# Regularity theory for a new class of fractional
parabolic stochastic evolution equations

Kristin Kirchner and Joshua Willems  Delft Institute of Applied Mathematics
Delft University of Technology
P.O. Box 5031
2600 GA Delft
The Netherlands<EMAIL_ADDRESS>Delft Institute of Applied Mathematics
Delft University of Technology
P.O. Box 5031
2600 GA Delft
The Netherlands<EMAIL_ADDRESS>
###### Abstract.

A new class of fractional-order stochastic evolution equations of the form
$(\partial_{t}+A)^{\gamma}X(t)=\dot{W}^{Q}(t)$, $t\in[0,T]$,
$\gamma\in(0,\infty)$, is introduced, where $-A$ generates a $C_{0}$-semigroup
on a separable Hilbert space $H$ and the spatiotemporal driving noise
$\dot{W}^{Q}$ is the formal time derivative of an $H$-valued cylindrical
$Q$-Wiener process. Mild and weak solutions are defined; these concepts are
shown to be equivalent and to lead to well-posed problems. Temporal and
spatial regularity of the solution process $X$ are investigated, the former
being measured by mean-square or pathwise smoothness and the latter by using
domains of fractional powers of $A$. In addition, the covariance of $X$ and
its long-time behavior are analyzed.

These abstract results are applied to the cases when $A:=L^{\beta}$ and
$Q:=\widetilde{L}^{-\alpha}$ are fractional powers of symmetric, strongly
elliptic second-order differential operators defined on (i) bounded Euclidean
domains or (ii) smooth, compact surfaces. In these cases, the Gaussian
solution processes can be seen as generalizations of merely spatial
(Whittle–)Matérn fields to space–time.

###### Key words and phrases:

Spatiotemporal Gaussian processes, Matérn covariance, nonlocal space–time
differential operators, mild solution, mean-square differentiability, strongly
continuous semigroups.

###### 2010 Mathematics Subject Classification:

Primary: 60G15, 60H15; secondary: 47D06, 35R11.

##  1\. Introduction

###  1.1. Motivation and background

Gaussian processes play an important role for modeling in spatial statistics.
Typical applications arise in the environmental sciences, where geographically
indexed data is collected, including climatology [3, 66], oceanography [8],
meteorology [38], and forestry [7, 43, 54]. More generally, hierarchical
models based on Gaussian processes have been used in various disciplines,
where spatially dependent (or spatiotemporal) data is recorded, such as
demography [28, 62], epidemiology [48], finance [33], and neuroimaging [56].

Since a Gaussian process $(X(j))_{j\in\mathcal{I}}$ is fully characterized by
its mean and its covariance function, _second-order-based approaches_ focus on
the construction of appropriate covariance classes. In the case that the index
set $\mathcal{I}$ is given by a spatial domain in the Euclidean space
$\mathcal{I}=\mathcal{D}\subseteq\mathbb{R}^{d}$, the _Matérn covariance
class_ [54] is an important and widely used model. The Matérn covariance
function is given by

$\varrho(x,y)=2^{1-\nu}\sigma^{2}[\Gamma(\nu)]^{-1}(\kappa\|x-y\|_{\mathbb{R}^{d}})^{\nu}K_{\nu}(\kappa\|x-y\|_{\mathbb{R}^{d}}),\quad
x,y\in\mathcal{D},$ (1.1)

where $K_{\nu}$ denotes the modified Bessel function of the second kind. It is
indexed by the three interpretable parameters
$\nu,\kappa,\sigma^{2}\in(0,\infty)$, which determine _smoothness_ ,
_correlation length_ and _variance_ of the process. It is this feature that
renders the Matérn class particularly suitable for making inference about
spatial data [69].

When considering _spatiotemporal_ phenomena, the following two difficulties
occur:

  1. 1.

It is desirable to control the properties of the stochastic process named
above (in particular, smoothness and correlation lengths) separately in space
and time. For this reason, considering (1.1) in $d+1$ dimensions is not
expedient and it is a difficult task to construct appropriate spatiotemporal
covariance models, see e.g. [22, 34, 36, 64, 65, 70].

  2. 2.

Second-order-based approaches require the factorization of, in general, dense
covariance matrices, causing computational costs which are cubic in the number
of observations. The two common assumptions imposed on spatiotemporal
covariance models to reduce the computational costs—separability
(factorization into merely spatial and temporal covariance functions) and
stationarity (invariance under translations)—have proven unrealistic in many
situations, see [23, 55, 70]. In particular, Stein [70] criticized the
behavior of separable covariance functions with respect to their
differentiability.

