# Higher order approximation of nonlinear SPDEs with additive space-time white
noise

Ana Djurdjevac, Máté Gerencsér, Helena Kremp  Freie Universität Berlin,
Arnimallee 6, 14195 Berlin, Germany  TU Wien, Wiedner Hauptstraße 8-10, 1040
Vienna, Austria<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
(Date: August 28, 2024)

###### Abstract.

We consider strong approximations of $1+1$-dimensional stochastic PDEs driven
by additive space-time white noise. It has long been proposed [DG01, JK08], as
well as observed in simulations, that approximation schemes based on samples
from the stochastic convolution, rather than from increments of the underlying
Wiener processes, should achieve significantly higher convergence rates with
respect to the temporal timestep. The present paper proves this. For a large
class of nonlinearities, with possibly superlinear growth, a temporal rate of
(almost) $1$ is proven, a major improvement on the rate $1/4$ that is known to
be optimal for schemes based on Wiener increments. The spatial rate remains
(almost) $1/2$ as it is standard in the literature.

Mathematics Subject Classification (2020): 60H15, 60H35

Keywords: stochastic PDEs, strong convergence rates, splitting scheme,
stochastic sewing

##  1\. Introduction

We consider stochastic reaction-diffusion equations of the form

(1.1) $\displaystyle\partial_{t}u=\Delta u+f(u)+\xi,$

with $(t,x)\in\mathbb{R}^{+}\times\mathbb{T}$, space-time white noise $\xi$, a
given nonlinearity $f:\mathbb{R}\to\mathbb{R}$ and initial condition $u_{0}$.
A common way to express the noise is to write $\xi$ as the (distributional)
derivative $\partial_{t}\partial_{x}W$ of a $2$-dimensional Brownian sheet
$W$.

Under some mild regularity assumption on $f$, existence and uniqueness of
solutions to (1.1) is classical. In the present work we are interested in full
discretisations of the equation. This question was first addressed in [Gyö99].
A finite difference in space, (explicit or implicit) Euler method in time was
studied based on sampling rectangular increments of $W$ on a grid with
meshsize $M^{-1}$ in time and $N^{-1}$ in space, and $L^{p}$ rate of
convergence of order $M^{-1/4}+N^{-1/2}$ was proved111To avoid obscuring the
overview of the literature with $\varepsilon$-s, for simplicity we do not
distinguish between rate $\alpha$ and rate $\alpha-\varepsilon$ for all
$\varepsilon>0$ in the mentioned results. . In [DG01] this rate is shown to be
optimal in the sense that the conditional variance of the solution at a point
given such samples of $W$ is lower bounded by a positive constant times
$M^{-1/4}+N^{-1/2}$. Similar upper and lower bounds are obtained in [BGJK20]
for schemes based on a Galerkin truncation of $W$ in space and then sampling
its increments in time.

In an attempt to overcome the order barrier $1/4$ with respect to the temporal
stepsize, [JK08] proposed a different scheme (already hinted at in [DG01,
Section 2.3]). The essential difference was the use of different functionals
of the noise: instead of sampling increments of $W$, they used samples from
the stochastic convolution with the semigroup generated by $\Delta$. The
stochastic convolution is a Gaussian process with explicitly known covariance,
so that its sampling is straightforward. [JK08] considered, instead of (1.1),
equations in a more abstract framework, viewing the nonlinearity as a function
$F:L^{2}(\mathbb{T})\to L^{2}(\mathbb{T})$ and the equation as a SDE on the
Hilbert space $L^{2}(\mathbb{T})$. Under certain regularity conditions on $F$,
the new scheme was shown to have a far superior rate of convergence $1$ with
respect to the time stepsize, a major improvement on the previous rate $1/4$.
However, one of the main assumptions therein turns out to be too restrictive
to allow for any(!) truly nonlinear SPDEs of the form (1.1):

######  Assumption 1.1.

([JK08, part of Asn. 2.4]) The map $F$ is Gateaux differentiable and there
exists constant $L>0$ such that for all $u\in L^{2}(\mathbb{T})$,
$v\in\mathrm{Dom}(1-\Delta)$ one has

(1.2)
$\displaystyle\big{\|}(1-\Delta)^{-1}F^{\prime}(u)(1-\Delta)v\|_{L^{2}(\mathbb{T})}\leq
L\|v\|_{L^{2}(\mathbb{T})}$

######  Proposition 1.2.

Let $f:\mathbb{R}\to\mathbb{R}$ be a differentiable function with bounded
derivative and let $F:L^{2}(\mathbb{T})\to L^{2}(\mathbb{T})$ be defined as
$F(u)(x)=f(u(x))$. Then $F$ satisfies 1.1 if and only if $f$ is affine linear.

######  Remark 1.3.

In [JK08] (and in a rather large portion of the literature) $F$ is in fact
assumed to be not only Gateaux, but Fréchet differentiable. Note that this
already excludes all truly nonlinear Nemytskii operators: If
$F:L^{2}(\mathbb{T})\to L^{2}(\mathbb{T})$ is defined as $F(u)(x)=f(u(x))$
with $f$ being a differentiable function with bounded derivative, then there
exists $u^{\star}\in L^{2}$ such that $F$ is Fréchet differentiable at
$u^{\star}$ if and only if $f$ is affine linear ([AP93, Proposition 2.8]).

The proof is fairly straightforward and is given in Section 3. In light of
Proposition 1.2, the problem of “overcoming of the order barrier” for (1.1)
remained open. Partial progress towards the conjectured rate $1$ has been made
in [Jen11] and [Wan20], who proved rate $1/2$ in time for globally Lipschitz
$f$ and cubic polynomial $f$ with negative leading order coefficient,
respectively. Some partial results hinting at the possibility of a higher rate
can be found in the theses [Kha15], [Sal15]. On the other side, temporal rate
$1$ is proved in [JKW11, BCH18] without imposing Assumption 1.1, but assuming
a strong coloring condition on the noise instead, falling well short of the
space-time white noise case. In yet another direction, [GS24] proved rate
$1/2$ in time (and even an improved rate $1$ in space) for (1.1) with
polynomial $f$ with odd degree and negative leading order coefficient, even
with just using Wiener increments, at the cost of measuring the error not in a
functional-, but rather a distributional space. While such distributional
norms are rather natural when dealing with higher dimensional stochastic
reaction-diffusion equations (see e.g. [MZ21]), in the $1$ dimensional case it
is desirable to bound the error in a genuine function space.

The aim of the present paper is to overcome all of the aforementioned caveats
of [JK08, Jen11, JKW11, Wan20, GS24]. We prove strong rate of convergence of
rate $1-\varepsilon$ in time for any $\varepsilon$. The spatial error remains
of order $N^{-1/2+\varepsilon}$ as common in the preceding literature. The
methods can be applied in considerable generality, both in terms of the
nonlinearity $f$ and the employed scheme. First, we consider $f$ that is
globally bounded and has globally bounded derivatives. We show the
aforementioned rate of convergence for a spectral Galerkin scheme in space and
accelerated exponential explicit Euler in time. In this case the global bounds
on $f$ allow simple a priori bounds as well as the application of Girsanov’s
theorem, which greatly simplifies the proof. The error estimate is uniform in
space-time in this case. Second, we consider $f$ that can grow polynomially,
obeying the one-sided Lipschitz condition. For this class of equations, simple
schemes like exponential explicit Euler or standard Euler are not suitable due
to the blow-up of such approximations, cf. [HJ12, BHJ+19]. Instead one
considers tamed schemes (see e.g. [BGJK23, BJ19, JP20, Wan20]), splitting
schemes (see e.g. [BCH18, BG19]) or implicit Euler schemes (see e.g. [LQ19]),
for which a priori bounds on the numerical solution can be derived (and
therefore the blow-up is avoided). We employ a splitting scheme for the
temporal discretisation and prove temporal rate $M^{-1+\varepsilon}$ with the
error measured uniform in time and $L^{2}$ in space. The precise statements
are formulated in Section 2 in Theorems 2.2 and 2.6.

The strategy leading to this improved rate relies on two key ingredients,
inspired by [GS24] and [BDG23]. The first important property to note is that
the errors strongly depend on the topology in which they are measured. To
illustrate this, consider the Ornstein-Uhlenbeck process $O$, that is, the
solution to (1.1) with $f=0$, $u_{0}=0$. It is well-known that $O$ is almost
$1/4$-Hölder continuous in time but no better. That is, for all
$\varepsilon>0$ there exist constants $c=c(\omega)>0$,
$C=C(\varepsilon,\omega)<\infty$ such that almost surely for all $0\leq s\leq
t\leq 1$

(1.3) $\displaystyle
c|t-s|^{1/4}\leq\|O_{t}-O_{s}\|_{L^{\infty}(\mathbb{T})}\leq
C|t-s|^{1/4-\varepsilon}.$

However, moving to weaker topologies the time regularity of $O$ increases: for
example, for any $\varepsilon>0$ there exists $C=C(\varepsilon,\omega)<\infty$
such that for all $0\leq s\leq t\leq 1$

(1.4) $\displaystyle\|O_{t}-O_{s}\|_{\mathscr{C}^{-1/2}(\mathbb{T})}\leq
C|t-s|^{1/2-\varepsilon},$

where $\mathscr{C}^{-1/2}(\mathbb{T})$ is a Besov-Hölder space of negative
regularity (see Section 3 below for details). Since the regularity properties
are naturally linked to rates of convergence of discretisations, one would
like to leverage improved temporal regularity estimates like (1.4) to obtain
improved temporal rates. This is the starting point to achieve $1/2$ temporal
rate in a distributional norm [GS24]. One key point of the present paper is to
use similar ideas but still end up with error bounds in a functional norm.
Note however that this idea seems to stop at rate $1/2$: one can not weaken
the topology further in (1.4) to improve the temporal regularity. Indeed, even
a single Fourier mode of $O$ is no better than $1/2$-Hölder continuous in
time.

To illustrate the other main ingredient, consider the time integral

(1.5) $\displaystyle
E_{M}:=\Big{|}\int_{0}^{T}P_{T-s}\big{(}f(O_{s})-f(O_{k_{M}(s)})\big{)}\,ds\Big{|}.$

Here $P$ is the heat kernel, $M$ is the number of timesteps in an equidistant
partition of the time horizon, and $k_{M}(s)$ is the last gridpoint before
$s$. In the error analysis, it turns out that $E_{M}$ determines the temporal
rate. However, efficient estimates for $E_{M}$ are highly nontrivial even if
$f\in C_{c}^{\infty}$. Using the triangle inequality, a global Lipschitz bound
on $f$, and (1.3), one easily obtains that $E_{M}\lesssim
M^{-1/4+\varepsilon}$, which is the classical error rate. It is not clear how
one could use (1.4) in order to improve the rate, since a truly nonlinear
function $f$ is not Lipschitz continuous with respect to the
$\mathscr{C}^{-1/2}(\mathbb{T})$-norm. Even if one could overcome this, the
resulting error bound would only be $M^{-1/2+\varepsilon}$. The main
improvement on estimating $E_{M}$ comes from not using the triangle
inequality. Indeed, a fundamental idea coming from the field of regularisation
by noise is that integrals along oscillatory processes enjoy a lot of
cancellations, which are lost when bringing the absolute value inside the
integral. For example, in the case when $f$ is merely a bounded measurable
function, a slight variation of [BDG23, Lemma 3.3.1] shows that $E_{M}\lesssim
M^{-1/2+\varepsilon}$, where the triangle inequality would give no rate
whatsoever. A robust approach to obtain such improved estimates is the
stochastic sewing strategy, originating from [Lê20] and introduced to treat
numerical analytic problems in [BDG21]. It is interesting to note however,
that for many regularisation by noise arguments the $1+1$-dimensional
stochastic heat equation behaves much like a fractional Brownian motion $B$
with Hurst parameter $1/4$, for which the best known rate for the analogue of
$E_{M}$, namely the error

(1.6) $\displaystyle\Big{|}\int_{0}^{T}f(B_{s})-f(B_{k_{M}(s)})\,ds\Big{|},$

is $3/4-\varepsilon$ ([BDG21, Lemma 4.1]) even in the case of $f\in
C^{\infty}_{c}$.

Therefore, while neither of the two methods are sufficient on their own to
obtain the desired rate, the aim of the present paper is to combine them in
such a way that leverages the advantages of both the distributional power
counting and the stochastic sewing, and yield the claimed temporal rate
$1-\varepsilon$. Since the expressions (1.5) and (1.6) differ by the semigroup
$P$ inside the integral, the heuristic goal is to use it to improve the rate
by lowering the spatial regularity where $f(O_{s})-f(O_{k_{M}(s)})$ is
estimated.

It is notable that our strong rate of $1$ for the temporal error even exceeds
the best known weak error rates for splitting schemes of the Allen-Cahn
equation, cf. [BG20], or for exponential Euler schemes of nonlinear heat
equations, cf. [Wan16], where a temporal weak rate of $1/2$ is proven.

Let us end with a couple of remarks and open questions. Unlike in the Wiener
increment sample case [DG01], for schemes based on sampling the stochastic
convolution we are not aware of lower error bounds. We expect that higher
temporal order than $M^{-1-\varepsilon}$ can not be achieved, which would show
optimality (up to $\varepsilon$) of our result. Even without the presence of a
spatial discretisation such a lower bound would be enlightening. In terms of
the temporal discretisations used, in the present paper we consider an
accelerated exponential explicit Euler and a splitting scheme. It would be
interesting to extend the methods to other approximations, e.g. to implicit
Euler or tamed schemes. Furthermore, it seems promising to pursue this
strategy for SPDEs whose nonlinearities do not only depend on the solution but
also on its gradient, e.g. the stochastic Burgers’ equation, whose
nonlinearity is $\partial_{x}(u^{2})$.

## Acknowledgments

The authors thank Arnulf Jentzen for bringing this interesting problem to
their attention, providing many useful references, and pointing out the
properties of Fréchet differentiability in Remark 1.3.
ADj gratefully acknowledges funding by Daimler and Benz Foundation as part of
the scholarship program for junior professors and postdoctoral researchers. MG
is funded by the European Union (ERC, SPDE, 101117125). Views and opinions
expressed are however those of the author(s) only and do not necessarily
reflect those of the European Union or the European Research Council Executive
Agency. Neither the European Union nor the granting authority can be held
responsible for them. HK was supported by the Austrian Science Fund (FWF)
P34992.

##  2\. Set-up and Statement

Let $(\Omega,\mathcal{F},\mathds{P})$ be a probability space. We fix a time
horizon $T>0$. The torus $\mathbb{T}$ is defined as
$\mathbb{T}=\mathbb{R}/\mathbb{Z}$. The space-time white noise $\xi$ is
defined as a mapping from the Borel sets $\mathcal{B}([0,T]\times\mathbb{T})$
into $L^{2}(\Omega)$, such that for any collection
$A_{1},\dots,A_{k}\in\mathcal{B}([0,T]\times\mathbb{T})$, the vector
$(\xi(A_{1}),\dots,\xi(A_{k}))$ is Gaussian with zero mean and covariance
$\mathds{E}[\xi(A_{i})\xi(A_{j})]=\lambda(A_{i}\cap A_{j})$, where $\lambda$
denotes the Lebesgue measure. We also fix a filtration
$\mathbb{F}=(\mathcal{F}_{t})_{t\in[0,T]}$, such that
$(\Omega,\mathcal{F},\mathbb{F},\mathds{P})$ is complete and such that for any
$t\in[0,T]$, $A\in\mathcal{B}([0,t]\times\mathbb{T})$,
$B\in\mathcal{B}([t,T]\times\mathbb{T})$, $\xi(A)$ is
$\mathcal{F}_{t}$-measurable and $\xi(B)$ is independent of $\mathcal{F}_{t}$.
An example for $\mathbb{F}$ would be the completed filtration generated by
$\xi$. The predictable $\sigma$-algebra on $\Omega\times[0,T]$ will be denoted
by $\mathcal{P}$. Stochastic integrals
$\int_{0}^{T}\int_{\mathbb{T}}g(s,y)\xi(ds,dy)$ against $\xi$ can be defined
for all $\mathcal{P}\times\mathcal{B}(\mathbb{T})$-measurable integrands
$g:\Omega\times[0,T]\times\mathbb{T}\to\mathbb{R}$ with $g\in
L^{2}(\Omega\times[0,T]\times\mathbb{T})$. We refer to [DPZ92] for more
details, but remark that for deterministic $f$ (which is the case used in the
large majority of the article), the stochastic integral is simply the unique
isometric and linear extension of the map $\mathbf{1}_{A}\mapsto\xi(A)$ to
$L^{2}([0,T]\times\mathbb{T})$.

For $k\in\mathbb{Z}$, denote the Fourier modes on $\mathbb{T}$ by
$e_{k}(x)=e^{-2\pi ikx}$. The set $(e_{k})_{k\in\mathbb{Z}}$ is an orthonormal
basis of $L^{2}(\mathbb{T},\mathbb{C})$. For $f\in
L^{1}(\mathbb{T},\mathbb{C})$, its Fourier transform is denoted by
$\mathcal{F}f(k)=\hat{f}(k)=\int_{\mathbb{T}}e^{-2\pi ikx}f(x)dx$,
$k\in\mathbb{Z}$. Denote by $(P_{t})_{t\geq 0}$ the heat semigroup on the
torus, that is

$P_{t}f=\mathcal{F}^{-1}(e^{-4\pi^{2}k^{2}t}\mathcal{F}f(k)).$

One can equivalently write $P_{t}f=p_{t}\ast f$, where

$p_{t}(x)=\sum_{k\in\mathbb{Z}}e^{-4\pi^{2}k^{2}t}e^{2\pi
ikx}=\frac{1}{\sqrt{4\pi t}}\sum_{m\in\mathbb{Z}}e^{-(x-m)^{2}/4t}.$

We consider the mild formulation of (1.1):

(2.1) $\displaystyle
u_{t}=P_{t}u_{0}+\int_{0}^{t}P_{t-s}\big{(}f(u_{s})\big{)}ds+\int_{0}^{t}\int_{\mathbb{T}}p_{t-s}(x-y)\xi(ds,dy).$

The stochastic convolution, also referred to as the Ornstein-Uhlenbeck (OU)
process, is denoted by

(2.2) $\displaystyle
O_{t}:=\int_{0}^{t}\int_{\mathbb{T}}p_{t-s}(x-y)\xi(ds,dy),\quad t\geq 0.$

The well-posedness of the mild formulation is classical under a one-sided
Lipschitz and polynomial growth assumption on $f$, see Proposition 3.3 below.

We start by setting up the result in the easier case of $f$ being bounded with
bounded derivatives up to order $2$. In this setting, several steps of the
proof are simplified. Introducing the key ideas in this case hopefully
benefits the reader in understanding the more general form. For now we work
under the following assumption.

######  Assumption 2.1.

  1. (a)

There exists a constant $K$ such that for all $i=0,1,2$ and all
$x\in\mathbb{R}$ one has

(2.3) $\displaystyle|\partial^{i}f(x)|\leq K,$

with the convention that $\partial^{0}f=f$.

  2. (b)

The initial condition $u_{0}$ is an $\mathcal{F}_{0}$-measurable random
variable with values in $\mathscr{C}^{1/2}(\mathbb{T})$ and for any
$p\in[1,\infty)$ there exists a constant $\mathcal{M}(p)$ such that
$\mathds{E}\|u_{0}\|_{\mathscr{C}^{1/2}}^{p}\leq\mathcal{M}(p)$.

We use the approximation scheme exactly as in [JK08], that is a spectral
Galerkin scheme in space and accelerated exponential Euler scheme in time,
defined as follows. For $N\in\mathbb{N}$, let $\Pi_{N}$ denote the orthogonal
projection from $L^{2}(\mathbb{T},\mathbb{C})$ to the subspace
$\text{span}(e_{k},\lvert k\rvert\leq N)$. Let
$\Delta_{N}:=\Delta\Pi_{N}=\Pi_{N}\Delta$ and let
$(P_{t}^{N}:=P_{t}\Pi_{N})_{t\geq 0}$ be the corresponding semigroup. As
before, one can write $P_{t}^{N}f=p_{t}^{N}\ast f$, where

$p_{t}^{N}(x)=\sum_{\lvert k\rvert\leq N}e^{-4\pi^{2}k^{2}t}e^{2\pi ikx}.$

One then first defines a spatial approximation $U^{N}$ as the solution of the
finite dimensional SDE

(2.4) $\displaystyle
U^{N}_{t}=P^{N}_{t}u_{0}+\int_{0}^{t}P_{t-s}^{N}f(U^{N}_{s})ds+O_{t}^{N},\quad
t\in[0,T],$

with the notation $O^{N}$ for the spatially discretised stochastic convolution

(2.5) $\displaystyle
O^{N}_{t}=\int_{0}^{t}\int_{\mathbb{T}}p^{N}_{t-s}(x-y)\xi(ds,dy).$

For $M\in\mathbb{N}$, let $h=T/M$ and consider the temporal gridpoints
$t_{k}=kh$ for $k=0,\ldots,M$. Let for $s\in[0,1]$, $k_{M}(s):=\lfloor
h^{-1}s\rfloor h$ be the last gridpoint before (or equal to) $s$. The
exponential Euler time discretisation of (2.4), that yields a space-time
discretisation of (2.1), is denoted by $V^{M,N}$ and defined inductively by
setting $V_{0}^{M,N}=\Pi_{N}u_{0}=:u_{0}^{N}$ and

$\displaystyle
V_{t_{k+1}}^{M,N}=P_{h}^{N}V_{t_{k}}^{M,N}+\Delta_{N}^{-1}(P_{h}^{N}-\operatorname{Id})\Pi_{N}(f(V_{t_{k}}^{M,N}))+O^{N}_{t_{k+1}}-P_{h}^{N}O_{t_{k}}^{N}$

for $k=0,\dots,M-1$. Here $\operatorname{Id}$ denotes the identity operator on
$L^{2}$. Alternatively (and often more conveniently) one can express $V^{M,N}$
also in a “mild” form

$\displaystyle V^{M,N}_{t_{k+1}}$
$\displaystyle=P_{t_{k+1}}^{N}u_{0}+\sum_{l=0}^{k}\int_{t_{l}}^{t_{l+1}}P_{t_{k+1}-s}^{N}f(V_{t_{l}}^{M,N})ds+O_{t_{k+1}}^{N}$
(2.6)
$\displaystyle=P_{t_{k+1}}^{N}u_{0}+\int_{0}^{t_{k+1}}P_{t_{k+1}-s}^{N}f(V_{k_{M}(s)}^{M,N})ds+O_{t_{k+1}}^{N},\quad
k=0,\dots,M-1,$

see e.g. [JK08, Sec. 5.(b)]. An advantage of the form (2) is that one can
easily extend it to arbitrary (i.e. not grid-)points $t\in[0,T]$, simply by
replacing each instance of $t_{k+1}$ on the right-hand side by $t$. We will
frequently use this extension.

