity of
$R$ and $S$ extend to fractional domain spaces and complex interpolation
spaces. Hence, contractivity on $Y$ often comes together with contractivity on
$X$.

The constant $C_{\mathrm{e}}$ appears exponentially in the above. In the
special case that $C_{F,X}=C_{G,X}=T=1$, and $p=2$, one can check that,
similarly to Theorem 4.4, this yields the numerically reasonable value
$C_{\mathrm{e}}=\sqrt{10}e^{5}\leq 470$.

###### Proof.

The assumptions of Theorems 4.3 and 4.4 hold, and thus the mild solution $U$
exists and the bound (6.5) holds.

By definition, $U(t_{j})=U^{j}=u_{0}$ for $j=0$. Let
$N\in\\{1,\ldots,N_{k}\\}$. Using (6.7), the discretisation error can be split
into three parts

$\displaystyle E(N)$ $\displaystyle\coloneqq\left\lVert\max_{1\leq j\leq
N}\|U(t_{j})-U^{j}\|\right\rVert_{p}$ $\displaystyle\leq\left\lVert\max_{1\leq
j\leq N}\|(S(t_{j})-R_{k}^{j})u_{0}\|\right\rVert_{p}$
$\displaystyle\phantom{\leq}+\left\lVert\max_{1\leq j\leq
N}\left\|\int_{0}^{t_{j}}S(t_{j}-s)F(s,U(s))\,\mathrm{d}s-k\sum_{i=0}^{j-1}R_{k}^{j-i}F(t_{i},U^{i})\right\|\right\rVert_{p}$
$\displaystyle\phantom{\leq}+\left\lVert\max_{1\leq j\leq
N}\left\|\int_{0}^{t_{j}}S(t_{j}-s)G(s,U(s))\,\mathrm{d}W_{H}(s)-\sum_{i=0}^{j-1}R_{k}^{j-i}G(t_{i},U^{i})\Delta
W_{i+1}\right\|\right\rVert_{p}$ $\displaystyle\eqqcolon M_{1}+M_{2}+M_{3}.$

Using convergence of $R$ of order $\alpha$ on $Y$ and the dominated
convergence theorem, we obtain

(6.9) $\displaystyle M_{1}$ $\displaystyle\leq
C_{\alpha}k^{\alpha}\|u_{0}\|_{L^{p}(\Omega;Y)}.$

To shorten the notation for the discrete terms, we introduce the piecewise
constant functions $F^{k}(s)\coloneqq F(t_{i},U^{i})$ and $G^{k}(s)\coloneqq
G(t_{i},U^{i})$ for $s\in[t_{i},t_{i+1}),0\leq i\leq N_{k}-1$ as well as
$S^{k}(s)\coloneqq R_{k}^{i}$ for $s\in(t_{i-1},t_{i}],1\leq i\leq N_{k}$.
This allows us to rewrite

$\displaystyle M_{2}$ $\displaystyle=\left\lVert\max_{1\leq j\leq
N}\left\|\int_{0}^{t_{j}}S(t_{j}-s)F(s,U(s))-S^{k}(t_{j}-s)F^{k}(s)\,\mathrm{d}s\right\|\right\rVert_{p}$
$\displaystyle\leq\left\lVert\sum_{i=0}^{N-1}\int_{t_{i}}^{t_{i+1}}\max_{1\leq
j\leq
N}\left\|S(t_{j}-s)[F(s,U(s))-F(s,U(t_{i}))]\right\|\,\mathrm{d}s\right\rVert_{p}$
$\displaystyle\phantom{\leq}+\left\lVert\sum_{i=0}^{N-1}\int_{t_{i}}^{t_{i+1}}\max_{1\leq
j\leq
N}\left\|S(t_{j}-s)[F(s,U(t_{i}))-F(t_{i},U(t_{i}))]\right\|\,\mathrm{d}s\right\rVert_{p}$
$\displaystyle\phantom{\leq}+\left\lVert\sum_{i=0}^{N-1}\int_{t_{i}}^{t_{i+1}}\max_{1\leq
j\leq
N}\left\|S(t_{j}-s)[F(t_{i},U(t_{i}))-F(t_{i},U^{i})]\right\|\,\mathrm{d}s\right\rVert_{p}$
$\displaystyle\phantom{\leq}+\left\lVert\int_{0}^{t_{N}}\max_{1\leq j\leq
N}\left\|[S(t_{j}-s)-S^{k}(t_{j}-s)]F^{k}(s)\right\|\,\mathrm{d}s\right\rVert_{p}$
$\displaystyle\eqqcolon M_{2,1}+M_{2,2}+M_{2,3}+M_{2,4}.$

Making use of Minkowski’s inequality in $L^{p}(\Omega)$, contractivity of
$(S(t))_{t\geq 0}$ and Lipschitz continuity of $\tilde{F}$, we derive the
bound

(6.10) $M_{2,3}\leq
C_{F,X}\sum_{i=0}^{N-1}\left\lVert\int_{t_{i}}^{t_{i+1}}\left\|U(t_{i})-U^{i}\right\|\,\mathrm{d}s\right\rVert_{p}\leq
C_{F,X}k\sum_{i=0}^{N-1}E(i)$

for $M_{2,3}$. Proceeding likewise for $M_{2,1}$, we obtain from Lemma 6.2
that

$\displaystyle M_{2,1}$ $\displaystyle\leq
C_{F,X}\sum_{i=0}^{N-1}\int_{t_{i}}^{t_{i+1}}(\mathbb{E}\left\|U(s)-U(t_{i})\right\|^{p})^{1/p}\,\mathrm{d}s$
$\displaystyle\leq
C_{F,X}\sum_{i=0}^{N-1}\int_{t_{i}}^{t_{i+1}}L_{1}(s-t_{i})+L_{2}(s-t_{i})^{1/2}+L_{3}(s-t_{i})^{\alpha}\,\mathrm{d}s$
$\displaystyle\leq
C_{F,X}\sum_{i=0}^{N-1}\left(\frac{L_{1}}{2}k^{2}+\frac{2L_{2}}{3}k^{3/2}+\frac{L_{3}}{\alpha+1}k^{\alpha+1}\right)$
(6.11)
$\displaystyle=C_{F,X}t_{N}\left(\frac{L_{1}}{2}k+\frac{2L_{2}}{3}k^{1/2}+\frac{L_{3}}{\alpha+1}k^{\alpha}\right).$