Owing to these problems, the class of _dynamical models_ has gained
popularity. The name originates from focusing on the dynamics of the
stochastic process which are described either by means of conditional
probability distributions or by representing the process as a solution of a
stochastic partial differential equation (SPDE). The latter approach was
originally proposed in the merely spatial case, motivated by the following
observation made by Whittle [73]: A stationary process
$(X(x))_{x\in\mathcal{D}}$ indexed by the entire Euclidean space
$\mathcal{D}=\mathbb{R}^{d}$ which solves the SPDE

$\bigl{(}\kappa^{2}-\Delta\bigr{)}^{\beta}X(x)=\mathcal{W}(x),\quad
x\in\mathcal{D},$ (1.2)

has a covariance function of Matérn type (1.1) with
$\nu=2\beta-\nicefrac{{d}}{{2}}$. Here, $\Delta$ denotes the Laplacian and
$\mathcal{W}$ is Gaussian white noise. This relation gave rise to the SPDE
approach proposed by Lindgren, Rue, and Lindström [50], where the SPDE (1.2)
is considered on a bounded domain $\mathcal{D}\subsetneq\mathbb{R}^{d}$ and
augmented with Dirichlet or Neumann boundary conditions. Besides enabling the
applicability of efficient numerical methods available for (S)PDEs, such as
finite element methods [11, 13, 14, 21, 40, 50] or wavelets [16, 39], this
approach has the advantage of allowing for

  1. (a)

nonstationary or anisotropic generalizations by replacing the operator
$\kappa^{2}-\Delta$ in (1.2) with more general strongly elliptic second-order
differential operators,

$(Lv)(x)=\kappa^{2}(x)v(x)-\nabla\cdot(a(x)\,\nabla v(x)),\quad
x\in\mathcal{D},$ (1.3)

where $\kappa\colon\mathcal{D}\to\mathbb{R}$ and
$a\colon\mathcal{D}\to\mathbb{R}_{\rm sym}^{d\times d}$ are functions [6, 11,
13, 14, 21, 35, 40, 50];

  2. (b)

more general domains, such as surfaces [15, 40] or manifolds [39].

In the SPDE (1.2) the fractional exponent $\beta$ defines the (spatial)
differentiability of its solution, see e.g. [21]. A realistic description of
spatiotemporal phenomena necessitates controllable differentiability in space
and time. This motivates to consider the space–time fractional SPDE model

$\begin{cases}\left(\partial_{t}+L^{\beta}\right)^{\gamma}X(t,x)=\dot{\mathcal{W}}(t,x),&\;t\in[0,T],\quad
x\in\mathcal{D},\\\ \hskip
48.36958ptX(0,x)=X_{0}(x),&\;x\in\mathcal{D},\end{cases}$ (1.4)

where $L$ in (1.3) is augmented with boundary conditions on
$\partial\mathcal{D}$, $(X_{0}(x))_{x\in\mathcal{D}}$ is the initial random
field, $\dot{\mathcal{W}}$ denotes space–time Gaussian white noise, and
$T\in(0,\infty)$ is the time horizon. Whenever $\beta=\gamma=1$, the SPDE
(1.4) simplifies to the stochastic heat equation and this spatiotemporal model
had already been mentioned in [50] and it was used for statistical inference
in [18, 67]. The novelty and sophistication of the SPDE model (1.4) lies in
the fractional power $\gamma\in(0,\infty)$ of the parabolic operator. Notably,
it is the interplay of the parameters $\beta$ and $\gamma$ that will
facilitate controlling spatial and temporal smoothness of the solution
process. For $\mathcal{D}=\mathbb{R}^{d}$, this has recently been
investigated via Fourier techniques in [49], see also [4, 19, 44].

Besides the aforementioned benefits of the SPDE approach and in contrast to
the SPDE $\bigl{(}\partial_{t}^{\gamma}+L^{\beta}\bigr{)}X=\dot{\mathcal{W}}$,
considered for instance in [17, 30], the SPDE model (1.4) furthermore exhibits
a long-time behavior resembling the spatial model (1.2).