######  Theorem 2.2.

Let 2.1 hold. Let $u$ be the unique mild solution of (1.1) and for any
$M,N\in\mathbb{N}$, let $V^{M,N}$ be as above. Then for any $\varepsilon>0$
and $p\in[1,\infty)$ there exists a constant
$C=C(T,\varepsilon,p,K,\mathcal{M})$222Here and below, whenever $\theta$ is
some collection of parameters, expressions of the form
$C=C(\theta,\mathcal{M})$ mean that there exists a $p^{*}$ depending on
$\theta$ such that the constant $C$ depends only on $\theta$ and
$\mathcal{M}(p^{*})$. such that

(2.7) $\displaystyle\big{(}\mathds{E}\sup_{t\in[0,T]}\lVert
u_{t}-V^{M,N}_{t}\rVert_{L^{\infty}(\mathbb{T})}^{p}\big{)}^{1/p}\leq
C\big{(}N^{-1/2+\varepsilon}+M^{-1+\varepsilon}\big{)}.$

The proof of Theorem 2.2 is given in Section 4.

In the second half of this article, we allow for superlinearily growing
nonlinearities $f$ that satisfy a one-sided Lipschitz condition. A prime
example is the Allen-Cahn nonlinearity $f(x)=x-x^{3}$. We now give the set up
for the more general formulation of the article, which is substantially more
technical. We start by the assumption on the nonlinearity.

######  Assumption 2.3.

  1. (a)

There exists a constant $K\geq 1$ and $m\geq 0$ such that for any $i=0,1,2,3$
and all $x\in\mathbb{R}$ one has

(2.8) $\displaystyle|\partial^{i}f(x)|\leq K(1+|x|^{(2m+1-i)\vee 0}),$

with the convention that $\partial^{0}f=f$, and furthermore for all
$x\in\mathbb{R}$ one has

(2.9) $\displaystyle\partial f(x)\leq K.$

  2. (b)

The initial condition $u_{0}$ is an $\mathcal{F}_{0}$-measurable random
variable with values in $\mathscr{C}^{2}(\mathbb{T})$ and for any
$p\in[1,\infty)$ there exists a constant $\mathcal{M}(p)$ such that
$\mathds{E}\|u_{0}\|_{\mathscr{C}^{2}}^{p}\leq\mathcal{M}(p)$.

######  Remark 2.4.

2.3 implies a local Lipschitz bound with polynomial growth and a global one-
sided Lipschitz bound. That is, for all $x,y\in\mathbb{R}$ one has

(2.10) $\displaystyle|f(x)-f(y)|$ $\displaystyle\leq
K(1+|x|^{2m}+|y|^{2m})|x-y|,$ (2.11) $\displaystyle(x-y)(f(x)-f(y))$
$\displaystyle\leq K|x-y|^{2}.$

For standard Euler or (accelerated) exponential Euler schemes for SPDEs with
superlinearily growing coefficients, a priori estimates are known to fail (cf.
[HJK11]). For our analysis, we consider the following splitting scheme:
$X^{M,N}_{0}=\Pi_{N}u_{0}=u_{0}^{N}$ and

(2.12) $\displaystyle Y_{t_{k}}^{M,N}$
$\displaystyle=\Phi_{h}(X_{t_{k}}^{M,N})$ $\displaystyle X_{t_{k+1}}^{M,N}$
$\displaystyle=P_{h}^{N}Y_{t_{k}}^{M,N}+O^{N}_{t_{k+1}}-P_{h}^{N}O^{N}_{t_{k}}$

for $k=0,\dots,M-1$ and $h=T/M$ and where $\Phi_{t}(z)$ solves

(2.13) $\displaystyle\partial_{t}\Phi(z)=f(\Phi(z)),\quad\Phi_{0}(z)=z$

and $O^{N}$ is the truncated Ornstein-Uhlenbeck process defined in (2.5).

######  Remark 2.5.

Often the ODE (2.13) admits an explicit solution. For example in the Allen-
Cahn case $f(x)=x-x^{3}$ one has

(2.14)
$\displaystyle\Phi_{t}(z)=\mathrm{sgn}(z)\frac{e^{t}}{\sqrt{z^{-2}-1+e^{2t}}}.$

This scheme corresponds to the semi-discrete splitting scheme considered in
[BCH18]. One can rewrite the scheme as a classical Euler scheme for an
auxiliary SPDE. To that aim, define the auxiliary function

(2.15) $\displaystyle g_{t}(z)=\frac{\Phi_{t}(z)-z}{t},\quad t>0,\quad
g_{0}(z):=f(z).$

Using the definition of $g_{h}$, $X^{M,N}$ can equivalently be written in the
form

(2.16) $\displaystyle
X^{M,N}_{t_{k+1}}=P_{h}^{N}X^{M,N}_{t_{k}}+hP_{h}^{N}g_{h}(X^{M,N}_{t_{k}})+O^{N}_{t_{k+1}}-P^{N}_{h}O^{N}_{t_{k}}.$

Equivalently, we can write a “mild” version of the approximation $X^{M,N}$ and
extend it to arbitrary points $t\in[0,T]$ as

(2.17) $\displaystyle
X_{t}^{M,N}=P_{t}^{N}u_{0}+\int_{0}^{t}P_{t-k_{M}(s)}^{N}g_{h}(X^{M,N}_{k_{M}(s)})ds+O_{t}^{N},$

which agrees with the inductive form (2.12) on the time grid points $t=t_{k}$,
$k=0,\ldots,M$.

######  Theorem 2.6.

Let 2.3 hold. Let $u$ be the unique mild solution of (1.1) and for any
$M,N\in\mathbb{N}$, let $X^{M,N}$ be as above. Then for any $\varepsilon>0$
and $p\in[1,\infty)$ there exists a constant
$C=C(T,\varepsilon,p,K,\mathcal{M})$ such that

(2.18) $\displaystyle\big{(}\mathds{E}\sup_{t\in[0,T]}\lVert
u_{t}-X^{M,N}_{t}\rVert_{L^{2}(\mathbb{T})}^{p}\big{)}^{1/p}\leq
C\big{(}N^{-1/2+\varepsilon}+M^{-1+\varepsilon}\big{)}.$

The proof of Theorem 2.6 is given in Section 6.

######  Remark 2.7.

Let $f$ satisfy 2.3 (a) with $m=0$. In this case $f$ is globally Lipschitz
continuous with at most linear growth. It is plausible to expect that Theorem
2.2 extends to this case and the splitting scheme is not necessary.

##  3\. Preliminaries

We introduce the Besov spaces as follows. Let $(\rho_{j})_{j\geq-1}$ be a
smooth dyadic partition of unity, i.e. a family of functions $\rho_{j}\in
C^{\infty}_{c}(\mathbb{R})$ for $j\geq-1$, such that

  * •

$\rho_{-1}$ and $\rho_{0}$ are non-negative even functions such that the
support of $\rho_{-1}$ is contained $B_{1/2}$, the ball of radius $1/2$ around
$0$, and the support of $\rho_{0}$ is contained in $B_{1}\setminus B_{1/4}$;

  * •

$\rho_{j}(x)=\rho_{0}(2^{-j}x)$, $x\in\mathbb{R}$, $j\geq 0$;

  * •

$\sum_{j=-1}^{\infty}\rho_{j}(x)=1$ for every $x\in\mathbb{R}$;

  * •

$\operatorname{supp}(\rho_{i})\cap\operatorname{supp}(\rho_{j})=\emptyset$ for
all $\lvert i-j\rvert>1$.

The existence of such a partition of unity is classical (see e.g. [BCD11]). We
denote by $\mathcal{S}^{\prime}(\mathbb{T})$ the space of Schwartz
distributions on the torus (i.e. the dual of
$\mathcal{S}(\mathbb{T}):=C^{\infty}(\mathbb{T})$). Note that for any
$u\in\mathcal{S}^{\prime}(\mathbb{T})$ its Fourier transform is meaningful,
and therefore one can define the operators (also known as Littlewood-Paley
blocks), $j\geq-1$,

(3.1) $\displaystyle\Delta_{j}:\mathcal{S}^{\prime}(\mathbb{T})\to
C^{\infty}(\mathbb{T})\qquad\Delta_{j}u=\mathcal{F}^{-1}\big{(}k\mapsto\rho_{j}(k)\mathcal{F}(u)(k)\big{)}.$

We then define the Besov spaces on the torus for $p,q\in[1,\infty]$,
$\theta\in\mathbb{R}$

(3.2) $\displaystyle
B^{\theta}_{p,q}:=\big{\\{}u\in\mathcal{S}^{\prime}(\mathbb{T}):\lVert
u\rVert_{B^{\theta}_{p,q}}=\lVert(2^{j\theta}\lVert\Delta_{j}u\rVert_{L^{p}(\mathbb{T})})_{j\geq-1}\rVert_{\ell^{q}}<\infty\big{\\}}.$

We introduce the shorthand $\mathscr{C}^{\theta}:=B^{\theta}_{\infty,\infty}$
for $\theta\in\mathbb{R}$. We collect the relevant properties of Besov spaces
below.

  * •

(Hölder spaces) If $\theta\in(0,1)$, then $\mathscr{C}^{\theta}$ coincides
with the space of $\theta$-Hölder continuous functions, (cf. [BCD11, Sec. 2.7,
Examples]).

  * •

(Embedding) If $1\leq p_{1}\leq p_{2}\leq\infty$, $1\leq q_{1}\leq
q_{2}\leq\infty$ and $\theta\in\mathbb{R}$, then $B^{\theta}_{p_{1},q_{1}}$ is
continuously embedded in $B_{p_{2},q_{2}}^{\theta-d(1/p_{1}-1/p_{2})}$ (cf.
[BCD11, Proposition 2.71]).

  * •

(Derivatives) If $u\in B^{\theta}_{p,q}$ and $n\in\mathbb{N}$,
$\lVert\partial^{n}u\rVert_{B^{s-n}_{p,q}}\leq\lVert u\rVert_{B^{s}_{p,q}}$.

  * •

(Products) If $\theta,\beta$ are such that $\theta+\beta>0$, then for any two
distributions $u\in\mathscr{C}^{\theta}$ and $v\in\mathscr{C}^{\beta}$ their
product $uv$ is well-defined and there exists a constant $C=C(\theta,\beta)$,
such that the bound

$\|uv\|_{\mathcal{C}^{\min(\theta,\beta)}}\leq
C\|u\|_{\mathcal{C}^{\theta}}\|v\|_{\mathcal{C}^{\beta}}$

holds (cf. [BCD11, Section 2]).

  * •

(Heat kernel bounds) If $\theta\geq 0$, $\tilde{\theta}\in[0,2]$, then there
exist constants $C_{1}(\theta)$ and $C_{2}(\tilde{\theta})$ such that for any
$\beta\in\mathbb{R}$, $u\in\mathcal{C}^{\beta}$, and $t\in(0,1]$, the bounds

(3.3) $\displaystyle\lVert P_{t}u\rVert_{\mathscr{C}^{\beta+\theta}}\leq
C_{1}t^{-\theta/2}\lVert
u\rVert_{\mathscr{C}^{\beta}},\quad\lVert(\operatorname{Id}-P_{t})u\rVert_{\mathscr{C}^{\beta-\tilde{\theta}}}\leq
C_{2}t^{\tilde{\theta}/2}\lVert u\rVert_{\mathscr{C}^{\beta}}$

hold (cf. [GIP15, Lemma A.7, A.8]).

For function spaces on $\mathbb{R}$, we only need the simple notion of
$C^{k}_{b}$ for $k=0,1,\ldots$, denoting the space of bounded measurable
functions whose distributional derivative up to order $k$ are essentially
bounded, equipped with the canonical norm (note in particular that elements of
$C^{0}_{b}$ are not assumed to be continuous). Moreover denote by
$(P_{t}^{\mathbb{R}})_{t\geq 0}$ the heat semigroup on $\mathbb{R}$, that is

(3.4) $\displaystyle P_{t}^{\mathbb{R}}f=p_{t}^{\mathbb{R}}\ast f,\qquad\qquad
p_{t}^{\mathbb{R}}(x)=\frac{1}{\sqrt{2\pi t}}e^{-x^{2}/2t}.$

The following estimate is rather immediate: for any $0\leq s\leq t\leq 1$ one
has

(3.5)
$\displaystyle\lVert(P_{t}^{\mathbb{R}}-P_{s}^{\mathbb{R}})u\rVert_{C^{0}_{b}}\leq|t-s|\lVert
u\rVert_{C^{2}_{b}}.$

For a Banach space $X$, $C_{T}X$ denotes the space of continuous functions in
time with values in $X$ equipped with the supremum norm. For $\gamma\in(0,1]$,
define

$C_{T}^{\gamma}X:=\big{\\{}u\in C_{T}X:\,\lVert
u\rVert_{C_{T}^{\gamma}X}:=\sup_{t\in[0,T]}\lVert u_{t}\rVert_{X}+\sup_{0\leq
s<t\leq T}\frac{\lVert
u_{t}-u_{s}\rVert_{X}}{(t-s)^{\gamma}}<\infty\big{\\}}.$

Below we use the notation $a\lesssim b$ if there exists a constant $C>0$, such
that $a\leq Cb$. The dependence of the constant $C$ will be clear from the
context. If we want to stress dependence of the constant $C$ on a parameter
$\tau$, we write $a\lesssim_{\tau}b$.

We collect a number of simple tools, starting with the proof of Proposition
1.2.

###### Proof of Proposition 1.2.

Note that if $f$ is affine, then 1.1 is clearly satisfied. To prove the
converse, with the notation $H^{s}=B^{s}_{2,2}$ notice that letting
$z=(1-\Delta)v\in L^{2}$, 1.1 implies that

(3.6) $\displaystyle\lVert F^{\prime}(u)z\rVert_{H^{-2}}\leq L\lVert
z\rVert_{H^{-2}},\quad\forall z,u\in L^{2}.$

Note that the Gateaux derivative of $F:u\mapsto f(u)$ is
$F^{\prime}:u\mapsto\big{(}z\mapsto f^{\prime}(u)z\big{)}$, so
$F^{\prime}(u)z$ is simply the product $f^{\prime}(u)z$ [AP93, Theorem 2.7].

We claim that (3.6) implies that $f^{\prime}$ is constant. Assume the contrary
and let $a,b$ such that $f^{\prime}(a)\neq f^{\prime}(b)$. Set
$u=\mathbf{1}_{[0,1/2]}a+\mathbf{1}_{(1/2,1)}b\in L^{2}(\mathbb{T})$, where we
identify the torus $\mathbb{T}$ with $[0,1)$ with periodic boundary
conditions. Clearly, $h:=f^{\prime}(u)\notin H^{1}$. Notice that for every
$k\in\mathbb{Z}$, $z_{k}:=k^{2}e_{k}$ has norm less than $1$ in $H^{-2}$.
Therefore (3.6) and $\lvert\langle hz_{k},1\rangle\rvert\leq\lVert
hz_{k}\rVert_{H^{-2}}=\sup_{\phi\,:\,\lVert\phi\rVert_{H^{2}}=1}\lvert\langle
hz_{k},\phi\rangle\rvert$ imply that

(3.7) $\displaystyle\|\partial h\|_{L^{2}}^{2}=\sum_{k\in\mathbb{Z}}|\langle
h,e_{k}\rangle|^{2}k^{2}=\sum_{0\neq k\in\mathbb{Z}}|\langle
h,k^{2}e_{k}\rangle|^{2}k^{-2}\leq L\sum_{0\neq k\in Z}k^{-2}<\infty,$

which is a contradiction. ∎

######  Proposition 3.1.

Let $(O_{t})_{t\geq 0}$ be as in (2.2). Let $p\in[1,\infty)$,
$\lambda\in(0,1)$ and $\varepsilon\in(0,1/2)$. Then there exists a constant
$C=C(p,T,\lambda,\varepsilon)$, such that for all $0\leq s\leq t\leq T$ one
has

(3.8) $\displaystyle\mathds{E}\lVert
O_{t}-O_{s}\rVert_{\mathscr{C}^{1/2-\lambda-\varepsilon}}^{p}\leq
C(t-s)^{p\lambda/2}.$

Furthermore, one has

(3.9) $\displaystyle\mathds{E}\lVert
O\rVert_{C_{T}^{\lambda/2}\mathscr{C}^{1/2-\lambda-\varepsilon}}^{p}\leq C.$

Furthermore if $\theta\in[0,1]$, then there exists a constant
$C=C(p,T,\theta,\varepsilon)$ such that for all $s,t\in[0,T]$ one has

(3.10)
$\displaystyle(\mathds{E}\sup_{r\in[0,T]}\lVert(P_{t+s}-P_{t})O_{r}\rVert_{\mathscr{C}^{-1/2+\varepsilon}}^{p})^{1/p}\leq
Ct^{-\theta/2-\varepsilon}s^{\frac{1+\theta}{2}}.$

###### Proof.

Note that (3.9) follows from (3.8), by virtue of Kolmogorov’s continuity
theorem (up to changing the constant $C$). Furthermore, (3.10) follows from
(3.9) and applying both bounds in (3.3).

To prove (3.8), we may and will assume that $p$ is sufficiently large. We then
show that for any $j\geq-1$,

(3.11)
$\displaystyle(\mathds{E}\lvert\Delta_{j}O_{t}(x)-\Delta_{j}O_{s}(x)\rvert^{p})^{1/p}\lesssim
2^{j(-1/2+\lambda)}\lvert t-s\rvert^{\lambda/2}.$

Indeed, from (3.11), raising to the $p$-th power, multiplying by
$2^{pj(1/2-\lambda-\varepsilon/2)}$, and summing over $j$, we get

(3.12) $\displaystyle\mathds{E}\lVert
O_{t}-O_{s}\rVert_{B^{1/2-\lambda-\varepsilon/2}_{p,p}}^{p}\lesssim(t-s)^{p\lambda/2}.$

Choosing $p$ sufficiently large, (3.8) follows by Besov embedding.

It is left to prove (3.11). Using Gaussian hypercontractivity, Itô’s isometry,
and Parsevel’s identity, we have that

$\displaystyle(\mathds{E}\lvert\Delta_{j}O_{t}(x)-\Delta_{j}O_{s}(x)\rvert^{p})^{2/p}$
$\displaystyle\lesssim\mathds{E}\lvert\Delta_{j}O_{t}(x)-\Delta_{j}O_{s}(x)\rvert^{2}$
$\displaystyle\lesssim\mathds{E}\bigg{\lvert}\int_{s}^{t}\int_{\mathbb{T}}\Delta_{j}p_{t-r}(x-y)\xi(dr,dy)\bigg{\rvert}^{2}$
$\displaystyle\quad+\mathds{E}\bigg{\lvert}\int_{0}^{s}\int_{\mathbb{T}}\Delta_{j}(p_{t-r}-p_{s-r})(x-y)\xi(dr,dy)\bigg{\rvert}^{2}$
$\displaystyle=\int_{s}^{t}\|\Delta_{j}p_{t-r}(x-\cdot)\|_{L^{2}(\mathbb{T})}^{2}\,dr$
$\displaystyle\qquad+\int_{0}^{s}\|\Delta_{j}(p_{t-r}-p_{s-r})(x-\cdot)\|_{L^{2}(\mathbb{T})}^{2}\,dr$
$\displaystyle=\int_{s}^{t}\sum_{k\in\mathbb{Z}}\rho_{j}(k)^{2}e^{-8\pi^{2}k^{2}(t-r)}dr$
$\displaystyle\quad+\int_{0}^{s}\sum_{k\in\mathbb{Z}}\rho_{j}(k)^{2}e^{-8\pi^{2}k^{2}(s-r)}(1-e^{-4\pi^{2}k^{2}(t-s)})^{2}dr.$

Integrating in time and using that $e^{-x}\leq\min(x^{-1},1)$ and
$1-e^{-x}\leq\min(x,x^{1/2},1)$ for $x\geq 0$, we obtain

$\displaystyle(\mathds{E}\lvert\Delta_{j}O_{t}(x)-\Delta_{j}O_{s}(x)\rvert^{p})^{2/p}$
$\displaystyle\lesssim\sum_{k\in\mathbb{Z}}\rho_{j}(k)^{2}\min(\lvert
t-s\rvert,\lvert k\rvert^{-2})$
$\displaystyle\quad+\sum_{k\in\mathbb{Z}}\rho_{j}(k)^{2}\min(\lvert
k\rvert^{-2},\lvert k\rvert^{-2}\lvert k\rvert^{2}\lvert t-s\rvert)$
$\displaystyle\lesssim 2^{j}\min(\lvert t-s\rvert,2^{-2j})$
$\displaystyle\lesssim 2^{j(2\lambda-1)}\lvert t-s\rvert^{\lambda}$

for any $\lambda\in[0,1]$, yielding (3.11). ∎

For the treatment of the temporal error, we use the stochastic sewing lemma,
originating from [Lê20]. We state here a weighted version (see [ABLM24,
DGL23]). The final conclusion of the lemma follows from [BFG22, Lemma 2.3].
Let

$[S,T]_{\leq}:=\\{(s,t)\mid S\leq s<t\leq T\\}$

and

$[S,T]^{\ast}_{\leq}:=\\{(s,t)\mid S\leq s<t<T,\,t-s\leq T-t\\}.$

For a function $\mathcal{A}$ of one variable and $s\leq t$, we write
$\mathcal{A}_{st}=\mathcal{A}_{t}-\mathcal{A}_{s}$ and for functions $A$ of
two variables and $s\leq u\leq t$, we denote $\delta
A_{sut}=A_{st}-A_{su}-A_{ut}$. Further, denote by $\mathds{E}_{s}$ the
conditional expectation with respect to $\mathcal{F}_{s}$.

######  Lemma 3.2.

Fix $p\geq 2$ and $0\leq S<T\leq 1$. Let $A:[S,T]_{\leq}\to L^{p}(\Omega)$ be
such that $A_{st}$ is $\mathcal{F}_{t}$-measurable for all
$(s,t)\in[S,T]_{\leq}$. Suppose that there exist
$\varepsilon_{1},\varepsilon_{2}>0,\delta_{1},\delta_{2}\geq 0$ and
$C_{1},C_{2}<\infty$ satisfying
$1/2+\varepsilon_{1}-\delta_{1}>0,\,1+\varepsilon_{2}-\delta_{2}>0$ and such
that for all $(s,t)\in[S,T]^{\ast}_{\leq}$, $u\in[s,t]$ the bounds

(3.13) $\displaystyle\lVert A_{st}\rVert_{L^{p}(\Omega)}$ $\displaystyle\leq
C_{1}(T-t)^{-\delta_{1}}\lvert t-s\rvert^{1/2+\varepsilon_{1}},$ (3.14)
$\displaystyle\lVert\mathds{E}_{s}[\delta A_{sut}]\rVert_{L^{p}(\Omega)}$
$\displaystyle\leq C_{2}(T-t)^{-\delta_{2}}\lvert
t-s\rvert^{1+\varepsilon_{2}}$

hold. Then there exists a unique $(\mathcal{F}_{t})_{t\in[S,T]}$-adapted
process $\mathcal{A}:[S,T]\to L^{p}(\Omega)$ such that $\mathcal{A}_{S}=0$ and
that there exist $K_{1},K_{2}$ such that for all $(s,t)\in[S,T]^{\ast}_{\leq}$
one has

(3.15) $\displaystyle\lVert\mathcal{A}_{st}-A_{st}\rVert_{L^{p}(\Omega)}$
$\displaystyle\leq K_{1}(T-t)^{-\delta_{1}}\lvert
t-s\rvert^{1/2+\varepsilon_{1}}+K_{2}(T-t)^{-\delta_{2}}\lvert
t-s\rvert^{1+\varepsilon_{2}},$ (3.16)
$\displaystyle\lVert\mathds{E}_{s}[\mathcal{A}_{st}-A_{st}]\rVert_{L^{p}(\Omega)}$
$\displaystyle\leq K_{2}(T-t)^{-\delta_{2}}\lvert
t-s\rvert^{1+\varepsilon_{2}}.$

Furthermore, there exists a constant $C$ depending only on
$p,\varepsilon_{1},\varepsilon_{2}$, such that the above bounds hold with
$K_{1}=CC_{1},K_{2}=CC_{2}$. Finally, there exists a constant $C^{\prime}$
depending only on $p,\varepsilon_{1},\varepsilon_{2},\delta_{1},\delta_{2}$,
such that for all $(s,t)\in[S,T]_{\leq}$ one has

(3.17) $\displaystyle\lVert\mathcal{A}_{st}\rVert_{L^{p}(\Omega)}$
$\displaystyle\leq C^{\prime}(C_{1}\lvert
t-s\rvert^{1/2+\varepsilon_{1}-\delta_{1}}+C_{2}\lvert
t-s\rvert^{1+\varepsilon_{2}-\delta_{2}}).$

Next, we give the result on the well-posedness of the reaction-diffusion
equation (1.1).