Analogously, uniform Hölder continuity yields

$\displaystyle M_{2,2}$
$\displaystyle\leq\sum_{i=0}^{N-1}\int_{t_{i}}^{t_{i+1}}\left\|F(s,U(t_{i}))-F(t_{i},U(t_{i}))\right\|_{L^{p}(\Omega;X)}\,\mathrm{d}s$
$\displaystyle\leq\sum_{i=0}^{N-1}\int_{t_{i}}^{t_{i+1}}(s-t_{i})^{\alpha}\,\mathrm{d}s\left\|[F(\cdot,U(t_{i}))]_{\alpha}\right\|_{p}$
(6.12)
$\displaystyle\leq\sum_{i=0}^{N-1}\frac{k^{\alpha+1}}{\alpha+1}C_{\alpha,F}=\frac{C_{\alpha,F}t_{N}}{\alpha+1}k^{\alpha}.$

Using the semigroup bound (2.4) together with the assumed convergence rate
$\alpha$ of $R$ on $Y$, the linear growth assumption and stability of $R$, we
obtain

$\displaystyle M_{2,4}$
$\displaystyle\leq\left\lVert\sum_{i=0}^{N-1}\int_{t_{i}}^{t_{i+1}}\left\|[S(t_{j}-s)-S(t_{j}-t_{i})]F(t_{i},U^{i})\right\|\,\mathrm{d}s\right\rVert_{p}$
$\displaystyle\phantom{\leq}+\left\lVert\sum_{i=0}^{N-1}\int_{t_{i}}^{t_{i+1}}\left\|\left[S(t_{j}-t_{i})-R_{k}^{j-i}\right]F(t_{i},U^{i})\right\|\,\mathrm{d}s\right\rVert_{p}$
$\displaystyle\leq
2C_{Y}\sum_{i=0}^{N-1}\left\lVert\int_{t_{i}}^{t_{i+1}}(s-t_{i})^{\alpha}\|F(t_{i},U^{i})\|_{Y}\,\mathrm{d}s\right\rVert_{p}$
$\displaystyle\phantom{\leq}+C_{\alpha}k^{\alpha}\sum_{i=0}^{N-1}\left\lVert\int_{t_{i}}^{t_{i+1}}\left\|F(t_{i},U^{i})\right\|_{Y}\,\mathrm{d}s\right\rVert_{p}$
$\displaystyle\leq\left(\frac{2C_{Y}}{\alpha+1}+C_{\alpha}\right)k^{\alpha+1}\sum_{i=0}^{N-1}\left(L_{F,Y}\left\lVert
1+\|U^{i}\|_{Y}\right\rVert_{p}+\|f(t_{i})\|_{L^{p}(\Omega;Y)}\right)$ (6.13)
$\displaystyle\leq\left(\frac{2C_{Y}}{\alpha+1}+C_{\alpha}\right)\big{(}L_{F,Y}K_{u_{0},f,g,Y}+\|f\|_{p,\infty,Y}\big{)}t_{N}k^{\alpha}.$

In conclusion from (6.1), (6.1), (6.10), and (6.1), $M_{2}$ is bounded by

$\displaystyle M_{2}$
$\displaystyle\leq\frac{C_{F,X}L_{1}}{2}t_{N}k+\frac{2C_{F,X}L_{2}}{3}t_{N}k^{1/2}+C_{2,\alpha}t_{N}k^{\alpha}+C_{F,X}k\sum_{i=0}^{N-1}E(i)$
(6.14)
$\displaystyle\leq\frac{C_{F,X}L_{1}}{2}t_{N}k+\frac{2C_{F,X}L_{2}}{3}t_{N}k^{1/2}+C_{2,\alpha}t_{N}k^{\alpha}+C_{F,X}\sqrt{t_{N}}\bigg{(}k\sum_{i=0}^{N-1}E(i)^{2}\bigg{)}^{1/2},$

where we have used the Cauchy-Schwarz inequality in the last line.

Let $\lfloor s\rfloor=\max\\{t_{i}:0\leq i\leq N_{k}-1,t_{i}\leq s\\}$. The
remaining term $M_{3}$ can be rewritten as

$\displaystyle M_{3}$ $\displaystyle=\left\lVert\max_{1\leq j\leq
N}\left\|\int_{0}^{t_{j}}S(t_{j}-s)G(s,U(s))-S^{k}(t_{j}-s)G^{k}(s)\,\mathrm{d}W_{H}(s)\right\|\right\rVert_{p}$
$\displaystyle\leq\left\lVert\max_{1\leq j\leq
N}\left\|\int_{0}^{t_{j}}S(t_{j}-s)[G(s,U(s))-G(s,U(\lfloor
s\rfloor)]\,\mathrm{d}W_{H}(s)\right\|\right\rVert_{p}$
$\displaystyle\phantom{\leq}+\left\lVert\max_{1\leq j\leq
N}\left\|\int_{0}^{t_{j}}S(t_{j}-s)[G(s,U(\lfloor s\rfloor))-G(\lfloor
s\rfloor,U(\lfloor s\rfloor)]\,\mathrm{d}W_{H}(s)\right\|\right\rVert_{p}$
$\displaystyle\phantom{\leq}+\left\lVert\max_{1\leq j\leq
N}\left\|\int_{0}^{t_{j}}S(t_{j}-s)[G(\lfloor s\rfloor,U(\lfloor
s\rfloor))-G^{k}(s)]\,\mathrm{d}W_{H}(s)\right\|\right\rVert_{p}$
$\displaystyle\phantom{\leq}+\left\lVert\max_{1\leq j\leq
N}\left\|\int_{0}^{t_{j}}[S(t_{j}-\lfloor
s\rfloor)-S(t_{j}-s)]G^{k}(s)\,\mathrm{d}W_{H}(s)\right\|\right\rVert_{p}$
$\displaystyle\phantom{\leq}+\left\lVert\max_{1\leq j\leq
N}\left\|\int_{0}^{t_{j}}[S(t_{j}-\lfloor
s\rfloor)-S^{k}(t_{j}-s)]G^{k}(s)\,\mathrm{d}W_{H}(s)\right\|\right\rVert_{p}$
$\displaystyle\eqqcolon M_{3,1}+M_{3,2}+M_{3,3}+M_{3,4}+M_{3,5}.$

