---
abstract: |
    We consider the effect of perturbations of $A$ on the solution to the following quasi-linear parabolic stochastic differential equation set in a <span style="font-variant:small-caps;">umd</span> Banach space $X$: $$\label{SDE_abstract}\tag{SDE}
    \left\{ \begin{aligned} dU(t) & = AU(t)\,dt +F(t,U(t))\,dt + G(t,U(t))\,dW_H(t), \quad t>0;\\
    U(0)&=x_0. \end{aligned}\right.$$ Here $A$ is the generator of an analytic $C_0$-semigroup on $X$, $G:[0,T]\times X\rightarrow {\mathcal L}(H,X_{\theta_G}^{A})$ and $F:[0,T]\times
    X\rightarrow X_{\theta_F}^{A}$ for some $\theta_G>-{\frac{1}{2}}$, $\theta_F>  -\frac{3}{2}+{\frac{1}{\tau}}$, where $\tau$ is the type of $X$. We assume $F$ and $G$ to satisfy certain global Lipschitz and linear growth conditions.

    Let $A_0$ denote the perturbed operator and $U_0$ the solution to with $A$ substituted by $A_0$. We provide estimates for ${\Vert}U - U_0 {\Vert}_{L^p(\Omega;C([0,T];X))}$ in terms of $D_{\delta}(A,A_0):={\Vert}R(\lambda:A)-R(\lambda:A_0){\Vert}_{{\mathcal L}(X^{A}_{\delta-1},X)}$. Here $\delta\in [0,1]$ is assumed to satisfy $0\leq \delta < \min\{\tfrac{3}{2}-{\tfrac{1}{\tau}}+\theta_F,\, {\tfrac{1}{2}}-{\tfrac{1}{p}}+\theta_G \}$.

    The work is inspired by the desire to prove convergence of space approximations of . In this article we prove convergence rates for the case that $A$ is approximated by its Yosida approximation.\
    [**Keywords:** ]{}[perturbations, stochastic differential equations, stochastic convolutions, stochastic partial differential equations, Yosida approximation]{}\
    [**MSC2010:** ]{} 46N40, 35R60, 35A30, 60H15
address:
- |
    Delft Institute of Applied Mathematics\
    Delft University of Technology\
    P.O. Box 5031\
    2600 GA Delft\
    The Netherlands
- |
    Pepartmnent of Mathematics and Informationtechnology\
    Montanuniversity Leoben\
    Fr. Josefstr. 18\
    8700 Leoben\
    Austria
author:
- Sonja Cox
- Erika Hausenblas
title: 'A perturbation result for quasi-linear stochastic differential equations in UMD Banach spaces.'
---

Introduction {#mnb}
============

In this article we consider the effect of perturbations of $A$ on the solution to the following stochastic differential equation set in a <span style="font-variant:small-caps;">umd</span> Banach space $X$: $$\label{SDE1}\tag{SDE}
\left\{ \begin{aligned} dU(t) & = AU(t)\,dt +F(t,U(t))\,dt + G(t,U(t))\,dW_H(t), \quad t>0;\\
U(0)&=x_0. \end{aligned}\right.$$ Here $A$ is the generator of an analytic $C_0$-semigroup on $X$, $G:[0,T]\times
X\rightarrow {\mathcal L}(H,X_{\theta_G}^{A})$ and $F:[0,T]\times X\rightarrow
X_{\theta_F}^{A}$ for some $\theta_G>-{\frac{1}{2}}$, $\theta_F>
-\frac{3}{2}+{\frac{1}{\tau}}$, where $\tau\in [1,2)$ is the type of $X$. We use $X_{\delta}^A$ to denote the fractional domain or extrapolation space corresponding to $A$. We assume $F$ and $G$ to satisfy certain global Lipschitz and linear growth conditions, see Section \[ss:setting\] below. The framework in which we consider is precisely the one for which existence and uniqueness of a solution has been proven in [@NVW08]. A typical example of a stochastic partial differential equation that fits into this framework is a one-dimensional parabolic stochastic partial differential equation driven by white noise.

The main motivation to study the effect of perturbations of $A$ on solutions to is the desire to prove convergence of certain numerical schemes for approximations in the space dimension. In fact, in [@CoxHau:11b] we demonstrate how the perturbation result proven in this article can be used to obtain pathwise convergence of certain Galerkin and finite element methods for in the case that $X$ is Hilbertian. Here we focus more on the theoretical aspects, and demonstrate how our perturbation result can be used to prove convergence of the solution processes if $A$ is replaced by its Yosida approximation.

With applications to numerical approximations in mind, we assume the perturbed equation to be set in a (possibly finite dimensional) closed subspace $X_0$ of $X$. We assume that there exists a bounded projection $P_0:X\rightarrow X_0$ such that $P_0(X)=X_0$. Let $i_{X_0}$ be the canonical embedding of $X_0$ in $X$ and let $A_0$ be a generator of an analytic $C_0$-semigroup $S_0$ on $X_0$. In the setting of numerical approximations, $A_0$ would be a suitable restriction of $A$ to the finite dimensional space $X_0$.

The perturbed equation we consider is the following stochastic differential equation: $$\label{SDE0_intro}\tag{SDE$_0$}
\left\{ \begin{aligned} dU^{(0)}(t) & =A_0U^{(0)}(t)\,dt +P_0F(t,U^{(0)}(t))\,dt \\
&\quad + P_0G(t,U^{(0)}(t))\,dW_H(t), \qquad\qquad  t>0;\\
U^{(0)}(0)& = P_0x_0. \end{aligned}\right.$$ Our main result, Theorem \[t:app\] below, states that if we have: $$\begin{aligned}
\label{defDAAintro}
D_{\delta}(A,A_0):=  {\Vert}R(\lambda_0:A)-i_{X_0} R(\lambda_0:A_0)P_0 {\Vert}_{{\mathcal L}(X_{\delta-1}^{A},X)} < \infty,\end{aligned}$$ for some $\delta\geq 0$ satisfying $$0\leq \delta < \min\{\tfrac{3}{2}-{\tfrac{1}{\tau}}+\theta_F,\, {\tfrac{1}{2}}+\theta_G \},$$ and $x_0\in L^p(\Omega;{\mathcal F}_0;X_{\delta}^{A})$ for $p\in (2,\infty)$ such that ${\frac{1}{p}}\leq {\frac{1}{2}}+\theta_G-\delta$, then there exists a solution to $\eqref{SDE0_intro}$ in $L^p(\Omega;C([0,T];X_0))$ and moreover: $$\begin{aligned}
{\Vert}U - i_{X_0}U^{(0)} {\Vert}_{L^p(\Omega;C([0,T];X))}&
\lesssim D_{\delta}(A,A_0)(1+{\Vert}x_0
{\Vert}_{L^p(\Omega;X_{\delta}^{A})}).
\end{aligned}$$ Note that if $\delta<1$ then a priori it is not obvious whether $D_{\delta}(A,A_0)$ is finite.

As a corollary of Theorem \[t:app\] we obtain an estimate in the Hölder norm provided we compensate for the initial values (see Corollary \[cor:Holder\] below), i.e., for $\lambda\in [0,{\frac{1}{2}})$ satisfying $$0\leq \lambda < \min\{\tfrac{3}{2}-{\tfrac{1}{\tau}}-(\delta-\theta_F){\vee}0, {\tfrac{1}{2}}-{\tfrac{1}{p}}-(\delta-\theta_G){\vee}0\},$$ we have: $$\begin{aligned}
{\Vert}U - Sx_0 - i_{X_0}(U^{(0)} - S_0P_0x_0) {\Vert}_{L^p(\Omega;C^{\lambda}([0,T];X))}&
\lesssim D_{\delta}(A,A_0)(1+{\Vert}x_0
{\Vert}_{L^p(\Omega;X_{\delta}^{A})}).\end{aligned}$$

Our results imply that if $(A_n)_{n\in\N}$ is a family of generators of analytic semigroups such that the resolvent of $A_n$ converges to the resolvent of $A$ in ${\mathcal L}(X_{\delta-1}^A,X)$ for some $\delta\in [0,1]$ (and $(A_n)_{n\in\N}$ is uniformly analytic), then the corresponding solution processes $U_n$ converge to the actual solution in $L^p(\Omega;C([0,T];X))$ and the convergence rate is given by $D_{\delta}(A,A_n)$.

In particular, we may apply Theorem \[t:app\] to the Yosida approximation of $A$. In this case we assume $\theta_F$ and $\theta_G$ are positive. The $n^{th}$ Yosida approximation of $A$ is given by $A_n=nAR(n:A)$, and we let $U^{(n)}$ denote the solution to where $A$ is substituted by $A_n$. By applying Theorem \[t:app\] we obtain that for $\eta\in [0,1]$ and $p\in (2,\infty)$ such that $$\eta<\min\{\tfrac{3}{2}-{\tfrac{1}{\tau}}+\theta_F,\tfrac{1}{2}-{\tfrac{1}{p}}+\theta_G \}$$ we have, assuming $x_0\in L^p(\Omega,\mathcal{F}_0;X_{\eta}^A)$: $$\begin{aligned}
{\Vert}U - U^{(n)}{\Vert}_{L^p(\Omega;C([0,T];X))}
&\lesssim  n^{-\eta}(1+{\Vert}x_0 {\Vert}_{L^p(\Omega;X_{\eta}^{A})}).\end{aligned}$$ See also Theorem \[t:yosida\].

It was proven in [@KunNeer11] that if $(A_n)_{n\in\N}$ is a family of generators of analytic semigroups such that the resolvent of $A_n$ converges to the resolvent of $A$ in the strong operator topology, then the corresponding solution processes $U_n$ converge to the actual solution in $L^p(\Omega;C([0,T];X))$. However, the approach taken in that article does not provide convergence rates and requires $\theta_F,\theta_G\geq 0$.

Another article in which approximations of solutions to are considered in the context of perturbations on $A$ is [@Brze:97]. In that article, it is assumed that $X$ is a <span style="font-variant:small-caps;">umd</span> space with martingale type 2. In Section 5 of that article the author considers approximations of $A$, $F$, $G$ and of the noise. Translated to our setting, the author assumes the perturbed operator $A_0$ to satisfy $X^{A_0}_{\theta_F}=X^{A}_{\theta_F}$ and $X^{A_0}_{\theta_G}=X^{A}_{\theta_G}$ (in particular, $X_0$ cannot be finite-dimensional).

A natural question to ask is how the type of perturbation studied here relates to the perturbations known in the literature. In [@DeschSchap:87], [@Jung:00], and [@Rob:77] (see also [@EngNa:00 Chapter III.3]) one has derived conditions for perturbations of $A$ that lead to an estimate of the type ${\Vert}S(t) - S_0(t){\Vert}_{{\mathcal L}(X)}=\mathcal{O}(t)$. In light of Proposition \[p:gbddSG\] below these results are comparable to our results if we were to take $\alpha=-1$. In particular, [@EngNa:00 Theorem III.3.9] gives precisely the same results as Proposition \[p:gbddSG\], but then for the case $\alpha=-1$ and $\beta=0$.

The proof of our perturbation result (Theorem \[t:app\]) requires regularity results for stochastic convolutions. As the convolution under consideration concerns the difference between two semigroups instead of a single semigroup, the celebrated factorization method of [@DKZ] fails. Therefore we prove a new result on the regularity of stochastic convolutions, see Lemma \[lem:stochConv\] below. This lemma in combination with some randomized boundedness results on $S-S_0P_0$ form the key ingredients of the proof Theorem \[t:app\].

The set-up of this article is as follows: Section \[sec:prelim\] contains the preliminaries; i.e., the necessary results on analytic semigroups, vector-valued stochastic integration theory, and $\gamma$-boundedness. In that section we also state the precise assumptions on $A$, $F$, and $G$ in , and prove the regularity results for (stochastic) convolutions that we need in the proof of Theorem \[t:app\]. In Section \[sec:app\] we prove Theorem \[t:app\] and in Section \[sec-yosida\] we prove convergence for the Yosida approximations.

Notation {#notation .unnumbered}
--------

For an operator $A$ on a Banach space $X$ we denote the resolvent set of $A$ by $\rho(A)$, i.e., $\rho(A)\subset \mathbb{C}$ is the set of all the complex numbers $\lambda\in \mathbb{C}$ for which $\lambda I-A$ is (boundedly) invertible. For $\lambda\in\rho(A)$ we denote the resolvent of $A$ in $\lambda$ by $R(\lambda:A)$, i.e., $R(\lambda:A)=(\lambda I -A)^{-1}$. The spectrum of $A$, i.e., the complement of $\rho(A)$ in $\C$, is denoted by $\sigma(A)$.

For $X, Y$ Banach spaces we let ${\mathcal L}(X,Y)$ be the Banach space of all bounded linear operators from $X$ to $Y$ endowed with the operator norm. For brevity we set ${\mathcal L}(X):={\mathcal L}(X,X)$.

For $T>0$ and $\theta>0$ we take the following definition for the Hölder norm: $$\begin{aligned}
{\Vert}f {\Vert}_{C^{\theta} ([0,T];Y)} &:= {\Vert}f(0){\Vert}_{Y} + \sup_{0\leq s<t\leq T} \frac{{\Vert}f(t)-f(s){\Vert}_{Y}}{(t-s)^{\theta}}.\end{aligned}$$

We write $A \lesssim B$ to express that there exists a constant $C>0$ such that $A\leq C B$, and we write $A\eqsim B$ if $A\lesssim B$ and $B\lesssim A$. Finally, for $X$ and $Y$ Banach spaces we write $X\simeq Y$ if $X$ and $Y$ are isomorphic as Banach spaces.

Preliminaries {#sec:prelim}
=============

Throughout this section $X$, $Y$, and $Y_{i}$, $i\in \{1,2\}$, will be used to denote Banach spaces and $H$ will denote a Hilbert space.

Analytic semigroups
-------------------

For $\delta \in [0,\pi]$ we define $$\Sigma_{\delta}:= \{ z\in \C\setminus\{0\} \,:\, |\arg(z)| < \delta\}.$$ We recall the definition of analytic $C_0$-semigroups [@Pazy:83 Chapter 2.5]:

Let $\delta\in (0,\frac{\pi}{2})$. A $C_0$-semigroup $(S(t))_{t\geq 0}$ on $X$ is called *analytic in* $\Sigma_{\delta}$ if

1.  $S$ extends to an analytic function $S:\Sigma_{\delta}\rightarrow {\mathcal L}(X)$;

2.  $S(z_1+z_2)=S(z_1)S(z_2)$ for $z_1,z_2\in \Sigma_{\delta}$;

3.  $\lim_{z\rightarrow 0; z\in \Sigma_\delta}S(z)x=x$ for all $x\in X$.

Typical examples of operators generating analytic $C_0$-semigroups are second-order elliptic operators. The theorem below is obtained from [@Pazy:83 Theorem 2.5.2] by straightforward adaptations and gives some characterizations of analytic $C_0$-semigroups that we shall need.

\[t:analchar\] Let $A$ be the generator of a $C_0$-semigroup $(S(t))_{t\geq0}$ on $X$. Let $\omega\in \R$ be such that $(e^{-\omega t}S(t))_{t\geq 0}$ is exponentially stable. The following statements are equivalent:

1.  \[analdef\] $S$ is an analytic $C_0$-semigroup on $\Sigma_{\delta}$ for some $\delta \in
    (0,\frac{\pi}{2})$ and for every $\delta'<\delta$ there exists a constant $C_{1,\delta'}$ such that ${\Vert}e^{-\omega z} S(z){\Vert}\leq C_{1,\delta'}$ for all $z\in \Sigma_{\delta'}$.

2.  \[analres\] There exists a $\theta \in (0,\frac{\pi}{2})$ such that $ \omega+\Sigma_{\frac{\pi}{2}+\theta}
    \subset \rho(A)$, and for every $\theta'\in (0,\theta)$ there exists a constant $C_{2,\theta'}>0$ such that: $$\begin{aligned}
    |\lambda-\omega| {\Vert}R(\lambda:A) {\Vert}\leq C_{2,\theta'}, \quad \textrm{for all } \lambda\in
    \omega+\Sigma_{\frac{\pi}{2}+\theta'}.\end{aligned}$$

3.  \[analdiff\] $S$ is differentiable for $t>0$ (in the uniform operator topology), $\frac{d}{dt}S=AS$, and there exists a constant $C_3$ such that: $$\begin{aligned}
    t{\Vert}A S(t){\Vert}\leq C_3 e^{\omega t}, \quad \textrm{for all } t>0.\end{aligned}$$

This theorem justifies the following definition:

\[d:uniftype\] Let $A$ be the generator of an analytic $C_0$-semigroup on $X$. We say that $A$ is *of type* $(\omega,\theta,K)$, where $\omega\in\R$, $\theta\in (0,\frac{\pi}{2})$ and $K>0$, if $\omega+\Sigma_{\frac{\pi}{2}+\theta} \subseteq \rho(A)$, $(e^{\omega t}S(t))_{t\geq 0}$ is exponentially stable, and $$|\lambda-\omega|{\Vert}R(\lambda:A) {\Vert}_{{\mathcal L}(X)} \leq K
\quad \textrm{for all } \lambda \in \omega+\Sigma_{\frac{\pi}{2}+\theta}.$$

\[r:analcont\] It follows from the aforementioned proof in [@Pazy:83] that the constants $\delta,  C_{1,\delta'}$; $\delta'\in (0,\delta)$, $C_{2,\theta'}$; $\theta'\in (0,\theta)$, and $C_{3}$ in Theorem \[t:analchar\] can be expressed explicitly in terms of $\omega$, $\theta$, and $K$; for example, we may take $C_3= \tfrac{K}{\pi \cos
\theta}$.

Note that if $A$ is the generator of an analytic $C_0$-semigroup of type $(\omega,\theta,K)$ then for all $\lambda \in
\omega(1+2(\cos\theta)^{-1})+\Sigma_{\frac{\pi}{2}+\theta}$ one has (noting that the choice of $\lambda$ implies $|\lambda|>2|\omega|$ and hence $|\lambda-\omega|>||\lambda|-|\omega||\geq {\frac{1}{2}}|\lambda|$): $$\begin{aligned}
\label{AResEst}
{\Vert}A R(\lambda:A){\Vert}_{{\mathcal L}(X)} = {\Vert}\lambda R(\lambda:A) - I {\Vert}\leq 1 + 2K.\end{aligned}$$

Let $A$ be the generator of an analytic semigroup of type $(\omega,\theta,K)$ on $X$. We define the extrapolation spaces of $A$ conform [@Pazy:83 Section 2.6]; i.e., for $\delta>0$ and $\lambda \in \C$ such that $\mathscr{R}e(\lambda)>\omega$ we define $X^{A}_{-\delta}$ to be the closure of $X$ under the norm ${\Vert}x{\Vert}_{X^{A}_{-\delta}} := {\Vert}(\lambda I -A)^{-\delta} x {\Vert}_X$. We also define the fractional domain spaces of $A$, i.e., for $\delta>0$ we define $X^{A}_{\delta}=D((\lambda I -A)^{\delta})$ and ${\Vert}x{\Vert}_{X^{A}_{\delta}} := {\Vert}(\lambda I
-A)^{\delta} x {\Vert}_X$. One may check that regardless of the choice of $\lambda$ the extrapolation spaces and the fractional domain spaces are uniquely determined up to isomorphisms: for $\delta>0$ one has $(\lambda I -A)^{\delta}(\mu
I -A)^{-\delta}\in {\mathcal L}(X)$ and: $${\Vert}(\lambda I -A)^{\delta}(\mu I -A)^{-\delta}{\Vert}_{{\mathcal L}(X)}\leq C(\omega,\theta,K,\lambda,\mu),$$ where $C(\omega,\theta,K,\lambda,\mu)$ denotes a constant depending only on $\omega,\theta,K,\lambda,$ and $\mu$. Moreover, for $\delta,\, \beta\in \R$ one has $(\lambda I-A)^{\delta}(\lambda I-A)^{\beta}=(\lambda I-A)^{\delta+\beta}$ on $X^A_{\gamma}$, where $\gamma= \max\{\beta,\delta+\beta\}$ (see [@Pazy:83 Theorem 2.6.8]).