######  Proposition 3.3.

Let 2.1 or 2.3 hold. Then there exists a unique mild solution $u$ to (1.1).
Moreover, for any $\lambda\in(0,1)$, $\varepsilon\in(0,1/2)$, $p\geq 1$ there
exists a constant $C=C(T,p,\lambda,\varepsilon,m,K)$ such that the solution
satisfies the bound

(3.18) $\displaystyle\mathds{E}\lVert
u\rVert_{C_{T}^{\lambda/2}\mathscr{C}^{1/2-\lambda-\varepsilon}(\mathbb{T})}^{p}\leq
C(1+\mathds{E}\lVert u_{0}\rVert_{\mathscr{C}^{1/2}}^{(2m+1)p}).$

###### Proof.

We give the proof under the assumption that $f\in C^{2}$ satisfies (2.8) and
(2.9) for $i=0,1,2$, which is in particular implied by 2.1 and 2.3. These
conditions on $f$ imply [Cer01, Hypothesis 6.2]. Furthermore, [Cer01,
Hypothesis 6.1] is satisfied by the operators $A=\Delta$ and
$Q=\operatorname{Id}$ with the spaces $H=L^{2}(\mathbb{T})$,
$E=C(\mathbb{T})$. Thus, the existence and uniqueness of a mild solution $u$
with paths in $C_{T}C(\mathbb{T})$ follows from [Cer01, Proposition 6.2.2], as
well as the bound, for any $q\geq 1$,

(3.19) $\displaystyle\mathds{E}\lVert
u\rVert_{C_{T}C(\mathbb{T})}^{q}\lesssim(1+\mathds{E}\lVert
u_{0}\rVert_{C(\mathbb{T})}^{q}).$

As an immediate consequence,

(3.20) $\displaystyle\mathds{E}\lVert
f(u)\rVert_{C_{T}C(\mathbb{T})}^{q}\lesssim(1+\mathds{E}\lVert
u_{0}\rVert_{C(\mathbb{T})}^{(2m+1)q}).$

As for the bound (3.18), note that this bound is shown for $O$ in (3.9), and
one also has immediately from (3.3)

$\displaystyle\|P_{t}u_{0}-P_{s}u_{0}\|_{\mathscr{C}^{1/2-\lambda-\varepsilon}}\lesssim|t-s|^{(\lambda+\varepsilon)/2}\|u_{0}\|_{\mathscr{C}^{1/2}},$

so it suffices to prove (3.18) for $v=u-O-P_{\cdot}u_{0}$. One can decompose
the increments of $v$ as

$\displaystyle
v_{t}-v_{s}=\int_{s}^{t}P_{t-r}f(u_{r})dr-\int_{0}^{s}P_{s-r}(\operatorname{Id}-P_{t-s})f(u_{r})dr.$

Using again the semigroup estimates (3.3), we obtain for $\lambda\in(0,1)$,
$\varepsilon\in(0,1/2)$

$\displaystyle\lVert
v_{t}-v_{s}\rVert_{\mathscr{C}^{1/2-\lambda-\varepsilon}}$
$\displaystyle\lesssim\int_{s}^{t}\lVert
P_{t-r}f(u_{r})\rVert_{\mathscr{C}^{1/2-\lambda-\varepsilon}}dr+\int_{0}^{s}\lVert
P_{s-r}(\operatorname{Id}-P_{t-s})f(u_{r})\rVert_{\mathscr{C}^{1/2-\lambda-\varepsilon}}dr$
$\displaystyle\lesssim\Big{(}\int_{s}^{t}(t-r)^{(\lambda/2+\varepsilon/2-1/4)\wedge
0}\,dr+\int_{0}^{s}(t-s)^{\lambda/2+\varepsilon/2}(s-r)^{-1/4}\,dr\Big{)}$
$\displaystyle\qquad\times\lVert f(u)\rVert_{C_{T}C(\mathbb{T})}$
$\displaystyle\lesssim(t-s)^{\lambda/2+\varepsilon/2}\lVert
f(u)\rVert_{C_{T}C(\mathbb{T})}.$

Taking $p$-th moment, using (3.20), and applying Kolmogorov’s continuity
theorem, we get the claimed bound (3.18). ∎

We give a version of Grönwall’s inequality that is repeatedly used in Section
4.

######  Proposition 3.4.

Let $V$ be a Banach space, $p\geq 1$, and take three processes $X,Y,Z$
belonging to $L^{p}(\Omega;C([0,T];V)).$ Assume furthermore that there exists
a Lipschitz continuous function $F$ on $V$ with Lipschitz constant $L_{1}$, a
family $(S(s,t))_{0\leq s\leq t\leq T}$ of uniformly bounded linear operators
on $V$ with uniform bound $L_{2}$ and such that $(s,t)\mapsto S(s,t)v$ is
measurable for any $v\in V$, and a measurable mapping $\tau:[0,T]\to[0,T]$
such that $\tau(s)\leq s$ and that the following equality holds for all $0\leq
t\leq T$:

(3.21) $\displaystyle
X_{t}-Y_{t}=Z_{t}+\int_{0}^{t}S(s,t)\big{(}F(X_{\tau(s)})-F(Y_{\tau(s)})\big{)}\,ds.$

Then there exists a constant $C=C(p,L_{1},L_{2},T)$ such that

(3.22) $\displaystyle\mathds{E}\sup_{t\in[0,T]}\|X_{t}-Y_{t}\|^{p}\leq
C\mathds{E}\sup_{t\in[0,T]}\|Z_{t}\|^{p}.$

##  4\. Proof of Theorem 2.2

We start by giving a brief overview of the proof of Theorem 2.2. Introduce
auxiliary processes $\tilde{U}^{N}$ and $\tilde{V}^{M,N}$. Let $\tilde{U}^{N}$
be defined as the solution of

(4.1) $\displaystyle\tilde{U}^{N}_{t}$
$\displaystyle=P^{N}_{t}u_{0}^{N}+\int_{0}^{t}P_{t-s}^{N}f(\tilde{U}^{N}_{s})ds+O_{t},\quad
t\in[0,T]$

and let $\tilde{V}^{M,N}$ be defined on the time grid points via the inductive
form: $\tilde{V}^{M,N}_{0}=\Pi_{N}u_{0}$ and

$\displaystyle\tilde{V}^{M,N}_{t_{k+1}}=P^{N}_{h}\tilde{V}^{M,N}_{t_{k}}+\Delta_{N}^{-1}(P^{N}_{h}-\operatorname{Id})\Pi_{N}(f(\tilde{V}^{M,N}_{t_{k}}))+O_{t_{k+1}}-P^{N}_{h}O_{t_{k}}$

for $k=0,\dots,M-1$, and for $t\in[0,T]$ via

(4.2) $\displaystyle\tilde{V}^{M,N}_{t}$
$\displaystyle=P^{N}_{t}u_{0}^{N}+\int_{0}^{t}P_{t-s}^{N}f(\tilde{V}^{M,N}_{k_{M}(s)})ds+O_{t}.$

First, we address the wellposedness for the equations for $U^{N}$ and
$\tilde{U}^{N}$ in the following proposition.

######  Proposition 4.1.

Let $f$ satisfy 2.1. Then there exist unique solutions $U^{N}$ and
$\tilde{U}^{N}$ to the equations (2.4) and (4.1). Moreover, for any
$\lambda\in(0,1)$, $\varepsilon\in(0,1/2)$, $p\geq 1$ there exists a constant
$C=C(T,p,\lambda,\varepsilon)$ such that the solutions satisfy

$\displaystyle\sup_{N\in\mathbb{N}}\mathds{E}\lVert
U^{N}\rVert_{C_{T}^{\lambda/2}\mathscr{C}^{1/2-\lambda-\varepsilon}(\mathbb{T})}^{p}+\sup_{N\in\mathbb{N}}\mathds{E}\lVert\tilde{U}^{N}\rVert_{C_{T}^{\lambda/2}\mathscr{C}^{1/2-\lambda-\varepsilon}(\mathbb{T})}^{p}\leq
C(1+\mathds{E}\lVert u_{0}\rVert_{\mathscr{C}^{1/2}}^{p}).$

###### Proof.

The argument is classical. Using the global Lipschitz bound on $f$ from 2.1,
that $\sup_{N}\lVert P_{t}^{N}u\rVert_{L^{\infty}}\leq\lVert
u\rVert_{L^{\infty}}$ for all $t\geq 0$ and applying the Banach fixed point
theorem, one finds a unique fixed point of the mild formulation of the
equations for $U^{N}$ and for $\tilde{U}^{N}$ in $C_{T}L^{\infty}$ if $T$ is
chosen small enough. Patching the solutions on subintervals together yields a
solution for any arbitrary time horizon $T>0$. Then plugging the solution back
in the mild formulation and using the semigroup estimates for $P^{N}$ instead
of $P$, one obtains the claimed regularity bounds (cf. in the proof of
Proposition 3.3). ∎

The error $u-V^{M,N}$ is decomposed into the spatial errors $u-U^{N}$,
$U^{N}-\tilde{U}^{N}$, $V^{M,N}-\tilde{V}^{M,N}$ and the temporal error
$\tilde{U}^{N}-\tilde{V}^{M,N}$. Bounding the Galerkin error $u-U^{N}$ by
order $N^{-1/2+\varepsilon}$ is fairly standard and is already done in e.g.
[JK08], although a noteworthy difference is that below we obtain estimates in
$L^{\infty}(\mathbb{T})$ instead of $L^{2}(\mathbb{T})$ (Lemmas 4.2 and 4.3).
The biggest novelty of the section comes from the treatment of the temporal
error in $\tilde{U}^{N}-\tilde{V}^{M,N}$, Lemma 4.4 below. These three lemmas
together imply Theorem 2.2.

######  Lemma 4.2.

Assume the setting of Theorem 2.2. Let $N\in\mathbb{N}$ and let
$U^{N},\tilde{U}^{N}$ be as in (2.4) and (4.1) and $V^{M,N},\tilde{V}^{M,N}$
be as in (2) and (4.2). Let $p\in[1,\infty)$ and $\varepsilon>0$. Then there
exists a constant $C=C(T,p,\varepsilon,K)$ such that the following bound holds

$\displaystyle\big{(}\mathds{E}\sup_{t\in[0,T]}\lVert\tilde{U}^{N}_{t}-U^{N}_{t}\rVert_{L^{\infty}}^{p}\big{)}^{1/p}+\big{(}\mathds{E}\sup_{t\in[0,T]}\lVert\tilde{V}^{M,N}_{t}-V^{M,N}_{t}\rVert_{L^{\infty}}^{p}\big{)}^{1/p}\leq
CN^{-1/2+\varepsilon}.$

###### Proof.

We first prove the bound for $\tilde{U}^{N}-U^{N}$. We have that

$\displaystyle\tilde{U}^{N}_{t}-U^{N}_{t}=\int_{0}^{t}P^{N}_{t-s}(f(\tilde{U}^{N}_{s})-f(U^{N}_{s}))ds+O_{t}-O_{t}^{N}.$

We will apply Proposition 3.4 with $X=\tilde{U}^{N}$, $Y=U^{N}$, $Z=O-O^{N}$,
$S(s,t)=P^{N}_{t-s}$, $\tau(s)=s$. Therefore, it suffices to bound
$\mathds{E}\sup_{t\in[0,T]}\|O_{t}-O^{N}_{t}\|_{L^{\infty}}^{p}$. The
difference of the truncated and full expansion of the Ornstein-Uhlenbeck
process we write as follows

$\displaystyle(O_{t}-O_{t}^{N})(x)=\int_{0}^{t}\int_{\mathbb{T}}(p_{t-s}-p^{N}_{t-s})(x-y)\xi(ds,dy),$

where

$\displaystyle(p_{t-s}-p^{N}_{t-s})(x)=\sum_{\lvert
k\rvert>N}e^{-4\pi^{2}k^{2}(t-s)}e^{2\pi ikx}.$

Then we obtain for $O_{r,t}=O_{t}-O_{r}$, $r\leq t$, and $O^{N}_{r,t}$
analoguously defined, that

$\displaystyle(O_{r,t}-O_{r,t}^{N})(x)$
$\displaystyle=\int_{r}^{t}\int_{\mathbb{T}}(p_{t-s}-p^{N}_{t-s})(x-y)\xi(ds,dy)$
$\displaystyle\qquad-\int_{0}^{r}\sum_{\lvert
k\rvert>N}e^{-4\pi^{2}k^{2}(r-s)}(1-e^{-4\pi^{2}k^{2}(t-r)})e^{2\pi
ik(x-y)}\xi(ds,dy)$

With the same steps as in the proof of Proposition 3.1 (that is, Gaussian
hypercontractivity, Itô’s isometry, and Parseval identity), we get

$\displaystyle(\mathds{E}\lvert\Delta_{j}(O_{r,t}-O_{r,t}^{N})(x)\rvert^{p})^{1/p}$
$\displaystyle\lesssim\bigg{(}\int_{r}^{t}\sum_{\lvert
k\rvert>N}\rho_{j}(k)^{2}e^{-8\pi^{2}k^{2}(t-s)}ds$
$\displaystyle\qquad+\int_{0}^{r}\sum_{\lvert
k\rvert>N}\rho_{j}(k)^{2}e^{-8\pi^{2}k^{2}(r-s)}(1-e^{-4\pi^{2}k^{2}(t-r)})^{2}ds\bigg{)}^{1/2}$
$\displaystyle\leq\bigg{(}\sum_{\lvert
k\rvert>N}\frac{1-e^{-8\pi^{2}k^{2}(t-r)}}{8\pi^{2}k^{2}}$
$\displaystyle\qquad+\sum_{\lvert
k\rvert>N}\frac{(1-e^{-8\pi^{2}k^{2}r})(1-e^{-4\pi^{2}k^{2}(t-r)})^{2}}{8\pi^{2}k^{2}}\bigg{)}^{1/2}$
$\displaystyle\lesssim\bigg{(}(t-r)^{\varepsilon}\sum_{\lvert
k\rvert>N}\frac{1}{k^{2-2\varepsilon}}\bigg{)}^{1/2}$ $\displaystyle\lesssim
N^{-1/2+\varepsilon^{\prime}}(t-r)^{\varepsilon^{\prime}/2},$

using that for any $\varepsilon^{\prime}\in[0,1]$, $1-e^{-x}\leq
x^{\varepsilon^{\prime}}$, $x\geq 0$. On the other hand, one has the trivial
uniform in $N$ bound

$\displaystyle(\mathds{E}\lvert\Delta_{j}(O_{r,t}-O_{r,t}^{N})(x)\rvert^{p})^{1/p}$
$\displaystyle\lesssim\bigg{(}\int_{r}^{t}\sum_{\lvert
k\rvert>N}\rho_{j}(k)^{2}e^{-8\pi^{2}k^{2}(t-s)}ds$
$\displaystyle\qquad+\int_{0}^{r}\sum_{\lvert
k\rvert>N}\rho_{j}(k)^{2}e^{-8\pi^{2}k^{2}(r-s)}(1-e^{-4\pi^{2}k^{2}(t-r)})^{2}ds\bigg{)}^{1/2}$
$\displaystyle\lesssim\bigg{(}\int_{r}^{t}2^{j}e^{-8\pi^{2}2^{2j}(t-s)}ds$
$\displaystyle\qquad+\int_{0}^{r}2^{j}e^{-8\pi^{2}2^{2j}(r-s)}(1-e^{-4\pi^{2}2^{2j}(t-r)})^{2}ds\bigg{)}^{1/2}$
$\displaystyle\lesssim 2^{-j/2}.$

Hence, interpolation between the two bounds yields that for any
$\varepsilon^{\prime\prime}\in(0,1]$,

(4.3)
$\displaystyle\big{(}\mathds{E}\lvert\Delta_{j}(O_{r,t}-O_{r,t}^{N})(x)\rvert^{p}\big{)}^{1/p}\lesssim
2^{-j\varepsilon^{\prime\prime}/2}N^{(-1/2+\varepsilon^{\prime})(1-\varepsilon^{\prime\prime})}(t-r)^{(1-\varepsilon^{\prime\prime})\varepsilon^{\prime}/2}.$

Choosing $\varepsilon^{\prime},\varepsilon^{\prime\prime}>0$ sufficiently
small and $p$ sufficiently large, we repeat the argument from (3.11) to (3.9),
to deduce from (4.3) the bound

(4.4) $\displaystyle\mathds{E}[\lVert
O-O^{N}\rVert_{C_{T}^{\varepsilon^{\prime\prime\prime}}\mathcal{C}^{\varepsilon^{\prime\prime\prime}}}^{p}]^{1/p}\lesssim
N^{-1/2+\varepsilon}$

with some small $\varepsilon^{\prime\prime\prime}>0$. Thus the claim follows
from (4.4) and Proposition 3.4.

The bound for $\tilde{V}^{M,N}-V^{M,N}$ is done analogously: since

(4.5)
$\displaystyle\tilde{V}^{M,N}_{t}-V^{M,N}_{t}=\int_{0}^{t}P_{t-s}^{N}(f(\tilde{V}^{M,N}_{k_{M}(s)}-f(V^{M,N}_{k_{M}(s)}))ds+O_{t}-O^{N}_{t}$

we can use Proposition 3.4 almost exactly as before, with the only difference
being that $\tau(s)=k_{M}(s)$. ∎

In the $L^{2}$-norm the following spatial error bound is known, see e.g.
[JK08, equation (5.2) for $\gamma=1/4-\varepsilon$,
$\lambda_{N}=\pi^{2}N^{2}$]. We prove it for the $L^{\infty}$-norm in the
following lemma.

######  Lemma 4.3.

Assume the setting of Theorem 2.2. Let $N\in\mathbb{N}$ and let $u,U^{N}$ be
as in (2.1) and (2.4). Let $p\geq 1$ and $\varepsilon\in(0,1/2)$. Then there
exists a constant $C=C(p,\varepsilon,K,\mathcal{M})$ such that the following
bound holds

$\displaystyle\big{(}\mathds{E}\sup_{t\in[0,T]}\lVert
u_{t}-U^{N}_{t}\rVert_{L^{\infty}}^{p}\big{)}^{1/p}\leq
CN^{-1/2+\varepsilon}.$

###### Proof.

We have that

$\displaystyle u_{t}-U^{N}_{t}$
$\displaystyle=(P_{t}-P^{N}_{t})u_{0}+\int_{0}^{t}P_{t-s}^{N}[f(u_{s})-f(U^{N}_{s})]ds$
$\displaystyle\qquad+\int_{0}^{t}(P_{t-s}-P^{N}_{t-s})f(u_{s})ds+O_{t}-O^{N}_{t}.$

First, note that

(4.6)
$\displaystyle\|(P_{t}-P^{N}_{t})u_{0}\|_{L^{\infty}}\leq\|u_{0}-\Pi_{N}u_{0}\|_{L^{\infty}}\lesssim
N^{-1/2+\varepsilon}\|u_{0}\|_{\mathcal{C}^{1/2}}.$

Next, using that $e^{-x}\leq x^{-2+\varepsilon}$, $x>0$,

$\displaystyle\int_{0}^{t}\lVert(P_{t-s}-P^{N}_{t-s})f(u_{s})\rVert_{L^{\infty}}ds$
$\displaystyle=\int_{0}^{t}\lVert(p_{t-s}-p^{N}_{t-s})\ast
f(u_{s})\rVert_{L^{\infty}}ds$ $\displaystyle\leq\int_{0}^{t}\lVert
p_{t-s}-p^{N}_{t-s}\rVert_{L^{2}}\lVert f(u_{s})\rVert_{L^{2}}ds$
$\displaystyle\lesssim\lVert
f\rVert_{L^{\infty}(\mathbb{R})}\int_{0}^{t}\sqrt{\sum_{\lvert
k\rvert>N}e^{-8\pi^{2}k^{2}(t-s)}}ds$ (4.7) $\displaystyle\lesssim\lVert
f\rVert_{L^{\infty}(\mathbb{R})}N^{-3/2+\varepsilon}\int_{0}^{t}(t-s)^{-1+\varepsilon/2}ds$
(4.8) $\displaystyle\lesssim\lVert
f\rVert_{L^{\infty}(\mathbb{R})}N^{-3/2+\varepsilon}.$

Then an application of Proposition 3.4 for
$Z_{t}=(P_{t}-P^{N}_{t})u_{0}+\int_{0}^{t}(P_{t-s}-P^{N}_{t-s})f(u_{s})ds+O_{t}-O^{N}_{t}$,
$\tau(s)=s$, $S(s,t)=P^{N}_{t-s}$ together with (4.6), (4), and (4.4), yields
the claim. ∎

As mentioned, the main effort of the section is devoted in proving rate
(almost) $1$ for the temporal error $\tilde{U}^{N}-\tilde{V}^{M,N}$,
formulated as follows.

######  Lemma 4.4.

Assume the setting of Theorem 2.2. Then for any $\varepsilon>0$ and $p\geq 1$
there exists a constant $C=C(\varepsilon,p,K,\mathcal{M})$ such that

$\displaystyle\big{(}\mathds{E}\sup_{t\in[0,T]}\lVert\tilde{U}^{N}_{t}-\tilde{V}^{M,N}_{t}\rVert_{L^{\infty}(\mathbb{T})}^{p}\big{)}^{1/p}\leq
CM^{-1+\varepsilon}.$

Before the proof of Lemma 4.4 we prove a number of temporal error estimates.
Our strategy is as follows. We decompose the error as

$\displaystyle\tilde{U}^{N}_{t}-\tilde{V}^{M,N}_{t}$
$\displaystyle=\int_{0}^{t}P^{N}_{t-s}[f(\tilde{U}^{N}_{s})-f(\tilde{V}^{M,N}_{k_{M}(s)})]ds$
(4.9)
$\displaystyle=\int_{0}^{t}P^{N}_{t-s}[f(\tilde{U}^{N}_{s})-f(\tilde{U}^{N}_{k_{M}(s)})]ds+\int_{0}^{t}P^{N}_{t-s}[f(\tilde{U}^{N}_{k_{M}(s)})-f(\tilde{V}^{M,N}_{k_{M}(s)})]ds$

The second term on the right-hand side of (4) is a buckling term (i.e. treated
by Grönwall’s lemma). The first term is the crucial one in determining the
temporal rate. First we bound this term when replacing $\tilde{U}^{N}$ by a
simpler process.