We bound each term individually. An application of the maximal inequality
Theorem 2.2, the Lipschitz continuity of $\tilde{G}$ and Lemma 6.2 result in

$\displaystyle M_{3,1}$
$\displaystyle\leq\left\lVert\sup_{t\in[0,t_{N}]}\left\|\int_{0}^{t}S(t-s)[G(s,U(s))-G(s,U(\lfloor
s\rfloor)]\,\mathrm{d}W_{H}(s)\right\|\right\rVert_{p}$ $\displaystyle\leq
B_{p}\left(\mathbb{E}\left(\sum_{i=0}^{N-1}\int_{t_{i}}^{t_{i+1}}\|G(s,U(s))-G(s,U(t_{i}))\|_{\mathcal{L}_{2}(H,X)}^{2}\,\mathrm{d}s\right)^{p/2}\right)^{1/p}$
$\displaystyle\leq
B_{p}C_{G,X}\bigg{(}\sum_{i=0}^{N-1}\int_{t_{i}}^{t_{i+1}}\left(\mathbb{E}\|U(s)-U(t_{i})\|^{p}\right)^{2/p}\,\mathrm{d}s\bigg{)}^{1/2}$
$\displaystyle\leq\sqrt{3}B_{p}C_{G,X}\bigg{(}\sum_{i=0}^{N-1}\int_{t_{i}}^{t_{i+1}}L_{1}^{2}(s-t_{i})^{2}+L_{2}^{2}(s-t_{i})+L_{3}^{2}(s-t_{i})^{2\alpha}\,\mathrm{d}s\bigg{)}^{1/2}$
$\displaystyle=\sqrt{3}B_{p}C_{G,X}\sqrt{t_{N}}\left(\frac{L_{1}^{2}}{3}k^{2}+\frac{L_{2}^{2}}{2}k+\frac{L_{3}^{2}}{2\alpha+1}k^{2\alpha}\right)^{1/2}$
(6.15)
$\displaystyle\leq\sqrt{3}B_{p}C_{G,X}\sqrt{t_{N}}\left(\frac{L_{1}}{\sqrt{3}}k+\frac{L_{2}}{\sqrt{2}}k^{1/2}+\frac{L_{3}}{\sqrt{2\alpha+1}}k^{\alpha}\right).$

Again invoking the maximal inequality, we conclude

$\displaystyle M_{3,2}$ $\displaystyle\leq
B_{p}\bigg{(}\sum_{i=0}^{N-1}\int_{t_{i}}^{t_{i+1}}\left\|\|G(s,U(t_{i}))-G(t_{i},U(t_{i}))\|_{\mathcal{L}_{2}(H,X)}\right\|_{p}^{2}\,\mathrm{d}s\bigg{)}^{1/2}$
(6.16) $\displaystyle\leq
B_{p}\bigg{(}\sum_{i=0}^{N-1}\int_{t_{i}}^{t_{i+1}}(s-t_{i})^{2\alpha}\,\mathrm{d}s\left\|[G(\cdot,U(t_{i}))]_{\alpha}\right\|_{p}^{2}\bigg{)}^{1/2}\leq\frac{B_{p}C_{\alpha,G}}{\sqrt{2\alpha+1}}\sqrt{t_{N}}k^{\alpha}$

from the uniform Hölder continuity of $G$. Proceeding analogously for
$M_{3,3}$ and then applying Minkowski’s inequality in $L^{p/2}(\Omega)$
results in

$\displaystyle M_{3,3}$
$\displaystyle\leq\left\lVert\sup_{t\in[0,t_{N}]}\left\|\int_{0}^{t}S(t-s)[G(\lfloor
s\rfloor,U(\lfloor
s\rfloor)-G^{k}(s)]\,\mathrm{d}W_{H}(s)\right\|\right\rVert_{p}$
$\displaystyle\leq
B_{p}C_{G,X}\left(\mathbb{E}\left(k\sum_{i=0}^{N-1}\|U(t_{i})-U^{i}\|^{2}\right)^{p/2}\right)^{1/p}$
$\displaystyle=B_{p}C_{G,X}k^{1/2}\left\|\sum_{l=0}^{N-1}\max_{0\leq j\leq
l}\|U(t_{j})-U^{j}\|^{2}\right\|_{p/2}^{1/2}$ $\displaystyle\leq
B_{p}C_{G,X}\sqrt{k}\bigg{(}\sum_{l=0}^{N-1}\left\|\max_{0\leq j\leq
l}\|U(t_{j})-U^{j}\|^{2}\right\|_{p/2}\bigg{)}^{1/2}$ (6.17)
$\displaystyle=B_{p}C_{G,X}\sqrt{k}\bigg{(}\sum_{l=0}^{N-1}E(l)^{2}\bigg{)}^{1/2}.$

Since $R$ is contractive on $Y$ by assumption, the conditions of Proposition
5.1 are fulfilled not only on $X$ but also on $Y$. Thus, we can use the
estimate (6.8). Together with the maximal inequality, the semigroup difference
bound (2.4), the ideal property of ${\mathcal{L}_{2}(H,X)}$, and linear growth
of $\tilde{G}$, this yields