Statement in Theorem \[t:analchar\] can be extended; from the proof of [@Pazy:83 Theorem 2.6.13] we obtain that for an analytic $C_0$-semigroup $S$ of type $(\omega,\theta,K)$ generated by $A$ one has, for $\delta>0$: $$\begin{aligned}
\label{analyticSG-Est}
{\Vert}S(t){\Vert}_{{\mathcal L}(X,X^A_{\delta})} & \leq 2\big(\tfrac{K}{\pi \cos \theta}\big)^{\lceil \delta \rceil} t^{-\delta} e^{\omega t}.\end{aligned}$$

The following interpolation result holds for the fractional domain spaces (see [@Pazy:83 Theorem 2.6.10]):

\[t:fracPowInt\] Let $A$ be the generator of an analytic $C_0$-semigroup on $X$ of type $(\omega,\theta,K)$. Let $\delta\in (0,1)$ and $\lambda\in \C$ such that $\mathscr{R}e(\lambda)>\omega$. Then for every $x\in D(A)$ we have: $$\begin{aligned}
{\Vert}(\lambda I - A)^{\delta} x {\Vert}& \leq 2(1+K){\Vert}x{\Vert}^{1-\delta}{\Vert}(\lambda I - A) x {\Vert}^{\delta}.\end{aligned}$$

For more properties of $X_{\delta}^{A}$, $\delta\in \R$, we refer to [@Pazy:83 Section 2.6].

Stochastic differential equations
---------------------------------

Throughout this section $X$ and $Y$ denote <span style="font-variant:small-caps;">umd</span> Banach spaces, and $(\Omega,(\mathcal{F}_t)_{t\geq 0},\P)$ denotes a probability space.

### Stochastic integration in <span style="font-variant:small-caps;">umd</span> Banach spaces

Before turning to stochastic differential equations, we recall the basics concerning stochastic integration in <span style="font-variant:small-caps;">umd</span> Banach spaces as presented in [@NVW07a]. Recall that the <span style="font-variant:small-caps;">umd</span> property is a geometric Banach space property that is satisfied by all Hilbert spaces and the ‘classical’ reflexive function spaces, e.g. the $L^p$-spaces and Sobolev spaces $W^{k,p}$ for $k\in \N$ and $p\in
(1,\infty)$. For the precise definition of the <span style="font-variant:small-caps;">umd</span> property and for a more elaborate treatment of spaces satisfying this property we refer to [@Bu3].

Fix $T>0$. An [*$H$-cylindrical Brownian motion over $(\Omega,(\mathcal{F}_t)_{t\geq
0},\P)$*]{} is a mapping $W_{H}:L^2(0,T;H)\rightarrow L^2(\Omega)$ with the following properties:

1.  for all $h\in L^2(0,T;H)$ the random variable $W_{H}(h)$ is Gaussian;

2.  for all $h_1,h_2\in L^2(0,T;H)$ we have $\E W_{H}(h_1)W_{H}(h_2)=\langle h_1,h_2\rangle$;

3.  for all $h\in H$ and all $t\in [0,T]$ we have that $W_H(1_{[0,t]}\otimes h)$ is $\mathcal{F}_t$-measurable;

4.  for all $h\in H$ and all $s,t\in [0,T]$, $s\leq t$ we have that $W_H(1_{[s,t]}\otimes h)$ is independent of $\mathcal{F}_s$.

Formally, an $H$-cylindrical Brownian motion can be thought of as a standard Brownian motion taking values in the Hilbert space $H$.

For the precise definition of the stochastic integral of a process $\Phi:[0,T]\times\Omega\rightarrow {\mathcal L}(H,X)$ with respect to $W_H$ we refer to [@NVW07a]. For our purposes it suffices to cite the characterization of such stochastically integrable processes in terms of the so-called $\gamma$-radonifying norm of the process.

Let ${\mathcal H}$ be a Hilbert space (we will take ${\mathcal H}=L^2(0,T;H)$ later on). The Banach space $\gamma({\mathcal H},X)$ is defined as the completion of ${\mathcal H}\otimes X$ with respect to the norm $$\Big{\Vert}\sum_{n=1}^N h_n\otimes x_n \Big{\Vert}_{\gamma({\mathcal H},X)}^2 := \E \Big{\Vert}\sum_{n=1}^N \gamma_n\otimes x_n \Big{\Vert}^2.$$ Here we assume that $(h_n)_{n=1}^N$ is an orthonormal sequence in ${\mathcal H}$, $(x_n)_{n=1}^N$ is a sequence in $X$, and $(\gamma_n)_{n=1}^N$ is a standard Gaussian sequence on some probability space. The space $\gamma(H,X)$ embeds continuously into ${\mathcal L}({\mathcal H},X)$ and it elements are referred to as the [*${\gamma}$-radonifying*]{} operators from ${\mathcal H}$ to $X$. For properties of this norm and further details we refer to the survey paper [@Nee-survey].

Let $(R,\mathcal{R},\mu)$ be a measure space. In that case $\gamma(R,H;X)$ is used as short-hand notation for $\gamma(L^2(R;H),X)$; in particular, $\gamma(0,T,H;X)$ is short-hand notation for $\gamma(L^2(0,T;H);X)$. We use $\gamma(0,T;X)$ to denote $\gamma(L^2(0,T),X)$. If $X$ is a Hilbert space, and $(R,\mathcal{R},\mu)$ is a $\sigma$-finite measure space, then $\gamma(R,H;X)= L^2(R,\mathcal{L}_2(H,X))$ where $\mathcal{L}_2(H,X)$ denotes the space of Hilbert-Schmidt operators from $H$ to $X$.

A process $\Phi:[0,\infty)\times \Omega\rightarrow {\mathcal L}(H,X)$ is called $H$-*strongly measurable* if for every $h\in H$ the process $\Phi h$ is strongly measurable. The process is called *adapted* if $\Phi h$ is adapted for each $h\in H$.

We cite [@NVW07a Theorem 3.6]:

\[t:stochint\] Let $p\in (1,\infty)$ and $T>0$ be fixed. For an $H$-strongly measurable adapted process $\Phi:(0,T)\times \Omega
\rightarrow {\mathcal L}(H,X)$ the following are equivalent:

1.  $\Phi$ is $L^p$-stochastically integrable with respect to $W_H$;\[l:sintble\]

2.  we have $\Phi^* x^* \in L^p(\Omega;L^2(0,T;H))$ for all $x^*\in X^*$ and there exists a (necessarily unique) $R_\Phi\in L^{p}(\Omega;
    {\gamma}(0,T,H;X))$ such that for all $x^*\in X^*$ we have $$R_\Phi^* x{^{\ast}}= \Phi^* x{^{\ast}}$$ in $L^p({\Omega};L^2(0,T;H))$.

In this situation one has $$\begin{aligned}
\label{BDG}
\E\, \sup_{0\leq t\leq T} \Big{\Vert}\int_{0}^t \Phi \,dW_H \Big{\Vert}_Y^p \eqsim_p
\E\, {\Vert}R_\Phi {\Vert}_{{\gamma}(0,T,H;Y)}^{p},\end{aligned}$$ the implied constants being independent of $\Phi$.

The inequalities in equation are referred to as *Burkholder-Davis-Gundy* inequalities.

From now on, if $\Phi$ is stochastically integrable, we shall simply use $\Phi$ to denote both a process and the (unique) $R_\Phi\in L^{p}(\Omega; \gamma(0,t,H;X))$ that satisfies $R_\Phi^* x{^{\ast}}= \Phi^* x{^{\ast}}$.

### The SDE {#ss:setting}

In Section \[sec:app\] we will prove a perturbation result for the following stochastic differential equation: $$\begin{aligned}
\label{SDE}\tag{SDE}
\left\{ \begin{array}{rcl} dU(t) &\! =\! & AU(t)\,dt + F(t,U(t))\,dt + G(t,U(t))\,dW_H(t);\quad t\in [0,T],\\
U(0)&\! =\!  &x_0. \end{array}\right.\end{aligned}$$ where $A$, $F$ and $G$ are assumed to satisfy conditions [[**(A)**]{}]{}, [[**(F)**]{}]{}, and [[**(G)**]{}]{} below. The assumptions on $F$ depend on the type of the Banach space $F$. The type of a Banach space is defined based on the behavior of moments of randomized sums, and it always takes values in the interval $[1,2]$. The greater the type, i.e., the closer to $2$, the more the space behaves like a Hilbert space: every Banach space has type at least $1$, and all Hilbert spaces have type $2$. The $L^p$-spaces have type $\min\{p,2\}$. We refer to [@LeTa:91] for a precise definition of type (and co-type) and further details. Both <span style="font-variant:small-caps;">umd</span> and type are preserved under Banach space isomorphisms.

[[**(A)**]{}]{}

:   $A$ generates an analytic $C_0$-semigroup on a <span style="font-variant:small-caps;">umd</span> Banach space $X$.

[[**(F)**]{}]{}

:   For some $\theta_F>  -1 + ({\frac{1}{\tau}}-\frac12)$, where $\tau$ is the type of $X$, the function $F:[0,T]\times X\rightarrow X_{\theta_F}$ is measurable in the sense that for all $x\in X$ the mapping $F(\cdot,x):[0,T]\rightarrow X_{\theta_F}$ is strongly measurable. Moreover, $F$ is uniformly Lipschitz continuous and uniformly of linear growth on $X$.

    That is to say, there exist constants $C_0$ and $C_1$ such that for all $t\in [0,T]$ and all $x,y\in X$: $$\begin{aligned}
    {\Vert}F(t,x) - F(t,y) {\Vert}_{X_{\theta_F}} & \leq C_0 {\Vert}x-y{\Vert}_{X}, \\ 
    {\Vert}F(t,x){\Vert}_{X_{\theta_F}} &\leq 
    C_1(1+{\Vert}x{\Vert}_{X}).\end{aligned}$$ The least constant $C_0$ such that the above holds is denoted by Lip$(F)$, and the least constant $C_1$ such that the above holds is denoted by $M(F)$.

[[**(G)**]{}]{}

:   For some $\theta_G>-{\frac{1}{2}}$, the function $G : [0,T]\times X\rightarrow {\mathcal L}(H,X_{\theta_G})$ is measurable in the sense that for all $h\in H$ and $x\in X$ the mapping $G(\cdot,x)h:[0,T]\rightarrow X_{\theta_G}$ is strongly measurable. Moreover, $G$ is uniformly $L^2_{\gamma}$-Lipschitz continuous and uniformly of linear growth on $X$.

    That is to say, there exist constants $C_0$ and $C_{1}$ such that for all $\alpha\in [0,{\frac{1}{2}})$, all $t\in [0,T]$, and all simple functions $\phi_1$, $\phi_2$, $\phi: [0,T]\to X$ one has: $$\begin{aligned}
    &{\Vert}s\mapsto (t-s)^{-\alpha}[G(s,\phi_1(s)) -G(s,\phi_2(s))] {\Vert}_{\gamma(0,t;H,X_{\theta_G})} \\
    &\qquad \qquad \le  C_0 {\Vert}s\mapsto (t-s)^{-\alpha}[\phi_1 -\phi_2] 
    {\Vert}_{L^2(0,t;X)\,\cap \,\gamma(0,t;X)}; \\
    &{\Vert}s\mapsto (t-s)^{-\alpha} G(s,\phi(s)){\Vert}_{\gamma(0,t;H,X_{\theta_G})}\\
    &\qquad \qquad  \leq C_1\big(1+ {\Vert}s\mapsto
    (t-s)^{-\alpha}\phi(s){\Vert}_{L^2(0,t;X)\,\cap\, \gamma(0,t;X)}\big).\end{aligned}$$The least constant $C_0$ such that the above holds is denoted by $\textrm{Lip}_{\gamma}(G)$, and the least constant $C_1$ such that the above holds is denoted by $M_{\gamma}(G)$.

If $Y_2$ is a type 2 space and $G:[0,T]\times Y_1\rightarrow \gamma(H,Y_2)$ is Lipschitz-continuous, uniformly in $[0,T]$, then $G$ is $L^2_{\gamma}$-Lipschitz continuous (see [@NVW08 Lemma 5.2]). More examples of $L^2_{\gamma}$-Lipschitz continuous operators can be found in [@NVW08].

### Existence and uniqueness

We recall an existence and uniqueness result for the problem . This result is formulated in a space of continuous, ‘weighted’ stochastically integrable processes which is defined as follows:

\[d:Vspace\] For $\alpha\geq 0$, $1\leq p<\infty$ and $0\leq a \leq b <
\infty$, we denote by ${V^{\alpha,p}_{\rm c}}([a,b]\times \Omega;Y)$ the space of adapted, continuous processes $\Phi:[a,b]\times \Omega\rightarrow
Y$ for which the following norm is finite: $$\begin{aligned}
{\Vert}\Phi{\Vert}_{{V^{\alpha,p}_{\rm c}}([a,b]\times \Omega;Y)} = {\Vert}\Phi {\Vert}_{L^p(\Omega;C([a,b];Y))} + \sup_{a\leq t\leq b}{\Vert}s\mapsto (t-s)^{-\alpha}\Phi(s){\Vert}_{L^p(\Omega;\gamma(a,t;Y))}.\end{aligned}$$

One easily checks that for $a\leq c<d\leq b$, $$\label{Vtransinv}
{\Vert}\Phi|_{[c,d]} {\Vert}_{ {V^{\alpha,p}_{\rm c}}([a,b]\times\Omega;X)} = {\Vert}\Phi|_{[c,d]} {\Vert}_{
{V^{{\alpha},p}_{\rm c}} ([c,d]\times \Omega;X)}.$$

Moreover, for $0\le\beta\le\alpha<\frac12$ and $\Phi \in {V^{\alpha,p}_{\rm c}}([a,b]\times \Omega;X)$ one has: $$\label{Vchange-of-alpha}
{\Vert}\Phi{\Vert}_{{V^{\beta,p}_{\rm c}} ([a,b]\times \Omega;X)} \leq (b-a)^{\alpha-\beta}{\Vert}\Phi{\Vert}_{{V^{\alpha,p}_{\rm c}}([a,b]\times \Omega;X)}.$$ Note also that we have ${V^{\alpha,p}_{\rm c}}([0,T]\times \Omega;X)
\subset L^p(\Omega;C([0,T];X))$. On the other hand, we have the following embedding (see [@NVW08 Lemma 3.3]):

\[lem:VEmbedHoelder\] Let $X$ be a Banach space with type $\tau$. Then for all $T>0$, ${\varepsilon}>0$ and $\alpha
\in [0,{\frac{1}{2}})$ there exists a constant $C$ such that for all $T_0\in [0,T]$ one has: $${V^{\alpha,p}_{\rm c}}([0,T_0]\times\Omega;X) \leq C L^p(\Omega;C^{{\frac{1}{\tau}}-{\frac{1}{2}}+{\varepsilon}} ([0,T_0];X)).$$

If $G:[0,T]\times X\rightarrow {\mathcal L}(H,X_{\theta_G})$ satisfies [[**(G)**]{}]{} and $\Phi_1,\Phi_2 \in {V^{\alpha,p}_{\rm c}}([0,T]\times \Omega;X)$ for some $p\geq 2$, then: $$\begin{aligned}\label{GLipschitzV2}
& \sup_{0\leq t\leq T}{\Vert}s\mapsto
(t-s)^{-\alpha}[G(s,\Phi_1(s))-G(s,\Phi_2(s))]{\Vert}_{L^p(\Omega;\gamma(0,t;X_{
\theta_G}))}\\
&\qquad \qquad \leq (1+T^{{\frac{1}{2}}-\alpha})\textrm{Lip}_{\gamma}(G){\Vert}\Phi_1-\Phi_2{\Vert}_{{V^{\alpha,p}_{\rm c}}([0,T]\times \Omega;X)},
\end{aligned}$$ and, for $\Phi \in {V^{\alpha,p}_{\rm c}}([0,T]\times \Omega;X)$: $$\begin{aligned}\label{GLipschitzV1}
&\sup_{0\leq t\leq T}{\Vert}s\mapsto
(t-s)^{-\alpha}G(s,\Phi(s)){\Vert}_{L^p(\Omega;\gamma(0,t;X_{\theta_G}))} \\
&\qquad \qquad  \leq (1+T^{{\frac{1}{2}}-\alpha})M_\gamma(G)\big( 1+{\Vert}\Phi
{\Vert}_{{V^{\alpha,p}_{\rm c}} ([0,T]\times \Omega;X)}\big).
\end{aligned}$$ The following existence and uniqueness result for solutions to is presented in [@NVW08 Theorem 6.2].

\[thm:NVW08\] Consider under the assumptions [[**(A)**]{}]{}, [[**(F)**]{}]{}, and [[**(G)**]{}]{}. Let $x_0\in L^p(\Omega,\mathcal{F}_0;X_{\eta})$ for $p\in (2,\infty)$ and $\eta>0$ satisfying $$0 \leq \eta < \min\{\tfrac{3}{2}-{\tfrac{1}{\tau}}+\theta_F,\, {\tfrac{1}{2}}-{\tfrac{1}{p}}+\theta_G \}.$$ Then for any $T>0$ and any $\alpha\in [0,{\frac{1}{2}})$ there exists a unique $U \in {V^{\alpha,p}_{\rm c}}([0,t]\times\Omega;X_{\eta}^{A})$ such that $s\mapsto S(t-s)G(s,U(s))$ is stochastically integrable for all $t\in[0,T]$, and $U$ satisfies: $$\begin{aligned}
\label{SDE_sol}
U(t)=S(t)x_0 + \int_{0}^{t} S(t-s)F(s,U(s))\,ds + \int_{0}^{t} S(t-s)G(s,U(s))\,dW_H(s)\end{aligned}$$ almost surely for all $t\in [0,T]$. Moreover: $$\begin{aligned}
\label{UinVestimate}
{\Vert}U {\Vert}_{ {V^{\alpha,p}_{\rm c}}([0,T]\times \Omega;X_{\eta}^{A})}  & \lesssim 1+{\Vert}x_0{\Vert}_{L^p(\Omega;X_{\eta}^{A})}.\end{aligned}$$

In [@NVW08] the authors assume $\theta_F\leq 0$ and $\theta_G\leq 0$ (and promptly refer to them as $-\theta_F$ and $-\theta_B$). However, one may check that the theorem remains valid for $\theta_F,\theta_G\geq 0$, which leads to extra space regularity of the solution (i.e., greater values for $\eta$ in $\eqref{UinVestimate}$).