######  Proposition 4.5.

Let 2.1 hold. Let $p\geq 1$ and $\varepsilon\in(0,1/2)$. Then there exists a
constant $C=C(T,p,\varepsilon,K,\mathcal{M})$ such that for all $0\leq s\leq
t\leq R\leq T$ it holds that

$\displaystyle\bigg{(}\mathds{E}\bigg{\lVert}\int_{s}^{t}P^{N}_{R-s}[f(O_{s}+P^{N}_{s}u_{0})-f(O_{k_{M}(s)}+P^{N}_{k_{M}(s)}u_{0})]ds\bigg{\rVert}_{L^{\infty}}^{p}\bigg{)}^{1/p}$
(4.10) $\displaystyle\leq CM^{-1+2\varepsilon}|t-s|^{1/4+\varepsilon/2}.$

###### Proof.

To simplify notation denote the shifted OU process by
$\tilde{O}_{t}:=O_{t}+P_{t}^{N}u_{0}$, $t\in[0,T]$. To prove the desired
estimate, it suffices to prove that for any $j\geq-1$, $x\in\mathbb{T}$,
$0\leq s\leq t\leq R\leq T$, one has

(4.11)
$\displaystyle\bigg{(}\mathds{E}\bigg{\lvert}\int_{s}^{t}\Delta_{j}P_{R-r}^{N}[f(\tilde{O}_{r})-f(\tilde{O}_{k_{M}(r)})](x)dr\bigg{\rvert}^{p}\bigg{)}^{1/p}\lesssim
M^{-1+2\varepsilon}2^{-j\varepsilon}(t-s)^{1/4+\varepsilon/2},$

Indeed, (4.11) implies

$\displaystyle\bigg{(}\mathds{E}\bigg{\lVert}\int_{s}^{t}P^{N}_{R-s}[f(\tilde{O}_{s})-f(\tilde{O}_{k_{M}(s)})]ds\bigg{\rVert}_{B^{\varepsilon}_{p,p}}^{p}\bigg{)}^{1/p}\lesssim
M^{-1+2\varepsilon}(t-s)^{1/4+\varepsilon/2},$

yielding the claim by using the embedding
$B^{\varepsilon}_{p,p}\hookrightarrow\mathscr{C}^{\varepsilon-1/p}\hookrightarrow
L^{\infty}$ for $p$ large enough.

To prove (4.11), we consider $j\geq-1$, $x\in\mathbb{T}$, $R\leq T$ fixed and
apply Lemma 3.2 to the germ, $0\leq s\leq t\leq R$,

(4.12) $\displaystyle
A_{st}=\mathds{E}_{s}\int_{s}^{t}\Delta_{j}P_{R-r}^{N}[f(\tilde{O}_{r})-f(\tilde{O}_{k_{M}(r)})](x)dr.$

We have that for $s<u<t$,

$\displaystyle\delta A_{sut}$
$\displaystyle=\mathds{E}_{s}\int_{u}^{t}\Delta_{j}P_{R-r}^{N}[f(\tilde{O}_{r})-f(\tilde{O}_{k_{M}(r)})](x)dr$
(4.13)
$\displaystyle\qquad-\mathds{E}_{u}\int_{u}^{t}\Delta_{j}P_{R-r}^{N}[f(\tilde{O}_{r})-f(\tilde{O}_{k_{M}(r)})](x)dr.$

Thus we obtain that $\mathds{E}_{s}[\delta A_{sut}]=0$. Therefore the
condition (3.14) is satisfied with $C_{2}=0$. We now verify (3.13). We claim
that, uniformly in $x,j$ and for all $0\leq u<t<R$ with $|t-u|\leq|R-t|$ one
has

(4.14) $\displaystyle\|A_{ut}\|_{L^{p}(\Omega)}\lesssim
2^{-j\varepsilon}M^{-1+2\varepsilon}(R-t)^{-1/4-\varepsilon/2}(t-u)^{1/2+\varepsilon}.$

First we consider the case $|t-u|\leq 3M^{-1}$. We are going to employ the
semigroup estimates and the regularity bounds for the OU process (3.9). Notice
that $\tilde{O}$ satisfies the bounds (3.9), (3.10) with $O$ replaced by
$\tilde{O}$, since by the semigroup estimates (3.3) one has for all
$s,t\in[0,T]$,

$\lVert
P_{t+s}u_{0}-P_{t}u_{0}\rVert_{\mathscr{C}^{-1/2+\varepsilon}}\lesssim_{\theta}s^{(1+\theta)/2}t^{-\theta/2-\varepsilon/2}\lVert
u_{0}\rVert_{\mathscr{C}^{1/2}}$

for any $\theta\in[0,1]$ and

$\lVert
P_{t+s}u_{0}-P_{t}u_{0}\rVert_{\mathscr{C}^{-1/2+\theta}}\lesssim_{\theta}s^{(1-\theta)/2}\lVert
u_{0}\rVert_{\mathscr{C}^{1/2}}$

for $\theta\in[0,1]$. Furthermore, for $\theta\in(0,1/2)$, $q\geq 1$ we can
bound the composition by

(4.15) $\displaystyle\mathds{E}\sup_{r\in[0,R]}\lVert
f^{\prime}(\lambda\tilde{O}_{r}+(1-\lambda)\tilde{O}_{k_{M}(r)})\rVert_{\mathscr{C}^{\theta}}^{q}\lesssim\mathds{E}\lVert
f^{\prime}\rVert_{C_{b}^{1}}^{q}\lVert\tilde{O}\rVert_{C_{T}\mathscr{C}^{\theta}}^{q}\lesssim_{q,\theta}1.$

Using these bounds, we obtain for $u,t$ such that $\lvert t-u\rvert\leq
3M^{-1}$,

$\displaystyle\bigg{(}\mathds{E}\bigg{\lvert}\int_{u}^{t}\Delta_{j}P_{R-r}^{N}[f(\tilde{O}_{r})-f(\tilde{O}_{k_{M}(r)})](x)dr\bigg{\rvert}^{p}\bigg{)}^{1/p}$
$\displaystyle\lesssim
2^{-j\varepsilon}\bigg{(}\mathds{E}\bigg{\lVert}\int_{u}^{t}P_{R-r}^{N}[f(\tilde{O}_{r})-f(\tilde{O}_{k_{M}(r)})]dr\bigg{\rVert}_{\mathscr{C}^{\varepsilon}}^{p}\bigg{)}^{1/p}$
$\displaystyle\lesssim
2^{-j\varepsilon}\int_{u}^{t}(R-r)^{-1/4-\varepsilon/2}$ $\displaystyle\hskip
5.69046pt\times\bigg{(}\mathds{E}\bigg{\lVert}\bigg{(}\int_{0}^{1}f^{\prime}(\lambda\tilde{O}_{r}+(1-\lambda)\tilde{O}_{k_{M}(r)})d\lambda\bigg{)}(\tilde{O}_{r}-\tilde{O}_{k_{M}(r)})\bigg{\rVert}_{\mathscr{C}^{-1/2}}^{p}\bigg{)}^{1/p}dr$
$\displaystyle\lesssim 2^{-j\varepsilon}(R-t)^{-1/4-\varepsilon/2}(t-u)$
$\displaystyle\hskip
5.69046pt\times\bigg{(}\mathds{E}\sup_{r\in[0,T]}\bigg{\lVert}\int_{0}^{1}f^{\prime}(\lambda\tilde{O}_{r}+(1-\lambda)\tilde{O}_{k_{M}(r)})d\lambda\bigg{\rVert}_{\mathscr{C}^{1/2-\varepsilon/2}}^{2p}\bigg{)}^{1/2p}$
$\displaystyle\hskip
5.69046pt\times\big{(}\mathds{E}\sup_{r\in[0,T]}\lVert(\tilde{O}_{r}-\tilde{O}_{k_{M}(r)})\rVert_{\mathscr{C}^{-1/2+\varepsilon}}^{2p}\big{)}^{1/2p}$
$\displaystyle\lesssim
2^{-j\varepsilon}(R-t)^{-1/4-\varepsilon/2}(t-u)(r-k_{M}(r))^{1/2-\varepsilon}$
$\displaystyle\lesssim
2^{-j\varepsilon}(R-t)^{-1/4-\varepsilon/2}(t-u)M^{-1/2+\varepsilon}$ (4.16)
$\displaystyle\lesssim
2^{-j\varepsilon}(R-t)^{-1/4-\varepsilon/2}(t-s)^{1/2+\varepsilon}M^{-1+2\varepsilon},$

using $\lvert t-u\rvert\leq 3M^{-1}$ in the last inequality. This shows that
(4.14) holds for $\lvert t-u\rvert\leq 3M^{-1}$.

Next, we consider $\lvert t-u\rvert>3M^{-1}$. Let $t^{\prime}$ be the second
smallest grid point bigger or equal to $u$. Note that this implies that for
any $r\geq t^{\prime}$ one has $k_{M}(r)-u\geq(r-u)/2$. Then we can decompose

$\displaystyle\int_{u}^{t}\Delta_{j}P_{R-r}^{N}[f(\tilde{O}_{r})-f(\tilde{O}_{k_{M}(r)})](x)dr$
$\displaystyle=\int_{u}^{t^{\prime}}\Delta_{j}P_{R-r}^{N}[f(\tilde{O}_{r})-f(\tilde{O}_{k_{M}(r)})](x)dr+\int_{t^{\prime}}^{t}\Delta_{j}P_{R-r}^{N}[f(\tilde{O}_{r})-f(\tilde{O}_{k_{M}(r)})])(x)dr,$

where the first summand we can cope with as above, because $(t^{\prime}-u)\leq
2M^{-1}$. The conditional expectation of the second summand, we rewrite as
follows

$\displaystyle\mathds{E}_{u}\bigg{[}\int_{t^{\prime}}^{t}\Delta_{j}P_{R-r}^{N}[f(\tilde{O}_{r})-f(\tilde{O}_{k_{M}(r)})](x)dr\bigg{]}$
(4.17)
$\displaystyle=\int_{t^{\prime}}^{t}\Delta_{j}P_{R-r}^{N}[(P^{\mathbb{R}}_{Q(r-u)}f)(P_{r-u}\tilde{O}_{u})-(P^{\mathbb{R}}_{Q(k_{M}(r)-u)}f)(P_{k_{M}(r)-u}\tilde{O}_{u})](x)dr$

using that

$\displaystyle\tilde{O}_{r}=P_{r-u}\tilde{O}_{u}+\int_{u}^{r}\int_{\mathbb{T}}p_{r-v}(\cdot-y)\xi(dv,dy)$

and that for random variables $X,Y$ with $X$ being
$\mathcal{F}_{u}$-measurable and $Y$ being independent of $\mathcal{F}_{u}$
and centered Gaussian with variance $\sigma$, we have that
$\mathds{E}_{u}[f(X+Y)]=(P^{\mathbb{R}}_{\sigma}f)(X)$, where $P^{\mathbb{R}}$
denotes the heat-semigroup on $\mathbb{R}$. Above we denote by
$Q(r-u)=\mathds{E}[(\int_{u}^{r}\int_{\mathbb{T}}p_{r-v}(x-y)\xi(dv,dy))^{2}]$
the variance, which only depends on the time distance $r-u$ due to
stationarity and also does not depend on $x$. Then we decompose further

$\displaystyle\int_{t^{\prime}}^{t}\Delta_{j}P_{R-r}^{N}[(P^{\mathbb{R}}_{Q(r-u)}f)(P_{r-u}\tilde{O}_{u})-(P^{\mathbb{R}}_{Q(k_{M}(r)-u)}f)(P_{k_{M}(r)-u}\tilde{O}_{u})](x)dr$
(4.18)
$\displaystyle=\int_{t^{\prime}}^{t}\Delta_{j}P_{R-r}^{N}[(P^{\mathbb{R}}_{Q(r-u)}f)(P_{r-u}\tilde{O}_{u})-(P^{\mathbb{R}}_{Q(r-u)}f)(P_{k_{M}(r)-u}\tilde{O}_{u})](x)dr$
(4.19)
$\displaystyle\quad+\int_{t^{\prime}}^{t}\Delta_{j}P_{R-r}^{N}[[(P^{\mathbb{R}}_{Q(r-u)}f)-(P^{\mathbb{R}}_{Q(k_{M}(r)-u)}f)](P_{k_{M}(r)-u}\tilde{O}_{u})](x)dr.$

For the first summand (4.18) we use that, similarly to (4.15) one has

(4.20)
$\displaystyle\mathds{E}\sup_{r\in[0,R]}\lVert(P^{\mathbb{R}}_{Q(r-u)}f)^{\prime}(\lambda
P_{r-u}\tilde{O}_{u}+(1-\lambda)P_{k_{M}(r)-u}\tilde{O}_{u})\rVert_{\mathscr{C}^{\theta}}^{q}\lesssim_{q,\theta}1.$

Using this and the bound (3.10) for $\tilde{O}$ we obtain

$\displaystyle\bigg{(}\mathds{E}\bigg{\lvert}\int_{t^{\prime}}^{t}\Delta_{j}P_{R-r}^{N}[(P^{\mathbb{R}}_{Q(r-u)}f)(P_{r-u}\tilde{O}_{u})-(P^{\mathbb{R}}_{Q(r-u)}f)(P_{k_{M}(r)-u}\tilde{O}_{u})](x)dr\bigg{\rvert}^{p}\bigg{)}^{1/p}$
$\displaystyle\lesssim
2^{-j\varepsilon}\bigg{(}\mathds{E}\bigg{\lVert}\int_{t^{\prime}}^{t}P_{R-r}^{N}\bigg{[}\bigg{(}\int_{0}^{t}(P^{\mathbb{R}}_{Q(r-u)}f)^{\prime}(\lambda
P_{r-u}\tilde{O}_{u}+(1-\lambda)P_{k_{M}(r)-u}\tilde{O}_{u})d\lambda\bigg{)}$
$\displaystyle\hskip
142.26378pt\times(P_{r-u}\tilde{O}_{u}-P_{k_{M}(r)-u}\tilde{O}_{u})\bigg{]}dr\bigg{\rVert}_{\mathscr{C}^{\varepsilon}}^{p}\bigg{)}^{1/p}$
$\displaystyle\lesssim
2^{-j\varepsilon}\int_{t^{\prime}}^{t}\bigg{(}(R-r)^{-1/4-\varepsilon/2}$
$\displaystyle\hskip
28.45274pt\times\bigg{(}\mathds{E}\bigg{\lVert}\int_{0}^{t}(P^{\mathbb{R}}_{Q(r-u)}f)^{\prime}(\lambda
P_{r-u}\tilde{O}_{u}+(1-\lambda)P_{k_{M}(r)-u}\tilde{O}_{u})d\lambda\bigg{\rVert}_{\mathscr{C}^{1/2-\varepsilon/2}}^{2p}\bigg{)}^{1/2p}\bigg{)}$
$\displaystyle\hskip 28.45274pt\times\big{(}(\mathds{E}\lVert
P_{r-u}\tilde{O}_{u}-P_{k_{M}(r)-u}\tilde{O}_{u}\rVert_{\mathscr{C}^{-1/2+\varepsilon}}^{2p})^{1/2p}\big{)}dr$
$\displaystyle\lesssim
2^{-j\varepsilon}\int_{t^{\prime}}^{t}(R-r)^{-1/4-\varepsilon/2}(r-k_{M}(r))^{1-2\varepsilon}(k_{M}(r)-u)^{-1/2+\varepsilon}dr$
$\displaystyle\lesssim
2^{-j\varepsilon}M^{-1+2\varepsilon}\int_{t^{\prime}}^{t}(R-r)^{-1/4-\varepsilon/2}(r-u)^{-1/2+\varepsilon}dr$
(4.21) $\displaystyle\lesssim
2^{-j\varepsilon}M^{-1+2\varepsilon}(R-t)^{-1/4-\varepsilon/2}(t-u)^{1/2+\varepsilon}$

where we used that $k_{M}(r)-u\geq(r-u)/2$ for $r\in[t^{\prime},t]$. For the
second summand (4.19), we use the following estimate on $Q$, where $u\leq
l\leq r$,

$\displaystyle Q(r-u)-Q(l-u)$
$\displaystyle=\mathds{E}\bigg{[}\bigg{(}\int_{u}^{r}\int
p_{r-t}(\cdot-y)\xi(dt,dy)\bigg{)}^{2}\bigg{]}-\mathds{E}\bigg{[}\bigg{(}\int_{u}^{l}\int
p_{l-t}(\cdot-y)\xi(dt,dy)\bigg{)}^{2}\bigg{]}$
$\displaystyle=\int_{u}^{r}\lVert
p_{r-t}\rVert_{L^{2}(\mathbb{T})}^{2}dt-\int_{u}^{l}\lVert
p_{l-t}\rVert_{L^{2}(\mathbb{T})}^{2}dt=\int_{l-u}^{r-u}\|p_{s}\|_{L^{2}(\mathbb{T})}^{2}\,ds$
$\displaystyle\leq\int_{l-u}^{r-u}\|p_{s}\|_{L^{1}(\mathbb{T})}\|p_{s}\|_{L^{\infty}(\mathbb{T})}\,ds\lesssim\int_{l-u}^{r-u}s^{-1/2}\,ds$
(4.22) $\displaystyle\lesssim(r-l)^{1-\varepsilon}(l-u)^{-1/2+\varepsilon}.$

The latter bound together with the heat kernel estimate (3.5) then yields

$\displaystyle\lVert(P^{\mathbb{R}}_{Q(r-u)}f)-(P^{\mathbb{R}}_{Q(k_{M}(r)-u)}f)\rVert_{L^{\infty}(\mathbb{R})}$
$\displaystyle\lesssim(Q(r-u)-Q(k_{M}(r)-u))$
$\displaystyle\lesssim(k_{M}(r)-u)^{-1/2+\varepsilon}(r-k_{M}(r))^{1-\varepsilon}.$

Hence we obtain

$\displaystyle\bigg{(}\mathds{E}\bigg{\lvert}\int_{t^{\prime}}^{t}\Delta_{j}P_{R-r}^{N}[[(P^{\mathbb{R}}_{Q(r-u)}f)-(P^{\mathbb{R}}_{Q(k_{M}(r)-u)}f)](P_{k_{M}(r)-u}O_{u}+P^{N}_{k_{M}(r)}u_{0})](x)dr\bigg{\rvert}^{p}\bigg{)}^{1/p}$
$\displaystyle\lesssim
2^{-j\varepsilon}\int_{t^{\prime}}^{t}(R-r)^{-\varepsilon/2}\lVert(P^{\mathbb{R}}_{Q(r-u)}f)-(P^{\mathbb{R}}_{Q(k_{M}(r)-u)}f)\rVert_{L^{\infty}(\mathbb{R})}dr$
$\displaystyle\lesssim
2^{-j\varepsilon}(R-t)^{-\varepsilon/2}\int_{t^{\prime}}^{t}(k_{M}(r)-u)^{-1/2+\varepsilon}(r-k_{M}(r))^{1-\varepsilon}dr$
(4.23) $\displaystyle\lesssim
2^{-j\varepsilon}M^{-1+\varepsilon}(R-t)^{-\varepsilon/2}(t-s)^{1/2+\varepsilon},$

using again that $k_{M}(r)-u\geq(r-u)/2$ for $r\in[t^{\prime},t]$. We
therefore get (4.14).