###  1.2. Contributions

We introduce a novel interpretation of (1.4) with $X_{0}=0$ as a fractional
parabolic stochastic evolution equation, and correspondingly define mild and
weak solutions for it. To this end, we first give a meaning to fractional
powers of an operator of the form $\partial_{t}+A$, where $-A$ generates a
$C_{0}$-semigroup. Generalizing the approach taken for $\gamma=1$ in [25,
Chapter 5], we prove that mild and weak solutions are equivalent under natural
assumptions, and we investigate their existence, uniqueness, regularity, and
covariance. Our main findings are that the problem (1.4) is well-posed, and
the properties of its solution $X$ with respect to smoothness and covariance
structure generalize those of the spatial Whittle–Matérn SPDE model (1.2) and
relate to the parameters $\beta,\gamma\in(0,\infty)$ in the desired way.
Restricting the analysis to a zero initial field is justified by our primary
interest in regularity related to the dynamics of (1.4) and the long-time
behavior of solutions.

In comparison with [9, 10, 51, 57, 71]—the only previous works on an equation
of the form $(\partial_{t}+L)^{\gamma}u=f$ known to the authors—the main
contributions of this work, besides considering a stochastic right-hand side,
are the fractional power $\beta$ in (1.4) and the method of proving regularity
using semigroups. As opposed to the extension approach in [9, 10, 51, 57, 71],
this setting does not require a Euclidean structure.

###  1.3. Outline

Preliminary notation and theory will be introduced in Section 2. In Section 3
we give a meaning to the parabolic operator $\partial_{t}+A$ and its
fractional powers in order to introduce well-defined mild and weak solutions
of (1.4) with ${X_{0}=0}$. Subsequently, we analyze these in terms of
spatiotemporal regularity. Section 4 is concerned with the covariance
structure of solutions. Finally, in Section 5 we apply our results to the
space–time Whittle–Matérn SPDE (1.4) considered on a bounded Euclidean domain
or on a surface. This article is supplemented by two appendices: Appendix A
contains several technical auxiliary results used in the proofs of Section 3.
Appendix B collects necessary definitions and results from functional
calculus.

##  2\. Preliminaries

###  2.1. Notation

The sets $\mathbb{N}:=\\{1,2,3,\dots\\}$ and
$\mathbb{N}_{0}:=\mathbb{N}\cup\\{0\\}$ denote the positive and non-negative
integers, respectively. We write $s\wedge t$ (or $s\vee t$) for the minimum
(or maximum) of two real numbers $s,t\in\mathbb{R}$. The real and imaginary
parts of a complex number $z\in\mathbb{C}$ are denoted by $\operatorname{Re}z$
and $\operatorname{Im}z$, respectively; its argument, denoted $\arg z$, takes
its values in $(-\pi,\pi]$. We write $\mathbf{1}_{D}$ for the indicator
function of a set $D$. The restriction of a function $f\colon D\to E$ to a
subset $D_{0}\subseteq D$ is denoted by $f|_{D_{0}}\colon D_{0}\to E$; the
image of $D_{0}$ under a linear mapping $T$ is written as $TD_{0}$. Given two
parameter sets $\mathscr{P},\mathscr{Q}$ and two mappings
$\mathscr{F},\mathscr{G}\colon\mathscr{P}\times\mathscr{Q}\to\mathbb{R}$, we
use the expression $\mathscr{F}(p,q)\lesssim_{q}\mathscr{G}(p,q)$ to indicate
that for each $q\in\mathscr{Q}$ there exists a constant ${C_{q}\in(0,\infty)}$
such that $\mathscr{F}(p,q)\leq C_{q}\,\mathscr{G}(p,q)$ for all
${p\in\mathscr{P}}$. We write $\mathscr{F}(p,q)\eqsim_{q}\mathscr{G}(p,q)$ if
both relations, $\mathscr{F}(p,q)\lesssim_{q}\mathscr{G}(p,q)$ and
$\mathscr{G}(p,q)\lesssim_{q}\mathscr{F}(p,q)$, hold simultaneously.

###  2.2. Banach spaces and operators

If not specified otherwise, $E$ or $F$ denote separable Banach spaces. We
instead write $H$ or $U$ if we work with separable Hilbert spaces and wish to
emphasize this. The scalar field $\mathbb{K}$ is either given by the real
numbers $\mathbb{R}$ or the complex numbers $\mathbb{C}$. A norm on $E$ will
be denoted by $\|\,\cdot\,\|_{E}$ and an inner product on $H$ by
$(\,\cdot\,,\,\cdot\,)_{H}$. We write $I$ for the identity operator. The
notation ${E\hookrightarrow F}$ indicates that $E$ is continuously embedded in
$F$, i.e., there exists a bounded injective map from $E$ to $F$. The dual
space of $E$ is denoted by $E^{*}$. We write $\overline{E_{0}}^{E}$ for the
closure of a subset $E_{0}\subseteq E$ with respect to the norm on $E$; the
superscript may be omitted when there is no risk of confusion. The Borel
$\sigma$-algebra of $E$ is denoted by $\mathcal{B}(E)$.