$\displaystyle M_{3,4}$
$\displaystyle\leq\left\lVert\sup_{t\in[0,t_{N}]}\left\|\int_{0}^{t}S(t-s)\left(\sum_{i=0}^{j-1}\mathbf{1}_{[t_{i},t_{i+1})}(s)[S(s-t_{i})-I]G(t_{i},U^{i})\right)\,\mathrm{d}W_{H}(s)\right\|\right\rVert_{p}$
$\displaystyle\leq
B_{p}\left(\mathbb{E}\left(\int_{0}^{t_{N}}\left\|\mathbf{1}_{[t_{i},t_{i+1})}(s)[S(s-t_{i})-I]G(t_{i},U^{i})\right\|_{\mathcal{L}_{2}(H,X)}^{2}\,\mathrm{d}s\right)^{p/2}\right)^{1/p}$
$\displaystyle\leq
2B_{p}C_{Y}\left(\mathbb{E}\left(\sum_{\ell=0}^{N-1}\int_{t_{\ell}}^{t_{\ell+1}}(s-t_{\ell})^{2\alpha}\left\|G(t_{\ell},U^{\ell})\right\|_{\mathcal{L}_{2}(H,Y)}^{2}\,\mathrm{d}s\right)^{p/2}\right)^{1/p}$
$\displaystyle\leq\frac{2B_{p}C_{Y}}{\sqrt{2\alpha+1}}\sqrt{t_{N}}k^{\alpha}\left\|\max_{0\leq
j\leq N-1}\left\|G(t_{j},U^{j})\right\|_{\mathcal{L}_{2}(H,Y)}\right\|_{p}$
(6.18)
$\displaystyle\leq\frac{2B_{p}C_{Y}}{\sqrt{2\alpha+1}}\big{(}L_{G,Y}K_{u_{0},f,g,Y}+|\\!|\\!|g|\\!|\\!|_{p,\infty,Y}\big{)}\sqrt{t_{N}}k^{\alpha}.$

Applying Proposition 2.3 with
$\Phi_{s}^{(j)}=\sum_{i=0}^{j-1}\mathbf{1}_{[t_{i},t_{i+1})}(s)[S(t_{j}-t_{i})-R_{k}^{j-i}]G(U^{i})$
to the remaining term, we conclude that

$\displaystyle M_{3,5}=\left(\mathbb{E}\max_{1\leq j\leq
N}\left\|\int_{0}^{t_{j}}\sum_{i=0}^{j-1}\mathbf{1}_{[t_{i},t_{i+1})}(s)[S(t_{j}-t_{i})-R_{k}^{j-i}]G(t_{i},U^{i})\,\mathrm{d}W_{H}(s)\right\|^{p}\right)^{1/p}$
$\displaystyle\leq
K_{p}\log(N)\left\lVert\left(\sum_{\ell=0}^{N-1}k\left(\max_{1\leq j\leq
N}\left\|[S(t_{j}-t_{\ell})-R_{k}^{j-l}]G(t_{\ell},U^{\ell})\right\|_{\mathcal{L}_{2}(H,X)}\right)^{2}\right)^{1/2}\right\rVert_{p}$
$\displaystyle\leq
K_{p}\log(N)\left(\mathbb{E}\left(\sum_{l=0}^{N-1}k\left(C_{\alpha}k^{\alpha}\left\|G(t_{\ell},U^{\ell})\right\|_{{\mathcal{L}_{2}(H,Y)}}\right)^{2}\right)^{p/2}\right)^{1/p}$
$\displaystyle\leq
K_{p}C_{\alpha}\sqrt{t_{N}}\log(N)k^{\alpha}\left\lVert\max_{0\leq j\leq
N-1}\left\|G(t_{j},U^{j})\right\|_{\mathcal{L}_{2}(H,Y)}\right\rVert_{p}$
(6.19) $\displaystyle\leq
K_{p}C_{\alpha}\big{(}L_{G,Y}K_{u_{0},f,g,Y}+|\\!|\\!|g|\\!|\\!|_{p,\infty,Y}\big{)}\sqrt{t_{N}}\log(N)k^{\alpha}$

using that $R$ approximates $S$ to order $\alpha$ on $Y$, the ideal property
of ${\mathcal{L}_{2}(H,X)}$, linear growth, and stability of $R$ on $Y$.
Combining the bounds (6.1) to (6.1), we deduce

$\displaystyle M_{3}$ $\displaystyle\leq
B_{p}C_{G,X}L_{1}\sqrt{t_{N}}k+\sqrt{\frac{3}{2}}B_{p}C_{G,X}L_{2}\sqrt{t_{N}}k^{1/2}+C_{3,\alpha}\sqrt{t_{N}}k^{\alpha}$
(6.20)
$\displaystyle\phantom{\leq}+C_{3,\log}\sqrt{t_{N}}\log(N)k^{\alpha}+B_{p}C_{G,X}\bigg{(}k\sum_{l=0}^{N-1}E(l)^{2}\bigg{)}^{1/2}.$

Having bounded each term individually in (6.9), (6.1) and (6.1), we conclude

$\displaystyle E(N)$ $\displaystyle\leq
C_{1}k+C_{2}k^{1/2}+C_{3}k^{\alpha}+C_{4}\log(N_{k})k^{\alpha}+C\bigg{(}k\sum_{l=0}^{N-1}E(l)^{2}\bigg{)}^{1/2},$

noting that $N\leq N_{k}$ and $t_{N}\leq T$. Thus, by the discrete version of
Gronwall’s Lemma 2.7

$\displaystyle
E(N)\leq(1+C^{2}t_{N})^{1/2}\mathrm{e}^{(1+C^{2}t_{N})/2}\left(C_{1}k+C_{2}k^{1/2}+C_{3}k^{\alpha}+C_{4}\log(N_{k})k^{\alpha}\right)$

follows. The desired error estimate is obtained for $N=N_{k}$. As $k\to 0$,
the terms with the lowest exponents dominate, i.e.

$E(N_{k})\lesssim
k^{1/2}+k+\log(N_{k})k^{\alpha}\lesssim\log(N_{k})k^{\min\\{\frac{1}{2},\alpha\\}},~{}~{}~{}(k\to
0).\qed$

######  Remark 6.5.

The result [20, Theorem 1.1] combines Hölder regularity in the $p$-th moment
and bounds on the pointwise strong error to obtain a uniform strong error.
Their effective method is based on a sophisticated application of the
Kolmogorov-Chentsov continuity theorem, as well as approximation arguments.
Let us refer to this method for obtaining uniform strong error estimates as
the Kolmogorov-Chentsov method. At first sight, one might think that the
result can be used to obtain the convergence rate of Theorem 6.4 up to an
arbitrary $\varepsilon>0$. Below, we point out what can precisely be achieved
via their method.