Therefore Lemma 3.2 finishes the proof of (4.11) provided we justify that

$\displaystyle\mathcal{A}_{t}=\int_{0}^{t}\Delta_{j}P^{N}_{R-r}[f(\tilde{O}_{r})-f(\tilde{O}_{k_{M}(r)})](x)dr.$

To this end, we need to verify (3.15) and (3.16), the latter of which is
trivial since $\mathds{E}_{s}(\mathcal{A}_{st}-A_{st})=0$. The former is
trivial for another reason: from the boundedness of $f$ one immediately gets
$|\mathcal{A}_{st}-A_{st}|\leq(t-s)\|f\|_{L^{\infty}}$. The proof is finished.
∎

######  Corollary 4.6.

Let 2.1 hold. Let $p\geq 1$ and $\varepsilon\in(0,1/4)$. Then there exists a
constant $C=C(T,p,\varepsilon,K,\mathcal{M})$ such that

(4.24)
$\displaystyle\bigg{(}\mathds{E}\sup_{R\in[0,T]}\bigg{\lVert}\int_{0}^{R}P^{N}_{R-s}[f(O_{s}+P^{N}_{s}u_{0})-f(O_{k_{M}(s)}+P^{N}_{k_{M}(s)}u_{0})]ds\bigg{\rVert}_{L^{\infty}}^{p}\bigg{)}^{1/p}\leq
CM^{-1+\varepsilon}.$

###### Proof.

We recall a small variation of Kolmogorov’s continuity theorem. Let
$(X_{t})_{t\in[0,T]}$ be a continuous stochastic process starting from $0$
with values in a Banach space $V$ and let $(S_{t})_{t\geq 0}$ be a continuous
semigroup of bounded linear operators on $V$. Then, if for some $p>0$,
$\alpha>0$, $C^{\prime}<\infty$ it holds for all $0\leq s\leq t\leq T$ that

(4.25) $\displaystyle\mathds{E}\|X_{t}-S_{t-s}X_{s}\|^{p}\leq
C^{\prime}|t-s|^{1+\alpha},$

then one has

(4.26) $\displaystyle\mathds{E}\sup_{t\in[0,T]}\|X_{t}\|^{p}\leq
C^{\prime\prime}C^{\prime},$

where $C^{\prime\prime}$ depends only on $p,\alpha,T$. It remains to notice
that by Proposition 6.5 the process

(4.27) $\displaystyle
X_{t}=\int_{0}^{t}P^{N}_{t-s}[f(O_{s}+P^{N}_{s}u_{0})-f(O_{k_{M}(s)}+P^{N}_{k_{M}(s)}u_{0})]ds,$

satisfies the above conditions with the semigroup $S=P^{N}$, $V=L^{\infty}$,
any $p\geq 4$, $\alpha=2\varepsilon$, and
$C^{\prime}=(CM^{-1+\varepsilon})^{p}$. ∎

######  Corollary 4.7.

Let 2.1 hold. Let $p\geq 1$ and $\varepsilon\in(0,1/2)$. Let $\tilde{U}^{N}$
be the solution of (4.1). Then there exists a constant
$C=C(T,p,\varepsilon,K,\mathcal{M})$ such that

(4.28)
$\displaystyle\bigg{(}\mathds{E}\sup_{R\in[0,T]}\bigg{\lVert}\int_{0}^{R}P^{N}_{R-s}[f(\tilde{U}^{N}_{s})-f(\tilde{U}^{N}_{k_{M}(s)})]ds\bigg{\rVert}_{L^{\infty}}^{p}\bigg{)}^{1/p}\leq
CM^{-1+\varepsilon}.$

###### Proof.

We follow the proof of Lemma 2.3.6 in [BDG23]. Define the probability measure
$\mathbb{Q}$ via

$\displaystyle\frac{d\mathbb{Q}}{d\mathds{P}}=\rho=\exp\bigg{(}-\int_{0}^{T}\int
f(\tilde{U}^{N}_{s}(y))\xi(dy,ds)-\frac{1}{2}\int_{0}^{T}\int\lvert
f(\tilde{U}^{N}_{s}(y))\rvert^{2}dyds\bigg{)}.$

Girsanov’s theorem ([DPZ92, Theorem 10.14]) gives that
$\xi(dy,ds)+f(\tilde{U}^{N}_{s}(y))dyds$ defines a space-time white noise
measure under $\mathbb{Q}$ independent of $\mathcal{F}_{0}$. This implies that
the law of $(\tilde{U}^{N}_{t})_{t\in[0,T]}$ under $\mathbb{Q}$ coincides with
the law of $(O_{t}+P_{t}^{N}u_{0})_{t\in[0,T]}$ under $\mathds{P}$. An easy
exercise shows that $\mathds{E}[\rho^{-1}]\lesssim C(\lVert
f\rVert_{L^{\infty}(\mathbb{R})})<\infty$. Therefore, defining a function $g$
on the space of continuous functions $Z$ by

(4.29) $\displaystyle
g(Z):=\sup_{R\in[0,T]}\bigg{\lVert}\int_{0}^{R}P^{N}_{R-s}[f(Z_{s})-f(Z_{k_{M}(s)})]ds\bigg{\rVert},$

we can bound

$\displaystyle\mathds{E}[\lvert
g(\tilde{U}^{N})\rvert^{p}]=\mathds{E}[\rho\rho^{-1}\lvert
g(\tilde{U}^{N})\rvert^{p}]$
$\displaystyle=\mathds{E}_{\mathbb{Q}}[\rho^{-1}\lvert
g(\tilde{U}^{N})\rvert^{p}]$
$\displaystyle\leq\mathds{E}_{\mathbb{Q}}[\rho^{-2}]^{1/2}\mathds{E}_{\mathbb{Q}}[\lvert
g(\tilde{U}^{N})\rvert^{2p}]^{1/2}$
$\displaystyle=\mathds{E}[\rho^{-1}]^{1/2}\mathds{E}[\lvert
g(O+P_{\cdot}u_{0}^{N})\rvert^{2p}]^{1/2}$
$\displaystyle\lesssim\mathds{E}[\lvert
g(O+P_{\cdot}u_{0}^{N})\rvert^{2p}]^{1/2}.$

The proof is finished by applying Corollary 4.6 with $2p$ in place of $p$ to
bound the right-hand side. ∎

We can now prove Lemma 4.4, which finalizes the proof of Theorem 2.2.

###### Proof of Lemma 4.4.

By (4), we can apply Proposition 3.4 with $X=\tilde{U}^{N}$,
$Y=\tilde{V}^{M,N}$,

(4.30) $\displaystyle
Z_{t}=\int_{0}^{t}P^{N}_{t-s}[f(\tilde{U}^{N}_{s})-f(\tilde{U}^{N}_{k_{M}(s)})]ds,$

$S(s,t)=P^{N}_{t-s}$, and $\tau(s)=k_{M}(s)$. Using Corollary 4.7 to bound
$Z$, we get the claim. ∎

##  5\. A priori bounds for superlinear $f$

In this section, we prepare for the error analysis in case of a superlinearily
growing nonlinearity $f$ that satisfies 2.3. One of the steps that become less
obvious (and, for too naive approximations simply impossible [BHJ+19]), is to
bound the approximations uniformly in $N,M$. The purpose of this section is to
have such a priori estimates on $X^{N,M}$ as well as a couple of related
processes.

Recall $\Phi_{h}$, $g_{h}$, and $X^{M,N}$ introduced in Section 2. We now
introduce some further auxiliary processes. First note that one can also view
$X^{M,N}$ as the standard Euler discretization for the SPDE with nonlinearity
$g_{h}$, cf. (2.17). The mild solution of the SPDE with nonlinearity $g_{h}$
is given by

(5.1) $\displaystyle
X^{h}_{t}=P_{t}u_{0}+\int_{0}^{t}P_{t-s}g_{h}(X^{h}_{s})ds+O_{t}.$

Furthermore we denote by $X^{h,N}$ its Galerkin approximation, that is,

(5.2) $\displaystyle
X^{h,N}_{t}=P^{N}_{t}u_{0}+\int_{0}^{t}P^{N}_{t-s}g_{h}(X^{h,N}_{s})ds+O^{N}_{t}.$

First, we derive bounds on the nonlinearities $g_{h}$, $\Phi_{h}$. Like $f$,
also $g_{h}$ enjoys a local Lipschitz condition, a polynomial growth and a
one-sided Lipschitz condition, but $\Phi_{h}$ is globally Lipschitz
continuous. The proof of the following lemma can be found in the Appendix.

######  Lemma 5.1.

Let $f$ satisfy 2.3 (a). Let $\Phi_{h}$, $g_{h}$ be given as in (2.13), (2.15)
Then there exist constants $C,\tilde{K}>0$ and $\tilde{m}\geq m$, depending
only on $K$ and $m$, such that for $i=0,1,2,3$ and for all $h\in[0,1]$, the
functions $\Phi_{h}$ and $g_{h}$ satisfy

$\displaystyle\lvert\Phi_{h}(x)-\Phi_{h}(y)\rvert$ $\displaystyle\leq
e^{Kh/2}\lvert x-y\rvert,\quad\forall x,y\in\mathbb{R}$
$\displaystyle\lvert\partial^{i}g_{h}(x)\rvert$
$\displaystyle\leq\tilde{K}(1+\lvert x\rvert^{2\tilde{m}+1-i}),\quad\forall
x\in\mathbb{R},\quad i=0,1,2,3$ $\displaystyle\partial g_{h}(x)$
$\displaystyle\leq K,\quad\forall x\in\mathbb{R}$ $\displaystyle\lvert
g_{h}(x)-g_{0}(x)\rvert$ $\displaystyle\leq Ch(1+\lvert
x\rvert^{4m+2}),\quad\forall x\in\mathbb{R}.$

In particular, $g_{h}$ satisfies

(5.3) $\displaystyle(g_{h}(x)-g_{h}(y))(x-y)$ $\displaystyle\leq
K(x-y)^{2},\quad\forall x,y\in\mathbb{R}$ (5.4) $\displaystyle\lvert
g_{h}(x)-g_{h}(y)\rvert$ $\displaystyle\leq\tilde{K}(1+\lvert
x\rvert^{2\tilde{m}}+\lvert y\rvert^{2\tilde{m}})\lvert x-y\rvert,\quad\forall
x,y\in\mathbb{R}.$

Given the uniform bounds on $g_{h}$, we obtain uniform bounds on the processes
$X^{h}$, $X^{h,N}$, formulated as follows.

######  Corollary 5.2.

Let 2.3 hold. Then there exist unique mild solutions $X^{h}$ and $X^{h,N}$ to
the equations (5.1) and (5.2), respectively. Moreover, for any $p\geq 1$,
$\varepsilon\in(0,1/2)$, $\lambda\in(0,1)$, there exists a constant
$C=C(T,m,K,\varepsilon,\lambda,p,\mathcal{M})$ such that the following bound
holds

$\displaystyle\sup_{M\in\mathbb{N}}\mathds{E}\lVert
X^{h}\rVert_{C_{T}^{\lambda/2}\mathscr{C}^{1/2-\lambda-\varepsilon}}^{p}+\sup_{M,N\in\mathbb{N}}\mathds{E}\lVert
X^{h,N}\rVert_{C_{T}^{\lambda/2}\mathscr{C}^{1/2-\lambda-\varepsilon}}^{p}\leq
C.$

###### Proof.

First note that the bound for $X^{h}$ follows directly from Proposition 3.3,
since by Lemma 5.1, $g_{h}$ satisfies 2.3 with constants uniform in $h$. The
proof for $X^{h,N}$ is almost identical: one only needs to note that [Cer01,
Hypothesis 6.1] is satisfied also by the operators $A=\Pi_{N}\Delta$,
$Q=\Pi_{N}$, with the spaces $H=L^{2}(\mathbb{T})$, $E=C(\mathbb{T})$. The
rest of the proof of Proposition 3.3 follows verbatim. ∎

The a priori bounds for the splitting scheme can be derived using that
$\Phi_{h}$ is globally Lipschitz and the uniform growth bound on $g_{h}$.

######  Proposition 5.3.

Let 2.3 hold. Let $p\geq 1$ and $X^{M,N}$ be as in (2.17). Then there exists a
constant $C=C(p,K,m,T,\mathcal{M})$ such that the following a priori bound
holds

$\displaystyle\sup_{M,N\in\mathbb{N}}\mathds{E}\sup_{k=0,\dots,M}\lVert
X^{M,N}_{t_{k}}\rVert_{L^{\infty}}^{p}\leq C.$

###### Proof.

We follow the proof of [BG19, Proposition 3.7]. Let
$R^{M,N}_{t_{l}}=X^{M,N}_{t_{l}}-O^{N}_{t_{l}}$, $l=0,\dots,M$. Using that
$P_{h}$ is bounded on $L^{\infty}$ with norm $1$, that $\Phi_{h}$ is globally
Lipschitz with Lipschitz constant $e^{Kh/2}$, $\Phi_{h}(z)-z=hg_{h}(z)$, and
the growth bound for $g_{h}$, we can write iteratively

$\displaystyle\lVert R^{M,N}_{t_{k+1}}\rVert_{L^{\infty}}$
$\displaystyle=\lVert
P_{h}^{N}(\Phi_{h}(R^{M,N}_{t_{k}}+O^{N}_{t_{k}})-\Phi_{h}(O^{N}_{t_{k}}))+P_{h}^{N}(\Phi_{h}(O^{N}_{t_{k}})-O_{t_{k}}^{N})\rVert_{L^{\infty}}$
$\displaystyle\leq e^{Kh/2}\lVert
R^{M,N}_{t_{k}}\rVert_{L^{\infty}}+h\tilde{K}^{p}(1+\lVert
O^{N}\rVert_{C_{T}L^{\infty}}^{(2\tilde{m}+1)})$ $\displaystyle\vdots$
$\displaystyle\leq
e^{(k+1)Kh/2}\lVert\Pi_{N}u_{0}\rVert_{L^{\infty}}+\tilde{K}\sum_{j=0}^{k}e^{jKh/2}h(1+\lVert
O\rVert_{C_{T}L^{\infty}}^{(2\tilde{m}+1)}).$

In the last inequality we also estimated $\lVert
O^{N}\rVert_{C_{T}L^{\infty}}^{p}=\lVert\Pi_{N}O\rVert_{C_{T}L^{\infty}}^{p}\leq\lVert
O\rVert_{C_{T}L^{\infty}}^{p}$. Note that $(k+1)h\leq Mh=T$. Therefore,

$\displaystyle\sup_{k=0,\dots,M}\lVert R^{M,N}_{t_{k}}\rVert_{L^{\infty}}$
$\displaystyle\leq e^{KT/2}(\lVert
u_{0}\rVert_{L^{\infty}}+Mh\tilde{K}(1+\lVert
O\rVert_{C_{T}L^{\infty}}^{(2\tilde{m}+1)}))$ $\displaystyle\leq C(1+\lVert
u_{0}\rVert_{L^{\infty}}+\lVert O\rVert_{C_{T}L^{\infty}}^{(2\tilde{m}+1)}).$

Taking $p$-th moment expectations and using the bounds on $O$ and $O^{N}$, we
obtain the claim. ∎

######  Corollary 5.4.

Let 2.3 hold. Let $X^{M,N}$ be as in (2.17). Then for any $p\geq 1$,
$\lambda\in(0,1)$, $\varepsilon\in(0,1/2)$, there exists a constant
$C=C(p,T,\lambda,\varepsilon,K,m,\mathcal{M})$ such that

(5.5) $\displaystyle\sup_{M,N\in\mathbb{N}}\mathds{E}\lVert
X^{M,N}\rVert_{C_{T}^{\lambda/2}\mathscr{C}^{1/2-\lambda-\varepsilon}}^{p}\leq
C.$

Let $R^{M,N}:=X^{M,N}-O^{N}$. Then for any $\alpha\in(0,2)$,
$\varepsilon\in(0,1-\alpha/2)$, there exists a constant
$C=C(p,T,\varepsilon,\alpha,K,m,\mathcal{M})$ such that,

$\displaystyle\sup_{M,N\in\mathbb{N}}\mathds{E}\lVert
R^{M,N}\rVert_{C_{T}^{1-\alpha/2-\varepsilon}\mathscr{C}^{\alpha}}^{p}\leq C.$

###### Proof.

The bound (5.5) follows from Proposition 5.3 in exactly the same way as in
Proposition 3.3 the bound (3.18) is derived from (3.19). To see the bound on
$R^{M,N}$ one can write

$\displaystyle R^{M,N}_{t}-R^{M,N}_{s}$
$\displaystyle=(P_{t}^{N}-P_{s}^{N})u_{0}+\int_{s}^{t}P^{N}_{t-k_{M}(r)}g_{h}(X^{M,N}_{k_{M}(r)})dr$
$\displaystyle\qquad+\int_{0}^{s}P_{s-k_{M}(r)}^{N}(P_{t-s}^{N}-\operatorname{Id})g_{h}(X^{M,N}_{k_{M}(r)})dr.$

We estimate each term by the semigroup estimates (3.3) as follows. First, we
easily see that

$\lVert(P_{t}-P_{s})u_{0}\rVert_{\mathscr{C}^{\alpha}}\lesssim\lvert
t-s\rvert^{1-\alpha/2}\lVert u_{0}\rVert_{\mathscr{C}^{2}}.$

Second, using $t-k_{M}(r)\geq t-r$, we have

$\displaystyle\bigg{\lVert}\int_{s}^{t}P^{N}_{t-k_{M}(r)}g_{h}(X^{M,N}_{k_{M}(r)})dr\bigg{\rVert}_{\mathscr{C}^{\alpha}}$
$\displaystyle\lesssim\int_{s}^{t}(t-k_{M}(r))^{-\alpha/2}\lVert
g_{h}(X^{M,N}_{k_{M}(r)})\rVert_{L^{\infty}}dr$
$\displaystyle\lesssim(t-s)^{1-\alpha/2}(1+\sup_{k}\lVert
X^{M,N}_{t_{k}}\rVert_{L^{\infty}}^{2\tilde{m}+1}).$

Third, using $s-k_{M}(r)\geq s-r$, we obtain

$\displaystyle\bigg{\lVert}\int_{0}^{s}P_{s-k_{M}(r)}^{N}(P_{t-s}^{N}-\operatorname{Id})g_{h}(X^{M,N}_{k_{M}(r)})dr\bigg{\rVert}_{\mathscr{C}^{\alpha}}$
$\displaystyle\lesssim\int_{0}^{s}(s-k_{M}(r))^{-1+\varepsilon}\lVert(P_{t-s}^{N}-\operatorname{Id})g_{h}(X^{M,N}_{k_{M}(r)})\rVert_{\mathscr{C}^{-(2-\alpha-2\varepsilon)}}dr$
$\displaystyle\lesssim(t-s)^{1-\alpha/2-\varepsilon}(1+\sup_{k}\lVert
X^{M,N}_{t_{k}}\rVert_{L^{\infty}}^{2\tilde{m}+1}).$

Taking $p$-th moment and using Proposition 5.3 gives the claim. ∎

##  6\. Proof of Theorem 2.6

In this section, we prove that the same error bound of
$M^{-1+\varepsilon}+N^{-1/2+\varepsilon}$ can be reached also in the
superlinearly growing case. There are several steps that become significantly
more involved. We mentioned and addressed the question of a priori bounds in
Section 5. The lack of global Lipschitz bound also requires changes in the
buckling, as the mild form of Grönwall’s lemma as in Proposition 3.4 is no
longer applicable. The buckling steps will be therefore performed by switching
to the variational formulation, where the one-sided Lipschitzness is easier to
exploit. Finally, the superlinear growth prevents us from appealing to
Girsanov’s theorem as in Section 4, which is then replaced by using a more
involved stochastic sewing argument.

######  Remark 6.1.

As mentioned, we will frequently switch to the the weak/variational form of
certain equations. Let us make this more precise. Recall (cf. e.g. [Bal77])
that if the functions $F,G\in C([0,T]\times\mathbb{T})$, $H\in C(\mathbb{T})$
satisfy for all $(t,x)\in[0,T]\times\mathbb{T}$ the mild formulation
F(t,x)=∫_Tp_t(x-y)H(y)dy+∫_0^t∫_Tp_t-s(x-y)G(s,y) ds dy, then $F$ is the
unique weak solution of ∂_t F=ΔF+G, F_0=H, and in particular it satisfies the
energy inequality ∂_t∥F(t,⋅)∥_L^2^2≤2⟨F(t,⋅),G(t,⋅)⟩, where
$\langle\cdot,\cdot\rangle$ is the $L^{2}(\mathbb{T})$ inner product. As a
consequence, for $p\geq 2$ one also has
∂_t∥F(t,⋅)∥_L^2^p≤p∥F(t,⋅)∥_L^2^p-2⟨F(t,⋅),G(t,⋅)⟩. The same conclusion holds
if the heat kernel $P$ is replaced by $P^{N}$.

Since we deal with growing nonlinearities, we introduce weighted some spaces.
For polynomial weights $\omega(x)=(1+\lvert x\rvert^{2})^{-\beta/2}$,
$\beta\geq 1$, we define for $k\in\mathbb{N}_{0}$ the weighted Hölder space

$C^{k}_{\omega}=\\{f\in\mathcal{S}^{\prime}\mid\omega f\in
C^{k}_{b}\\},\quad\lVert f\rVert_{C^{k}_{\omega}}:=\lVert\omega
f\rVert_{C^{k}_{b}}.$

One can easily show the following semigroup estimate on the weighted space:
for any $0\leq s\leq t\leq 1$ one has

(6.1)
$\displaystyle\lVert(P_{t}^{\mathbb{R}}-P_{s}^{\mathbb{R}})u\rVert_{C^{0}_{\omega}}\lesssim\lvert
t-s\rvert\lVert u\rVert_{C^{2}_{\omega}}.$

Notice that, since $g_{h}$ and its derivatives of order $i=1,2,3$ satisfy a
polynomial growth bound, there exists an appropriate weight $\omega$, such
that $g_{h}\in C^{3}_{\omega}$. For the remainder of the article we fix such a
weight.

The error between the true solution $u$ and the numerical solution $X^{M,N}$
can be decomposed as follows.

$\displaystyle\mathds{E}[\sup_{t\in[0,T]}\lVert
u_{t}-X^{M,N}_{t}\rVert_{L^{2}}^{p}]$
$\displaystyle\lesssim\mathds{E}[\sup_{t\in[0,T]}\lVert
u_{t}-X^{h}_{t}\rVert_{L^{2}}^{p}]+\mathds{E}[\sup_{t\in[0,T]}\lVert
X^{h}_{t}-X^{h,N}_{t}\rVert_{L^{2}}^{p}]$
$\displaystyle\qquad+\mathds{E}[\sup_{t\in[0,T]}\lVert
X^{h,N}_{t}-X^{M,N}_{t}\rVert_{L^{2}}^{p}].$

The first term $u-X^{h}$ is the error coming from replacing $f$ by $g_{h}$ in
the continuum equation. It is estimated in Lemma 6.3. The second term
$X^{h}-X^{h,N}$ is the error of the spectral Galerkin approximation of
$X^{h}.$ This is bounded in Lemma 6.2. The third term $X^{h,N}-X^{M,N}$ is the
most crucial contribution to the error and the most challenging to bound by a
quantity of order $M^{-1+\varepsilon}+N^{-1/2+\varepsilon}$. This is the
content of Lemma 6.4. These three estimates together yield the main result,
Theorem 2.6.