Moreover, in [@NVW08] the authors assume $\alpha>{\frac{1}{p}}-\theta_G$, this assumption can be dropped: existence of a solution in ${V^{\alpha,p}_{\rm c}}([0,t]\times\Omega;X)$ for any $\alpha\in [0,{\frac{1}{2}})$ follows by equation . Uniqueness of a solution for any $\alpha\in [0,{\frac{1}{2}})$ follows by observing that if $\Phi \in {V^{\alpha,p}_{\rm c}}([0,t]\times\Omega;X_{\eta}^{A})$ for some $\alpha\in [0,{\frac{1}{2}})$, and $\Phi$ satisfies , then $\Phi \in
{V^{\beta,p}_{\rm c}}([0,t]\times\Omega;X_{\eta}^{A})$ for $0\leq \beta<\alpha+{\frac{1}{2}}+\theta_G-\eta$ (this follows by the proof of Proposition \[prop:stochConvV\]).

Randomized boundedness
----------------------

For vector-valued stochastic integrals, the concept of $\gamma$-boundedness plays the same role as uniform boundedness does for ordinary integrals: the Kalton-Weis multiplier theorem (Proposition \[prop:KW\] below) allows one to estimate terms out of a stochastic integral, provided they are $\gamma$-bounded. Note that any $\gamma$-bounded set of operators is automatically uniformly bounded, and the reverse holds if $X$ is a Hilbert space.

A family $\mathscr{B}\subset {\mathcal L}(X,Y)$ is called [*${\gamma}$-bounded*]{} if there exists a constant $C$ such that for all $N\ge 1$, all $x_1,\dots, x_N\in X$, and all $B_1,\dots, B_N\in \mathscr{B}$ we have: $$\E \Big{\Vert}\sum_{n=1}^N {\gamma}_n B_n x_n\Big{\Vert}_Y^2 \le C^2 \E \Big{\Vert}\sum_{n=1}^N {\gamma}_n x_n\Big{\Vert}_X^2.$$ The least admissible constant $C$ is called the ${\gamma}$-bound of $\mathscr{B}$, notation: ${\gamma}_{[X,Y]}(\mathscr{B})$.

The following lemma is a direct consequence of the Kahane contraction principle:

\[lem:cont\] If $\mathscr{B}\subset {\mathcal L}(X,Y)$ is ${\gamma}$-bounded and $M>0$ then $M\mathscr{B}:= \{ aB: a\in [-M,M], B\in
\mathscr{B}\}$ is ${\gamma}$-bounded with ${\gamma}_{[X,Y]}(M\mathscr{B}) \leq M {\gamma}_{[X,Y]}(\mathscr{B})$.

The following proposition, which is a variation of a result of Weis [@Weis01 Proposition 2.5], gives a sufficient condition for ${\gamma}$-boundedness.

\[prop:integr-der\] Let $f:[0,T]\rightarrow {\mathcal L}(X,Y)$ be a function such that for all $x\in X$ the function $t\mapsto f(t)x$ is continuously differentiable. Suppose $g\in L^1(0,T)$ is such that for all $t\in (0,T)$: $$\begin{aligned}
{\Vert}\tfrac{d}{dt}f(t)x {\Vert}_Y & \leq g(t){\Vert}x{\Vert}_X,\quad \textrm{for all } x\in X.\end{aligned}$$ Then the set $\mathscr{R}: = \{f(t): \ t\in (0,T)\}$ is ${\gamma}$-bounded in $\mathscr{L}(X,Y)$ and $${\gamma}_{[X,Y]}(\mathscr{R}) \le {\Vert}f(0+){\Vert}+ {\Vert}g{\Vert}_{L^1(0,T)}.$$

The following ${\gamma}$-multiplier result, due to Kalton and Weis [@KalWei07] (see also [@Nee-survey]), establishes a relation between stochastic integrability and ${\gamma}$-boundedness.

\[prop:KW\] Suppose $X$ does not contain a closed subspace isomorphic to $c_0$. Suppose $M:(0,T)\to {\mathcal L}(X,Y)$ is a strongly measurable function with $\gamma$-bounded range $\mathcal{M}=\{M(t): \ t\in (0,T)\}$. If $\Phi\in  {\gamma}(0,T,H;X)$ then $M\Phi\in {\gamma}(0,T,H;Y)$ and: $${\Vert}M\Phi{\Vert}_{{\gamma}(0,T,H;Y)} \leq {\gamma}_{[X,Y]}(\mathcal{M})\,
{\Vert}\Phi{\Vert}_{{\gamma}(0,T,H;X)}.$$

The assumption that $X$ should not contain a copy of $c_0$ can be avoided, provided one replaces ${\gamma}(0,T,H;X)$ by ${\gamma}_\infty(L^2(0,T;H),X)$, the space of all ${\gamma}$-summing operators from $L^2(0,T;H)$ to $X$. We refer to [@Nee-survey] for more details. In all applications in this paper, $X$ is a <span style="font-variant:small-caps;">umd</span> space and therefore does not contain a copy of $c_0$.

Finally, we recall the following $\gamma$-boundedness estimate for analytic semigroups (see e.g. [@NVW08 Lemma 4.1]).

\[lem:analyticRbound\] Let $X$ be a Banach space and let $A$ be the generator of an analytic $C_0$-semigroup $S$ of type $(\omega,\theta,K)$ on $X$. Then for all $0\leq \delta<\alpha$ and $T>0$ there exists a constant $C$ depending on $S$ only in terms of $\omega$, $\theta$, and $K$, such that for all $t\in (0,T]$ the set $\mathscr{S}_{\alpha,t} = \{s^\alpha S(s): \ s\in [0,t]\}$ is ${\gamma}$-bounded in ${\mathcal L}(X,X_\delta^A)$ and we have $$\gamma_{[X,X_\delta^A]}(\mathscr{S}_{\alpha,t})\leq C t^{\alpha-\delta}, \quad t\in (0,T].$$

Note that the constant $C$ in the lemma above *may* depend on $T$.

Estimates for (stochastic) convolutions
---------------------------------------

In this section we provide the estimates for (stochastic) convolutions necessary to derive the perturbation result given in Theorem \[t:app\]. In order to avoid confusion further on, we shall use $Y_1$ and $Y_2$ to denote <span style="font-variant:small-caps;">umd</span> Banach spaces in this section.

The following lemma is proven in [@coxNeer:11]. It is an adaptation of [@NVW08 Proposition 4.5].

\[lem:h1\] Let $(R,\mathcal{R},\mu)$ be a finite measure space and $(S,\mathcal{S},\nu)$ a $\sigma$-finite measure space. Let $\Phi_1:[0,T]\times \Omega \rightarrow {\mathcal L}(H,Y_1)$, let $\Phi_2\in
L^1(R;{\mathcal L}(Y_1,Y_2))$, and let $f\in L^{\infty}(R\times [0,T];L^2(S))$. If $\Phi_1$ is $L^p$-stochastically integrable for some $p\in (1,\infty)$, then $$\begin{aligned}
& \Big{\Vert}s\mapsto \int_{0}^{T} \int_R f(r,u)(s)\Phi_2(r)\Phi_1(u)\, d\mu(r) 
\,dW_H(u)\Big{\Vert}_{L^p(\Omega;\gamma(S;Y_2))}\\
& \qquad \qquad\lesssim  \operatorname*{ess\,sup}_{(r,u)\in R\times
[0,T]} {\Vert}f(r,u){\Vert}_{L^2(S)} {\Vert}\Phi_2 {\Vert}_{L^1(R,{\mathcal L}(Y_1,Y_2))}  {\Vert}\Phi_1
{\Vert}_{L^p(\Omega;\gamma(0,T;H,Y_1))},\end{aligned}$$ with implied depending only on $p$, $Y_1$, $Y_2$, provided the right-hand side is finite.

To our knowledge, most regularity results for stochastic convolutions are based on the factorization method introduced in [@DaKwaZab87]. The result below is merely based on the regularity of the convolving functions.

\[lem:stochConv\] Let $T>0$, $p\in [1,\infty)$ and $\eta> 0$. Suppose the process $\Phi\in L^p(\Omega;\gamma(0,T,H;Y_1))$ is adapted to $({\mathcal F}_t)_{t\geq 0}$ and satisfies: $$\begin{aligned}
\sup_{0\leq t\leq T}{\Vert}s\mapsto (t-s)^{-\eta}\Phi(s){\Vert}_{L^p(\Omega;\gamma(0,t,H;Y_1))}<\infty.\end{aligned}$$

Let $\Psi:[0,T]\rightarrow {\mathcal L}(Y_1,Y_2)$ be such that $\Psi x$ is continuously differentiable on $(0,T)$ for all $x\in
Y_1$. Suppose moreover there exists a $g\in L^{1}(0,T)$ and $0 \leq  \theta < \eta$ such that $$\begin{aligned}
  v^\theta {\Vert}\tfrac{d}{dv}\Psi(v) x{\Vert}_{Y_2} + \theta v^{\theta-1} {\Vert}\Psi(v) x{\Vert}_{Y_2}  & \leq g(v){\Vert}x{\Vert}_{Y_1}, \quad \textrm{for all } x\in Y_1.\end{aligned}$$ Then the stochastic convolution process $$t\mapsto \int_{0}^{t}\Psi(t-s)\Phi(s)\,dW_H(s)$$ is well-defined and $$\begin{aligned}
& \Big{\Vert}t\mapsto \int_{0}^{t} \Psi(t-s)\Phi(s)\,dW_H(s) \Big{\Vert}_{C^{\eta-\theta}([0,T];L^p(\Omega;Y_2))} \\
& \qquad \qquad  \leq  2\bar{C}_p {\Vert}g {\Vert}_{L^{1}(0,T)}\sup_{0\leq t\leq T}{\Vert}s\mapsto
(t-s)^{-\eta}\Phi(s){\Vert}_{L^p(\Omega;\gamma(0,t,H;Y_1))},\end{aligned}$$ where $\bar{C}_p$ is the constant in the Burkholder-Davis-Gundy inequality for the $p^{\textrm{\scriptsize{th}}}$ moment, for the space $Y_1$, see equation .

Before proving the Lemma, observe that the corollary below follows directly from Kolmogorov’s continuity criterion (see Theorem I.2.1 in Revuz and Yor).

\[cor:stochConv\] Let the setting be as in Lemma \[lem:stochConv\] and assume in addition that ${\frac{1}{p}}<\eta-\theta$. Let $0<\beta
< \eta-\theta-{\frac{1}{p}}$. There exists a modification of the stochastic convolution process $t\mapsto \int_{0}^{t}\Psi(t-s)\Phi(s)\,dW_H(s)$, which we shall denote by $\Psi\diamond \Phi$, such that: $$\begin{aligned}
& {\Vert}\Psi\diamond \Phi {\Vert}_{L^p(\Omega;C^{\beta}([0,T];Y_2))} \\
& \qquad \qquad\leq \tilde{C} {\Vert}g {\Vert}_{L^{1}(0,T)}\sup_{0\leq t\leq T}{\Vert}s\mapsto
(t-s)^{-\eta}\Phi(s){\Vert}_{L^p(\Omega;\gamma(0,t,H;Y_1))},\end{aligned}$$ where $\tilde{C}$ depends only on $\eta$, $\beta$ and $p$ and $\bar{C}_p$.

By Proposition \[prop:integr-der\] and assumption it follows that $\{s^{\theta}\Psi(s):s\in [0,T] \}$ is $\gamma$-bounded. Thus by the Kalton-Weis multiplier Theorem (see Proposition \[prop:KW\]), and the fact that $$\sup_{0\leq t\leq T}{\Vert}s\mapsto (t-s)^{-\eta}\Phi(s){\Vert}_{L^p(\Omega;\gamma(0,t,H;Y_1))}<\infty,$$ it follows that $s\mapsto \Psi(t-s)\Phi(s)1_{s\in[0,t]}\in L^p(\Omega;\gamma(0,t,H;Y_2))$ for all $t\in [0,T]$. By Theorem \[t:stochint\] this process is stochastically integrable.

In what follows we let $\frac{d}{dv}$ denote the derivative with respect to the strong operator topology. By the triangle inequality we have: $$\begin{aligned}\label{hoelderLp}
&\Big{\Vert}\int_{0}^{t} \Psi(t-u)\Phi(u)\,dW_H(u)  -  \int_{0}^{s} \Psi(s-u)\Phi(u)\,dW_H(u) \Big{\Vert}_{L^p(\Omega;Y_2)}\\
& \qquad \qquad \leq \Big{\Vert}\int_{0}^{s} [\Psi(t-u)-\Psi(s-u)]\Phi(u)\,dW_H(u) \Big{\Vert}_{L^p(\Omega;Y_2)} \\
& \qquad \qquad \quad + \Big{\Vert}\int_{s}^{t} \Psi(t-u)\Phi(u)\,dW_H(u) \Big{\Vert}_{L^p(\Omega;Y_2)}\\
& \qquad\qquad  = \Big{\Vert}\int_{0}^{s} \int_{s-u}^{t-u}\tfrac{d}{dv}\Psi(v) \,dv \Phi(u)\,dW_H(u) \Big{\Vert}_{L^p(\Omega;Y_2)}\\
& \qquad \qquad \quad + \Big{\Vert}\int_{s}^{t} (t-u)^{-\theta}\int_{0}^{t-u}\tfrac{d}{dv}[v^{\theta}\Psi(v) ]\,dv \Phi(u)\,dW_H(u)
\Big{\Vert}_{L^p(\Omega;Y_2)}.
\end{aligned}$$ We now wish to apply the stochastic Fubini theorem (see [@CoxGor:11 Lemma 2.7], [@NV:06]). Consider $\Upsilon:[0,s]\times [0,t]\rightarrow {\mathcal L}(H,Y)$ defined by $\Upsilon(u,v)= 1_{\{s-u\leq v\leq t-u\}}\tfrac{d}{dv}\Psi(v)\Phi(u)$. As $\tfrac{d}{dv}\Psi$ is strongly continuous and $\Phi$ is $H$-strongly measurable, we have that $\Upsilon$ is $H$-strongly measurable. Moreover, as $\Phi$ is adapted it follows that $\Upsilon_{v}:=\Upsilon(\cdot,v)$ is adapted for almost all $v\in [0,t]$. Finally, we have that $\Upsilon\in L^1(0,t;\gamma(0,s,H;Y_2))$ by assumption: $$\begin{aligned}
{\Vert}\Upsilon(\cdot,v){\Vert}_{\gamma(0,s,H;Y_2)} \leq v^{\eta-\theta}g(v){\Vert}u\mapsto (s-u)^{-\eta}\Phi(u){\Vert}_{\gamma(0,s,H;Y_1)},\end{aligned}$$ where we use that $v\geq s-u$ on $\textrm{supp}(\Upsilon)$.

Note that stochastic Fubini theorem in [@CoxGor:11 Lemma 2.7], [@NV:06] requires $\Upsilon_v$ to be progressive. However, it suffices to assume that $\Upsilon_v$ is adapted, see [@Ver:11]. Thus the conditions necessary to apply the stochastic Fubini theorem are satisfied, and we have: $$\label{mainReg1}
\begin{aligned}
&\Big{\Vert}\int_{0}^{s} \int_{s-u}^{t-u} \tfrac{d}{dv}[\Psi(v)\Phi(u)]\,dv\,dW_H(u) \Big{\Vert}_{L^p(\Omega;Y_2)}\\
& \quad = \Big{\Vert}\int_{0}^{t} \int_{(s-v){\vee}0}^{(t-v){\wedge}s}\tfrac{d}{dv}[\Psi(v)\Phi(u)]\,dW_H(u)
\,dv\Big{\Vert}_{L^p(\Omega;Y_2)}\\
& \quad \leq \int_0^{t} v^{-\theta} g(v) \Big{\Vert}\int_{(s-v){\vee}0}^{(t-v){\wedge}s} \Phi(u)\,dW_H(u)
\Big{\Vert}_{L^p(\Omega;Y_1)}\,dv\\
& \quad \leq \bar{C}_p \int_0^{t} v^{-\theta} g(v) \big{\Vert}1_{[(s-v){\vee}0,(t-v){\wedge}s]}\Phi
\big{\Vert}_{L^p(\Omega;\gamma(0,t; Y_1))}\,dv\\
& \quad \leq \bar{C}_p \int_{0}^{t} v^{-\theta} g(v) [(t-s){\wedge}v]^{\eta}\big{\Vert}u\mapsto ([((t-v){\wedge}s)-u]^{-\eta}\Phi(u))\big{\Vert}_{L^p(\Omega;\gamma(0,t; Y_1))}\,dv\\
& \quad \leq \bar{C}_p (t-s)^{\eta-\theta} \int_{0}^{t} g(v)\,dv \sup_{t\in [0,T]}\big{\Vert}u\mapsto
(t-u)^{-\eta}\Phi(u)\big{\Vert}_{L^p(\Omega;\gamma(0,t; Y_1))}.
\end{aligned}$$ For the final term in one may also check that the conditions of the stochastic Fubini hold and thus: $$\begin{aligned}
&  \Big{\Vert}\int_{s}^{t} (t-u)^{-\theta}\int_{0}^{t-u}\tfrac{d}{dv}[v^{\theta}\Psi(v)\Phi(u)]\,dv\,dW_H(u)
\Big{\Vert}_{L^p(\Omega;Y_2)}\\
& \qquad \qquad \leq \int_{0}^{t-s} g(v) \Big{\Vert}\int_{s}^{t-v} (t-u)^{-\theta}\Phi(u)\,dW_H(u) \Big{\Vert}_{L^p(\Omega;Y_1)}\,dv\\
& \qquad \qquad \leq \bar{C}_p \int_{0}^{t-s}  g(v) \big{\Vert}u\mapsto 1_{[s,t-v]}(u) (t-u)^{-\theta}\Phi(u)
\big{\Vert}_{L^p(\Omega;\gamma(0,t,H;F_1))}\,dv\\
& \qquad \qquad \leq \bar{C}_p (t-s)^{\eta-\theta} {\Vert}g{\Vert}_{L^1(0,T)} \sup_{t\in [0,T]}\big{\Vert}u\mapsto
(t-u)^{-\eta}\Phi(u)\big{\Vert}_{L^p(\Omega;\gamma(0,t; Y_1))}.\end{aligned}$$ By inserting the two estimates above in we obtain that $$\begin{aligned}
&\Big{\Vert}\int_{0}^{t} \Psi(t-u)\Phi(u)\,dW_H(u) - \int_{0}^{s} \Psi(s-u)\Phi(u)\,dW_H(u) \Big{\Vert}_{L^p(\Omega;Y_2))}\\
& \qquad\qquad  \leq 2\bar{C}_p(t-s)^{\eta-\theta}{\Vert}g{\Vert}_{L^1(0,T)} \sup_{t\in [0,T]}\big{\Vert}u\mapsto
(t-u)^{-\eta}\Phi(u)\big{\Vert}_{L^p(\Omega;\gamma(0,t; Y_1))},\end{aligned}$$ which completes the proof as $0\leq s<t\leq T$ where chosen arbitrarily.