Suppose that $R$ approximates $S$ to order $1/2$, a pointwise strong error
estimate of rate $1/2$ has already been established, and Assumption 6.1 holds
for fixed $p\in[2,\infty)$ and $\alpha=1/2$. This means that the fixed data
$(u_{0},f,g)$ is assumed to have certain $L^{p}(\Omega)$-integrability. We
will check what type of rate the Kolmogorov-Chentsov method yields for

${\rm E}_{k}^{p,\infty}\coloneqq\bigg{\lVert}\max_{0\leq j\leq
N_{k}}\|U(t_{j})-U^{j}\|\bigg{\rVert}_{p},$

and compare it to the rate ${\rm E}_{k}^{p,\infty}\leq C_{p}k^{1/2}\log(T/k)$
we obtained in Theorem 6.4. We distinguish between three cases.

  1. (a)

Integrability of data in $L^{2}(\Omega)$: In this case, the Kolmogorov-
Chentsov method does not apply, so no convergence rate is obtained.

  2. (b)

Integrability of data in $L^{p}(\Omega)$ for a fixed $p\in(2,\infty)$: the
Kolmogorov-Chentsov method gives ${\rm E}_{k}^{p,\infty}\leq
C_{\gamma,p}k^{\gamma-1/p}$ for any $\gamma\in(1/p,1/2)$.

  3. (c)

Integrability of data in $L^{p}(\Omega)$ for all $p\in(2,\infty)$: the
Kolmogorov-Chentsov method gives ${\rm E}_{k}^{p,\infty}\leq
C_{\gamma,p}k^{\gamma}$ for any $\gamma\in(0,1/2)$.

In the last case, there is an arbitrarily small difference in the error rate.
We can obtain this error rate under the assumption that the data is
$L^{p}(\Omega)$-integrable for a fixed $p\in[2,\infty)$. In the case one has
this for all $p<\infty$, one needs to choose a very large $p$ in the
Kolmogorov-Chentsov method to get close to the desired rate, which in turn
produces large constants in the rate estimate.

###  6.2. The exponential Euler method

We analyse the time discretisation error for the special case $R_{k}\coloneqq
S(k)$ known as the exponential Euler method. Obviously, the exponential Euler
method is contractive for contractive semigroups. Furthermore, several terms
in the error analysis vanish for exponential Euler, since
$S(t_{j})-R_{k}^{j}=S(t_{j})-S(k)^{j}=0$ by the semigroup property. In
particular, the logarithmic correction factor is not needed for this scheme.

######  Corollary 6.6 (Exponential Euler).

Suppose that Assumption 6.1 holds for some $\alpha\in(0,1]$ and
$p\in[2,\infty)$. Let $A$ be the generator of a $C_{0}$-contraction semigroup
$(S(t))_{t\geq 0}$ on both $X$ and $Y$. Suppose that $Y\hookrightarrow
D_{A}(\alpha,\infty)$ continuously if $\alpha\in(0,1)$ or $Y\hookrightarrow
D(A)$ continuously if $\alpha=1$. Let $u_{0}\in
L_{\mathcal{F}_{0}}^{p}(\Omega;Y)$. Consider the exponential Euler method
$R\coloneqq S$ for time discretisation. Denote by $U$ the mild solution of
(6.1) and by $(U^{j})_{j=0,\ldots,N_{k}}$ the temporal approximations as
defined in (6.6). Then for $N_{k}\geq 2$

$\left\lVert\max_{0\leq j\leq N_{k}}\|U(t_{j})-U^{j}\|\right\rVert_{p}\leq
C_{\mathrm{S},\mathrm{e}}\left(C_{\mathrm{S},1}k+C_{\mathrm{S},2}k^{1/2}+C_{\mathrm{S},3}k^{\alpha}\right)$

with constants $C_{\mathrm{S},\mathrm{e}}\coloneqq C_{\mathrm{e}}$,
$C_{\mathrm{S},1}\coloneqq C_{1}$, $C_{\mathrm{S},2}\coloneqq C_{2}$ as in
Theorem 6.4, $C_{\mathrm{S},3}\coloneqq
C_{\mathrm{S},2,\alpha}T+C_{\mathrm{S},3,\alpha}T^{1/2}$,
$C_{\mathrm{S},3,\alpha}\coloneqq C_{3,\alpha}$, and

$\displaystyle C_{\mathrm{S},2,\alpha}$
$\displaystyle\coloneqq\frac{1}{\alpha+1}\left(C_{F,X}L_{3}+C_{\alpha,F}+2C_{Y}\big{(}L_{F,Y}K_{u_{0},f,g,Y}+\|f\|_{L^{p}(\Omega;C([0,T];Y))}\big{)}\right),$

where $C_{3,\alpha}$ is as defined in Theorem 6.4, $L_{3}$ as in Lemma 6.2,
$K_{u_{0},f,g,Y}$ as in (6.8), $C_{Y}$ denotes the embedding constant of $Y$
into $D_{A}(\alpha,\infty)$ or $D(A)$, and $B_{p}$ is the constant from
Theorem 2.2.

In particular, the approximations $(U^{j})_{j}$ converge at rate
$\min\\{\alpha,\frac{1}{2}\\}$ as $k\to 0$.

###### Proof.

Adopt the notation from the proof of Theorem 6.4. Contractivity of $R$ on $X$
and $Y$ is immediate from contractivity of $S$ on these spaces. Since
$S(t_{j})-R_{k}^{j}=0$ for any $j\in\\{0,\ldots,N_{k}\\}$, the terms $M_{1}$
and $M_{3,5}$ vanish. Moreover, the second term in $M_{2,4}$ vanishes so that

$\displaystyle M_{2,4}$
$\displaystyle\leq\frac{2C_{Y}}{\alpha+1}\big{(}L_{F,Y}K_{u_{0},f,g,Y}+\|f\|_{p,\infty,Y}\big{)}t_{N}k^{\alpha}.$

Combining the individual bounds for the remaining terms, the estimate follows
from a discrete Gronwall argument as in the proof of Theorem 6.4. The
logarithmic correction factor vanishes due to $M_{3,5}=0$. ∎

######  Remark 6.7.

Adding a term that is quadratic in the Wiener increment to the exponential
Euler discretisation yields the _Milstein scheme_ , which has been found to
give good convergence properties [44]. In the parabolic case (i.e., $A$ self-
adjoint and with compact resolvent), [44, Thm. 1] yields convergence of rate
arbitrarily close to $1$ in the cases of additive noise or multiplicative
noise satisfying a commutativity condition, which has been removed in
subsequent work [69]. An extension of these results for the Milstein scheme to
the hyperbolic case has been raised as a direction for future research in
[44], which, to the best of our knowledge, remains open. Moreover, in [44,
69], the pointwise strong error is analysed, from which a pathwise uniform
convergence rate can only be obtained at the price of deteriorating the
convergence rate, as discussed Remark 6.5.