######  Lemma 6.2.

Let 2.3 hold. Then for any $p\geq 1$, $\varepsilon\in(0,1/2)$, there exists a
constant $C=C(T,\varepsilon,p,m,K,\mathcal{M})$ such that

$\displaystyle\mathds{E}\sup_{t\in[0,T]}\lVert
X^{h}_{t}-X^{h,N}_{t}\rVert_{L^{2}}^{p}\leq CN^{-p(1/2-\varepsilon)}.$

###### Proof.

Let $R^{h}:=X^{h}-O$ and $R^{h,N}:=X^{h,N}-O^{N}$. Then we clearly have

(6.2) $\displaystyle\lVert
X^{h}_{t}-X^{h,N}_{t}\rVert_{L^{2}}^{p}\lesssim\lVert
R^{h}_{t}-R^{h,N}_{t}\rVert_{L^{2}}^{p}+\lVert
O_{t}-O^{N}_{t}\rVert_{L^{2}}^{p}.$

The second term is bounded in (4.4). For the first term, we use Remark 6.1
with $R^{h}-R^{h,N}$, $H=u_{0}-\Pi_{N}u_{0}$, and
$G=g_{h}(X^{h})-\Pi_{N}(g_{h}(X^{h,N}))$. We get that for $p\geq 2$

$\displaystyle\lVert R^{h}_{t}-R^{h,N}_{t}\rVert_{L^{2}}^{p}$
$\displaystyle\leq\lVert u_{0}-\Pi_{N}u_{0}\rVert_{L^{2}}^{p}$
$\displaystyle\qquad+p\int_{0}^{t}\lVert
R^{h}_{s}-R^{h,N}_{s}\rVert_{L^{2}}^{p-2}\langle
g_{h}(X^{h}_{s})-\Pi_{N}(g_{h}(X^{h,N}_{s})),R^{h}_{s}-R^{h,N}_{s}\rangle ds.$

Smuggling in $g_{h}(X^{h,N}_{s})$, using the definition of $R^{h},R^{h,N}$ and
the bounds for $g_{h}$ from Lemma 5.1, as well as Cauchy-Schwartz inequality
followed by Young’s inequality for $p_{1}=p/(p-1)$ and $p_{2}=p$, we arrive at

$\displaystyle\lVert R^{h}_{t}-R^{h,N}_{t}\rVert_{L^{2}}^{p}$
$\displaystyle\leq\lVert u_{0}-\Pi_{N}u_{0}\rVert_{L^{2}}^{p}$
$\displaystyle\qquad+p\int_{0}^{t}\lVert
R^{h}_{s}-R^{h,N}_{s}\rVert_{L^{2}}^{p-2}\langle
g_{h}(X^{h}_{s})-g_{h}(X^{h,N}_{s}),X^{h}_{s}-X^{h,N}_{s}\rangle ds$
$\displaystyle\qquad-p\int_{0}^{t}\lVert
R^{h}_{s}-R^{h,N}_{s}\rVert_{L^{2}}^{p-2}\langle
g_{h}(X^{h}_{s})-g_{h}(X^{h,N}_{s}),O_{s}^{N}-O_{s}\rangle ds$
$\displaystyle\qquad+p\int_{0}^{t}\lVert
R^{h}_{s}-R^{h,N}_{s}\rVert_{L^{2}}^{p-2}\langle
g_{h}(X^{h,N}_{s})-\Pi_{N}(g_{h}(X^{h,N}_{s})),R^{h}_{s}-R^{h,N}_{s}\rangle
ds$ $\displaystyle\lesssim\lVert u_{0}-\Pi_{N}u_{0}\rVert_{L^{2}}^{p}$
$\displaystyle\qquad+\int_{0}^{t}\lVert
R^{h}_{s}-R^{h,N}_{s}\rVert_{L^{2}}^{p-2}\lVert
X^{h}_{s}-X^{h,N}_{s}\rVert_{L^{2}}^{2}ds$ $\displaystyle\qquad+(1+\lVert
X^{h}\rVert_{C_{T}L^{\infty}}^{2\tilde{m}}+\lVert
X^{h}\rVert_{C_{T}L^{\infty}}^{2\tilde{m}})$ $\displaystyle\hskip
85.35826pt\times\int_{0}^{t}\lVert
R^{h}_{s}-R^{h,N}_{s}\rVert_{L^{2}}^{p-2}\lVert
X^{h}_{s}-X^{h,N}_{s}\rVert_{L^{2}}\lVert O_{t}-O^{N}_{t}\rVert_{L^{2}}ds$
$\displaystyle\qquad+\int_{0}^{t}\lVert
R^{h}_{s}-R^{h,N}_{s}\rVert_{L^{2}}^{p}ds+\int_{0}^{t}\lVert
g_{h}(X^{h,N}_{s})-\Pi_{N}(g_{h}(X^{h,N}_{s}))\rVert_{L^{2}}^{p}ds.$

Using (6.2) and that $\lVert\Pi_{N}u-u\rVert_{L^{2}}\lesssim N^{-\alpha}\lVert
u\rVert_{H^{\alpha}}$, $\alpha\geq 0$, we further obtain

$\displaystyle\lVert R^{h}_{t}-R^{h,N}_{t}\rVert_{L^{2}}^{p}$
$\displaystyle\lesssim N^{-\alpha p}\lVert
u_{0}\rVert_{H^{\alpha}}+\int_{0}^{t}\lVert
R^{h}_{s}-R^{h,N}_{s}\rVert_{L^{2}}^{p}ds$ $\displaystyle\qquad+(1+\lVert
X^{h}\rVert_{C_{T}L^{\infty}}^{2\tilde{m}}+\lVert
X^{h,N}\rVert_{C_{T}L^{\infty}}^{2\tilde{m}})\int_{0}^{t}\lVert
R^{h}_{s}-R^{h,N}_{s}\rVert_{L^{2}}^{p-2}\lVert
O_{t}-O^{N}_{t}\rVert_{L^{2}}^{2}ds$ $\displaystyle\qquad+(1+\lVert
X^{h}\rVert_{C_{T}L^{\infty}}^{2\tilde{m}}+\lVert
X^{h,N}\rVert_{C_{T}L^{\infty}}^{2\tilde{m}})\int_{0}^{t}\lVert
R^{h}_{s}-R^{h,N}_{s}\rVert_{L^{2}}^{p-1}\lVert
O_{t}-O^{N}_{t}\rVert_{L^{2}}ds$ $\displaystyle\qquad+N^{-p\alpha}\lVert
g_{h}(\tilde{X}^{h,N})\rVert_{C_{T}H^{\alpha}}^{p}.$

Applying again twice Young’s inequality for $p_{1}=p/(p-1)$ and $p_{2}=p$ and
for $p_{1}=p/(p-2)$ and $p_{2}=p/2$ yields

$\displaystyle\lVert R^{h}_{t}-R^{h,N}_{t}\rVert_{L^{2}}^{p}$
$\displaystyle\lesssim N^{-\alpha p}\lVert
u_{0}\rVert_{H^{\alpha}}+\int_{0}^{t}\lVert
R^{h}_{s}-R^{h,N}_{s}\rVert_{L^{2}}^{p}ds$ $\displaystyle\qquad+(1+\lVert
X^{h}\rVert_{C_{T}L^{\infty}}^{2\tilde{m}}+\lVert
X^{h,N}\rVert_{C_{T}L^{\infty}}^{2\tilde{m}})^{p}\lVert
O-O^{N}\rVert_{C_{T}L^{\infty}}^{p}$ $\displaystyle\qquad+N^{-p\alpha}\lVert
g_{h}(\tilde{X}^{h,N})\rVert_{C_{T}H^{\alpha}}^{p}.$

We choose $\alpha\in(0,1/2)$ and $\varepsilon^{\prime}\in(0,1/2-\alpha)$.
Recalling the embedding $\mathcal{C}^{\beta}\subset H^{\beta^{\prime}}$ for
$\beta>\beta^{\prime}$ and using the bound for $g^{\prime}_{h}$ from Lemma
5.1, we obtain that

$\displaystyle\lVert g_{h}(X^{h,N})\rVert_{C_{T}H^{\alpha}}\lesssim\lVert
g_{h}(X^{h,N})\rVert_{C_{T}\mathscr{C}^{\alpha+\varepsilon^{\prime}}}\lesssim(1+\lVert
X^{h,N}\rVert_{C_{T}L^{\infty}}^{2\tilde{m}})\lVert
X^{h,N}\rVert_{C_{T}\mathscr{C}^{\alpha+\varepsilon^{\prime}}}.$

Hence Grönwall’s inequality together with (4.4) and (6.2) yield

$\displaystyle\lVert X^{h}_{t}-X^{h,N}_{t}\rVert_{L^{2}}^{p}$
$\displaystyle\lesssim N^{-p\alpha}\lVert
u_{0}\rVert_{\mathscr{C}^{1/2}}+N^{-p\alpha}\lVert
O\rVert_{C_{T}\mathscr{C}^{\alpha+\varepsilon^{\prime}}}^{p}(1+\lVert
X^{h}\rVert_{C_{T}L^{\infty}}^{2\tilde{m}}+\lVert
X^{h,N}\rVert_{C_{T}L^{\infty}}^{2\tilde{m}})^{p}$
$\displaystyle\qquad+N^{-p\alpha}(1+\lVert
X^{h,N}\rVert_{C_{T}L^{\infty}}^{2\tilde{m}})^{p}\lVert
X^{h,N}\rVert_{C_{T}\mathscr{C}^{\alpha+\varepsilon^{\prime}}}^{p}.$

The claim thus follows from the moment bounds from Corollary 5.2 and the
arbitrariness of $\alpha\in(0,1/2)$.

∎

######  Lemma 6.3.

Let 2.3 hold and let $p\geq 1$. Then there exists a constant
$C=C(T,K,m,p,\mathcal{M})$ such that

$\displaystyle\mathds{E}\sup_{t\in[0,T]}\lVert
u_{t}-X^{h}_{t}\rVert_{L^{2}}^{p}\leq CM^{-p}.$

###### Proof.

We use Remark 6.1 with $F=u-X^{h}$, $G=f(u)-g_{h}(X^{h})$, and $H=0$. Using
the energy estimate and the bounds on $g_{h}$ and $g_{h}-g_{0}$ from Lemma
5.1, we obtain for $p\geq 2$, $t\in[0,T]$,

$\displaystyle\lVert u_{t}-X^{h}_{t}\rVert_{L^{2}}^{p}$ $\displaystyle\leq
p\int_{0}^{t}\lVert u_{s}-X^{h}_{s}\rVert_{L^{2}}^{p-2}\langle
u_{s}-X^{h}_{s},g_{h}(u_{s})-g_{h}(X^{h}_{s})\rangle ds$
$\displaystyle\qquad+p\int_{0}^{t}\lVert
u_{s}-X^{h}_{s}\rVert_{L^{2}}^{p-2}\langle
u_{s}-X^{h}_{s},g_{0}(u_{s})-g_{h}(u_{s})\rangle ds$
$\displaystyle\lesssim\int_{0}^{t}\lVert
u_{s}-X^{h}_{s}\rVert_{L^{2}}^{p}ds+h(1+\lVert
u\rVert_{C_{T}L^{\infty}}^{4m+2})\int_{0}^{t}\lVert
u_{s}-X^{h}_{s}\rVert_{L^{2}}^{p-1}ds.$

The claim thus follows from applying Young’s inequality with $p_{1}=p$ and
$p_{2}=p/(p-1)$ to the second term, using Grönwall’s inequality, taking
expectations, and using the a priori bounds from Proposition 3.3. ∎

######  Lemma 6.4.

Let 2.3 hold. Then for any $p\geq 1$, $\varepsilon\in(0,1/4)$, there exists a
constant $C=C(T,\varepsilon,p,K,\mathcal{M})$ such that the following bound
holds

$\displaystyle\mathds{E}\sup_{t\in[0,T]}\lVert
X^{h,N}_{t}-X^{M,N}_{t}\rVert_{L^{2}}^{p}\leq
C\big{(}M^{p(-1+\varepsilon)}+N^{p(-1/2+\varepsilon)}\big{)}.$

To prove the lemma, a key estimate is the following analogue of Proposition
4.5.

######  Proposition 6.5.

Let 2.3 hold. Then for any $p\geq 1$, $\varepsilon\in(0,1/4)$, there exists a
constant $C=C(T,\varepsilon,p,K,\mathcal{M})$ such that the following bound
holds

$\displaystyle\mathds{E}\sup_{t\in[0,T]}\bigg{\lVert}\int_{0}^{t}P^{N}_{t-s}[g_{h}(X^{M,N}_{s})-g_{h}(X^{M,N}_{k_{M}(s)})]ds\bigg{\rVert}_{L^{\infty}}^{p}\leq
CM^{p(-1+\varepsilon)}.$

We first give the proof of Lemma 6.4 using Proposition 6.5.

###### Proof of Lemma 6.4.

Writing $X^{h,N}=R^{h,N}+O^{N}$ and $X^{M,N}=R^{M,N}+O^{N}$, we have that

(6.3) $\displaystyle X^{h,N}_{t}-X^{M,N}_{t}=R^{h,N}_{t}-R^{M,N}_{t}$
$\displaystyle=\int_{0}^{t}P^{N}_{t-s}g_{h}(X^{h,N}_{s})ds-\int_{0}^{t}P^{N}_{t-k_{M}(s)}g_{h}(X^{M,N}_{k_{M}(s)})ds$
(6.4)
$\displaystyle=\int_{0}^{t}P^{N}_{t-s}[g_{h}(X^{h,N}_{s})-g_{h}(X^{M,N}_{s})]ds$
(6.5)
$\displaystyle\quad+\int_{0}^{t}P^{N}_{t-s}[g_{h}(X^{M,N}_{s})-g_{h}(X^{M,N}_{k_{M}(s)})]ds$
(6.6)
$\displaystyle\quad-\int_{0}^{t}[P^{N}_{t-k_{M}(s)}-P_{t-s}^{N}]g_{h}(X^{M,N}_{k_{M}(s)})ds$

The last term (6.6) is easy to bound with the semigroup estimates, using also
the uniform growth bound on $g_{h}$ from Lemma 5.1:

$\displaystyle\bigg{\lVert}\int_{0}^{t}[P^{N}_{t-k_{M}(s)}-P_{t-s}^{N}]g_{h}(X^{M,N}_{k_{M}(s)})ds\bigg{\rVert}_{L^{\infty}}$
$\displaystyle=\bigg{\lVert}\int_{0}^{t}P^{N}_{t-s}(P_{s-k_{M}(s)}^{N}-\operatorname{Id})g_{h}(X^{M,N}_{k_{M}(s)})ds\bigg{\rVert}_{L^{\infty}}$
$\displaystyle\lesssim\int_{0}^{t}(s-k_{M}(s))^{1-\varepsilon}(t-s)^{-1+\varepsilon}\lVert
g_{h}(X^{M,N}_{k_{M}(s)})\rVert_{L^{\infty}}ds$ (6.7) $\displaystyle\lesssim
M^{-1+\varepsilon}(1+\lVert X^{M,N}\rVert_{C_{T}L^{\infty}}^{2\tilde{m}+1}).$

The first term (6.4) is a buckling term. However, due to the growth of the
Lipschitz constant of $g_{h}$, we cannot apply Grönwall’s inequality in the
mild form and we switch again to the variational formulation. Define

$\displaystyle\bar{R}^{M,N}_{t}=\int_{0}^{t}P^{N}_{t-s}g_{h}(X^{M,N}_{s})ds$

and use Remark 6.1 with $F=R^{h,N}-\bar{R}^{M,N}$,
$G=g_{h}(X^{h,N})-g_{h}(X^{M,N})$, $H=0$. From the energy inequality and
adding and subtracting appropriate terms we get for any $p\geq 2$

$\displaystyle\lVert R^{h,N}_{t}-\bar{R}^{M,N}_{t}\rVert_{L^{2}}^{p}$
$\displaystyle\leq p\int_{0}^{t}\lVert
R^{h,N}_{s}-\bar{R}^{M,N}_{s}\rVert_{L^{2}}^{p-2}\langle
g_{h}(X^{h,N}_{s})-g_{h}(X^{M,N}_{s}),R^{h,N}_{s}-R^{M,N}_{s}\rangle ds$
$\displaystyle\quad+p\int_{0}^{t}\lVert
R^{h,N}_{s}-\bar{R}^{M,N}_{s}\rVert_{L^{2}}^{p-2}\langle
g_{h}(X^{h,N}_{s})-g_{h}(X^{M,N}_{s}),R^{M,N}_{s}-\bar{R}^{M,N}_{s}\rangle ds$
(6.8) $\displaystyle\quad+\mbox{$\displaystyle p\int_{0}^{t}\lVert
R^{h,N}_{s}-\bar{R}^{M,N}_{s}\rVert_{L^{2}}^{p-2}\langle(\Pi_{N}-\operatorname{Id})(g_{h}(X^{h,N}_{s})-g_{h}(X^{M,N}_{s})),R^{h,N}_{s}-\bar{R}^{M,N}_{s}\rangle
ds.$}$

Keeping in mind that $X^{h,N}-X^{M,N}=R^{h,N}-R^{M,N}$, from the bounds for
$g_{h}$ from Lemma 5.1 one therefore has

$\displaystyle\lVert R^{h,N}_{t}-\bar{R}^{M,N}_{t}\rVert_{L^{2}}^{p}$
$\displaystyle\lesssim\int_{0}^{t}\lVert
R^{h,N}_{s}-\bar{R}^{M,N}_{s}\rVert_{L^{2}}^{p-2}\lVert
R^{h,N}_{s}-R^{M,N}_{s}\rVert_{L^{2}}^{2}ds$ $\displaystyle\quad+(1+\lVert
X^{h,N}\rVert_{C_{T}L^{\infty}}^{2\tilde{m}}+\lVert
X^{M,N}\rVert_{C_{T}L^{\infty}}^{2\tilde{m}})$ $\displaystyle\hskip
56.9055pt\times\int_{0}^{t}\lVert
R^{h,N}_{s}-\bar{R}^{M,N}_{s}\rVert_{L^{2}}^{p-2}\lVert
R^{h,N}_{s}-R^{M,N}_{s}\rVert_{L^{2}}\lVert
R^{M,N}_{s}-\bar{R}^{M,N}_{s}\rVert_{L^{2}}ds$
$\displaystyle\quad+\int_{0}^{t}\lVert
R^{h,N}_{s}-\bar{R}^{M,N}_{s}\rVert_{L^{2}}^{p-1}\lVert(\Pi_{N}-\operatorname{Id})(g_{h}(X^{h,N})-g_{h}(X^{M,N}_{s}))\rVert_{L^{2}}ds.$

Utilising Young’s inequality to separate all terms, we get

$\displaystyle\lVert R^{h,N}_{t}-\bar{R}^{M,N}_{t}\rVert_{L^{2}}^{p}$
$\displaystyle\lesssim\int_{0}^{t}\lVert
R^{h,N}_{s}-\bar{R}^{M,N}_{s}\rVert_{L^{2}}^{p}ds+\int_{0}^{t}\lVert
R^{h,N}_{s}-R^{M,N}_{s}\rVert_{L^{2}}^{p}ds$ $\displaystyle\quad+[1+\lVert
X^{M,N}\rVert_{C_{T}L^{\infty}}^{2\tilde{m}}+\lVert
X^{h,M}\rVert_{C_{T}L^{\infty}}^{2\tilde{m}}]^{p}\int_{0}^{t}\lVert
R^{M,N}_{s}-\bar{R}^{M,N}_{s}\rVert_{L^{\infty}}^{p}ds$
$\displaystyle\quad+\int_{0}^{t}\lVert(\Pi_{N}-\operatorname{Id})(g_{h}(X^{h,N})-g_{h}(X^{M,N}_{s}))\rVert_{L^{2}}^{p}ds.$

The very last term is bounded exactly as in the proof of Lemma 6.2. Take
$\alpha\in(0,1/2)$ and $\varepsilon^{\prime}\in(0,1/2-\alpha)$ as therein.
Doing so and applying Grönwall’s lemma thus yields that

$\displaystyle\lVert R^{h,N}_{t}-\bar{R}^{M,N}_{t}\rVert_{L^{2}}^{p}$
$\displaystyle\lesssim\int_{0}^{t}\lVert
R^{h,N}_{s}-R^{M,N}_{s}\rVert_{L^{2}}^{p}ds$ $\displaystyle\qquad+[1+\lVert
X^{M,N}\rVert_{C_{T}L^{\infty}}^{2\tilde{m}}+\lVert
X^{h,N}\rVert_{C_{T}L^{\infty}}^{2\tilde{m}}]^{p}\int_{0}^{t}\lVert
R^{M,N}_{s}-\bar{R}^{M,N}_{s}\rVert_{L^{\infty}}^{p}\,ds$
$\displaystyle\qquad+\mbox{$\displaystyle
N^{-\alpha}(1+\|X^{h,N}\|_{C_{T}L^{\infty}}^{2\tilde{m}}+\|X^{M,N}\|_{C_{T}L^{\infty}}^{2\tilde{m}})(\|X^{h,N}\|_{C_{T}\mathscr{C}^{\alpha+\varepsilon^{\prime}}}+\|X^{M,N}\|_{C_{T}\mathscr{C}^{\alpha+\varepsilon^{\prime}}}).$}$

Let us abbreviate by $K_{*}$ random variables such that all of their moments
are bounded by the a priori bounds Corollary 5.2 and Corollary 5.4. They may
change from line to line. So for example the above inequality can be written
as

(6.9) $\displaystyle\lVert
R^{h,N}_{t}-\bar{R}^{M,N}_{t}\rVert_{L^{2}}^{p}\lesssim\int_{0}^{t}\lVert
R^{h,N}_{s}-R^{M,N}_{s}\rVert_{L^{2}}^{p}ds+K_{*}\int_{0}^{t}\lVert
R^{M,N}_{s}-\bar{R}^{M,N}_{s}\rVert_{L^{\infty}}^{p}\,ds+K_{*}N^{-\alpha}.$

Notice that the error $R^{M,N}_{t}-\bar{R}^{M,N}_{t}$ can be decomposed into
the term in (6.5) and the term in (6.6). Thus, since (6.5) is precisely the
term that is bounded in Proposition 6.5 and (6.6) is bounded in (6.7), we have
that

(6.10) $\displaystyle\mathds{E}(E_{*})^{q}:=\mathds{E}\sup_{t\in[0,T]}\lVert
R^{M,N}_{t}-\bar{R}^{M,N}_{t}\rVert_{L^{\infty}}^{q}\lesssim
M^{q(-1+\varepsilon/2)}$

for any $q\geq 1$. Putting everything together: using (6.7) and (6.9) in (6.3)
and using (6.10) we get

$\displaystyle\|R^{h,N}_{t}-R^{M,N}_{t}\|_{L^{2}}^{p}\lesssim
K_{*}\big{(}M^{p(-1+\varepsilon)}+N^{-\alpha
p}+E_{*}^{p}\big{)}+\int_{0}^{t}\lVert
R^{h,N}_{s}-R^{M,N}_{s}\rVert_{L^{2}}^{p}ds.$

By Grönwall’s inequality we get

$\displaystyle\sup_{t\in[0,T]}\|R^{h,N}_{t}-R^{M,N}_{t}\|_{L^{2}}^{p}\lesssim
K_{*}\big{(}M^{p(-1+\varepsilon)}+N^{-\alpha p}+E_{*}^{p}\big{)},$

and taking expectations and applying Hölder’s inequality to handle
$\mathds{E}(K_{\ast}E_{\ast}^{p})$, we finally get

$\displaystyle\mathds{E}\sup_{t\in[0,T]}\|R^{h,N}_{t}-R^{M,N}_{t}\|_{L^{2}}^{p}\lesssim
M^{p(-1+\varepsilon)}+N^{-\alpha p}$

as claimed. ∎

###### Proof of Proposition 6.5.

The proof of the proposition relies on stochastic sewing in the form of Lemma
3.2, similarly to Proposition 4.5, however, starting with a somewhat more
complicated germ $A_{s,t}$. Nevertheless, some of the steps remain unchanged
from the proof of Proposition 4.5, in these cases we shall simply refer back.

Again due to an application of a version of Kolmogorov’s continuity theorem
stated in the proof of Corollary 4.7 and Besov embeddings, it suffices to
prove that for any $j\geq-1$, $x\in\mathbb{T}$, $0\leq s\leq t\leq R\leq T$,
one has

$\displaystyle\mathds{E}\Big{|}\int_{s}^{t}\Delta_{j}P^{N}_{R-r}[g_{h}(R^{M,N}_{r}+O_{r}^{N})-g_{h}(R^{M,N}_{k_{M}(r)}+O_{k_{M}(r)}^{N})]dr\Big{|}^{p}$
(6.11) $\displaystyle\leq C2^{-jp\varepsilon}M^{p(-1+2\varepsilon)}\lvert
t-s\rvert^{1/4+\varepsilon/2}.$

To prove (4.11), we consider $j\geq-1$, $x\in\mathbb{T}$, $R\leq T$ fixed and
apply Lemma 3.2 with the germ

$\displaystyle
A_{st}=\int_{s}^{t}\mathds{E}_{s}[\Delta_{j}P^{N}_{R-r}(g_{h}(\mathds{E}_{s}R^{M,N}_{r}+O_{r}^{N})-g_{h}(\mathds{E}_{s}R^{M,N}_{k_{M}(r)}+O_{k_{M}(r)}^{N}))](x)dr.$

We aim to prove that for all $0\leq s<u<t<R$ such that $|t-s|\leq|R-t|$, one
has

(6.12) $\displaystyle\|A_{ut}\|_{L^{p}(\Omega)}\lesssim
2^{-j\varepsilon}M^{-1+2\varepsilon}(R-t)^{-1/4-\varepsilon/2}(t-u)^{1/2+\varepsilon}$

and

(6.13) $\displaystyle\|\mathds{E}_{s}\delta
A_{s,u,t}\|_{L^{p}(\Omega)}\lesssim
2^{-j\varepsilon}M^{-1+2\varepsilon}(R-t)^{-1/4-\varepsilon/2}(t-s)^{1+\varepsilon}.$

To mimic the steps of the proof of Proposition 4.5, we recall some analogues
of the basic bounds used therein. Let us now use the shorthand
$\mathcal{R}_{u,r}=\mathds{E}_{u}R^{M,N}_{r}$.