We write ${T\in\mathscr{L}(E;F)}$ if the linear operator $T\colon E\to F$ is
bounded. Whenever $E=F$, we abbreviate $\mathscr{L}(E):=\mathscr{L}(E;E)$, and
this convention holds also for all other spaces of operators to be introduced.
The space $\mathscr{L}(E;F)$ is rendered a Banach space when equipped with the
usual operator norm; the space of Hilbert–Schmidt operators
$\mathscr{L}_{2}(U;H)\subseteq\mathscr{L}(U;H)$ is a Hilbert space with
respect to the inner product
$(T,S)_{\mathscr{L}_{2}(U;H)}:=\sum_{j\in\mathbb{N}}(Te_{j},Se_{j})_{H}$,
where $(e_{j})_{j\in\mathbb{N}}$ is any orthonormal basis for $U$. We write
${T^{*}\in\mathscr{L}(F^{*};E^{*})}$ for the adjoint operator of
$T\in\mathscr{L}(E;F)$. In the case that $T\in\mathscr{L}(U;H)$, we identify
$U^{*}=U$ and $H^{*}=H$ via the Riesz maps, so that
$T^{*}\in\mathscr{L}(H;U)$. An operator $T\in\mathscr{L}(H)$ is said to be
self-adjoint if $T^{*}=T$, non-negative if $(Tx,x)_{H}\geq 0$ holds for all
$x\in H$, and strictly positive if there exists a constant
$\theta\in(0,\infty)$ such that $(Tx,x)_{H}\geq\theta\|x\|_{H}^{2}$ holds for
all $x\in H$.

A linear operator $A$ on $E$ with domain $\mathsf{D}(A)$ is denoted by
${A\colon\mathsf{D}(A)\subseteq E\to E}$ and its range by $\mathsf{R}(A)$. We
call $A$ closed if its graph $\mathsf{G}(A):={\\{(x,Ax):x\in\mathsf{D}(A)\\}}$
is closed with respect to the graph norm
$\|(x,Ax)\|_{\mathsf{G}(A)}:=\|x\|_{E}+\|Ax\|_{E}$, and densely defined if
$\mathsf{D}(A)$ is dense in $E$. The definition
$\|x\|_{\mathsf{D}(A)}:=\|(x,Ax)\|_{\mathsf{G}(A)}$ yields a norm on
$\mathsf{D}(A)$. If $\mathsf{G}(A)\subseteq\mathsf{G}(\widetilde{A})$ for
another linear operator $\widetilde{A}$ on $E$, then $\widetilde{A}$ is called
an extension of $A$ and we write $A\subseteq\widetilde{A}$. If
$\overline{\mathsf{G}(A)}$ is the graph of a linear operator, then we call
this operator the closure of $A$, denoted $\overline{A}$.

###  2.3. Function spaces

Let a measure space $(S,\mathcal{S},\mu)$ be given. We abbreviate the phrases
“almost everywhere” and “almost all” by “a.e.” and “a.a.”, respectively.

We say that a function $f\colon S\to E$ is strongly measurable if it is the
$\mu$-a.e. limit of measurable simple functions. For $p\in[1,\infty]$, the
Bochner space of (equivalence classes of) strongly measurable, $p$-integrable
functions is denoted by $L^{p}(S;E)$. It is equipped with the norm

$\|f\|_{L^{p}(S;E)}:=\begin{cases}\left(\int_{S}\|f(t)\|_{E}^{p}\mathop{}\\!\mathrm{d}\mu(t)\right)^{\nicefrac{{1}}{{p}}}&\text{if
$p\in[1,\infty)$},\\\ \operatorname{ess\,sup}_{t\in S}\|f(t)\|_{E}&\text{if
$p=\infty$},\end{cases}$

where $\operatorname{ess\,sup}$ denotes the essential supremum. The norm on
$L^{2}(S;H)$ is induced by the inner product
$(f,g)_{L^{2}(S;H)}:=\int_{S}(f(t),g(t))_{H}\mathop{}\\!\mathrm{d}\mu(t)$.