###  6.3. Error estimates on the full time interval

In this subsection, we will extend the error estimates of Theorem 6.4 and
Corollary 6.6 to the full time interval by using a suitable Hölder regularity
of the paths of the mild solution.

The following simple deterministic result provides a way to connect the
uniform error to the error on the grid. Given a non-decreasing function
$\Phi:[0,T]\to[0,\infty)$ such that $\Phi\neq 0$ on $(0,T]$ we say that $u\in
C^{\Phi}([0,T];X)$ if $u:[0,T]\to X$ is continuous and

$[u]_{C^{\Phi}([0,T];X)}=\sup_{0\leq s<t\leq
T}\frac{\|u(t)-u(s)\|}{\Phi(t-s)}<\infty.$

Moreover, we set
$\|u\|_{C^{\Phi}([0,T];X)}\coloneqq\|u\|_{\infty}+[u]_{C^{\Phi}([0,T];X)}$. We
shall be particularly interested in the function
$\Phi(r)=r^{\alpha}(1+\log(\frac{T}{r}))^{1/2}$ for $r\in(0,T]$ for some
$\alpha>0$ and $\Phi(0)=0$ in the following.

######  Lemma 6.8 (Decomposition of the error on the full time interval).

Let $u\in C^{\Phi}([0,T];X)$ for a non-decreasing function
$\Phi:[0,T]\to[0,\infty)$ such that $\Phi\neq 0$ on $(0,T]$. Let
$\Pi\subseteq[0,T]$ be a finite time grid, and denote by $\tilde{u}:\Pi\to X$
an approximation of $u$, which is extended to $[0,T]$ by setting
$\tilde{u}(t)\coloneqq\tilde{u}(\lfloor t\rfloor_{\Pi})$ for $t\notin\Pi$,
where $\lfloor t\rfloor_{\Pi}\coloneqq\max\\{s\in\Pi:s\leq t\\}$. Then it
holds that

$\sup_{t\in[0,T]}\|u(t)-\tilde{u}(t)\|\leq\Phi(h)\cdot\|u\|_{C^{\Phi}([0,T];X)}+\sup_{t\in\Pi}\|u(t)-\tilde{u}(t)\|$

for the maximal time step
$h\coloneqq\sup_{t\in[0,T]}\operatorname{dist}(t,\Pi)$.

###### Proof.

For $t\in[0,T]$ we can write

$\displaystyle\|u(t)-\tilde{u}(t)\|$ $\displaystyle\leq\|u(t)-u(\lfloor
t\rfloor_{\Pi})\|+\|u(\lfloor t\rfloor_{\Pi})-\tilde{u}(t)\|$
$\displaystyle\leq\|u\|_{C^{\Phi}([0,T];X)}\cdot\Phi(t-\lfloor
t\rfloor_{\Pi})+\sup_{s\in\Pi}\|u(s)-\tilde{u}(s)\|,$

which implies the required result. ∎

From the above, we see that to estimate the uniform error on $[0,T]$, we need
an (optimal) Hölder regularity result for the mild solution $U$ to (6.1). To
obtain such a result, the main difficulty lies in estimating the stochastic
convolution.

######  Lemma 6.9 (Path regularity of stochastic convolutions).

Let $X,Y$ be Hilbert spaces such that $Y\hookrightarrow X$ continuously. Let
$A$ be the generator of a $C_{0}$-contraction semigroup $(S(t))_{t\geq 0}$ on
both $X$ and $Y$. Suppose that $Y\hookrightarrow D_{A}(\alpha,\infty)$ holds
for some $\alpha\in(0,1/2]$. Let $q\in(2,\infty]$ be such that
$\frac{1}{2}-\frac{1}{q}=\alpha$ and let $2\leq p<p_{0}<\infty$. Suppose that

$g\in L^{p}(\Omega;L^{2}(0,T;\mathcal{L}_{2}(H,Y)))\cap
L^{p_{0}}(\Omega;L^{q}(0,T;\mathcal{L}_{2}(H,X)))$

and define $J_{g}:\Omega\times[0,T]\to X$ as the stochastic convolution

$J_{g}(t)=\int_{0}^{t}S(t-s)g(s)dW_{H}(s).$

Then one has $J_{g}\in L^{p}(\Omega;C^{\Psi}([0,T];X))$ for
$\Psi:(0,T]\to(0,\infty),\Psi(r)\coloneqq
r^{\alpha}(1+\log(\frac{T}{r}))^{1/2}$ and there exist constants
$C_{p},C_{\alpha,p,p_{0},T}\geq 0$ such that

$\displaystyle\|J_{g}\|_{L^{p}(\Omega;C^{\Psi}([0,T];X))}\leq
C_{p}\|g\|_{L^{p}(\Omega;L^{2}(0,T;\mathcal{L}_{2}(H,Y)))}+C_{\alpha,p,p_{0},T}\|g\|_{L^{p_{0}}(\Omega;L^{q}(0,T;\mathcal{L}_{2}(H,X)))}.$

By a simple rescaling, the result extends to quasi-contraction semigroups.
Moreover, from the proof below one can see that a certain Orlicz integrability
in $\Omega$ is sufficient for $g$. Note that the above path regularity is
optimal for $q=\infty$. Indeed, Lévy’s modulus of continuity theorem for a
scalar Brownian motion states that a.s.

$\limsup_{h\downarrow
0}\sup_{t\in[0,1-h]}\frac{B(t+h)-B(t)}{\sqrt{2h\log(1/h)}}=1,$

which shows that $\Psi$ cannot be replaced by a “better” function.