Note that by Jensen’s inequality and Corollary 5.4 one has for any $q\geq 1$

(6.14)
$\displaystyle\big{(}\mathds{E}\lVert\mathcal{R}_{s,r}-\mathcal{R}_{s,k_{M}(r)}\rVert_{L^{\infty}}^{q}\big{)}^{1/q}$
$\displaystyle\leq\big{(}\mathds{E}\lVert
R^{M,N}_{r}-R^{M,N}_{k_{M}(r)}\rVert_{L^{\infty}}^{q}\big{)}^{1/q}\lesssim_{q}(r-k_{M}(r))^{1-\varepsilon}.$

This yields the analogue of (3.9) to bound the difference of the two arguments
of the nonlinearity:

(6.15)
$\displaystyle\big{(}\mathds{E}\lVert\mathcal{R}_{u,r}+O_{r}^{N}-(\mathcal{R}_{u,k_{M}(r)}+O^{N}_{k_{M}(r)})\rVert_{\mathscr{C}^{-1/2+\varepsilon}}^{q}\big{)}^{1/q}\lesssim_{q}(r-k_{M}(r))^{1/2-\varepsilon}.$

Similarly, one has the bound, if $k_{M}(r)>u$,

$\displaystyle\big{(}\mathds{E}\lVert\mathcal{R}_{u,r}+P_{r-u}O^{N}_{u}-(\mathcal{R}_{u,k_{M}(r)}+P_{k_{M}(r)-u}O^{N}_{u})\rVert_{\mathscr{C}^{-1/2+\varepsilon}}^{q}\big{)}^{1/q}$
(6.16)
$\displaystyle\lesssim_{q}(r-k_{M}(r))^{1-2\varepsilon}(k_{M}(r)-u)^{-1/2+\varepsilon},$

where we employed (6.14) and (3.10) with $\theta=1-4\varepsilon$, $r-k_{M}(r)$
in place of $s$, and $k_{M}(r)-u$ in place of $t$. The bound (4.15) is
replaced by

$\displaystyle\mathds{E}\sup_{r\in[0,R]}\lVert
g_{h}^{\prime}(\lambda(\mathcal{R}_{u,r}+O^{N}_{r})+(1-\lambda)(\mathcal{R}_{u,k_{M}(r)}+O^{N}_{k_{M}(r)}))\rVert_{\mathscr{C}^{\theta}}^{q}$
(6.17) $\displaystyle\lesssim\mathds{E}\big{(}(1+\lVert
X^{M,N}\rVert_{C_{T}L^{\infty}}^{2\tilde{m}-1})\lVert
X^{M,N}\rVert_{C_{T}\mathscr{C}^{\theta}}\big{)}^{q}\lesssim_{q,\theta}1.$

for $\theta\in(0,1/2)$, which follows using the identification of the Besov
space $\mathscr{C}^{\theta}$ with the space of $\theta$-Hölder continuous
functions and the bound for $g_{h}^{\prime\prime}$ from Lemma 5.1.

To prove (6.12), we distinguish $\lvert t-u\rvert\leq 3M^{-1}$ and $\lvert
t-u\rvert>3M^{-1}$ as before. In the case of $\lvert t-u\rvert\leq 3M^{-1}$,
we estimate similar as in (4.16) replacing $\tilde{O}_{\cdot}$ by
$\mathcal{R}_{u,\cdot}+O_{\cdot}^{N}$ and use (6.15) and (6) to obtain the
desired estimate.

In the case of $\lvert t-u\rvert>3M^{-1}$, we introduce $t^{\prime}$ as before
and decompose the time integral $\int_{u}^{t}$ into $\int_{u}^{t^{\prime}}$
and $\int_{t^{\prime}}^{t}$, where $\int_{u}^{t^{\prime}}$ is treated as in
the case before and $\int_{t^{\prime}}^{t}$ is left to bound. Recall that for
$r\geq t^{\prime}$ we have $k_{M}(r)-u\geq(r-u)/2$ and that
$\mathcal{R}_{u,r}$ and $\mathcal{R}_{u,k_{M}(r)}$ are
$\mathcal{F}_{u}$-measurable. Thus we have, similarly to (4.17) that the time
integral from $t^{\prime}$ to $t$ equals

$\displaystyle\int_{t^{\prime}}^{t}\Delta_{j}P_{R-u}\mathds{E}_{u}[(g_{h}(\mathcal{R}_{u,r}+O^{N}_{r})-g_{h}(\mathcal{R}_{u,k_{M}(r)}+O^{N}_{k_{M}(r)}))]dr$
$\displaystyle=\int_{u}^{t}\Delta_{j}P_{R-u}(P^{\mathbb{R}}_{Q^{N}(r-u)}g_{h})(\mathcal{R}_{u,r}+P_{r-u}O^{N}_{u})$
$\displaystyle\hskip
142.26378pt-(P^{\mathbb{R}}_{Q^{N}(k_{M}(r)-u)}g_{h})(\mathcal{R}_{u,k_{M}(r)}+P_{k_{M}(r)-u}O^{N}_{u})dr$
$\displaystyle=\int_{t^{\prime}}^{t}\Delta_{j}P_{R-r}^{N}[(P^{\mathbb{R}}_{Q^{N}(r-u)}g_{h})(\mathcal{R}_{u,r}+P_{r-u}O^{N}_{u})$
$\displaystyle\hskip
142.26378pt-(P^{\mathbb{R}}_{Q^{N}(r-u)}g_{h})(\mathcal{R}_{u,k_{M}(r)}+P_{k_{M}(r)-u}O^{N}_{u})](x)dr$
$\displaystyle\quad+\mbox{$\displaystyle\int_{t^{\prime}}^{t}\Delta_{j}P_{R-r}^{N}[[(P^{\mathbb{R}}_{Q^{N}(r-u)}g_{h})-(P^{\mathbb{R}}_{Q^{N}(k_{M}(r)-u)}g_{h})](\mathcal{R}_{u,k_{M}(r)}+P_{k_{M}(r)-u}O^{N}_{u})](x)dr$},$
$\displaystyle=:I_{1}+I_{2},$

where

$\displaystyle
Q^{N}(t-s)=\mathds{E}\bigg{(}\int_{s}^{t}\int_{\mathbb{T}}p^{N}_{t-r}(x-y)\xi(dr,dy)\bigg{)}^{2}.$

We are left to bound the two terms $I_{1}$ and $I_{2}$. To bound the term
$I_{1}$ we proceed just as in (4.21), but using (6) instead of (3.10) and the
bound

$\displaystyle\mathds{E}\lVert(P^{\mathbb{R}}_{Q^{N}(r-u)}g_{h})^{\prime}(\lambda(P_{r-u}O^{N}_{u}+\mathcal{R}_{u,r})+(1-\lambda)(P_{k_{M}(r)-u}O^{N}_{u}+\mathcal{R}_{u,k_{M}(r)}))\rVert_{\mathscr{C}^{\theta}}^{q}\lesssim_{\theta,q}1$

for $\theta\in(0,1/2)$, $q\geq 1$, in place of (4.20). This gives
(E—I_1—^p)^1/p≲2^-jεM^-1+2ε (R-t)^-1/4-ε/2(t-u)^1/2+ε. To bound $I_{2}$ we
replace the bound (4) by the following:

$\displaystyle\mathds{E}$
$\displaystyle\bigg{\lvert}\int_{t^{\prime}}^{t}\Delta_{j}P_{R-r}^{N}[[(P^{\mathbb{R}}_{Q^{N}(r-u)}g_{h})-(P^{\mathbb{R}}_{Q^{N}(k_{M}(r)-u)}g_{h})](\mathcal{R}_{u,k_{M}(r)}+P_{k_{M}(r)-u}O^{N}_{u})](x)dr\bigg{\rvert}^{p}$
$\displaystyle\lesssim 2^{-j\varepsilon
p}\mathds{E}\bigg{(}\int_{t^{\prime}}^{t}(R-r)^{-\varepsilon/2}$
$\displaystyle\quad\qquad\qquad\times\mbox{$\displaystyle\lVert[(P^{\mathbb{R}}_{Q^{N}(r-u)}g_{h})-(P^{\mathbb{R}}_{Q^{N}(k_{M}(r)-u)}g_{h})](\mathcal{R}_{u,k_{M}(r)}+P_{k_{M}(r)-u}O^{N}_{u})\rVert_{L^{\infty}}dr$}\bigg{)}^{p}$
$\displaystyle\lesssim 2^{-j\varepsilon
p}\mathds{E}\bigg{(}\int_{t^{\prime}}^{t}(R-r)^{-\varepsilon/2}\lVert(P^{\mathbb{R}}_{Q^{N}(r-u)}g_{h})-(P^{\mathbb{R}}_{Q^{N}(k_{M}(r)-u)}g_{h})\rVert_{C^{0}_{\omega}}$
$\displaystyle\hskip
184.9429pt\times\lVert\omega^{-1}(\mathcal{R}_{u,k_{M}(r)}+P_{k_{M}(r)-u}O^{N}_{u})\rVert_{L^{\infty}}dr\bigg{)}^{p}.$

Then we further estimate using (6.1) and (4) (which also holds true for
$Q^{N}$ instead of $Q$),

$\displaystyle\lVert(P^{\mathbb{R}}_{Q^{N}(r-u)}g_{h})-(P^{\mathbb{R}}_{Q^{N}(k_{M}(r)-u)}g_{h})\rVert_{C^{0}_{\omega}}$
$\displaystyle\lesssim(Q^{N}(r-u)-Q^{N}(k_{M}(r)-u))\lVert
g_{h}\rVert_{C^{2}_{\omega}}$
$\displaystyle\lesssim(k_{M}(r)-u)^{-1/2+\varepsilon}(r-k_{M}(r))^{1-\varepsilon}\lVert
g_{h}\rVert_{C^{2}_{\omega}}$

and

$\displaystyle\lVert\omega^{-1}(\mathcal{R}_{u,k_{M}(r)}+P_{k_{M}(r)-u}O^{N}_{u})\rVert_{L^{\infty}}\lesssim(1+\lVert
O^{N}\rVert_{C_{T}L^{\infty}}^{2}+\lVert
R^{M,N}\rVert_{C_{T}L^{\infty}}^{2})^{\beta/2}.$

Together with the a priori bounds from Corollary 5.4, we get
(E—I_2—^p)^1/p≲2^-jεM^-1+ε (R-t)^-ε/2(t-u)^1/2+ε. The proof of (6.12) is
therefore finished.

It remains to prove (6.13). Before proceeding, we derive some bounds on
$\mathcal{R}_{\cdot,\cdot}$. Recall that for any norm $|\cdot|$, any $q\geq
1$, and any random variables $X,Y$, where $Y$ is $\mathcal{F}_{s}$-measurable,
one has $(\mathds{E}|X-\mathds{E}_{s}X|^{q})^{1/q}\leq
2(\mathds{E}|X-Y|^{q})^{1/q}$. Therefore we can estimate, using Corollary 5.4,

(6.18) $\displaystyle\big{(}\mathds{E}\lVert
R^{M,N}_{r}-\mathcal{R}_{s,r}\rVert_{L^{\infty}}^{q}\big{)}^{1/q}$
$\displaystyle\leq 2\big{(}\mathds{E}\lVert
R^{M,N}_{r}-R^{M,N}_{s}\rVert_{L^{\infty}}^{q}\big{)}^{1/q}\lesssim_{q}\lvert
r-s\rvert^{1-\varepsilon}.$

As a consequence, by the triangle inequality, for $s\leq u$

(6.19)
$\displaystyle(\mathds{E}\lVert\mathcal{R}_{s,r}-\mathcal{R}_{u,r}\rVert_{L^{\infty}}^{q})^{1/q}\lesssim_{q}|r-s|^{1-\varepsilon}.$

Similarly, we can bound for $s\leq u$

(6.20)
$\displaystyle(\mathds{E}\lVert\mathcal{R}_{s,r}-\mathcal{R}_{u,r}\rVert_{\mathscr{C}^{1/2-\varepsilon/2}}^{q})^{1/q}\lesssim_{q}\lvert
r-s\rvert^{3(1-\varepsilon)/4},$

using again the a priori estimates, Corollary 5.4 with
$\alpha=1/2-\varepsilon$. Next, we claim that333Notice that simply
substituting $k_{M}(r)$ in place of $r$ in (6.19) does not give the required
order, since one may have $|k_{M}(r)-s|\gg|r-s|$.

(6.21) $\displaystyle\mathds{E}\big{(}\lVert
R^{M,N}_{k_{M}(r)}-\mathcal{R}_{s,k_{M}(r)}\rVert_{L^{\infty}}^{q}\big{)}^{1/q}\lesssim_{q}\lvert
r-s\rvert^{1-\varepsilon}.$

Indeed, to see this, we make the following case separation as in [BDG22, Proof
of Lemma 4.7]: If $s\geq k_{M}(r)-M^{-1}$ then
$\mathcal{R}_{s,k_{M}(r)}=\mathds{E}_{s}R^{M,N}_{k_{M}(r)}=R^{M,N}_{k_{M}(r)}$
and the left-hand side vanishes, such that the bound is trivial. If $s\leq
k_{M}(r)-M^{-1}$, then $|k_{M}(r)-s|\leq|r-s|$ and the bound follows from
(6.18) plugging $k_{M}(r)$ in place of $r$.
Moreover, using triangle inequality and (6.21) we obtain for $s\leq u$,

(6.22)
$\displaystyle(\mathds{E}\lVert\mathcal{R}_{s,k_{M}(r)}-\mathcal{R}_{u,k_{M}(r)}\rVert_{L^{\infty}}^{q})^{1/q}\lesssim_{q}|r-s|^{1-\varepsilon}.$

Now we move on to prove (6.13), for which we have to bound the term

$\displaystyle\mathds{E}_{s}\delta
A_{sut}=\Delta_{j}\int_{u}^{t}\mathds{E}_{s}\mathds{E}_{u}P^{N}_{R-r}\big{(}$
$\displaystyle
g_{h}(\mathcal{R}_{s,r}+O^{N}_{r})-g_{h}(\mathcal{R}_{s,k_{M}(r)}+O^{N}_{k_{M}(r)})$
$\displaystyle-(g_{h}(\mathcal{R}_{u,r}+O^{N}_{r})-g_{h}(\mathcal{R}_{u,k_{M}(r)}+O^{N}_{k_{M}(r)}))\big{)}(x)dr.$

Let us write

$\displaystyle
g_{h}(\mathcal{R}_{s,r}+O^{N}_{r})-g_{h}(\mathcal{R}_{s,k_{M}(r)}+O^{N}_{k_{M}(r)})-(g_{h}(\mathcal{R}_{u,r}+O^{N}_{r})-g_{h}(\mathcal{R}_{u,k_{M}(r)}+O^{N}_{k_{M}(r)}))$
$\displaystyle=(\mathcal{R}_{s,r}-\mathcal{R}_{u,r})\int_{0}^{1}g^{\prime}_{h}(\lambda(\mathcal{R}_{s,r}+O^{N}_{r})+(1-\lambda)(\mathcal{R}_{u,r}+O^{N}_{r}))d\lambda$
$\displaystyle\quad+(\mathcal{R}_{s,k_{M}(r)}-\mathcal{R}_{u,k_{M}(r)})$
$\displaystyle\hskip
85.35826pt\times\mbox{$\displaystyle\int_{0}^{1}g^{\prime}_{h}(\lambda(\mathcal{R}_{s,k_{M}(r)}+O^{N}_{k_{M}(r)})+(1-\lambda)(\mathcal{R}_{u,k_{M}(r)}+O^{N}_{k_{M}(r)}))d\lambda$}$
$\displaystyle=(\mathcal{R}_{s,r}-\mathcal{R}_{u,r})\int_{0}^{1}\bigg{(}g^{\prime}_{h}(\lambda(\mathcal{R}_{s,r}+O^{N}_{r})+(1-\lambda)(\mathcal{R}_{u,r}+O^{N}_{r}))$
$\displaystyle\hskip 99.58464pt-\mbox{$\displaystyle
g^{\prime}_{h}(\lambda(\mathcal{R}_{s,k_{M}(r)}+O^{N}_{k_{M}(r)})+(1-\lambda)(\mathcal{R}_{u,k_{M}(r)}+O^{N}_{k_{M}(r)}))$}d\lambda\bigg{)}$
$\displaystyle\quad+(\mathcal{R}_{s,k_{M}(r)}-\mathcal{R}_{s,r}+\mathcal{R}_{u,r}-\mathcal{R}_{u,k_{M}(r)})$
$\displaystyle\hskip
85.35826pt\times\mbox{$\displaystyle\int_{0}^{1}g^{\prime}_{h}(\lambda(\mathcal{R}_{s,k_{M}(r)}+O^{N}_{k_{M}(r)})+(1-\lambda)(\mathcal{R}_{u,k_{M}(r)}+O^{N}_{k_{M}(r)}))d\lambda$}$
$\displaystyle=:J_{1}+J_{2}.$

By the bounds for $g_{h}$ from Lemma 5.1, as well as using (6.14) and (6.19),
(6.22), we obtain for $J_{2}$ the following estimate

$\displaystyle\bigg{(}\mathds{E}\bigg{\lvert}\Delta_{j}\int_{u}^{t}\mathds{E}_{s}[P^{N}_{R-r}\mathds{E}_{u}[J_{2}]](x)dr\bigg{\rvert}^{p}\bigg{)}^{1/p}$
$\displaystyle\lesssim
2^{-j\varepsilon}(R-t)^{-\varepsilon/2}(1+\mathds{E}\lVert
R^{M,N}\rVert_{C_{T}L^{\infty}}^{2\tilde{m}p}+\mathds{E}\lVert
O^{N}\rVert_{C_{T}L^{\infty}}^{2\tilde{m}p})^{1/2p}$ $\displaystyle\hskip
85.35826pt\times\bigg{(}\mathds{E}\bigg{(}\int_{u}^{t}\lVert\mathcal{R}_{s,k_{M}(r)}-\mathcal{R}_{s,r}+\mathcal{R}_{u,r}-\mathcal{R}_{u,k_{M}(r)}\rVert_{L^{\infty}}dr\bigg{)}^{2p}\bigg{)}^{1/2p}$
$\displaystyle\lesssim
2^{-j\varepsilon}(R-t)^{-\varepsilon/2}\int_{u}^{t}\min((r-k_{M}(r))^{1-\varepsilon/2},(r-s)^{1-\varepsilon/2})dr$
$\displaystyle\lesssim
2^{-j\varepsilon}(R-t)^{-\varepsilon/2}(t-s)^{1+\varepsilon/2}M^{-1+\varepsilon},$

where the last bound follows by the interpolation $\min(a,b)\leq
a^{1-\theta}b^{\theta}$ for $\theta=\varepsilon/(1-\varepsilon)\in(0,1)$,
$a,b\geq 0$.

To estimate $J_{1}$ we distinguish again the cases $\lvert t-u\rvert\leq
3M^{-1}$ and $\lvert t-u\rvert>3M^{-1}$ like for $A_{ut}$. In the first case,
the estimate is carried out as in (4.21) using (6.15) in place of (3.9).
Notice that (6.20) gives an additional factor $(r-s)^{3(1-\varepsilon)/4}$,
and thus we obtain that in the case of $\lvert t-u\rvert\leq 3M^{-1}$, (E|Δ_j
∫_u^tE_s[P^N_R-rE_u[J_1]](x)dr|^p)^1/p≲2^-jεM^-1+ε (R-t)^-ε/2(t-u)^5/4+ε/4. If
$\lvert t-u\rvert>3M^{-1}$, we introduce $t^{\prime}$ as before, in particular
$k_{M}(r)\geq u\geq s$. The integral from $u$ to $t^{\prime}$ is treated as in
the case $|t-u|\leq 3M^{-1}$. For the integral from $t^{\prime}$ to $t$ we
have that, using that the factor $\mathcal{R}_{s,r}-\mathcal{R}_{u,r}$ is
$\mathcal{F}_{u}$-measurable,

$\displaystyle\bigg{(}\mathds{E}\bigg{\lvert}\Delta_{j}\int_{t^{\prime}}^{t}\mathds{E}_{s}[P^{N}_{R-r}\mathds{E}_{u}[J_{1}]](x)dr\bigg{\rvert}^{p}\bigg{)}^{1/p}$
$\displaystyle\lesssim
2^{-j\varepsilon}(R-t)^{-1/4-\varepsilon/2}\bigg{(}\mathds{E}\int_{t^{\prime}}^{t}\lVert\mathcal{R}_{s,r}-\mathcal{R}_{u,r}\rVert_{\mathscr{C}^{1/2-\varepsilon/2}}^{p}$
$\displaystyle\quad\times\int_{0}^{1}\Big{\lVert}(P^{\mathbb{R}}_{Q^{N}(r-u)}g_{h}^{\prime})(\lambda\mathcal{R}_{s,r}+(1-\lambda)\mathcal{R}_{u,r}+P_{r-u}O^{N}_{u})$
$\displaystyle\qquad\quad\mbox{$\displaystyle-(P^{\mathbb{R}}_{Q^{N}(k_{M}(r)-u)}g_{h}^{\prime})(\lambda\mathcal{R}_{s,k_{M}(r)}+(1-\lambda)\mathcal{R}_{u,k_{M}(r)}+P_{k_{M}(r)-u}O^{N}_{u})$}_{\mathscr{C}^{-1/2+\varepsilon}}^{p}d\lambda
dr\Big{\rVert}^{1/p}.\bigg{)}$

Using similar steps as for estimating the time integral from $t^{\prime}$ to
$t$ above, now applied for $g_{h}^{\prime}\in C^{2}_{\omega}$ instead of
$g_{h}$, to bound