In the setting of the lemma above one may also take $g\in L^{q'}(0,T)$ and $\Phi$ such that $$\int_{0}^{T}\big{\Vert}s \mapsto (t-s)^{-\eta}\Phi(s)\big{\Vert}_{L^p(\Omega;\gamma(0,t; Y_1))}^q dt<\infty,$$ where $q\in [1,\infty]$; ${\frac{1}{q}}+{\frac{1}{q'}}=1$. In that case one obtains: $$\begin{aligned}
& \Big{\Vert}t\mapsto \int_{0}^{t} \Psi(t-s)\Phi(s)\,dW_H(s) \Big{\Vert}_{C^{\eta-\theta}([0,T];L^p(\Omega;Y_2))} \\
& \qquad \qquad \leq (3+2^{\eta})\bar{C}_p {\Vert}g {\Vert}_{L^{q'}(0,T)} \Big(\int_{0}^{T}\big{\Vert}s\mapsto (t-s)^{-\eta}\Phi(s)\big{\Vert}_{L^p(\Omega;\gamma(0,t,H;Y_1))}^{q}\,dt\Big)^{{\frac{1}{q}}}.\end{aligned}$$ We omit the proof because it requires significantly more space, and we do not need this result in what follows (for the extended proof, see [@cox:thesis Lemma A.9]).

Based on the above two lemmas, we obtain the following result for stochastic convolutions in the ${V^{\alpha,p}_{\rm c}}$-norm:

\[prop:stochConvV\] Let the setting be as in Lemma \[lem:stochConv\] and assume in addition that ${\frac{1}{p}}< \eta-\theta$. Let $\alpha\in [0,{\frac{1}{2}})$. Then $\Psi\diamond \Phi\in {V^{\alpha,p}_{\rm c}}([0,T]\times\Omega;Y_2)$. Moreover, there exists a constant $C$ such that for all $T_0\in [0,T]$ we have: $$\begin{aligned}
{\Vert}\Psi\diamond \Phi {\Vert}_{{V^{\alpha,p}_{\rm c}}([0,T_0]\times\Omega;Y_2)} & \leq C  {\Vert}g {\Vert}_{L^{1}(0,T)} \sup_{0\leq t\leq
T_0}{\Vert}s\mapsto (t-s)^{-\eta}\Phi(s){\Vert}_{L^p(\Omega;\gamma(0,t;Y_1))}.\end{aligned}$$

For the norm estimate in $L^p(\Omega;C([0,T_0];Y_2))$ we apply Corollary \[cor:stochConv\] with $\beta>0$ such that $0\leq \beta+\theta < \eta - {\frac{1}{p}}$. For the estimate in the weighted $\gamma$-norm fix $t\in [0,T_0]$. We apply Lemma \[lem:h1\] with $\Phi_1(u)=(t-u)^{-\eta}\Phi(u)1_{0\leq u < t}$, $\Phi_2(r) = \frac{d}{dr}[r^{\theta}\Psi(r)]$, $R=[0,t]$ and $f(r,u)(s)= (t-s)^{-\alpha}(s-u)^{-\theta}(t-u)^{\eta}1_{0\leq r< s-u}1_{0\leq u < t}$. From Lemma \[lem:h1\] it follows that: $$\begin{aligned}
& \Big{\Vert}s\mapsto (t-s)^{-\alpha}\int_{0}^{s}\Psi(s-u)\Phi(u)\,dW_H(u)\Big{\Vert}_{L^p(\Omega;\gamma(0,t;Y_2))} \\
&\qquad\qquad  \lesssim t^{{\frac{1}{2}}+\eta-\alpha-\theta}{\Vert}g{\Vert}_{L^1(0,T)}{\Vert}s\mapsto
(t-s)^{-\eta}\Phi(s){\Vert}_{L^p(\Omega;\gamma(0,t;Y_1))}.\end{aligned}$$ Taking the supremum over $t\in[0,T_0]$ and using that $\eta-\theta>0$ (whence $T_0^{{\frac{1}{2}}+\eta-\alpha-\theta}\leq T^{{\frac{1}{2}}+\eta-\alpha-\theta}$) we arrive at the desired result.

For deterministic convolutions we have the following:

\[prop:detHolder\] Suppose $\Phi\in L^p(\Omega;L^{\infty}(0,T;Y_1))$ for some $p\in [1,\infty)$. Let $\Psi:[0,T]\rightarrow {\mathcal L}(Y_1,Y_2)$ be such that $\Psi x$ is continuously differentiable on $(0,T)$ for all $x\in Y_1$ Suppose moreover there exists a $g\in L^{1}(0,T)$ and a $\theta\in [0,1]$ such that for all $v\in (0,T)$ we have: $$\begin{aligned}
v^\theta {\Vert}\tfrac{d}{dv}\Psi(v) x{\Vert}_{Y_2} + \theta v^{\theta-1} {\Vert}\Psi(v) x{\Vert}_{Y_2}  & \leq g(v){\Vert}x{\Vert}_{Y_1}, \quad \textrm{for all } x\in Y_1.\end{aligned}$$ Then there exists a constant $C$ such that for all $T_0\in [0,T]$ we have, almost surely: $$\begin{aligned}
{\Vert}\Psi*\Phi{\Vert}_{C^{1-\theta} ([0,T_0];Y_2)} & \leq C {\Vert}g {\Vert}_{L^1(0,T_0)} {\Vert}\Phi{\Vert}_{L^{\infty}(0,T_0;Y_1)}.\end{aligned}$$

By Lemma \[lem:VEmbedHoelder\], with ${\varepsilon}=\frac{3}{2}-{\frac{1}{\tau}}T-\theta$, we obtain the following corollary:

\[cor:detConvV\] Let the setting be as in Proposition \[prop:detHolder\]. Assume in addition that $Y_2$ has type $\tau$, and let $0\leq \theta < \frac{3}{2}-{\frac{1}{\tau}}$. Then for $\alpha\in [0,{\frac{1}{2}})$ and $p\in [1,\infty)$ there exists a constant $C$ such that for $T_0 \in [0,T]$ one has: $$\begin{aligned}
{\Vert}\Psi * \Phi {\Vert}_{{V^{\alpha,p}_{\rm c}}([0,T_0]\times\Omega;Y_2)} \leq C {\Vert}g {\Vert}_{L^{1}(0,T_0)} {\Vert}\Phi{\Vert}_{L^p(\Omega;L^{\infty}(0,T_0;Y_1))}.\end{aligned}$$

Observe that we have, for $0\leq s< t\leq T_0$: $$\begin{aligned}\label{hoelderC}
&\Big{\Vert}\int_{0}^{t} \Psi(t-u)\Phi(u,\omega)\,du  -  \int_{0}^{s} \Psi(s-u)\Phi(u,\omega)\,du \Big{\Vert}_{Y_2} \\
& \qquad  \qquad \leq \Big{\Vert}\int_{0}^{s} \int_{s-u}^{t-u} \tfrac{d}{dv}[\Psi(v)\Phi(u,\omega)]\,dv\,du \Big{\Vert}_{Y_2}\\
& \qquad \qquad\quad + \Big{\Vert}\int_{s}^{t} (t-u)^{-\theta}\int_{0}^{t-u}\tfrac{d}{dv}[v^{\theta}\Psi(v) \Phi(u,\omega)]\,dv
\,du
\Big{\Vert}_{Y_2}.
\end{aligned}$$ Now $$\begin{aligned}
& \Big{\Vert}\int_{0}^{s} \int_{s-u}^{t-u}  \tfrac{d}{dv}[\Psi(v)\Phi(u,\omega)]\,dv\,du \Big{\Vert}_{Y_2}\\
& \qquad \qquad \leq \int_{0}^{s} \int_{(s-v){\vee}s}^{(t-v){\wedge}s} \,du (s-v)^{-\theta}g(v) \,dv {\Vert}\Phi(\omega){\Vert}_{L^{\infty}(0,t;Y_1)}\\
& \qquad \qquad \leq (t-s)^{1-\theta}\int_{0}^{t} g(v) \,dv {\Vert}\Phi(\omega){\Vert}_{L^{\infty}(0,t;Y_1)},
\end{aligned}$$ where we used that $\int_{(s-v){\vee}s}^{(t-v){\wedge}s} \,du \leq (t-s){\wedge}v$. Furthermore, we have $$\begin{aligned}
&\Big{\Vert}\int_{s}^{t} (t-u)^{-\theta}\int_{0}^{t-u}\tfrac{d}{dv}[v^{{\varepsilon}}\Psi(v)] \Phi(u,\omega)\,dv \,du \Big{\Vert}_{Y_2} \\
&\qquad \qquad \leq
(1-\theta)^{-1}(t-s)^{1-\theta}\Big{\Vert}\int_{0}^{t} g(v) \,dv {\Vert}\Phi(\omega){\Vert}_{L^{\infty}(0,T;Y_1)}.
\end{aligned}$$ Inserting these two estimates in completes the proof.

A perturbation result {#sec:app}
=====================

In this section we shall prove the perturbation theorem announced in the introduction. Consider with $A$, $F$ and $G$ satisfying [[**(A)**]{}]{}, [[**(F)**]{}]{}, and [[**(G)**]{}]{}. Keeping in mind possible applications in approximations of solutions to stochastic partial differential equations, we let $X_0$ be a (possibly finite-dimensional) closed subspace of $X$. We assume there exists a bounded projection $P_0:X\rightarrow X_0$ such that $P_0(X)=X_0$. Let $i_{X_0}$ represent the canonical embedding of $X_0$ into $X$ (note however that we shall omit $i_{X_0}$ when it is clear from the context).

Let $A_0$ be the generator of an analytic $C_0$-semigroup $S_0$ on $X_0$. For $t\geq 0$ define $\tilde{S}_0\in {\mathcal L}(X)$ by $
\tilde{S}_0(t):=i_{X_0}S_0(t) P_0,$ this defines a *degenerate* $C_0$-semigroup, i.e., $\tilde{S}_0$ satisfies the semigroup property but $\tilde{S}_0(0)=i_{X_0}P_0$ (which is clearly not the identity unless $X_0=X$).

Let $x_0\in L^p(\Omega,{\mathcal F}_0;X)$ (where $p>2$ satisfies ${\frac{1}{p}}\leq {\frac{1}{2}}+\theta_G$) and let $U$ be the solution to as provided by Theorem \[thm:NVW08\].

\[t:app\] Let $\omega\geq 0$, $\theta\in(0,\frac{\pi}{2})$ and $K>0$ be such that $A$ and $A_0$ are both of type $(\omega,\theta,K)$. Suppose there exist $\delta\in [0,1]$ and $p\in (2,\infty)$ satisfying $$0\leq \delta < \min\{\tfrac{3}{2}-{\tfrac{1}{\tau}}+\theta_F,\, {\tfrac{1}{2}}-{\tfrac{1}{p}}+\theta_G\}$$ such that for some $\lambda_0\in \rho(A)$ we have: $$\begin{aligned}
\label{app:ass}
D_{\delta}(A,A_0):=  {\Vert}R(\lambda_0:A)-i_{X_0} R(\lambda_0:A_0)P_0 {\Vert}_{{\mathcal L}(X_{\delta-1}^{A},X)} < \infty.\end{aligned}$$ Suppose $x_0\in L^p(\Omega,{\mathcal F}_0;X_{\delta}^{A})$ and $y_0 \in L^p(\Omega,{\mathcal F}_0;X)$.

For any $\alpha\in [0,{\frac{1}{2}})$ there exists a unique process $U^{(0)}\in {V^{\alpha,p}_{\rm c}}([0,T_0]\times\Omega;X_0)$ such that $s\mapsto 1_{[0,t]}S_0(t-s)P_0G(s,U^{(0)}(s))$ is stochastically integrable for all $t\in [0,T]$ and for all $t\in [0,T]$ we have: $$\begin{aligned}\label{pert.sol}
U^{(0)}(t) = S_0(t-s)P_0 y_0 & + \int_{0}^{T} S_0(t-s)P_0F(s,U^{(0)}(s))\,ds\\
& + \int_{0}^{t}
S_0(t-s)P_0 G(s,U^{(0)}(s))\,dW_H(s), \quad \textrm{a.s.}
\end{aligned}$$ Moreover: $$\begin{aligned}\label{pert_est}
& {\Vert}U - i_{X_0}U^{(0)} {\Vert}_{{V^{\alpha,p}_{\rm c}}([0,T]\times \Omega;X)} \\
& \qquad\qquad \lesssim  {\Vert}x_0 - y_0 {\Vert}_{L^p(\Omega;X)} + D_{\delta}(A,A_0)(1+{\Vert}x_0 {\Vert}_{L^p(\Omega;X_{\delta}^{A})}).
\end{aligned}$$ The implied constant depends on $X_0$ only in terms of ${\Vert}P_0{\Vert}_{{\mathcal L}(X,X_0)}$, on $A$ and $A_0$ only in terms of $1+D_{\delta}(A,A_0)$, $\omega$, $\theta$ and $K$, and on $F$ and $G$ only in terms of their Lipschitz and linear growth constants Lip$(F)$, $\textrm{Lip}_{\gamma}(G)$, $M(F)$ and $M_{\gamma}(G)$.

To prove Theorem \[t:app\] we need a proposition concerning the $\gamma$-boundedness of $S-\tilde{S}_0$. The proof of this proposition is postponed to the end of this section.

\[p:gbddSG\] Let $A$, $A_0$ be as introduced above, i.e., $A$ generates an analytic semigroup on $X$ and $A_0$ generates an analytic semigroup on $X_0$. Let $\omega\geq 0, \theta\in (0,\frac{\pi}{2})$ and $K>0$ be such that $A$ and $A_0$ are of type $(\omega,\theta,K)$. Suppose there exists a $\lambda_0\in \C$, $\mathscr{R}e(\lambda_0) > \omega$, and $\delta\in \R$ such that $D_{\delta}(A,A_0)<\infty$, where $D_{\delta}(A,A_0)$ is as defined in . Set $$\omega'=\omega+|\lambda_0-\omega|(\cos\theta)^{-1}.$$ Then for all $\beta \in \R$ such that $\beta\in [\delta-1,\delta]$ one has: $$\begin{aligned}
\label{p:gbddsg:unif1}
\sup_{t\in [0,\infty)} t^{\delta-\beta} e^{-\omega' t} {\Vert}S(t) - \tilde{S}_0(t){\Vert}_{{\mathcal L}(X_{\beta}^{A},X)} & \lesssim D_{\delta}(A,A_0),\end{aligned}$$ and $$\begin{aligned}
\label{p:gbddsg:unif2}
\sup_{t\in [0,\infty)} t^{\delta-\beta+1}e^{-\omega' t} {\Vert}\tfrac{d}{dt}S(t)-\tfrac{d}{dt}\tilde{S}_0(t) {\Vert}_{{\mathcal L}(X_{\beta}^{A},X)}
\lesssim D_{\delta}(A,A_0),\end{aligned}$$ with implied constants depending only on ${\Vert}P_0{\Vert}_{{\mathcal L}(X,X_0)}$, $\omega$, $\theta$, $K$, $\delta-\beta$.

Moreover, for all $\alpha > \delta-\beta$ we have, for $t\in [0,T]$: $$\begin{aligned}
\gamma_{[X_{\beta}^{A},X]}\left(\left\{  s^{\alpha} [ S(s) - \tilde{S}_0(s)]; \, 0\leq s\leq t\right\}\right) &
\lesssim t^{\alpha+\beta-\delta} D_{\delta}(A,A_0),\end{aligned}$$ with implied constant depending only on ${\Vert}P_0{\Vert}_{{\mathcal L}(X,X_0)}$, $\omega$, $\theta$, $K$, $\delta-\beta$, and $T$.

We split the proof into several parts.

Part 1. {#part-1. .unnumbered}
-------

In order to prove existence and uniqueness of $U^{(0)}\in
{V^{\alpha,p}_{\rm c}}([0,T_0]\times\Omega;X_{0})$ satisfying it suffices, by Theorem \[thm:NVW08\], to prove that there exist $\eta_{F}>-\frac{3}{2}+{\frac{1}{\tau}}$ and $\eta_{G}>-{\frac{1}{2}}+{\frac{1}{p}}$ such that $P_0 F:[0,T]\times X\rightarrow X_{0,\eta_{F}}^{A_0}$ is Lipschitz continuous and of linear growth and $P_0
G:[0,T]\times X\rightarrow
\gamma(H,X_{0,\eta_{G}}^{A_0})$ is $L^2_{\gamma}$-Lipschitz continuous and of linear growth. If $\theta_F\geq 0$ then clearly we may take $\eta_{F}=0$, and we have Lip$(P_0F)\leq {\Vert}P_0{\Vert}_{{\mathcal L}(X,X_0)}\textrm{Lip}(F)$, $M(P_0F)\leq {\Vert}P_0{\Vert}_{{\mathcal L}(X,X_0)}M(F)$. The same goes for $\theta_G\geq 0$.

Now suppose $\theta_F<0$. Recall the following representation of negative fractional powers of an operator $A$ generating an analytic semigroup $S$ of type $(\omega,\theta,K)$ (see [@Pazy:83 Chapter 2.6]): $$(\lambda I-A)^{\eta}={\frac{1}{\Gamma(-\eta)}}\int_0^{\infty}t^{-\eta-1}e^{-\lambda t}S(t)dt, \quad  \eta<0,\
\mathscr{R}e(\lambda)>\omega.$$ Let $\bar{\omega}>\omega'$, where $\omega'$ is as in Proposition \[p:gbddSG\]. From the representation above and Proposition \[p:gbddSG\] it follows that for $\beta\in [\delta-1,\delta]$, $\eta<\beta-\delta$ and $x\in X$ we have: $$\begin{aligned}
& {\Vert}P_0 x {\Vert}_{X_{0,\eta}^{A_0}} \eqsim {\Vert}((\bar{\omega} I - A_0)^{\eta}P_0 x {\Vert}_{X} = \Big{\Vert}{\frac{1}{\Gamma(-\eta)}}\int_0^{\infty}t^{-\eta-1}e^{-\bar{\omega} t}\tilde{S}_0(t) x dt\Big{\Vert}_{X}\\
&\qquad   \leq {\frac{1}{\Gamma(-\eta)}}\int_0^{\infty}t^{-\eta-1}e^{-\bar{\omega} t}{\Vert}(S(t)-\tilde{S}_0(t))x{\Vert}_{X}dt  \\
&\qquad  \quad + {\frac{1}{\Gamma(-\eta)}}\Big{\Vert}\int_0^{\infty}t^{-\eta-1}e^{-\bar{\omega} t}S(t) x dt \Big{\Vert}_{X}\\
&\qquad \lesssim D_{\delta}(A,A_0)\int_0^{\infty}t^{-\eta-1+\beta-\delta}e^{-(\bar{\omega}-\omega') t} dt {\Vert}x {\Vert}_{X_{\beta}^{A}} + {\Vert}(\bar{\omega} I -
A)^{\eta }x{\Vert}_X,\end{aligned}$$ with implied constants depending on $X_0$ only in terms of ${\Vert}P_0{\Vert}_{{\mathcal L}(X,X_0)}$ and on $A$ and $A_0$ only in terms of $\omega,\theta,$ and $K$. Thus for $\beta\in [\delta-1,\delta]$, $\eta<\beta-\delta$ we have: $$\begin{aligned}\label{fracspaceEmbed}
{\Vert}P_0 x {\Vert}_{X_{0,\eta}^{A_0}} & \eqsim {\Vert}(\bar{\omega} I - A_0)^{\eta}P_0 x {\Vert}_{X} \\
& \lesssim (1+D_{\delta}(A,A_0)){\Vert}(\bar{\omega} I - A)^{\beta} x {\Vert}_{X} \eqsim (1+D_{\delta}(A,A_0)){\Vert}x {\Vert}_{X_{\beta}^{A}},
\end{aligned}$$ with implied constants depending on $X_0$ only in terms of ${\Vert}P_0{\Vert}_{{\mathcal L}(X,X_0)}$ and on $A$ and $A_0$ only in terms of $\omega,\theta,$ and $K$.