###### Proof of Lemma 6.9.

For $0\leq s<t\leq T$, we can write

$\displaystyle\|J_{g}(t)-J_{g}(s)\|$
$\displaystyle\leq\Big{\|}(S(t-s)-I)\int_{0}^{s}S(s-r)g(r)\,\mathrm{d}W_{H}(r)\Big{\|}+\Big{\|}\int_{s}^{t}S(t-r)g(r)\,\mathrm{d}W_{H}(r)\Big{\|}$
$\displaystyle\eqqcolon T_{1}(t,s)+T_{2}(t,s).$

For $T_{1}$ we can write

$T_{1}(t,s)\leq\|S(t-s)-I\|_{\mathcal{L}(Y,X)}\Big{\|}\int_{0}^{s}S(s-r)g(r)\,\mathrm{d}W_{H}(r)\Big{\|}_{Y}\leq
c(t-s)^{\alpha}\|J_{g}(s)\|_{Y}$

for some $c\geq 0$. Therefore, by Theorem 2.2 we obtain

$\displaystyle\bigg{\|}\sup_{0\leq s<t\leq
T}\frac{T_{1}(t,s)}{\Psi(t-s)}\bigg{\|}_{p}\leq c\bigg{\|}\sup_{0\leq s<t\leq
T}\frac{\|J_{g}(s)\|_{Y}}{(1+\log(\frac{T}{t-s}))^{1/2}}\bigg{\|}_{L^{p}(\Omega)}\leq
cB_{p}\|g\|_{L^{p}(\Omega;L^{2}(0,T;\mathcal{L}_{2}(H,Y)))}.$

For $T_{2}$ we use the dilation result of [63, Theorem I.7.1] (cf. [34]). We
can find a Hilbert space $\widetilde{X}$, a contractive injection
$Q:X\to\widetilde{X}$, a contractive projection $P:\widetilde{X}\to X$, and a
unitary $C_{0}$-group $(G(t))_{t\in\mathbb{R}}$ on $\widetilde{X}$ such that
$S(t)=PG(t)Q$ for $t\geq 0$. Thus, we can write

$\displaystyle
T_{2}(t,s)=\Big{\|}\int_{s}^{t}PG(t-r)Qg(r)\,\mathrm{d}W_{H}(r)\Big{\|}_{X}\leq\Big{\|}\int_{s}^{t}G(-r)Qg(r)\,\mathrm{d}W_{H}(r)\Big{\|}_{\widetilde{X}}=\|I(t)-I(s)\|_{\widetilde{X}},$

where $I(t)\coloneqq\int_{0}^{t}G(-r)Qg(r)\,\mathrm{d}W_{H}(r)$. Then by [61,
(2.12) and Theorem 3.2(vi)] we have $I\in
L^{p}(\Omega;C^{|\cdot|^{\alpha}|\log(\cdot)|^{1/2}}([0,T];\tilde{X}))$ and
thus by boundedness of $|\log(\cdot)|^{1/2}(1+\log(\frac{T}{\cdot}))^{-1/2}$
on $(0,T]$ also $I\in L^{p}(\Omega;C^{\Psi}([0,T];\tilde{X}))$. Moreover,
there are constants $c_{\alpha,T},C_{\alpha,p,p_{0},T}\geq 0$ such that

$\displaystyle\|I\|_{L^{p}(\Omega;C^{\Psi}([0,T];\tilde{X}))}$
$\displaystyle\leq
c_{\alpha,T}\|I\|_{L^{p}(\Omega;B^{\alpha}_{\Phi_{2},\infty}(0,T;\widetilde{X}))}$
$\displaystyle\leq
C_{\alpha,p,p_{0},T}\|G(-r)Qg(r)\|_{L^{p_{0}}(\Omega;L^{q}(0,T;\mathcal{L}_{2}(H,\widetilde{X})))}$
$\displaystyle\leq
C_{\alpha,p,p_{0},T}\|g\|_{L^{p_{0}}(\Omega;L^{q}(0,T;\mathcal{L}_{2}(H,X)))},$

where $B_{\Phi_{2},\infty}^{\alpha}(0,T;\tilde{X})$ denotes the Besov-Orlicz
space corresponding to $\Phi_{2}(x)\coloneqq\exp(x^{2})-1$, cf. [61, Section
2.3] for the definition. It follows that

$\displaystyle\bigg{\|}\sup_{0\leq s<t\leq
T}\frac{T_{2}(t,s)}{\Psi(t-s)}\bigg{\|}_{p}$
$\displaystyle\leq\|I\|_{L^{p}(\Omega;C^{\Psi}([0,T];\tilde{X}))}\leq
C_{\alpha,p,p_{0},T}\|g\|_{L^{p_{0}}(\Omega;L^{q}(0,T;\mathcal{L}_{2}(H,X)))}.$

Now the required estimate follows by combining the estimates for $T_{1}$ and
$T_{2}$. ∎

######  Remark 6.10.

For analytic semigroups on $X$, the result of Lemma 6.9 even holds if merely
$g\in L^{p_{0}}(\Omega;L^{q}(0,T;\mathcal{L}_{2}(H,X)))$, and even $J_{g}\in
L^{p}(\Omega;B^{\alpha}_{\Phi_{2},\infty}(0,T;X))$ (see [61, Theorem 5.1]). In
particular, the space $Y$ and contractivity of $S$ are not needed. We do not
know if one can take $p_{0}=p$ in Lemma 6.9, even in the analytic case. Also,
we do not know if the above Besov regularity of $J_{g}$ holds in the non-
analytic case.

Sharp path regularity results such as the one of Lemma 6.9 play an important
role in obtaining convergence rates for numerical schemes for SPDEs. In
particular, recent other applications of [61] to numerics include [29, 56, 72,
73]. Below, we apply Lemma 6.9 to obtain additional information on the
numerical approximation in the Kato setting, and it seems to be the first of
its kind for hyperbolic equations.

After these preparations, we can now prove the required path regularity of the
mild solution.