$\displaystyle\bigg{(}\mathds{E}\int_{t^{\prime}}^{t}\lVert(P^{\mathbb{R}}_{Q^{N}(r-u)}g_{h}^{\prime})(\lambda\mathcal{R}_{s,r}+(1-\lambda)\mathcal{R}_{u,r}+P_{r-u}O^{N}_{u})$
$\displaystyle\hskip
17.07182pt\mbox{$\displaystyle-(P^{\mathbb{R}}_{Q^{N}(k_{M}(r)-u)}g_{h}^{\prime})(\lambda\mathcal{R}_{s,k_{M}(r)}+(1-\lambda)\mathcal{R}_{u,k_{M}(r)}+P_{k_{M}(r)-u}O^{N}_{u})$}\rVert_{\mathscr{C}^{-1/2+\varepsilon}}^{2p}dr\bigg{)}^{1/2p}$

for any fixed $\lambda\in[0,1]$, as well as (6.20) and (6), we arrive at

$\displaystyle\bigg{(}\mathds{E}\bigg{\lvert}\Delta_{j}\int_{t^{\prime}}^{t}\mathds{E}_{s}[P^{N}_{R-r}\mathds{E}_{u}[J_{1}]](x)dr\bigg{\rvert}^{p}\bigg{)}^{1/p}$
$\displaystyle\lesssim
2^{-j\varepsilon}(t-s)^{3(1-\varepsilon)/4}(R-t)^{-1/4-\varepsilon/2}\int_{t^{\prime}}^{t}(r-k_{M}(r))^{1-2\varepsilon}(k_{M}(r)-u)^{-1/2+\varepsilon}dr$
$\displaystyle\lesssim
2^{-j\varepsilon}(t-s)^{5/4+\varepsilon/4}(R-t)^{-1/4-\varepsilon/2}M^{-1+2\varepsilon}.$

The estimates for $J_{1}$ and $J_{2}$ together yield (6.13). Hence an
application of Lemma 3.2 together with (6.12) and (6.13) yields (6), provided
we justify that

(6.23)
$\displaystyle\mathcal{A}_{t}=\int_{0}^{t}\Delta_{j}P^{N}_{R-r}[g_{h}(R^{M,N}_{r}+O_{r}^{N})-g_{h}(R^{M,N}_{k_{M}(r)}+O_{k_{M}(r)}^{N})](x)dr.$

To this end, we need to verify (3.15) and (3.16). We do so in two steps: we
show the bounds with $\mathcal{A}_{st}-A_{st}$ replaced by
$\mathcal{A}_{st}-\tilde{A}_{st}$ and by $\tilde{A}_{st}-A_{st}$, where the
intermediate process $\tilde{A}_{st}$ is defined by

$\displaystyle\tilde{A}_{s,t}=\int_{s}^{t}\mathds{E}_{s}[\Delta_{j}P^{N}_{R-r}(g_{h}(R^{M,N}_{r}+O^{N}_{r})-g_{h}(R^{M,N}_{k_{M}(r)}+O^{N}_{k_{M}(r)}))](x)dr.$

The difference $\mathcal{A}_{st}-\tilde{A}_{st}$ is treated similarly to the
argument concluding the proof of Proposition 4.5: (3.16) is satisfied with
$K_{2}=0$, and

$\displaystyle\big{(}\mathds{E}\big{\lvert}\mathcal{A}_{st}-\tilde{A}_{st}\big{\rvert}^{p}\big{)}^{1/p}\leq
4(t-s)(1+\mathds{E}\lVert
X^{M,N}\rVert_{C_{T}L^{\infty}}^{p(2\tilde{m}+1)})^{1/p},$

verifying (3.15). The verification of (3.15) for $\tilde{A}_{st}-A_{st}$ is
identical. As for (3.16), we clearly have

$\displaystyle(\mathds{E}\lvert A_{st}-\tilde{A}_{st}\rvert^{p})^{1/p}$
$\displaystyle\lesssim(t-s)(1+\mathds{E}\lVert
X^{M,N}\rVert_{C_{T}L^{\infty}}^{2p(2\tilde{m}+1)})^{1/2p}\ $
$\displaystyle\times\sup_{r\in[s,t]}\Big{(}\big{(}\mathds{E}\lVert
R^{M,N}_{r}-\mathcal{R}_{s,r}\rVert_{L^{\infty}}^{2p}\big{)}^{1/2p}+\big{(}\mathds{E}\lVert
R^{M,N}_{k_{M}(r)}-\mathcal{R}_{s,k_{M}(r)}\rVert_{L^{\infty}}^{2p}\big{)}^{1/2p}\Big{)}$
$\displaystyle\lesssim(t-s)^{2-\varepsilon}$

where we used (6.18) and (6.21). This verifies (3.16) for
$\tilde{A}_{st}-A_{st}$, which proves (6.23) and brings the proof to an end. ∎

## Appendix A

###### Proof of Lemma 5.1.

We have that

$\displaystyle\Phi_{h}(z)$ $\displaystyle=z+\int_{0}^{h}f(\Phi_{s}(z))ds,\quad
h\geq 0$ $\displaystyle g_{h}(z)$
$\displaystyle=\frac{1}{h}\int_{0}^{h}f(\Phi_{s}(z))ds,\quad h>0.$

Thus, we obtain

$\displaystyle(\Phi_{h}(x)-\Phi_{h}(y))^{2}$
$\displaystyle=(x-y)^{2}+2\int_{0}^{h}(f(\Phi_{s}(x))-f(\Phi_{s}(y)))(\Phi_{s}(x)-\Phi_{s}(y))ds$
$\displaystyle\leq(x-y)^{2}+2K\int_{0}^{h}(\Phi_{s}(x)-\Phi_{s}(y))^{2}ds$

and hence with Grönwall’s inequality

$\displaystyle(\Phi_{h}(x)-\Phi_{h}(y))^{2}$ $\displaystyle\leq
e^{Kh}(x-y)^{2}.$

This implies that $z\mapsto\Phi_{h}(z)$ is Lipschitz continuous with Lipschitz
constant $L(h)=e^{Kh/2}$, that satisfies $L(h)^{1/h}=e^{K/2}$. Since
$\Phi_{h}$ is Lipschitz, we moreover obtain that

$\lvert\partial_{z}\Phi_{s}(z)\rvert\leq e^{Ks/2}\leq e^{Kh/2}$

for $s\in[0,h]$. Moreover we have that, using Young’s inequality

$\displaystyle\Phi_{h}(z)^{2}$
$\displaystyle=z^{2}+2\int_{0}^{h}f(\Phi_{s}(z))\Phi_{s}(z)ds$
$\displaystyle=z^{2}+2\int_{0}^{h}(f(\Phi_{s}(z))-f(0))\Phi_{s}(z)ds+2\int_{0}^{h}f(0)\Phi_{s}(z)ds$
$\displaystyle\leq
z^{2}+2K\int_{0}^{h}\Phi_{s}(z)^{2}ds+hf(0)^{2}+\int_{0}^{h}\Phi_{s}(z)^{2}ds,$

such that Grönwall’s inequality implies that

$\displaystyle\sup_{s\in[0,h]}\lvert\Phi_{s}(z)\rvert^{2}\leq
e^{(2K+1)h}(h\lvert f(0)\rvert^{2}+\lvert z\rvert^{2})$

and thus

(A.1) $\displaystyle\sup_{s\in[0,h]}\lvert\Phi_{s}(z)\rvert\leq
e^{(2K+1)h/2}(h^{1/2}\lvert f(0)\rvert+\lvert z\rvert)\leq
Ke^{(2K+1)h/2}(1+\lvert z\rvert),$

for $h\leq 1$ and using that $\lvert f(0)\rvert\leq K$. Furthermore, we have
that

$\displaystyle(\partial_{z}^{2}\Phi_{h}(z))^{2}$
$\displaystyle=1+2\int_{0}^{h}(\partial
f)(\Phi_{s}(z))(\partial_{z}^{2}\Phi_{s}(z))^{2}ds$
$\displaystyle\qquad+2\int_{0}^{h}(\partial^{2}f)(\Phi_{s}(z))(\partial_{z}\Phi_{s}(z))^{2}\partial_{z}^{2}\Phi_{s}(z)ds$

and thus, using Young’s inequality, the bounds on $f$ and (A.1), as well as
the Lipschitz bound of $\Phi_{s}$, there exist constants
$C^{\prime}(K,m),C(K,m)>1$, such that

$\displaystyle(\partial_{z}^{2}\Phi_{h}(z))^{2}$ $\displaystyle\leq
1+2K\int_{0}^{h}(\partial_{z}^{2}\Phi_{s}(z))^{2}ds+\int_{0}^{h}((\partial^{2}f)(\Phi_{s}(z))(\partial_{z}\Phi_{s}(z))^{2})^{2}ds$
$\displaystyle\qquad+\int_{0}^{h}(\partial_{z}^{2}\Phi_{s}(z))^{2}ds$
$\displaystyle\leq
1+(2K+1)\int_{0}^{h}(\partial_{z}^{2}\Phi_{s}(z))^{2}ds+hC^{\prime}(K,m)e^{hC(K,m)}(1+\lvert
z\rvert^{2(2m-1)}).$

Thus Grönwall’s inequality implies that for possibly different constants and
$h\leq 1$,

(A.2)
$\displaystyle\sup_{s\in[0,h]}\lvert\partial^{2}_{z}\Phi_{s}(z)\rvert\leq
C^{\prime}(K,m)e^{C(K,m)h}(1+\lvert z\rvert^{2m-1}).$

The bound for $\partial_{z}^{3}\Phi_{h}(z)$ follows in a similar fashion using
(A.2) and

$\displaystyle(\partial_{z}^{3}\Phi_{h}(z))^{2}$
$\displaystyle=1+2\int_{0}^{h}(\partial
f)(\Phi_{s}(z))(\partial_{z}^{3}\Phi_{s}(z))^{2}ds$
$\displaystyle\qquad+6\int_{0}^{h}(\partial^{2}f)(\Phi_{s}(z))\partial_{z}\Phi_{s}(z)\partial_{z}^{2}\Phi_{s}(z)\partial_{z}^{3}\Phi_{s}(z)ds$
$\displaystyle\qquad+2\int_{0}^{h}(\partial^{3}f)(\Phi_{s}(z))(\partial_{z}\Phi_{s}(z))^{3}\partial_{z}^{3}\Phi_{s}(z)ds$

and we obtain, for possibly different constants $C^{\prime}(K,m),C(K,m)$ and
$\tilde{m}\geq m$, $h\leq 1$,

(A.3)
$\displaystyle\sup_{s\in[0,h]}\lvert\partial^{3}_{z}\Phi_{s}(z)\rvert\leq
C^{\prime}(K,m)e^{C(K,m)h}(1+\lvert z\rvert^{2\tilde{m}-3}).$

We have that

$\displaystyle\partial_{z}g_{h}(z)$
$\displaystyle=\frac{1}{h}\int_{0}^{h}(\partial
f)(\Phi_{s}(z))\partial_{z}\Phi_{s}(z)ds,$
$\displaystyle\partial^{2}_{z}g_{h}(z)$
$\displaystyle=\frac{1}{h}\int_{0}^{h}(\partial^{2}f)(\Phi_{s}(z))(\partial_{z}\Phi_{s}(z))^{2}ds+\frac{1}{h}\int_{0}^{h}(\partial
f)(\Phi_{s}(z))\partial_{z}^{2}\Phi_{s}(z))ds,$
$\displaystyle\partial^{3}_{z}g_{h}(z)$
$\displaystyle=\frac{1}{h}\int_{0}^{h}(\partial^{3}f)(\Phi_{s}(z))(\partial_{z}\Phi_{s}(z))^{3}ds+3\frac{1}{h}\int_{0}^{h}(\partial^{2}f)(\Phi_{s}(z))(\partial_{z}^{2}\Phi_{s}(z))\partial_{z}\Phi_{s}(z)ds$
$\displaystyle\qquad+\frac{1}{h}\int_{0}^{h}(\partial
f)(\psi_{s}(z))\partial_{z}^{3}\Phi_{s}(z))ds.$

Together with (A.1), (A.2) and (A.3) we thus obtain

$\displaystyle\lvert g_{h}(z)\rvert\leq
K(1+\sup_{s\in[0,1]}\lvert\Phi_{s}(z)\rvert^{2m+1})\leq
K^{2}e^{(2K+1)h/2}(1+\lvert z\rvert^{2m+1})$

with $K^{2}e^{(2K+1)h/2}\leq K^{2}e^{(2K+1)/2}$ for $h\in[0,1]$ and

$\displaystyle\lvert\partial_{z}g_{h}(z)\rvert\leq
Ke^{Kh/2}(1+\sup_{s\in[0,1]}\lvert\Phi_{s}(z)\rvert^{2m})\leq
C^{\prime}(K,m)e^{hC(K,m)}(1+\lvert z\rvert^{2m}).$

Using (A.2), (A.3), we find for the higher order derivatives, $i=2,3$, that
there exists constants $\tilde{m}\geq m$, $K(h,m)>1$, such that

$\displaystyle\lvert\partial^{i}_{z}g_{h}(z)\rvert$ $\displaystyle\leq
K(h,m)(1+\lvert z\rvert^{2\tilde{m}-i}),$

where $K(h,m)\leq\tilde{K}$ for $h\in[0,1]$ for a constant $\tilde{K}>1$.
Moreover, since $\partial_{h}\partial_{z}\Phi_{h}(z)=(\partial
f)(\Phi_{h}(z))\partial_{z}\Phi_{h}(z)$ by Grönwall’s inequality and $\partial
f\leq K$, we have that

$\displaystyle\partial_{z}\Phi_{h}(z)\leq e^{\int_{0}^{h}(\partial
f)(\Phi_{s}(z))ds}\leq e^{hK}$

and thus

$\displaystyle\partial_{z}g_{h}(z)=\frac{1}{h}(\partial_{z}\Phi_{h}(z)-1)\leq\frac{1}{h}(e^{hK}-1)\leq
K.$

Finally, we can bound using the bounds on $f$ and (A.1)

$\displaystyle\lvert g_{h}(z)-g_{0}(z)\rvert=\lvert g_{h}(z)-f(z)\rvert$
$\displaystyle=\frac{1}{h}\bigg{\lvert}\int_{0}^{h}f(\Phi_{s}(z))-f(z)ds\bigg{\rvert}$
$\displaystyle\leq K^{2}e^{Kh/2}(1+\lvert
z\rvert^{2m+1})\sup_{s\in[0,h]}\lvert\Phi_{s}(z)-z\rvert$

and

$\displaystyle\lvert\Phi_{s}(z)-z\rvert\leq sK^{2}e^{Kh/2}(1+\lvert
z\rvert^{2m+1}).$

Together we thus obtain

$\displaystyle\lvert g_{h}(z)-g_{0}(z)\rvert\leq hK^{4}e^{Kh}(1+\lvert
z\rvert^{2(2m+1)}),\quad h\in[0,1].$

∎

## References

  * [AP93] A. Ambrosetti and G. Prodi. A primer of nonlinear analysis. No. 34. Cambridge University Press, Cambridge, 1993.
  * [ABLM24] S. Athreya, O. Butkovsky, K. Lê, and L. Mytnik. Well-posedness of stochastic heat equation with distributional drift and skew stochastic heat equation. _Communications on Pure and Applied Mathematics_ 77, no. 5, (2024), 2708–2777. doi:https://doi.org/10.1002/cpa.22157.
  * [Bal77] J. M. Ball. Shorter notes: Strongly continuous semigroups, weak solutions, and the variation of constants formula. _Proceedings of the American Mathematical Society_ 63, no. 2, (1977), 370–373. doi:10.2307/2041821.
  * [BCD11] H. Bahouri, J.-Y. Chemin, and R. Danchin. _Fourier Analysis and Nonlinear Partial Differential Equations_. Grundlehren der mathematischen Wissenschaften. Springer Berlin Heidelberg, 2011. doi:10.1007/978-3-642-16830-7.
  * [BCH18] C.-E. Bréhier, J. Cui, and J. Hong. Strong convergence rates of semidiscrete splitting approximations for the stochastic Allen–Cahn equation. _IMA Journal of Numerical Analysis_ 39, no. 4, (2018), 2096–2134. doi:10.1093/imanum/dry052.
  * [BDG21] O. Butkovsky, K. Dareiotis, and M. Gerencsér. Approximation of SDEs: a stochastic sewing approach. _Probability Theory and Related Fields_ 181, (2021), 975 – 1034. doi:10.1007/s00440-021-01080-2.
  * [BDG22] O. Butkovsky, K. Dareiotis, and M. Gerencsér. Strong rate of convergence of the euler scheme for sdes with irregular drift driven by levy noise. _arXiv preprint arXiv:2204.12926_ (2022).
  * [BDG23] O. Butkovsky, K. Dareiotis, and M. Gerencsér. Optimal rate of convergence for approximations of SPDEs with nonregular drift. _SIAM Journal on Numerical Analysis_ 61, no. 2, (2023), 1103–1137. doi:10.1137/21m1454213.
  * [BFG22] C. Bellingeri, P. K. Friz, and M. Gerencsér. Singular paths spaces and applications. _Stochastic Analysis and Applications_ 40, no. 6, (2022), 1126–1149. doi:10.1080/07362994.2021.1988641.
  * [BG19] C.-E. Bréhier and L. Goudenège. Analysis of some splitting schemes for the stochastic Allen-Cahn equation. _Discrete and Continuous Dynamical Systems - B_ 24, no. 8, (2019), 4169–4190. doi:10.3934/dcdsb.2019077.
  * [BG20] C.-E. Bréhier and L. Goudenège. Weak convergence rates of splitting schemes for the stochastic Allen–Cahn equation. _BIT Numerical Mathematics_ 60, no. 3, (2020), 543–582. doi:10.1007/s10543-019-00788-x
  * [BGJK20] S. Becker, B. Gess, A. Jentzen, and P. Kloeden. Lower and upper bounds for strong approximation errors for numerical approximations of stochastic heat equations. _BIT Numerical Mathematics_ 60, (2020), 1057––1073. doi:10.1007/s10543-020-00807-2.
  * [BGJK23] S. Becker, B. Gess, A. Jentzen, and P. E. Kloeden. Strong convergence rates for explicit space-time discrete numerical approximations of stochastic Allen-Cahn equations. _Stochastics and Partial Differential Equations: Analysis and Computations_ 11, no. 1, (2023), 211–268. doi:10.1007/s40072-021-00226-6.
  * [BJ19] S. Becker and A. Jentzen. Strong convergence rates for nonlinearity-truncated Euler-type approximations of stochastic Ginzburg–Landau equations. _Stochastic Processes and their Applications_ 129, no. 1 (2019), 28–69. https://doi.org/10.1016/j.spa.2018.02.008
  * [BHJ+19] M. Beccari, M. Hutzenthaler, A. Jentzen, R. Kurniawan, F. Lindner, and D. Salimova. Strong and weak divergence of exponential and linear-implicit Euler approximations for stochastic partial differential equations with superlinearly growing nonlinearities. _arXiv preprint arXiv:1903.06066_ (2019).
  * [Cer01] S. Cerrai. _Second Order PDE’s in Finite and Infinite Dimension: A Probabilistic Approach_. No. Nr. 1762 in Lecture Notes in Mathematics. Springer, 2001. doi:10.1007/B80743.
  * [DG01] A. M. Davie and J. G. Gaines. Convergence of numerical schemes for the solution of parabolic stochastic partial differential equations. _Mathematics of Computation_ 70, no. 233, (2001), 121–135. doi:10.1090/s0025-5718-00-01224-2.
  * [DGL23] K. Dareiotis, M. Gerencsér, and K. Lê. Quantifying a convergence theorem of Gyöngy and Krylov. _The Annals of Applied Probability_ 33, no. 3, (2023), 2291–2323. doi:10.1214/22-aap1867.
  * [DPZ92] G. Da Prato and J. Zabczyk. _Stochastic equations in infinite dimensions_ , vol. 44 of _Encyclopedia of Mathematics and its Applications_. Cambridge University Press, Cambridge, 1992. doi:10.1017/CBO9780511666223.
  * [GIP15] M. Gubinelli, P. Imkeller, and N. Perkowski. Paracontrolled distributions and singular PDEs. _Forum of Mathematics, Pi_ 3, (2015), e6. doi:10.1017/fmp.2015.2.
  * [GS24] M. Gerencsér and H. Singh. Strong convergence of parabolic rate 1 of discretisations of stochastic Allen-Cahn-type equations. _Transactions of the American Mathematical Society_ 377, no. 03, (2024), 1851–1881. doi:10.1090/tran/9029.
  * [Gyö99] I. Gyöngy. Lattice Approximations for Stochastic Quasi-Linear Parabolic Partial Differential Equations driven by Space-Time White Noise II. _Potential Analysis_ 11, no. 1, (1999), 1–37. doi:10.1023/a:1008699504438.
  * [HJ12] M. Hutzenthaler and A. Jentzen. Numerical approximations of stochastic differential equations with non-globally Lipschitz continuous coefficients. _Memoirs of the American Mathematical Society_ 236(2012). doi:10.1090/memo/1112.
  * [HJK11] M. Hutzenthaler, A. Jentzen, and P. E. Kloeden. Strong and weak divergence in finite time of euler’s method for stochastic differential equations with non-globally Lipschitz continuous coefficients. _Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences_ 467, (2011), 1563 – 1576. doi:https://www.jstor.org/stable/29792808.
  * [Jen11] A. Jentzen. Higher order pathwise numerical approximations of SPDEs with additive noise. _SIAM Journal on Numerical Analysis_ 49, no. 2, (2011), 642–667. doi:10.1137/080740714.
  * [JK08] A. Jentzen and P. E. Kloeden. Overcoming the order barrier in the numerical approximation of stochastic partial differential equations with additive space–time noise. _Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences_ 465, no. 2102, (2008), 649–667. doi:10.1098/rspa.2008.0325.
  * [JKW11] A. Jentzen, P. Kloeden, and G. Winkel. Efficient simulation of nonlinear parabolic spdes with additive noise. _The Annals of Applied Probability_ 21, no. 3, (2011), 908–950. doi:http://www.jstor.org/stable/23033359.
  * [JP20] A. Jentzen and P. Pušnik. Strong convergence rates for an explicit numerical approximation method for stochastic evolution equations with non-globally Lipschitz continuous nonlinearities. _IMA Journal of Numerical Analysis_ 40, no. 2, (2020), 1005–1050. https://doi.org/10.1093/imanum/drz009
  * [Kha15] S. Khalid. A contribution to the error analysis of the accelerated exponential Euler scheme, 2015. Master’s thesis.
  * [Lê20] K. Lê. A stochastic sewing lemma and applications. _Electronic Journal of Probability_ 25, (2020), 1–55. doi:doi:10.1214/20-EJP442.
  * [LQ19] Z. Liu and Z. Qiao. Strong approximation of monotone stochastic partial differential equations driven by white noise. _IMA Journal of Numerical Analysis_ 40, no. 2, (2019), 1074–1093. doi:10.1093/imanum/dry088.
  * [MZ21] T. Ma and R. C. Zhu. Convergence rate for Galerkin approximation of the stochastic Allen—Cahn equations on 2d torus. _Acta Mathematica Sinica, English Series_ 37, no. 3, (2021), 471–490. doi:10.1007/s10114-020-9367-4.
  * [Sal15] D. Salimova. A further contribution to the error analysis of the accelerated exponential Euler scheme, 2015. Master’s thesis.
  * [Wan16] X. Wang. Weak error estimates of the exponential Euler scheme for semi-linear SPDEs without Malliavin calculus. _Discrete and Continuous Dynamical Systems_ 36 no. 1, (2016), 481–497. doi: 10.3934/dcds.2016.36.481
  * [Wan20] X. Wang. An efficient explicit full-discrete scheme for strong approximation of stochastic Allen–Cahn equation. _Stochastic Processes and their Applications_ 130, no. 10, (2020), 6271–6299. doi:10.1016/j.spa.2020.05.011.