Note that by assumption we have $\theta_F> -\frac{3}{2}+{\frac{1}{\tau}} + \delta \geq \delta -1$. Hence one can pick $\eta_{F}$ such that $-\frac{3}{2}+{\frac{1}{\tau}}<\eta_{F}<\theta_F-\delta$. By it follows that $P_0 F:[0,T]\times
X\rightarrow
X_{0,\eta_{F}}^{A_0}$ is Lipschitz continuous and $$\begin{aligned}
\label{eq:LipP0F}
\textrm{Lip}(P_0F)& \lesssim (1+D(A,A_0))\textrm{Lip}(F); & M(P_0F)\lesssim (1+D(A,A_0))M(F),\end{aligned}$$ with implied constant depending on $X_0$ only in terms of ${\Vert}P_0{\Vert}_{{\mathcal L}(X,X_0)}$ and on $A$ and $A_0$ only in terms of $\omega,\theta,$ and $K$.

Similarly, if $\theta_G<0$ there exists a $\eta_{G}$ such that $-{\frac{1}{2}}+{\frac{1}{p}}<\eta_{G}<\theta_G-\delta$ such that $P_0 G:[0,T]\times X\rightarrow
\gamma(H,X_{0,\eta_{G}}^{A_0})$ is $L^2_{\gamma}$-Lipschitz continuous and $$\begin{aligned}
\label{eq:LipP0G}
\textrm{Lip}_{\gamma}(P_0G)& \lesssim (1+D(A,A_0))\textrm{Lip}_{\gamma}(G); & M_{\gamma}(P_0G)\lesssim (1+D(A,A_0))M_{\gamma}(G),\end{aligned}$$ with implied constant depending on $X_0$ only in terms of ${\Vert}P_0{\Vert}_{{\mathcal L}(X,X_0)}$ and on $A$ and $A_0$ only in terms of $\omega,\theta,$ and $K$.

Part 2. {#part-2. .unnumbered}
-------

Define $\tilde{U}^{(0)} = i_{X_0}U^{(0)}$ and observe that if $U^{(0)}$ satisfies , then $\tilde{U}^{(0)}$ satisfies: $$\begin{aligned}
\tilde{U}^{(0)}(t) = \tilde{S}_0(t-s) y_0 &+ \int_{0}^{T} \tilde{S}_0(t-s)F(s,\tilde{U}^{(0)}(s))\,ds\\
& + \int_{0}^{t} \tilde{S}_0(t-s)G(s,\tilde{U}^{(0)}(s))\,dW_H(s), \quad \textrm{a.s.}\end{aligned}$$ Let $T_0\in [0,T]$ be fixed. By the above we have: $$\label{pert.split}
\begin{aligned}
& {\Vert}U - \tilde{U}^{(0)}{\Vert}_{{V^{\alpha,p}_{\rm c}}([0,T_0]\times\Omega;X)} \\
& \qquad \leq {\Vert}(S-\tilde{S}_0)x_0
{\Vert}_{{V^{\alpha,p}_{\rm c}}([0,T_0]\times\Omega;X)} + {\Vert}\tilde{S}_0(x_0-y_0) {\Vert}_{{V^{\alpha,p}_{\rm c}}([0,T_0]\times\Omega;X)}\\
& \qquad\quad + \Big{\Vert}t\mapsto \int_0^{t}\tilde{S}_0(t-s)[F(s,U(s))-F(s,\tilde{U}^{(0)}(s))]\,ds
\Big{\Vert}_{{V^{\alpha,p}_{\rm c}}([0,T_0]\times\Omega;X)}\\
& \qquad\quad + \Big{\Vert}t\mapsto \int_0^{t}[S(t-s)-\tilde{S}_0(t-s)]F(s,U(s))\,ds
\Big{\Vert}_{{V^{\alpha,p}_{\rm c}}([0,T_0]\times\Omega;X)}\\
& \qquad\quad + \Big{\Vert}t\mapsto \int_0^{t}\tilde{S}_0(t-s)[G(s,U(s))-G(s,\tilde{U}^{(0)}(s))]\,dW_H(s)
\Big{\Vert}_{{V^{\alpha,p}_{\rm c}}([0,T_0]\times\Omega;X)}\\
& \qquad\quad + \Big{\Vert}t\mapsto \int_0^{t}[S(t-s)-\tilde{S}_0(t-s)]G(s,U(s))\,dW_H(s)
\Big{\Vert}_{{V^{\alpha,p}_{\rm c}}([0,T_0]\times\Omega;X)}.
\end{aligned}$$

Let $\eta_{F}$ and $\eta_{G}$ be as defined in part 1. Let ${\varepsilon}>0$ be such that $$\begin{aligned}
{\varepsilon}&\leq 1-2\alpha;\\
{\varepsilon}&<\min\{\tfrac{3}{2}-{\tfrac{1}{\tau}}+\eta_{F},{\tfrac{1}{2}}-{\tfrac{1}{p}}+\eta_{G}\}.\end{aligned}$$ It follows that ${\varepsilon}+ \delta < \min\{\tfrac{3}{2}-{\tfrac{1}{\tau}}+\theta_{F},{\tfrac{1}{2}}-{\tfrac{1}{p}}+\theta_{G}\}$. By equation we may assume, without loss of generality, that $\alpha={\frac{1}{2}}-{\varepsilon}/2$.

We will estimate each of the six terms on the right-hand side of in parts 2a-2f below. In part 2c and 2e we keep track of the dependence on $T_0$, for the other parts this is not necessary.

### Part 2a. {#part-2a. .unnumbered}

By Proposition \[p:gbddSG\] with $\beta=\delta$ there exists an ${\mathcal M}>0$ depending on $X_0$ only in terms of ${\Vert}P_0{\Vert}_{{\mathcal L}(X,X_0)}$, and on $A$ and $A_0$ only in terms om $\omega$, $\theta$ and $K$, such that $$\begin{aligned}
\sup_{t\in [0,T_0]}{\Vert}S(t) - \tilde{S}_0(t){\Vert}_{{\mathcal L}(X_{\delta}^{A},X)}\leq {\mathcal M}D_{\delta}(A,A_0);\\
\gamma_{[X_{\delta}^{A},X]}\{t^{{\varepsilon}/2}(S(t)-\tilde{S}_0(t))\,:\,t\in[0,T_0]\}\leq {\mathcal M}D_{\delta}(A,A_0).\end{aligned}$$ Thus by Proposition \[prop:KW\] we have $$\begin{aligned}
& {\Vert}(S-\tilde{S}_0)x_0{\Vert}_{{V^{\alpha,p}_{\rm c}}([0,T_0]\times\Omega;X)} \\
& \quad \leq {\mathcal M}D_{\delta}(A,A_0)\big[\sup_{t\in[0,T_0]}{\Vert}s\mapsto
(t-s)^{-\alpha}s^{-{\varepsilon}/2}x_0{\Vert}_{L^p(\Omega;\gamma(0,t;X_{\delta}^{A}))}+ {\Vert}x_0{\Vert}_{L^p(\Omega;X_{\delta}^{A})}\big].\end{aligned}$$ For $f\in L^2(0,t)$ and $x\in L^p(\Omega;X_{\delta}^{A})$ we have $${\Vert}f\otimes
x{\Vert}_{L^p(\Omega;\gamma(0,t;X_{\delta}^{A}))}={\Vert}f{\Vert}_{L^2(0,t)}{\Vert}x{\Vert}_{L^p(\Omega;X_{\delta}^{A})}.$$ Thus, recalling that $\alpha={\frac{1}{2}}-{\varepsilon}/2$, we have: $$\begin{aligned}
&\sup_{t\in[0,T_0]}{\Vert}s\mapsto
(t-s)^{-\alpha}s^{-{\varepsilon}/2}x_0{\Vert}_{L^p(\Omega;\gamma(0,t;X_{\delta}^{A}))} \\
&\qquad \qquad  \leq {\Vert}s\mapsto
(1-s)^{-\alpha}s^{-{\varepsilon}/2} {\Vert}_{L^2(0,1)} {\Vert}x_0{\Vert}_{L^p(\Omega;X_{\delta}^{A})} \leq C_{{\varepsilon}} {\Vert}x_0{\Vert}_{L^p(\Omega;X^A_{\delta})},\end{aligned}$$ where $C_{{\varepsilon}}$ is a constant depending only on ${\varepsilon}$, and we used that $\alpha={\frac{1}{2}}-{\varepsilon}/2$. Hence $$\label{pert.h1}
\begin{aligned}
{\Vert}(S-\tilde{S}_0)x_0{\Vert}_{{V^{\alpha,p}_{\rm c}}([0,T_0]\times\Omega;X)} & \leq {\mathcal M}D_{\delta}(A,A_0)(1+C_{{\varepsilon}}){\Vert}x_0{\Vert}_{L^p(\Omega;X_{\delta}^{A})}.
\end{aligned}$$

### Part 2b. {#part-2b. .unnumbered}

By assumption (see Remark \[r:analcont\]) there exists an ${\mathcal M}$ depending only on ${\Vert}P_0{\Vert}_{{\mathcal L}(X,X_0)}$, $\omega,\theta,$ and $K$ and $T$ such that we have that $\sup_{t\in [0,T]}{\Vert}\tilde{S}_0(t){\Vert}_{{\mathcal L}(X,X_0)} \leq {\mathcal M}$. Moreover, by Lemma \[lem:analyticRbound\] we may pick ${\mathcal M}$ such that in addition we have that $\gamma_{[X,X]}\{t^{{\varepsilon}/2}\tilde{S}_0(t)\,:\,t\in[0,T_0]\}\leq
{\mathcal M}.$ Thus by the same argument as in part 2a we have: $$\label{pert.h2}
\begin{aligned}
{\Vert}\tilde{S}_0(x_0-y_0) {\Vert}_{{V^{\alpha,p}_{\rm c}}([0,T_0]\times\Omega;X)} & \leq {\mathcal M}(1+C_{{\varepsilon}}) {\Vert}x_0 -y_0 {\Vert}_{L^p(\Omega;X)}.
\end{aligned}$$

### Part 2c. {#part-2c. .unnumbered}

Recall that $\eta_{F}\leq 0$. By equation there exists an ${\mathcal M}$ depending only on $\omega$, $\theta$, $K$ and $T$ such that for all $t\in [0,T]$ we have: $$\begin{aligned}
& t^{-\eta_{F}+{\varepsilon}} {\Vert}\tfrac{d}{dt}S_0(t) x{\Vert}_{{\mathcal L}(X_{0,\eta_{F}}^{A_0},X)} + ({\varepsilon}-\eta_{F}) t^{-\eta_{F}+{\varepsilon}-1} {\Vert}S_{0}(t) x{\Vert}_{{\mathcal L}(X_{0,\eta_{F}}^{A_0},X)} \\
& \qquad \qquad= t^{-\eta_{F}+{\varepsilon}} {\Vert}\tfrac{d}{dt}S_0(t) x{\Vert}_{{\mathcal L}(X_{0,\eta_{F}}^{A_0},X_0)} + ({\varepsilon}-\eta_{F}) t^{-\eta_{F}+{\varepsilon}-1} {\Vert}S_{0}(t) x{\Vert}_{{\mathcal L}(X_{0,\eta_{F}}^{A_0},X_0)} \\
& \qquad \qquad\leq {\mathcal M}t^{-1+{\varepsilon}}.\end{aligned}$$ By Corollary \[cor:detConvV\] with $Y_1=X_{0,\eta_{F}}^{A_0}$, $Y_2=X$, $$\Phi(s) =
P_0[F(s,U(s))-F(s,\tilde{U}^{(0)}(s))],$$ $\Psi(s)=S_0(s)$, $\theta = -\eta_{F}+{\varepsilon}$ and $g(v)={\mathcal M}v^{-1+{\varepsilon}}$, it follows that: $$\label{pert.h3}
\begin{aligned}
&\Big{\Vert}t\mapsto \int_0^{t}\tilde{S}_0(t-s)[F(s,U(s))-F(s,\tilde{U}^{(0)}(s))]\,ds
\Big{\Vert}_{{V^{\alpha,p}_{\rm c}}([0,T_0]\times\Omega;X)}\\
& \qquad \qquad\lesssim T_0^{{\varepsilon}}{\Vert}P_0[F(\cdot,U)-F(\cdot,\tilde{U}^{(0)})]{\Vert}_{L^p(\Omega;L^{\infty}(0,T_0;X_{0,\eta_{F}}^{A_0}))}\\
& \qquad \qquad\leq T_0^{{\varepsilon}}\textrm{Lip}(P_0F){\Vert}U-\tilde{U}^{(0)}{\Vert}_{L^p(\Omega;L^{\infty}(0,T_0;X))}\\
& \qquad \qquad\lesssim T_0^{{\varepsilon}}(1+D_{\delta}(A,A_0))\textrm{Lip}(F){\Vert}U-\tilde{U}^{(0)}{\Vert}_{L^p(\Omega;L^{\infty}(0,T_0;X))},
\end{aligned}$$ where the second-last estimate follows by Lipschitz-continuity of $P_0F$ and the final estimate follows by . Note that the implied constants are independent of $T_0$, and depend on $X_0$ only in terms of ${\Vert}P_0{\Vert}_{{\mathcal L}(X,X_0)}$ and on $A_0$ only in terms of $\omega$, $\theta$ and $K$.

### Part 2d. {#part-2d. .unnumbered}

By Proposition \[p:gbddSG\] with $\beta= \theta_F{\wedge}\delta\in[\delta-1,\delta]$ we have that there exists a constant ${\mathcal M}$ depending only on ${\Vert}P_0{\Vert}_{{\mathcal L}(X,X_0)}$, $\omega,\theta,K$, $(\delta-\theta_F){\vee}0$ and $T$ such that for all $t\in[0,T]$ we have: $$\begin{aligned}
& t^{(\delta-\theta_F)^+ +{\varepsilon}}{\Vert}\tfrac{d}{dt}[S(t)-\tilde{S}_0(t)]{\Vert}_{{\mathcal L}(X_{\theta_F{\wedge}\delta}^{A},X)}\\
& \qquad +((\delta-\theta_F)^+ +{\varepsilon})t^{(\delta-\theta_F)^+ +{\varepsilon}-1}{\Vert}S(t)-\tilde{S}_0(t){\Vert}_{{\mathcal L}(X_{\theta_F{\wedge}\delta}^{A},X)}  \leq {\mathcal M}D_{\delta}(A,A_0) t^{-1+{\varepsilon}}.\end{aligned}$$ Thus by Corollary \[cor:detConvV\] with $Y_1=X_{\theta_F{\wedge}\delta}^{A}$, $Y_2=X$, $\Phi(s)= F(s,U(s))$, $\Psi(s)= S(s)-\tilde{S}_0(s)$, $\theta= (\delta-\theta_F)^{+}+{\varepsilon}$, $g(v)={\mathcal M}D_{\delta}(A,A_0) v^{-1+{\varepsilon}}$, we obtain: $$\label{pert.h4}
\begin{aligned}
&\Big{\Vert}t\mapsto \int_0^{t}[S(t-s)-\tilde{S}_0(t-s)]F(s,U(s))\,ds \Big{\Vert}_{{V^{\alpha,p}_{\rm c}}([0,T_0]\times\Omega;X)}\\
&\qquad\qquad \lesssim D_{\delta}(A,A_0){\Vert}F(\cdot,U){\Vert}_{L^p(\Omega;L^{\infty}(0,T_0;X_{\theta_F{\wedge}\delta}^{A}))}\\
&\qquad \qquad\leq D_{\delta}(A,A_0)M(F){\Vert}U{\Vert}_{L^p(\Omega;L^{\infty}(0,T_0;X))} \\
&\qquad \qquad\lesssim D_{\delta}(A,A_0)M(F)(1+{\Vert}x_0{\Vert}_{L^p(\Omega;X)}),
\end{aligned}$$ where the penultimate estimate follows by the linear growth condition on $F$ and the final estimate by . Note that the implied constants are independent of $T_0$, and depend on $X_0$ only in terms of ${\Vert}P_0{\Vert}_{{\mathcal L}(X,X_0)}$, and on $A$ and $A_0$ only in terms of $\omega$, $\theta$ and $K$.

### Part 2e. {#part-2e. .unnumbered}

Recall that $\eta_{G}\leq 0$. By equation there exists an ${\mathcal M}$ depending only on $\omega$, $\theta$, $K$ and $T$ such that for all $t\in [0,T]$ we have: $$\begin{aligned}
& t^{-\eta_{G}+{\varepsilon}/2} {\Vert}\tfrac{d}{dt} S_0(t){\Vert}_{{\mathcal L}(X_{0,-\eta_{G}}^{A},X_0)}+ ({\varepsilon}/2-\eta_{G})t^{-\eta_{G}+{\varepsilon}/2-1} {\Vert}S_0(t){\Vert}_{{\mathcal L}(X_{0,-\eta_{G}}^{A},X_0)} \\
& \qquad \qquad \leq {\mathcal M}t^{-1+{\varepsilon}/2}.\end{aligned}$$ By applying Proposition \[prop:stochConvV\] with $Y_1=X_{0,\eta_{G}}^{A_0}$, $Y_2=X$, $\Psi(s)= S_0(s)$, $\eta=\alpha$, $\alpha=\alpha$, $\theta=-\eta_{G}+{\varepsilon}/2$ and $g(v)= {\mathcal M}v^{-1+{\varepsilon}/2}$ and $$\Phi(s)=
P_0[G(s,U(s))-G(s,\tilde{U}^{(0)}(s))]$$ we obtain: $$\label{pert.h5}
\begin{aligned}
&\Big{\Vert}t\mapsto \int_0^{t}\tilde{S}_0(t-s)[G(s,U(s))-G(s,\tilde{U}^{(0)}(s))]\,dW_H(s)
\Big{\Vert}_{{V^{\alpha,p}_{\rm c}}([0,T_0]\times\Omega;X)}\\
&\qquad \lesssim T_0^{{\frac{1}{2}}{\varepsilon}}\sup_{0\leq t\leq T_0}{\Vert}s\mapsto
(t-s)^{-\alpha}P_0[G(s,U(s))-G(s,\tilde{U}^{(0)}(s))]{\Vert}_{L^p(\Omega;\gamma(0,t;X_{0,\eta_{G}}^{A_0}))}\\
&\qquad \lesssim T_0^{{\frac{1}{2}}{\varepsilon}}(1+D_{\delta}(A,A_0))\textrm{Lip}_{\gamma}(G){\Vert}U-\tilde{U}^{(0)}{\Vert}_{{V^{\alpha,p}_{\rm c}}([0,T_0]\times\Omega;X)},
\end{aligned}$$ where the final estimate follows from estimates and . Note that the implied constants are independent of $T_0$, and depend on $X_0$ only in terms of ${\Vert}P_0{\Vert}_{{\mathcal L}(X,X_0)}$, and on $A$ and $A_0$ only in terms of $\omega$, $\theta$ and $K$.