Now let $S$ be an interval $S:=J\subseteq\mathbb{R}$, equipped with the Borel
$\sigma$-algebra and the Lebesgue measure. The space of continuous functions
from $J$ to $E$ will be denoted by $C(J;E)$ or $C^{0,0}(J;E)$ and be endowed
with the supremum norm. For $\alpha\in(0,1]$, we consider the space
$C^{0,\alpha}(J;E)$ of $\alpha$-Hölder continuous functions with norm

$\|f\|_{C^{0,\alpha}(J;E)}:=|f|_{C^{0,\alpha}(J;E)}+\|f\|_{C(J;E)},\;\text{where}\;\;|f|_{C^{0,\alpha}(J;E)}:=\sup_{t,s\in
J,\,t\neq s}\tfrac{\|f(t)-f(s)\|_{E}}{|t-s|^{\alpha}}$

is the $\alpha$-Hölder seminorm. For $n\in\mathbb{N}_{0}$ and $0\leq\alpha\leq
1$, the space $C^{n,\alpha}(J;E)$ consists of functions whose $n$th derivative
exists and belongs to $C^{0,\alpha}(J;E)$. On this space we use the norm
$\|f\|_{C^{n,\alpha}(J;E)}:=\|f^{(n)}\|_{C^{0,\alpha}(J;E)}+\sum_{k=0}^{n-1}\|f^{(k)}\|_{C(J;E)}$,
where $f^{(k)}$ denotes the $k$th derivative of $f$. Moreover, we define
$C^{\infty}(J;E):=\bigcap_{n\in\mathbb{N}}C^{n,0}(J;E)$. We say that $f\in
C^{n,\alpha}(J;E)$ is compactly supported if the support of $f$, defined by
$\operatorname{supp}f:=\overline{\\{t\in J:f(t)\neq 0\\}}^{J}$, is compact.
The space consisting of such functions is denoted by $C_{c}^{n,\alpha}(J;E)$.
If $f$ vanishes at a point $t\in J$, then we use the notation ${f\in
C^{n,\alpha}_{0,\\{t\\}}(J;E)}$. The spaces $C_{c}^{\infty}(J;E)$ and
$C_{0,\\{t\\}}^{\infty}(J;E)$ are defined analogously.

For an open interval $J$, we say that $u\in L^{2}(J;E)$ belongs to
$H^{1}(J;E)$ if there exists a function $v\in L^{2}(J;E)$ such that
$\int_{J}v(t)\phi(t)\mathop{}\\!\mathrm{d}t=-\int_{J}u(t)\phi^{\prime}(t)\mathop{}\\!\mathrm{d}t$
holds for all $\phi\in C_{c}^{\infty}(J;\mathbb{R})$. The function
${\partial_{t}u:=v}$ is called the weak derivative of $u$ and the norm on
$H^{1}(J;E)$ is
$\|u\|_{H^{1}(J;E)}:=\bigl{(}\|u\|_{L^{2}(J;E)}^{2}+\|\partial_{t}u\|_{L^{2}(J;E)}^{2}\bigr{)}^{\nicefrac{{1}}{{2}}}$.
The completion of $C_{c}^{\infty}((0,\infty);E)$ with respect to
$\|\,\cdot\,\|_{H^{1}(0,\infty;E)}$ defines the space
$H^{1}_{0,\\{0\\}}(0,\infty;E)$. Elements of $H^{1}_{0,\\{0\\}}(J;E)$ are
restrictions of functions in $H^{1}_{0,\\{0\\}}(0,\infty;E)$ to
${J\subseteq(0,\infty)}$.

Whenever the function space contains functions mapping to $E=\mathbb{R}$, we
omit the codomain, e.g., we write $L^{p}(S):=L^{p}(S;\mathbb{R})$ for the
Lebesgue spaces.

###  2.4. Vector-valued stochastic processes

Throughout this article, $(\Omega,\mathcal{F},\mathbb{P})$ denotes a complete
probability space that is equipped with a normal filtration
$(\mathcal{F}_{t})_{t\geq 0}$, i.e., $\mathcal{F}_{0}$ contains all elements
$B\in\mathcal{F}$ with $\mathbb{P}(B)=0$ and
$\mathcal{F}_{t}=\bigcap_{s>t}\mathcal{F}_{s}$ for all $t\geq 0$. Statements
which hold $\mathbb{P}$-almost surely are marked with “$\mathbb{P}$-a.s.”.