######  Proposition 6.11 (Path regularity of the mild solution).

Suppose that Assumption 6.1 holds for some $\alpha\in(0,1/2]$ and
$p\in[2,\infty)$. Let $p_{0}\in(p,\infty)$ and $q\in(2,\infty]$ be such that
$\frac{1}{2}-\frac{1}{q}=\alpha$, and suppose that $f,g$, and $u_{0}$
additionally satisfy

$f\in L^{p_{0}}(\Omega;L^{1}(0,T;X)),\ \ g\in
L^{p_{0}}(\Omega;L^{q}(0,T;\mathcal{L}_{2}(H,X))),\ \ \text{and}\ \ u_{0}\in
L_{\mathcal{F}_{0}}^{p_{0}}(\Omega;X)\cap L_{\mathcal{F}_{0}}^{p}(\Omega;Y).$

Let $A$ be the generator of a $C_{0}$-contraction semigroup $(S(t))_{t\geq 0}$
on both $X$ and $Y$. Suppose that $Y\hookrightarrow D_{A}(\alpha,\infty)$
continuously. Let $\Psi:(0,T]\to(0,\infty)$ be given by $\Psi(r)\coloneqq
r^{\alpha}(1+\log(\frac{T}{r}))^{1/2}$. Then the mild solution to (6.1)
satisfies $U\in L^{p}(\Omega;C^{\Psi}([0,T];X))$ and there exists a constant
$C$ depending on $(T,p,p_{0},\alpha,\tilde{F},\tilde{G},\;X,Y)$ such that

$\displaystyle\|U\|_{L^{p}(\Omega;C^{\Psi}([0,T];X))}\leq C\big{(}1$
$\displaystyle+\|u_{0}\|_{L^{p}(\Omega;Y)}+\|f\|_{p,\infty,Y}+|\\!|\\!|g|\\!|\\!|_{p,\infty,Y}$
$\displaystyle+\|u_{0}\|_{L^{p_{0}}(\Omega;X)}+\|f\|_{p_{0},1,X}+|\\!|\\!|g|\\!|\\!|_{{p_{0}},q,X}\big{)}.$

###### Proof.

The mild solution formula (6.2) yields an initial value term, a difference of
deterministic convolutions, and a stochastic version of the latter. The first
two can be estimated as in the proof of Lemma 6.2, resulting in an upper bound
of the form

$c(1+\|u_{0}\|_{L^{p}(\Omega;Y)}+\|f\|_{p,\infty,Y}+|\\!|\\!|g|\\!|\\!|_{p,2,Y})$

for some $c\geq 0$ depending on $T$. To the remaining term, we apply Lemma 6.9
and note that

$\displaystyle|\\!|\\!|G(\cdot,U(\cdot))|\\!|\\!|_{p,2,Y}$ $\displaystyle\leq
L_{G,Y}C_{u_{0},f,g,Y}+|\\!|\\!|g|\\!|\\!|_{p,\infty,Y},$
$\displaystyle|\\!|\\!|G(\cdot,U(\cdot))|\\!|\\!|_{p_{0},q,X}$
$\displaystyle\leq
T^{1/q}|\\!|\\!|G(\cdot,U(\cdot))|\\!|\\!|_{p_{0},\infty,X}+|\\!|\\!|g|\\!|\\!|_{p_{0},q,X}\leq
T^{1/q}C_{G,X}\tilde{C}_{u_{0},f,g,X}+|\\!|\\!|g|\\!|\\!|_{p_{0},q,X}$
$\displaystyle\lesssim
1+\|u_{0}\|_{L^{p_{0}}(\Omega;X)}+\|f\|_{p_{0},1,X}+|\\!|\\!|g|\\!|\\!|_{p_{0},q,X},$

where $\tilde{C}_{u_{0},f,g,X}$ is defined as $C_{u_{0},f,g,X}$ in (6.4) with
$p$ replaced by $p_{0}$. ∎

Consequently, we can now “upgrade” Theorem 6.4 to an estimate on the full time
interval.

######  Theorem 6.12 (Uniform error on the full interval for general schemes).

Suppose that Assumption 6.1 holds for some $\alpha\in(0,1/2]$ and
$p\in[2,\infty)$. Let $A$ be the generator of a $C_{0}$-contraction semigroup
$(S(t))_{t\geq 0}$ on both $X$ and $Y$. Let $(R_{k})_{k>0}$ be a time
discretisation scheme which is contractive on $X$ and $Y$. Assume $R$
approximates $S$ to order $\alpha$ on $Y$. Suppose that $Y\hookrightarrow
D_{A}(\alpha,\infty)$ continuously. Let $p_{0}\in(p,\infty)$ and
$q\in(2,\infty]$ be such that $\frac{1}{2}-\frac{1}{q}=\alpha$, and suppose
that $f,g$, and $u_{0}$ have additional integrability as $X$-valued processes

$f\in L^{p_{0}}(\Omega;L^{1}(0,T;X)),\ \ g\in
L^{p_{0}}(\Omega;L^{q}(0,T;\mathcal{L}_{2}(H,X))),\ \ \text{and}\ \ u_{0}\in
L_{\mathcal{F}_{0}}^{p_{0}}(\Omega;X)\cap L_{\mathcal{F}_{0}}^{p}(\Omega;Y).$

Denote by $U$ the mild solution of (6.1) and by $(U^{j})_{j=0,\ldots,N_{k}}$
the temporal approximations as defined in (6.6). Define the piecewise constant
extension $\tilde{U}:[0,T]\to L^{p}(\Omega;X)$ by $\tilde{U}(t)\coloneqq
U^{j}$ for $t\in[t_{j},t_{j+1})$, $0\leq j\leq N_{k}-1$, and
$\tilde{U}(T)\coloneqq U^{N_{k}}$. Then for all $N_{k}\geq 2$ there is a
constant $C\geq 0$ depending on $(u_{0},T,p,p_{0},\alpha,F,G,\;X,Y)$ such that

$\bigg{\|}\sup_{t\in[0,T]}\|U(t)-\tilde{U}(t)\|\bigg{\|}_{p}\leq
C\Big{(}1+\log\Big{(}\frac{T}{k}\Big{)}\Big{)}k^{\alpha}.$

###### Proof.

The error bound follows from applying Lemma 6.8 with
$\Phi=(\cdot)^{\alpha}(1+\log(\frac{T}{\cdot}))^{1/2}$ in combination with
Theorem 6.4 and Proposition 6.11 to bound the first and second term obtained
from the proposition, respectively. ∎

Invoking Corollary 6.6 instead of Theorem 6.4, we obtain an analogous bound
for exponential Euler with the square root of a logarithmic factor.

######  Corollary 6.13 (Uniform error on the interval for exponential Euler).

Suppose that the conditions of Theorem 6.12 hold and that $R_{k}=S(k)$. Then