### Part 2f. {#part-2f. .unnumbered}

By Proposition \[p:gbddSG\] with $\beta= \theta_G{\wedge}\delta\in [\delta-1,\delta]$ we have that there exists a constant ${\mathcal M}$ depending only on ${\Vert}P_0{\Vert}_{{\mathcal L}(X,X_0)}$, $\omega,\theta,K$, $(\delta-\theta_G){\vee}0$ and $T$ such that for all $t\in[0,T]$ we have: $$\begin{aligned}
& t^{(\delta-\theta_G)^{+}+{\varepsilon}/2}{\Vert}\tfrac{d}{dt} [S(t)-\tilde{S}_0(t)]{\Vert}_{{\mathcal L}(X_{\theta_G{\wedge}\delta}^{A},X)}\\
& \qquad \qquad+ ((\delta-\theta_G)^{+}+{\varepsilon}/2)t^{(\delta-\theta_G)^{+}+{\varepsilon}/2-1}{\Vert}S(t)-\tilde{S}_0(t){\Vert}_{{\mathcal L}(X_{\theta_G{\wedge}\delta}^{A},X)} \\
& \qquad \leq {\mathcal M}D_{\delta}(A,A_0)t^{-1+{\varepsilon}/2}.\end{aligned}$$ Thus by Proposition \[prop:stochConvV\] with $Y_1=X_{\theta_G{\wedge}\delta}^{A}$, $Y_2=X$, $\Phi(s)= G(s,U(s))$, $\Psi(s)= S(s)-\tilde{S}_0(s)$, $\eta=\alpha$, $\alpha=\alpha$, $\theta=(\delta-\theta_G)^+ +{\varepsilon}/2$ and $g(v)= {\mathcal M}D_{\delta}(A,A_0)v^{-1+{\varepsilon}/2}$ we obtain: $$\label{pert.h6}
\begin{aligned}
&\Big{\Vert}t\mapsto \int_0^{t}[S(t-s)-\tilde{S}_0(t-s)]G(s,U(s))\,dW_H(s) \Big{\Vert}_{{V^{\alpha,p}_{\rm c}}([0,T_0]\times\Omega;X)}\\
&\qquad\qquad \lesssim D_{\delta}(A,A_0)\sup_{0\leq t\leq T_0}{\Vert}s\mapsto
(t-s)^{-\alpha}G(s,U(s)){\Vert}_{L^p(\Omega;\gamma(0,t;X_{\theta_G{\wedge}\delta}^{A}))}\\
&\qquad\qquad \leq D_{\delta}(A,A_0)M_{\gamma}(G){\Vert}U{\Vert}_{{V^{\alpha,p}_{\rm c}}([0,T_0]\times\Omega;X)}\\
&\qquad\qquad \lesssim D_{\delta}(A,A_0)M_{\gamma}(G)\big(1+{\Vert}x_0
{\Vert}_{L^p(\Omega;X)}\big),
\end{aligned}$$ where the penultimate line follows by estimate . Note that the implied constants are independent of $T_0$, and depend on $X_0$ only in terms of ${\Vert}P_0{\Vert}_{{\mathcal L}(X,X_0)}$ and on $A$ and $A_0$ only in terms of $\omega$, $\theta$ and $K$.

### Part 2g. {#part-2g. .unnumbered}

Inserting - in we obtain that there exists a constant $C>0$ independent of $x_0$ and $y_0$, depending on $X_0$ only in terms of ${\Vert}P_0{\Vert}_{{\mathcal L}(X,X_0)}$, on $A$ and $A_0$ only in terms of $1+D_{\delta}(A,A_0)$, $\omega$, $\theta$ and $K$, and on $F$ and $G$ only in terms of their Lipschitz and linear growth constants Lip$(F)$, $\textrm{Lip}_{\gamma}(G)$, $M(F)$ and $M_{\gamma}(G)$, such that for all $T_0\in [0,T]$ one has: $$\begin{aligned}
& {\Vert}U - \tilde{U}^{(0)}{\Vert}_{{V^{\alpha,p}_{\rm c}}([0,T_0]\times\Omega;X)} \\
& \qquad \qquad \leq CT_{0}^{{\frac{1}{2}}{\varepsilon}}{\Vert}U - U^{(0)}{\Vert}_{{V^{\alpha,p}_{\rm c}}([0,T_0]\times\Omega;X)} \\
& \qquad \qquad \quad  + C\Big({\Vert}x_0-y_0 {\Vert}_{L^p(\Omega;X)}+D_{\delta}(A,A_0)\big(1+{\Vert}x_0{\Vert}_{L^p(\Omega;X_{\delta}^{A})}\big)\Big).\end{aligned}$$ Setting $T_0= [2C]^{-2/{\varepsilon}}$ we obtain: $$\label{T0est}
\begin{aligned}
& {\Vert}U - \tilde{U}^{(0)}{\Vert}_{{V^{\alpha,p}_{\rm c}}([0,T_0]\times\Omega;X)} \\
& \qquad \qquad \leq 2C\Big({\Vert}x_0-y_0
{\Vert}_{L^p(\Omega;X)}+D_{\delta}(A,A_0)\big(1+{\Vert}x_0{\Vert}_{L^p(\Omega;X_{\delta}^{A})}\big)\Big).
\end{aligned}$$

Part 3. {#part-3. .unnumbered}
-------

Let $t_0\geq 0$, $z\in L^p(\Omega,{\mathcal F}_{t_0};X)$, $T>0$ and $\alpha\in [0,{\frac{1}{2}})$. By $U(z,t_0,\cdot)$, we denote the (unique) process in ${V^{\alpha,p}_{\rm c}}([t_0,t_0+T]\times\Omega;X)$ satisfying, for $s\in [t_0,t_0+T]$: $$\begin{aligned}
U(z,t_0,s)=S(t-t_0)z& +\int_{t_0}^{t}S(t-t_0-s)F\big(U(z,t_0,s)\big)\,ds \\
&+ \int_{t_0}^{t}S(t-t_0-s)G\big(U(z,t_0,s)\big)\,dW_H(s) \quad \textrm{a.s.}\end{aligned}$$ The process $U^{(0)}(z,t_0,\cdot)$ is defined analogously.

From the proof of it follows that for any $x\in L^p(\Omega,{\mathcal F}_{t_0};X_{\delta}^{A})$ and $y\in
L^p(\Omega,{\mathcal F}_{t_0};X)$ we have: $$\label{T0est.shift}
\begin{aligned}
&{\Vert}U(x,t_0,\cdot) - U^{(0)}(y,t_0,\cdot){\Vert}_{{V^{\alpha,p}_{\rm c}}([t_0,t_0+T_0]\times\Omega;X)} \\
&\qquad\qquad  \leq 2C\Big({\Vert}x-y {\Vert}_{L^p(\Omega;X)}+D_{\delta}(A,A_0)[1+{\Vert}x{\Vert}_{L^p(\Omega;X_{\delta}^{A})}]\Big),
\end{aligned}$$ with $C$ as in .

Part 4. {#part-4. .unnumbered}
-------

Throughout this section one may check that the implied constants always depend $X_0$ only in terms of ${\Vert}P_0{\Vert}_{{\mathcal L}(X,X_0)}$, on $A$ and $A_0$ only in terms of $1+D_{\delta}(A,A_0)$, $\omega$, $\theta$ and $K$, and on $F$ and $G$ only in terms of their Lipschitz and linear growth constants Lip$(F)$, $\textrm{Lip}_{\gamma}(G)$, $M(F)$ and $M_{\gamma}(G)$, even when this is not mentioned explicitly.

By uniqueness of the solution to it follows that for any $0\leq s_0\leq t_0\leq t$ and any $x,y\in
L^p(\Omega,{\mathcal F}_{s_0};X)$ one has: $$U(x,s_0,t)=U\big(U(x,s_0,t_0),t_0,t\big) \ \ \textrm{and}\ \
U^{(0)}(y,s_0,t)=U^{(0)}\big(U^{(0)}(y,s_0,t_0),t_0,t\big).$$

Let $j\in \N$. By the embedding ${V^{\alpha,p}_{\rm c}}([0,T_0]\times\Omega;X) \hookrightarrow L^{\infty}(0,T_0;L^p(\Omega;X))$ and estimate with $x=U(x_0,0,(j-1)T_0)$ and $y=U^{(0)}(y_0,0,(j-1)T_0)$, we obtain: $$\label{recur2}\begin{aligned}
& {\Vert}U(x_0,0,jT_0)-U^{(0)}(y_0,0,jT_0){\Vert}_{L^p(\Omega;X)} \\
& \qquad \qquad= \Big{\Vert}U\Big(U\big(x_0,0,(j-1)T_0\big),(j-1)T_0,T_0\Big)\\
& \qquad \qquad\qquad\quad -U^{(0)}\Big(U^{(0)}\big(y_0,0,(j-1)T_0\big),(j-1)T_0,T_0\Big) \Big{\Vert}_{L^p(\Omega;X)}\\
& \qquad \qquad\lesssim {\Vert}U\big(x_0,0,(j-1)T_0\big)-U^{(0)}\big(y_0,0,(j-1)T_0\big){\Vert}_{L^p(\Omega;X)}\\
& \qquad \qquad\quad + D_{\delta}(A,A_0)\big[1+{\Vert}U\big(x_0,0,(j-1)T_0\big){\Vert}_{L^p(\Omega;X^{A}_{\eta})}\big],\end{aligned}$$ with implied constant independent of $j$ and $n$ and the ‘initial values’ $U(x_0,0,(j-1)T_0)$ and $U^{(0)}(y_0,0,(j-1)nT_0)$.

By equation it follows that: $$\begin{aligned}\label{supLpest}
\sup_{1\leq j\leq \lfloor T/T_0 \rfloor} {\Vert}U(x_0,0,jT_0) {\Vert}_{L^p(\Omega;X^{A}_{\eta})} & \leq \sup_{s\in [0,T]}{\Vert}U(x_0,0,s) {\Vert}_{L^p(\Omega;X^{A}_{\eta})}\\
 &  \lesssim 1+{\Vert}x_0 {\Vert}_{L^p(\Omega;X^{A}_{\eta})}.
\end{aligned}$$

Thus for $j=1,\ldots,\lceil T/T_0 \rceil$ we obtain the following relation from : $$\begin{aligned}
& {\Vert}U(x_0,0,jT_0)-U^{(0)}(y_0,0,jT_0){\Vert}_{L^p(\Omega;X)} \\
& \qquad \qquad  \lesssim {\Vert}U(x_0,0,(j-1)T_0)-U^{(0)}(y_0,0,(j-1)T_0){\Vert}_{L^p(\Omega;X)}\\
& \qquad \qquad \quad + D_{\delta}(A,A_0)(1+{\Vert}x_0 {\Vert}_{L^p(\Omega;X^{A}_{\eta})}).\end{aligned}$$ Note that $T_0$ depends on $X_0$ only in terms of ${\Vert}P_0{\Vert}_{{\mathcal L}(X,X_0)}$, on $A$ and $A_0$ only in terms of $1+D_{\delta}(A,A_0)$, $\omega$, $\theta$ and $K$, and on $F$ and $G$ only in terms of Lip$(F)$, $\textrm{Lip}_{\gamma}(G)$, $M(F)$, and $M_{\gamma}(G)$. By induction we obtain, for $j=1,\ldots,\lceil T/T_0 \rceil$: $$\label{recur3}\begin{aligned}
&{\Vert}U(x_0,jT_0)-U^{(0)}(y_0,jT_0){\Vert}_{L^p(\Omega;X)} \\
& \qquad\qquad  \lesssim {\Vert}x_0 - y_0 {\Vert}_{L^p(\Omega;X)} + D_{\delta}(A,A_0)(1+{\Vert}x_0
{\Vert}_{L^p(\Omega;X^{A}_{\eta})}).\end{aligned}$$

Fix $j\in \N,$ $j< \lceil T/T_0\rceil$. Set $$x=U(x_0,0,(j-1)T_0)\quad \textrm{and} \quad
y=U^{(0)}(y_0,0,(j-1)T_0)$$ in to obtain, using and : $$\begin{aligned}
&\big{\Vert}U\big(U(x_0,0,(j-1)T_0),(j-1)T_0,\cdot\big)\\
& \qquad \qquad \quad -U^{(0)}\big(U^{(0)}(y_0,(j-1)T_0),(j-1)T_0,\cdot \big)\big{\Vert}_{{V^{\alpha,p}_{\rm c}}([(j-1)T_0,jT_0]\times
\Omega;X)} \\
& \qquad \qquad\lesssim   {\Vert}U(x_0,0,(j-1)T_0)-U^{(0)}(y_0,0,(j-1)T_0){\Vert}_{L^p(\Omega;X)} \\
&\qquad\qquad \quad + D_{\delta}(A,A_0) \big(1+{\Vert}U(x_0,0,(j-1)T_0) {\Vert}_{L^p(\Omega;X^{A}_{\eta})}\big)\\
& \qquad \qquad\lesssim {\Vert}x_0 - y_0 {\Vert}_{L^p(\Omega;X)} + D_{\delta}(A,A_0)(1+{\Vert}x_0 {\Vert}_{L^p(\Omega;X^{A}_{\eta})}),\end{aligned}$$ with implied constants independent of $j$.

Due to inequality we thus obtain: $$\begin{aligned}
&{\Vert}U(x_0,0,\cdot) - U^{(0)}(y_0,0,\cdot) {\Vert}_{{V^{\alpha,p}_{\rm c}}([0,T]\times \Omega;X)} \\
& \qquad \leq \Big{\Vert}\sum_{j=1}^{\lceil T/T_0 \rceil} \Big( U\big[U\big(x_0,0,(j-1)T_0\big),(j-1)T_0,\cdot\big]\\
& \qquad \qquad \quad -U^{(0)}\big[U^{(0)}\big(y_0,0,(j-1)T_0\big),(j-1)T_0,\cdot \big] \Big)1_{[(j-1)T_0,jT_0)}
\Big{\Vert}_{{V^{\alpha,p}_{\rm c}}([0,T]\times \Omega;X)}\\
& \qquad  \leq   \sum_{j=1}^{\lceil T/T_0 \rceil} \Big{\Vert}U\big[U\big(x_0,0,(j-1)T_0\big),(j-1)T_0,\cdot\big]\\
& \qquad \qquad \quad -U^{(0)}\big[U^{(0)}\big(y_0,0,(j-1)T_0\big),(j-1)T_0,\cdot \big]
\Big{\Vert}_{{V^{\alpha,p}_{\rm c}}([(j-1)T_0,jT_0]\times \Omega;X)}\\
&\qquad \lesssim \sum_{j=1}^{\lceil T/T_0 \rceil} {\Vert}x_0 - y_0 {\Vert}_{L^p(\Omega;X)} +D_{\delta}(A,A_0)(1+{\Vert}x_0
{\Vert}_{L^p(\Omega;X^{A}_{\eta})})\\
&\qquad \lesssim {\Vert}x_0 - y_0 {\Vert}_{L^p(\Omega;X)} + D_{\delta}(A,A_0)(1+{\Vert}x_0 {\Vert}_{L^p(\Omega;X^{A}_{\eta})}).\end{aligned}$$

It remains to provide a proof for Proposition \[p:gbddSG\]. For that purpose, we first prove the following lemma. Given the lemma, the proof of Proposition \[p:gbddSG\] basically follows the lines of known proofs concerning comparison of semigroups, see [@EngNa:00 Chapter III.3.b]. For notational simplicity we define the pseudo-resolvent $$\begin{aligned}
\label{pseudoR2}
R(\lambda:\tilde{A}_0)&:=i_{X_0}R({\lambda}:A_0)P_0,\quad {\lambda}\in \omega+\Sigma_{\frac{\pi}{2}+\theta}\end{aligned}$$ (we leave it to the reader to verify the resolvent identity).

\[l:extend\] Let the setting be as in Proposition \[p:gbddSG\]. Then for all ${\lambda}\in \omega'+\Sigma_{\frac{\pi}{2}+\theta}$ we have: $$\begin{aligned}
&{\Vert}R({\lambda}:A)  - R({\lambda}:\tilde{A}_0)  {\Vert}_{{\mathcal L}(X^{A}_{\beta},X)} \\
& \qquad \qquad \leq C_{\omega,\theta,K,P_0} |\lambda-\omega|^{\delta-\beta-1} {\Vert}R(\lambda_0:A)-R({\lambda}_0:\tilde{A}_0)
{\Vert}_{{\mathcal L}(X_{\delta-1}^{A},X)},
\end{aligned}$$ where $C_{\omega,\theta,K,P_0}$ is a constant depending only on $\omega,\theta,K$ and ${\Vert}P_0{\Vert}_{{\mathcal L}(X,X_0)}$.

Using only the resolvent identity and the definition of $R(\lambda:\tilde{A}_0)$ (see ) one may verify that the following identity holds: $$\label{resid}
\begin{aligned}
& R({\lambda}:A) - R({\lambda}:\tilde{A}_0) \\
&\qquad\qquad
=[I+(\lambda_0-\lambda)R(\lambda:\tilde{A}_0)][R(\lambda_0:A)-R(\lambda_0:\tilde{A}_0)](\lambda_0-A)R(\lambda:A).
\end{aligned}$$ Moreover, one may check that: $$\begin{aligned}
\omega'+\Sigma_{\frac{\pi}{2}+\theta}\subset \big(\omega+\Sigma_{\frac{\pi}{2}+\theta}\big)\bigcap \big\{\lambda\in \C:
|\lambda-\omega|\geq |\lambda_0-\omega|\big\}.\end{aligned}$$ Therefore one has, for $\lambda \in \omega'+\Sigma_{\frac{\pi}{2}+\theta}$, $${\Vert}I+ (\lambda_0-\lambda) R(\lambda:\tilde{A}_0){\Vert}_{{\mathcal L}(X)} \leq 1 +
\tfrac{|\lambda_0-\lambda|}{|\lambda-\omega|}K{\Vert}P_0{\Vert}_{{\mathcal L}(X,X_0)} \leq 1+2K{\Vert}P_0{\Vert}_{{\mathcal L}(X,X_0)}.$$ From one obtains: $$\label{Ressplit}
\begin{aligned}
& {\Vert}R({\lambda}:A)  - R({\lambda}:\tilde{A}_0)  {\Vert}_{{\mathcal L}(X^{A}_{\beta},X)} \leq (1+2K{\Vert}P_0{\Vert}_{{\mathcal L}(X,X_0)})\\
& \qquad \qquad \times {\Vert}R(\lambda_0:A)-R(\lambda_0:\tilde{A}_0) {\Vert}_{{\mathcal L}(X_{\delta-1}^{A},X)} {\Vert}(\lambda_0-A)R(\lambda:A) {\Vert}_{{\mathcal L}(X^{A}_{\beta},X^{A}_{\delta-1})}.
\end{aligned}$$ Let $\bar{\lambda}\in \C$, $\Re e(\bar{\lambda})>\omega$ be such that $|\bar{\lambda}-\lambda_0 | \leq 2| \bar{\lambda} - \omega | $ (if $\Re e(\lambda_0)>\omega$ one may simply pick $\bar{\lambda}=\lambda_0$). For $\eta\in\R$ and $x\in X_{\eta}^{A}$ set ${\Vert}x_0{\Vert}_{X_{\eta}^A}:= {\Vert}(\bar{\lambda}-A)^{\eta}x{\Vert}_{X}.$ Then: $$\begin{aligned}
{\Vert}(\lambda_0-A)R(\lambda:A) {\Vert}_{{\mathcal L}(X^{A}_{\beta},X^{A}_{\delta-1})} & = {\Vert}(\bar{\lambda}-A)^{\delta-\beta-1}(\lambda_0-A)R(\lambda:A) {\Vert}_{{\mathcal L}(X)}\\
& \leq \big(1+\tfrac{|\bar{\lambda}-\lambda_0|}{|\bar{\lambda}-\omega|}K\big){\Vert}(\bar{\lambda}-A)^{\delta-\beta}R(\lambda:A) {\Vert}_{{\mathcal L}(X)}\\
& \leq (1+2K){\Vert}(\bar{\lambda}-A)^{\delta-\beta}R(\lambda:A) {\Vert}_{{\mathcal L}(X)}.\end{aligned}$$

If $\delta-\beta =1$ then: $$\begin{aligned}
{\Vert}(\bar{\lambda}-A)^{\delta-\beta}R(\lambda:A) {\Vert}_{{\mathcal L}(X)} & = {\Vert}(\bar{\lambda}-A)R(\lambda:A) {\Vert}_{{\mathcal L}(X)}
\leq 1+2K.\end{aligned}$$ If $\delta-\beta=0$ then: $$\begin{aligned}
{\Vert}(\bar{\lambda}-A)^{\delta-\beta}R(\lambda:A) {\Vert}_{{\mathcal L}(X)} = {\Vert}R(\lambda:A) {\Vert}_{{\mathcal L}(X)} \leq
K|\lambda-\omega|^{-1}.\end{aligned}$$ For $\delta-\beta\in (0,1)$ we have, by Theorem \[t:fracPowInt\], $$\begin{aligned}
{\Vert}(\bar{\lambda}-A)^{\delta-\beta}R(\lambda:A) {\Vert}_{{\mathcal L}(X)}
& \leq 2(1+K){\Vert}R(\lambda:A) {\Vert}_{{\mathcal L}(X)}^{1+\beta-\delta} {\Vert}(\bar{\lambda}-A) R(\lambda:A)
{\Vert}_{{\mathcal L}(X)}^{\delta-\beta}\\
& \leq 2(1+K)(1+2K)^{\delta-\beta} K^{1+\beta-\delta}|\lambda - \omega |^{\delta-\beta-1}\\
&  \leq 2(1+2K)^2|\lambda - \omega |^{\delta-\beta-1}.\end{aligned}$$ Substituting this into one obtains: $$\begin{aligned}
& {\Vert}R({\lambda}:A)  - R({\lambda}:\tilde{A}_0)  {\Vert}_{{\mathcal L}(X^{A}_{\beta},X)} \\
& \qquad \leq  2(1+2K)^4 {\Vert}P_0{\Vert}_{{\mathcal L}(X,X_0)}|\lambda - \omega |^{\delta-\beta-1}{\Vert}R(\lambda_0:A)-R(\lambda_0:\tilde{A}_0)
{\Vert}_{{\mathcal L}(X_{\delta -1}^{A},X)}.\end{aligned}$$

Let $\omega'$ be as defined in Lemma \[l:extend\]. For brevity set ${\varepsilon}= \delta-\beta$. First of all observe that $$\begin{aligned}
\lim_{s\downarrow 0} s^{{\varepsilon}}{\Vert}[S(s)-\tilde{S}_0(s)] {\Vert}_{{\mathcal L}(X_{\beta}^{A},X)} =0.\end{aligned}$$ Fix $\theta'\in (0,\theta)$. It follows from [@Pazy:83 Theorem 1.7.7], that one has, for all $t>0$: $$\begin{aligned}
S(t) &= {\tfrac{1}{2\pi i}}\int_{\omega'+\Gamma_{\theta'}} e^{\lambda t}R(\lambda:A) d\lambda;\end{aligned}$$ where $\Gamma_{\theta'}$ is the path composed from the two rays $r e^{i(\frac{\pi}{2}+\theta')}$ and $r
e^{-i(\frac{\pi}{2}+\theta')}$, $0\leq r <\infty$, and is oriented such that $\Im m(\lambda)$ increases along $\Gamma_{\theta'}$. As $\omega'\geq \omega$, the integral is well-defined as ${\mathcal L}(X)$-valued Bochner integral, and for $t>0$ one has: $$\begin{aligned}
\tfrac{d}{dt}S(t) &= {\tfrac{1}{2\pi i}}\int_{\omega'+\Gamma_{\theta'}} \lambda e^{\lambda t}R(\lambda:A) d\lambda;\end{aligned}$$ the integral again being well-defined as ${\mathcal L}(X)$-valued Bochner integrals (see also the proof of [@Pazy:83 Theorem 2.5.2]). Analogous identities hold for $\tilde{S}_0$ and $R(\lambda:\tilde{A}_0)$.

First let us assume that ${\varepsilon}\in (0,1)$. Below we shall apply Lemma \[l:extend\], observing that for $r\in [0,\infty)$ we have $$|\omega'+re^{\pm i (\frac{\pi}{2}+\theta')}-\omega | \geq K_{\theta} r,$$ where $K_{\theta}$ is a constant depending only on $\theta$. Note that we use the coordinate transform $\lambda=\omega'+re^{\pm i (\frac{\pi}{2}+\theta')}$. For $s>0$ we have: $$\begin{aligned}
&{\Vert}S(s)-\tilde{S}_0(s) {\Vert}_{{\mathcal L}(X_{\beta}^{A},X)} =\Big{\Vert}{\tfrac{1}{2\pi i}}\int_{\Gamma_{\theta'}} e^{\lambda
s}[R(\lambda:A)-R(\lambda:\tilde{A}_0)] d\lambda \Big{\Vert}_{{\mathcal L}(X_{\beta}^{A},X)}\\
&\qquad\qquad   \leq {\tfrac{1}{2\pi}}  \int_{0}^{\infty} \big|e^{-i(\frac{\pi}{2}+\theta')+(\omega'+r e^{-i(\frac{\pi}{2}+\theta')})s
}\big|\\
&\qquad\qquad \qquad \times \big{\Vert}R(\omega'+re^{-i(\frac{\pi}{2}+\theta')}:A)-R(\omega'+re^{-i(\frac{\pi}{2}+\theta')}:\tilde{A}_0)\big{\Vert}_{{\mathcal L}(X_{\beta}^{A},X)} \,dr \\
&\qquad\qquad  \quad + {\tfrac{1}{2\pi}}  \int_{0}^{\infty} \big| e^{i(\frac{\pi}{2}+\theta')+(\omega'+re^{i(\frac{\pi}{2}+\theta')})s
}\big|\\
&\qquad\qquad \qquad \times \big{\Vert}R(\omega'+re^{i(\frac{\pi}{2}+\theta')}:A)-R(\omega'+re^{i(\frac{\pi}{2}+\theta')}:\tilde{A}_0)\big{\Vert}_{{\mathcal L}(X_{\beta}^{A},X)} \,dr
\\
&\qquad\qquad  \leq {\tfrac{1}{\pi}} C_{\omega,\theta,K,P_0}K_{\theta} D_{\delta}(A,A_0)e^{\omega's} \int_{0}^{\infty} r^{{\varepsilon}-1}e^{-rs \sin\theta' }    \,dr \\
&\qquad\qquad   ={\tfrac{1}{\pi}} C_{\omega,\theta,K,P_0}K_{\theta} D_{\delta}(A,A_0)  [s\sin\theta']^{-{\varepsilon}}e^{\omega's}\int_{0}^{\infty} u^{{\varepsilon}-1}
e^{-u }
du  \\
&\qquad\qquad   = \tfrac{\Gamma({\varepsilon})}{\pi} [\sin\theta']^{-{\varepsilon}} C_{\omega,\theta,K,P_0}D_{\delta}(A,A_0) s^{-{\varepsilon}}e^{\omega's}.\end{aligned}$$

For ${\varepsilon}= 0$ one may avoid the singularity in $0$ in the usual way: for $s>0$ given we integrate over $$\omega'+\Gamma_{\theta',s}= (\omega'+\Gamma^{(1)}_{\theta',s})\cup (\omega'+\Gamma^{(2)}_{\theta',s})\cup(\omega'+\Gamma^{(3)}_{\theta',s}),$$ where $\Gamma^{(1)}_{\theta',s}$ and $\Gamma^{(2)}_{\theta',s}$ are the rays $r e^{i(\frac{\pi}{2}+\theta')}$ and $r
e^{-i(\frac{\pi}{2}+\theta')}$, $s^{-1} \leq r <\infty$, and $\Gamma^{(3)}_{\theta',s}= s^{-1}e^{i\phi}$, $\phi\in[-\frac{\pi}{2}-\theta',\frac{\pi}{2}+\theta']$. This leads to the following estimate: $$\begin{aligned}
&{\Vert}S(s)-\tilde{S}_0(s) {\Vert}_{{\mathcal L}(X_{\beta}^{A},X)} =\Big{\Vert}{\tfrac{1}{2\pi i}}\int_{\omega'+\Gamma_{\theta',s}} e^{\lambda
s}[R(\lambda:A)-R(\lambda:\tilde{A}_0)] d\lambda \Big{\Vert}_{{\mathcal L}(X_{\beta}^{A},X)}\\
&\qquad \qquad \leq C_{\omega,\theta,K,P_0}K_{\theta}D_{\delta}(A,A_0) e^{\omega's}\Big[{\tfrac{1}{\pi}} \int_{s^{-1}}^{\infty} r^{-1}e^{-rs \sin\theta' }  
\,dr +
e \Big]\\
&\qquad \qquad \leq C_{\omega,\theta,K,P_0}K_{\theta}D_{\delta}(A,A_0) \big[[\pi \sin\theta']^{-1} e^{-\sin\theta'} + e\big]e^{\omega's} \\
& \qquad \qquad \leq 2
[\sin\theta']^{-1}C_{\omega,\theta,K,P_0}K_{\theta}D_{\delta}(A,A_0)e^{\omega's}.\end{aligned}$$ Recalling that ${\varepsilon}= \delta-\beta$ this proves the uniform boundedness estimate of .

Similarly to the above, for ${\varepsilon}\in [0,1]$ and $s>0$ we have: $$\begin{aligned}
& {\Vert}\tfrac{d}{ds}S(s)-\tfrac{d}{ds}\tilde{S}_0(s) {\Vert}_{{\mathcal L}(X_{\beta}^{A},X)}  =\Big{\Vert}{\tfrac{1}{2\pi
i}}\int_{\Gamma_{\omega'+\theta'}} \lambda e^{\lambda s}[R(\lambda:A)-R(\lambda:\tilde{A}_0)] d\lambda \Big{\Vert}_{{\mathcal L}(X_{\beta}^{A},X)}\\
&\qquad  \leq   {\tfrac{1}{2\pi}}e^{\omega's} \int_{0}^{\infty} r e^{- rs \sin\theta' }\big{\Vert}R(re^{-i(\frac{\pi}{2}+\theta')}:A)-R(re^{-i(\frac{\pi}{2}+\theta')}:\tilde{A}_0)\big{\Vert}_{{\mathcal L}(X_{\beta}^{A},X)}
\,dr \\
&\qquad  \quad +{\tfrac{1}{2\pi}} e^{\omega's}\int_{0}^{\infty} re^{- rs \sin\theta' }\big{\Vert}R(re^{i(\frac{\pi}{2}+\theta')}:A)-R(re^{i(\frac{\pi}{2}+\theta')}:\tilde{A}_0) \big{\Vert}_{{\mathcal L}(X_{\beta}^{A},X)}
\,dr\\
& \qquad ={\tfrac{1}{\pi}} C_{\omega,\theta,K,P_0}K_{\theta}D_{\delta}(A,A_0)  [s\sin\theta']^{-1-{\varepsilon}} e^{\omega's}\int_{0}^{\infty} u^{{\varepsilon}}
e^{-u }
 du \\
&\qquad  = \tfrac{{\varepsilon}\Gamma({\varepsilon})}{\pi} [\sin\theta']^{-1-{\varepsilon}} C_{\omega,\theta,K,P_0}K_{\theta}D_{\delta}(A,A_0) s^{-1-{\varepsilon}} e^{\omega's} .\end{aligned}$$ Recalling that ${\varepsilon}= \delta-\beta$ this proves the uniform boundedness estimate of .

Concerning the $\gamma$-boundedness estimates, fix $\alpha >{\varepsilon}$. By Proposition \[prop:integr-der\] one has: $$\begin{aligned}
\gamma_{[X_{\beta}^{A},X]}\left(\{ s^{\alpha} [S(s)-\tilde{S}_0(s)]:s\in [0,t]\}\right) &  \leq \int_{0}^{t} \big{\Vert}\tfrac{d}{ds} \big(s^{\alpha} [S(s)-\tilde{S}_0(s)]\big) \big{\Vert}_{{\mathcal L}(X_{\beta}^{A},X)} \,ds  \\
& \leq \int_{0}^{t}  \alpha s^{\alpha-1}  \big{\Vert}S(s)-\tilde{S}_0(s) \big{\Vert}_{{\mathcal L}(X_{\beta}^{A},X)}\,ds \\
&  \quad  + \int_{0}^{t} s^{\alpha}\big{\Vert}\tfrac{d}{ds}S(s)-\tfrac{d}{ds}\tilde{S}_0(s) \big{\Vert}_{{\mathcal L}(X_{\beta}^{A},X)} \,ds.
\end{aligned}$$

Substituting and into the above one obtains that there exists a constant $C$ depending only on $\omega$, $\theta$, $K$, ${\varepsilon}=\delta-\beta$, and ${\Vert}P_0{\Vert}_{{\mathcal L}(X,X_0)}$ such that: $$\begin{aligned}
\gamma_{[X_{\beta}^{A},X]}\left(\{ s^{\alpha} [S(s)-\tilde{S}_0(s)]:s\in [0,t]\}\right)
& \leq C D_{\delta}(A,A_0) \int_{0}^{t}  e^{\omega' s} s^{\alpha-1-{\varepsilon}}e^{\bar{\omega t}}
 \,ds  \\
& \leq C e^{(\omega' T){\vee}0} t^{\alpha-{\varepsilon}} D_{\delta}(A,A_0),\end{aligned}$$ as $\alpha>{\varepsilon}$.

\[cor:Holder\] Let the setting be as in Theorem \[t:app\]. Let $\lambda \in [0,{\frac{1}{2}})$ satisfy $$\begin{aligned}
0\leq \lambda < \min\{1-(\delta-\theta_F){\vee}0, {\tfrac{1}{2}}-{\tfrac{1}{p}}-(\delta-\theta_G){\vee}0\}.\end{aligned}$$ Suppose $x_0\in L^p(\Omega,{\mathcal F}_0;X_{\delta}^{A})$ and $y_0 \in L^p(\Omega,{\mathcal F}_0;X)$, then: $$\begin{aligned}
& {\Vert}U - Sx_0 - i_{X_0}(U^{(0)} - S_0P_0y_0) {\Vert}_{L^p(\Omega;C^{\lambda}([0,T];X))}\\
& \qquad \lesssim {\Vert}x_0-y_0{\Vert}_{L^p(\Omega,X)} + D_{\delta}(A,A_0)(1+{\Vert}x_0
{\Vert}_{L^p(\Omega;X_{\delta}^{A})}),\end{aligned}$$ with implied constant depending on $X_0$ only in terms of ${\Vert}P_0{\Vert}_{{\mathcal L}(X,X_0)}$, on $A$ and $A_0$ only in terms of $1+D_{\delta}(A,A_0)$, $\omega$, $\theta$ and $K$, and on $F$ and $G$ only in terms of their Lipschitz and linear growth constants Lip$(F)$, $\textrm{Lip}_{\gamma}(G)$, $M(F)$, and $M_{\gamma}(G)$.

As before, we write: $$\label{pert.holder}
\begin{aligned}
& {\Vert}U - Sx_0 - i_{X_0}(U^{(0)} - S_0P_0y_0) {\Vert}_{L^p(\Omega;C^{\lambda}([0,T];X))} \\
& \qquad = \Big{\Vert}t\mapsto \int_0^{t}\tilde{S}_0(t-s)[F(s,U(s))-F(s,\tilde{U}^{(0)}(s))]\,ds
\Big{\Vert}_{L^p(\Omega;C^{\lambda}([0,T];X))}\\
& \quad\qquad + \Big{\Vert}t\mapsto \int_0^{t}[S(t-s)-\tilde{S}_0(t-s)]F(s,U(s))\,ds
\Big{\Vert}_{L^p(\Omega;C^{\lambda}([0,T];X))}\\
& \quad\qquad + \Big{\Vert}t\mapsto \int_0^{t}\tilde{S}_0(t-s)[G(s,U(s))-G(s,\tilde{U}^{(0)}(s))]\,dW_H(s)
\Big{\Vert}_{L^p(\Omega;C^{\lambda}([0,T];X))}\\
& \quad\qquad + \Big{\Vert}t\mapsto \int_0^{t}[S(t-s)-\tilde{S}_0(t-s)]G(s,U(s))\,dW_H(s)
\Big{\Vert}_{L^p(\Omega;C^{\lambda}([0,T];X))}.
\end{aligned}$$

For the first and second term on the right-hand side of we apply Proposition \[prop:detHolder\]. Note that as before we may pick $\eta_F,\eta_G\leq 0$ such that $\eta_F<\theta_F-\delta$ and $\eta_G<\theta_G-\delta$ and $$\lambda <\min\{1+\eta_F,{\tfrac{1}{2}}-{\tfrac{1}{p}}+\eta_G\}.$$ Our choice of $Y_1$, $Y_2$, $\Phi$, $\Psi$ is the same as in part 2c, respectively 2d, of the proof of Theorem \[t:app\], whereas we set $\theta=1-\lambda$. This leads to the following estimates: $$\begin{aligned}
& \Big{\Vert}t\mapsto \int_0^{t}\tilde{S}_0(t-s)[F(s,U(s))-F(s,\tilde{U}^{(0)}(s))]\,ds
\Big{\Vert}_{L^p(\Omega;C^{\lambda}([0,T];X))}\\
& \qquad  \qquad\lesssim {\Vert}U - \tilde{U}^{(0)}(s) {\Vert}_{L^{p}(\Omega;L^{\infty}(0,T;X))}\leq {\Vert}U - \tilde{U}^{(0)}(s) {\Vert}_{{V^{\alpha,p}_{\rm c}}([0,T]\times \Omega;X)},\end{aligned}$$ and $$\begin{aligned}
&\Big{\Vert}t\mapsto \int_0^{t}[S(t-s)-\tilde{S}_0(t-s)]F(s,U(s))\,ds
\Big{\Vert}_{L^p(\Omega;C^{\lambda}([0,T];X))}\\
& \qquad \qquad \lesssim D_{\delta}(A,A_0){\Vert}U {\Vert}_{L^{p}(\Omega;L^{\infty}(0,T;X))} \lesssim D_{\delta}(A,A_0)\big(1 + {\Vert}x_0{\Vert}_{L^p(\Omega,X)}\big).\end{aligned}$$

For the third and fourth term on the right-hand side of we apply Corollary with $\beta=\lambda$ and $\alpha\in (0,{\frac{1}{2}})$ such that $\alpha>\lambda +{\frac{1}{p}}+ \eta_G$. The choice of $Y_1$, $Y_2$, $\Phi$ and $\Psi$ is as in parts 2e and 2f of the proof of Theorem \[t:app\]. This leads to: $$\begin{aligned}
& \Big{\Vert}t\mapsto \int_0^{t}\tilde{S}_0(t-s)[G(s,U(s))-G(s,\tilde{U}^{(0)}(s))]\,dW_H(s)
\Big{\Vert}_{L^p(\Omega;C^{\lambda}([0,T];X))}\\
& \qquad \qquad \lesssim {\Vert}U - \tilde{U}^{(0)}(s) {\Vert}_{{V^{\alpha,p}_{\rm c}}([0,T]\times \Omega;X)},\end{aligned}$$ and $$\begin{aligned}
& \Big{\Vert}t\mapsto \int_0^{t}[S(t-s)-\tilde{S}_0(t-s)]G(s,U(s))\,dW_H(s)
\Big{\Vert}_{L^p(\Omega;C^{\lambda}([0,T];X))}\\
& \qquad \qquad \lesssim  D_{\delta}(A,A_0) {\Vert}U {\Vert}_{{V^{\alpha,p}_{\rm c}}([0,T]\times \Omega;X)} \lesssim D_{\delta}(A,A_0)\big(1 + {\Vert}x_0{\Vert}_{L^p(\Omega,X)}\big).\end{aligned}$$

Combining these estimates with Theorem \[t:app\] gives the desired result. It goes without saying that all the implied constants above depend on $X_0$ only in terms of ${\Vert}P_0{\Vert}_{{\mathcal L}(X,X_0)}$, on $A$ and $A_0$ only in terms of $1+D_{\delta}(A,A_0)$, $\omega$, $\theta$ and $K$, and on $F$ and $G$ only in terms of their Lipschitz and linear growth constants Lip$(F)$, $\textrm{Lip}_{\gamma}(G)$, $M(F)$, and $M_{\gamma}(G)$.

Yosida approximations {#sec-yosida}
=====================

Consider under the assumptions [[**(A)**]{}]{}, [[**(F)**]{}]{}, and [[**(G)**]{}]{} with the additional assumption that $\theta_F,\theta_G\geq 0$. We define $A_n:= nA R(n:A)$ to be the $n^{\textrm{th}}$ Yosida approximation of $A$. Let $U$ to denote the solution to with operator $A$ and initial data $x_0\in L^p(\Omega,{\mathcal F}_0;X)$ and, for $n\in \N$, let $U^{(n)}$ to denote the solution to with operator $A_n$ instead of $A$ and initial data $y_{0}\in L^p(\Omega,{\mathcal F}_0;X)$.

\[t:yosida\] For any $\eta\in [0,1]$ and $p\in (2,\infty)$ such that $$\eta<\min\{\tfrac{3}{2}-{\tfrac{1}{\tau}}+\theta_F,\tfrac{1}{2}-{\tfrac{1}{p}}+\theta_G \}$$ and any $\alpha\in
[0,{\frac{1}{2}})$ we have, assuming $y_0\in L^p(\Omega,\mathcal{F}_0;X_{\eta}^A)$: $$\begin{aligned}
{\Vert}U - U^{(n)}{\Vert}_{{V^{\alpha,p}_{\rm c}}([0,T]\times \Omega;X)}
&\lesssim  {\Vert}x_0 - y_0 {\Vert}_{L^p(\Omega;X)} + n^{-\eta}(1+{\Vert}y_0
{\Vert}_{L^p(\Omega;X_{\eta}^{A})}),\end{aligned}$$ with implied constants independent of $n$, $x_0$ and $y_0$.

The following corollary is a direct consequence of the Borel-Cantelli lemma and the above theorem (see Corollary [@Kloeneuen Lemma 2.1]):

\[c:yosida\_pathwise\] Let $\eta>0$ and $p\in (2,\infty)$ be such that $$\eta+{\tfrac{1}{p}}<\min\{\tfrac{3}{2}-{\tfrac{1}{\tau}}+\theta_F,\tfrac{1}{2}-{\tfrac{1}{p}}+\theta_G,1 \}$$ and assume $y_0=x_0\in L^{p}(\Omega,\mathcal{F}_0;X_{\eta}^A)$. Then there exists a random variable $\chi\in L^0(\Omega)$ such that for all $n\in \N$: $$\begin{aligned}
{\Vert}U - U^{(n)}{\Vert}_{C([0,T];{\mathcal H})} &\leq \chi n^{-\eta}.\end{aligned}$$

To prove Theorem \[t:yosida\] we shall need the following lemma:

\[l:fracpowYosida\] Let $\beta \in [0,1]$. Then there exists a constant $K'$ such that for all $n\geq 2\omega$ and all $x\in X^{A}_{\beta}$ one has: $$\begin{aligned}
{\Vert}(2\omega I -A_n)^{\beta} x {\Vert}& \leq K'{\Vert}(2\omega I -A)^{\beta} x {\Vert}.\end{aligned}$$

Observe that $$\begin{aligned}\label{omega-An}
2\omega I -A_n  = [(n+2\omega)I-4\omega^2 R(2\omega:A)](2\omega I - A)R(n:A).
\end{aligned}$$ Thus for $x\in D(A)$ and $n\geq 2\omega$ we have: $$\begin{aligned}
{\Vert}(2\omega I-A_n)x {\Vert}&\leq {\Vert}[(n+2\omega)I-4\omega^2 R(2\omega:A)] R(n:A) {\Vert}_{{\mathcal L}(X)} {\Vert}(2\omega
I -A)x {\Vert}\\
& \leq [K\tfrac{n+2\omega}{n-\omega}+K^2\tfrac{4\omega}{n-\omega}]{\Vert}(2\omega I -A)x {\Vert}\leq 4K(1+K){\Vert}(2\omega I -A)x
{\Vert}.\end{aligned}$$ This proves the lemma for $\beta=1$. For $\beta=0$ the lemma is trivial. For $\beta\in (0,1)$ we need two extra observations.

First of all, for $s>\omega$ and $\beta\in (0,1)$ we have, by definition, (see [@Pazy:83 Section 2.6]): $$\begin{aligned}
(s I-A)^{-\beta} x & =\tfrac{\sin (\pi\beta)}{\pi} \int_{0}^{\infty} t^{-\beta} ((t+s)I - A)^{-1} x dt,\end{aligned}$$ and hence $$\label{fracPowEst1}
\begin{aligned}
{\Vert}(s I-A)^{-\beta}{\Vert}_{{\mathcal L}(X)} & \leq K\tfrac{\sin (\pi\beta)}{\pi} \int_{0}^{\infty} t^{-\beta} (t+s-\omega)^{-1}  dt
\\
& \leq K \tfrac{\sin (\pi\beta)}{\pi}  \Big[ (s-\omega)^{-1} \int_{0}^{s-\omega} t^{-\beta} dt +
\int_{s-\omega}^{\infty}
t^{-1-\beta} dt \Big]\\
& = K\tfrac{\sin (\pi \beta)}{\pi \beta (1-\beta)} (s-\omega)^{-\beta}.
\end{aligned}$$

Secondly, let $\mu,\lambda\in \omega+\Sigma_{\frac{\pi}{2}+\theta}$. We have: $${\Vert}e^{-(\lambda I -A)R(\mu:A)t}
{\Vert}_{{\mathcal L}(X)}=e^{-t}{\Vert}e^{(\mu-\lambda) R(\mu,A)t}{\Vert}_{{\mathcal L}(X)}\leq e^{-t+\frac{|\mu-\lambda|}{|\mu-\omega|}Kt}.$$ Now suppose $n\geq 2\omega(1+4K)$, $\lambda=2\omega$, $\mu=\frac{n\lambda}{\lambda+n}=\frac{2\omega n}{2\omega+n}$. In that case one may check that $ \tfrac{|\mu-\lambda|}{|\mu-\omega|}K \leq {\tfrac{1}{2}},$ and thus that for $\beta\in (0,1)$: $$\begin{aligned}
{\Vert}[-(2\omega I -A)R(\tfrac{2\omega n}{2\omega+n}:A)]^{-\beta} {\Vert}_{{\mathcal L}(X)} & = \Big{\Vert}{\tfrac{1}{\Gamma(\beta)}}\int_{0}^{\infty}
t^{\beta-1}e^{-(2\omega I -A)R(\frac{2\omega n}{2\omega+n}:A)t}dt \Big{\Vert}_{{\mathcal L}(X)}\\
& \leq 2^{\beta}.
\end{aligned}$$ As $[-(2\omega I -A)R(\tfrac{2\omega n}{2\omega+n}:A)]^{-\beta} \in {\mathcal L}(X)$ for any $n\geq 2\omega$, it follows that there exists a constant $M>0$ such that for all $n\geq 2\omega$: $$\begin{aligned}
\label{fracPowEst2}
{\Vert}[-(2\omega I -A)R(\tfrac{2\omega n}{2\omega+n}:A)]^{-\beta} {\Vert}_{{\mathcal L}(X)} & \leq M.\end{aligned}$$

For $\beta\in (0,1)$ and $x\in X^A_{\beta}$ we have, by standard theory on functional calculus (see [@Haase]), equation \[omega-An\], and the estimates and : $$\begin{aligned}
{\Vert}(2\omega I -A_n)^{\beta} x {\Vert}&= {\Vert}(n+2\omega)^{\beta}(\tfrac{2\omega n}{2\omega+n}I-A)^{\beta}(n I -A)^{-\beta}x {\Vert}\\
& \leq (n+2\omega)^{\beta} {\Vert}\big[-(2\omega I-A)R(\tfrac{2\omega n}{2\omega+n}:A)\big]^{-\beta}{\Vert}_{{\mathcal L}(X)} \\
& \quad \times {\Vert}(2\omega I -A)^{\beta}x {\Vert}{\Vert}(n I-A)^{-\beta}{\Vert}_{{\mathcal L}(X)}\\
& \leq 4\tfrac{\sin (\pi \beta)}{\pi \beta (1-\beta)} KM {\Vert}(2\omega I -A)^{\beta}x {\Vert}.\end{aligned}$$

Without loss of generality we may assume $\omega\geq 0$. In order to apply Theorem \[t:app\], we must prove that $A_n$, $n\geq 2\omega$, are of uniform type, i.e., that there exist $\bar{\omega}\in \R, \bar{\theta}\in
(0,\frac{\pi}{2})$ and $\bar{K}>0$ such that $A_n$ is of type $(\bar{\omega}, \bar{\theta},\bar{K})$ for all $n\geq
2\omega$. Fix $n\geq 2\omega$. One checks that: $$\begin{aligned}
\label{yosres}
R(\lambda:A_n) & = (n+\lambda)^{-1}(n-A)R(\tfrac{\lambda n}{n+\lambda}:A)\end{aligned}$$ whenever $\frac{\lambda n}{n+\lambda} \in \omega + \Sigma_{\frac{\pi}{2}+\theta}$. Define $f:\C\rightarrow \C$; $f(z)=\frac{nz}{n-z}$. From it follows that $\lambda\in \rho(A)$ if and only if $f(\lambda)\in
\rho(A_n)$. By standard theory on Möbius transforms we have that $$f\big(\{\omega+\Sigma_{\frac{\pi}{2}+\theta}\}\big) = \C \setminus \big( D_{1}\cap D_{2}\big),$$ where $D_1$ and $D_2$ are both closed disks with radius $\frac{n^2}{2(n-\omega)\cos \theta}$; the center of $D_1$ is in $\frac{n}{2(n-\omega)}(2\omega-n, \tan(\theta))$ and the center of $D_2$ is is in $\frac{n}{2(n-\omega)}(2\omega-n,-\tan(\theta))$. The boundaries of these disks intersect each other on the real axis at the points $-n$ and $\frac{n\omega}{n-\omega}$. The angle at intersection is $\pi-2\theta$. As $n\geq 2\omega$ we have $\frac{n\omega}{n-\omega}\leq 2\omega$ and thus $\rho(A_n)\subset 2\omega+\Sigma_{\frac{\pi}{2}+\theta}$. It remains to prove the desired estimate on the resolvent.

Using one may check that for $\lambda \in 2\omega+\Sigma_{\frac{\pi}{2}+\theta}$ we have: $$\begin{aligned}
\label{YosidaResComp}
R(\lambda:A)-R(\lambda:A_n) & = -(\lambda+n)^{-1}A^2R(\tfrac{\lambda n}{n+\lambda}:A)R(\lambda:A).\end{aligned}$$ Thus by we have, for $\lambda \in  \omega(1+2(\cos\theta)^{-1}) +\Sigma_{\frac{\pi}{2}+\theta}$: $$\begin{aligned}
{\Vert}R(\lambda:A)-R(\lambda:A_n) {\Vert}_{{\mathcal L}(X)} & \leq (1+2K)^2|\lambda+n|^{-1} \leq (1+2K)^2|\lambda-\omega|^{-1}.\end{aligned}$$ The final estimate follows from the fact that by standard theory on Möbius transforms we have that $\tfrac{|\lambda-\omega|}{|\lambda+n|}\leq 1$ for $\lambda \in \omega+\Sigma_{\frac{\pi}{2}+\theta}$. In conclusion we have, for $\lambda \in  \omega(1+2(\cos\theta)^{-1}) +\Sigma_{\frac{\pi}{2}+\theta}$: $$\begin{aligned}
{\Vert}R(\lambda:A_n) {\Vert}_{{\mathcal L}(X)} & \leq {\Vert}R(\lambda:A){\Vert}_{{\mathcal L}(X)}+ {\Vert}R(\lambda:A)-R(\lambda:A_n) {\Vert}_{{\mathcal L}(X)}  \\
& \leq [K+(1+2K)^2]|\lambda-\omega|^{-1}.\end{aligned}$$ This proves that $A_n$ is of type $(\omega(1+2(\cos\theta)^{-1}),\theta,K+(1+2K)^2)$ for all $n\geq 2\omega$.

It also follows from that if we take, for example, $\lambda_0= \omega(1+2(\cos\theta)^{-1})$, then we have, for $n\geq 2\omega$: $$\begin{aligned}
{\Vert}R(\lambda_0:A)-R(\lambda_0:A_n){\Vert}_{{\mathcal L}(X)} & \leq (1+2K)^{2}n^{-1}.\end{aligned}$$ In other words, for all $n\in \N$ condition in Theorem \[t:app\] is satisfied with $\delta=1$ and $\lambda_0= \omega(1+2(\cos\theta)^{-1})$. In particular we can apply Theorem \[t:app\] to obtain the desired result for the case $\theta_F>-\frac{1}{2}+{\frac{1}{\tau}}$, where $\tau$ is the type of $X$, and $\theta_G> {\frac{1}{2}}+{\frac{1}{p}}$. Concerning the dependence on $1+D(A,A_n)$ of the implied constant in , note that $1+D(A,A_n)$ is uniformly bounded in $n$, both from above and away from $0$.

In order to get the desired result for general $\theta_F,\theta_G\geq 0$ we consider the difference $R(\lambda_0:A)-R(\lambda_0:A_n)$ in the ${\mathcal L}(X_{\delta-1}^{A_n},X)$-norm. (Note that if $A$ is unbounded then $R(\lambda_0:A)-R(\lambda_0:A_n)\notin {\mathcal L}(X^{A}_{\delta-1},X)$ for any $\delta<1$.) For $n\geq
\omega(1+2(\cos\theta)^{-1})$ we have, by , that ${\Vert}(2\omega I-A_n) {\Vert}_{{\mathcal L}(X)}\leq
2n(1+K)$. Thus by Theorem \[t:fracPowInt\] we have, for $\delta \in (0,1)$: $$\begin{aligned}
{\Vert}(2\omega I - A_n)^{1-\delta} x {\Vert}& \leq 2(1+2K) {\Vert}x {\Vert}^{\delta} {\Vert}A_n x{\Vert}^{1-\delta} \\
& \leq 2^{2-\delta}(1+2K)^{2-\delta} n^{1-\delta} {\Vert}x {\Vert}.\end{aligned}$$ It follows that for $\delta \in [0,1)$ we have: $$\begin{aligned}
{\Vert}R(\lambda_0:A)-R(\lambda_0:A_n) {\Vert}_{{\mathcal L}(X^{A_n}_{\delta-1},X)} & \leq 2^{2-\delta}(1+2K)^{4-\delta} n^{-\delta}.\end{aligned}$$

We are now ready to apply Theorem \[t:app\]. First of all observe that by Lemma \[l:fracpowYosida\] we have that $F:[0,T]\times X\rightarrow X_{\theta_F}^{A_n}$ is Lipschitz continuous and of linear growth for all $n\geq 2\omega$ with Lipschitz and growth constants independent of $n$, and $G:[0,T]\times X\rightarrow \gamma(H,X_{\theta_G}^{A_n})$ is $L^2_{\gamma}$-Lipschitz continuous and of linear growth for all $n\geq
2\omega$ with Lipschitz and growth constants independent of $n$. Also, $1+D(A,A_n)$ is uniformly bounded in $n$.

Fix $\eta\in [0,1]$ such that $\eta<\min\{\frac{3}{2}-{\frac{1}{\tau}}+\theta_F,\frac{1}{2}-{\frac{1}{p}}+\theta_G \}$ and suppose $y_0\in
L^p(\Omega,\mathcal{F}_0;X_{\eta}^{A})$. It follows from Theorem \[t:app\] with $\delta=\eta$, but with $A_n$ playing the role of $A$ and $A$ playing the role of $A_0$, that: $$\begin{aligned}
{\Vert}U - U^{(n)}{\Vert}_{{V^{\alpha,p}_{\rm c}}([0,T]\times \Omega;X)} &\lesssim  {\Vert}x_0 - y_0{\Vert}_{L^p(\Omega;X)}
+ n^{-\eta}(1+{\Vert}y_0 {\Vert}_{L^p(\Omega;X_{\eta}^{A})}),\end{aligned}$$ with implied constants independent of $n$, $x_0$ and $y_0$.

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