We call every strongly measurable function $Z\colon\Omega\to E$ a (vector-
valued) random variable, and the expectation of $Z\in L^{1}(\Omega;E)$ is
defined as the Bochner integral
$\mathbb{E}[Z]:=\int_{\Omega}Z(\omega)\mathop{}\\!\mathrm{d}\mathbb{P}(\omega)$.
An $E$-valued stochastic process $X=(X(t))_{t\in[0,T]}$ indexed by the
interval $[0,T]$, $T\in(0,\infty)$, is called integrable if
${(X(t))_{t\in[0,T]}\subseteq L^{p}(\Omega;E)}$ holds for $p=1$, and square-
integrable if this inclusion is true for $p=2$. It is said to be predictable
if it is strongly measurable as a mapping from $[0,T]\times\Omega$ to $E$,
where the former set is equipped with the $\sigma$-algebra generated by the
family

$\\{(s,t]\times F_{s}:0\leq s<t\leq
T,\,F_{s}\in\mathcal{F}_{s}\\}\cup\\{\\{0\\}\times
F_{0}:F_{0}\in\mathcal{F}_{0}\\}.$

Given another $E$-valued process
$\widetilde{X}:=(\widetilde{X}(t))_{t\in[0,T]}$, we call $\widetilde{X}$ a
modification of $X$, provided that $\mathbb{P}(X(t)=\widetilde{X}(t))=1$ holds
for all $t\in[0,T]$. Moreover, $X$ and $\widetilde{X}$ are said to be
indistinguishable if $\mathbb{P}(\forall t\in[0,T]:X(t)=\widetilde{X}(t))=1$.

For a self-adjoint strictly positive operator $Q\in\mathscr{L}(H)$,
$(W^{Q}(t))_{t\geq 0}$ denotes a cylindrical $Q$-Wiener process with respect
to $(\mathcal{F}_{t})_{t\geq 0}$ which takes its values in $H$, cf. [52,
Proposition 2.5.2]; if $Q=I$, we omit the superscript and call $(W(t))_{t\geq
0}$ a cylindrical Wiener process.

##  3\. Analysis of the fractional stochastic evolution equation

The aim of this section is to define and analyze solutions to the following
stochastic evolution equation of the general fractional order
$\gamma\in(0,\infty)$:

$(\partial_{t}+A)^{\gamma}X(t)=\dot{W}^{Q}(t),\quad t\in[0,T],\qquad X(0)=0.$
(3.1)

We interpret this as an abstraction of (1.4) with $X_{0}=0$. As noted in the
introduction, we restrict the discussion to a zero initial field, since we are
primarily interested in properties resulting from the dynamics of the SPDE
(1.4), respectively (3.1), and the long-time behavior for $0\ll T<\infty$ of
its solution. We also note that imposing non-zero boundary data for fractional
problems is, in general, highly non-trivial, see e.g. the recent works [1, 5]
on the fractional Laplacian.

In Subsection 3.1 we investigate the _parabolic operator_ $\mathcal{B}$, which
is defined as the closure of the sum operator $\partial_{t}+A$ on an
appropriate domain. In particular, we consider the $C_{0}$-semigroup generated
by $-\mathcal{B}$, which is used to define fractional powers
$\mathcal{B}^{\gamma}$ for $\gamma\in\mathbb{R}$. Interpreting the expression
$(\partial_{t}+A)^{\gamma}$ appearing in (3.1) as $\mathcal{B}^{\gamma}$, we
use this result to define mild solutions in Subsection 3.2. In this part, we
furthermore introduce a weak solution concept for (3.1), and prove equivalence
of the two solution concepts as well as existence and uniqueness of mild and
weak solutions. Spatiotemporal regularity of solutions is the subject of
Subsection 3.3.

###  3.1. The parabolic operator and its fractional powers

In this subsection we define the parabolic operator $\mathcal{B}$ and
fractional powers $\mathcal{B}^{\gamma}$. We start by formulating several
assumptions on the linear operator $A$, to which we shall refer throughout the
remainder of this work. For an overview of the theory of $C_{0}$-semigroups,
we refer the reader to [31] or [61]. The complexification of a normed space or
operator is indicated by the subscript $\mathbb{C}$; see Subsection B.2.1 in
Appendix B for details.

######  Assumption 3.1.

Let $H$ be a separable Hilbert space over the real scalar field $\mathbb{R}$.
We assume that the linear operator ${A\colon\mathsf{D}(A)\subseteq H\to H}$
satisfies

  1. (i)

$-A$ generates a $C_{0}$-semigroup $(S(t))_{t\geq 0}$.

Sometimes we additionally require one or more of the following conditions:

  1. (ii)

$(S(t))_{t\geq 0}$ is (uniformly) bounded analytic, i.e., the mapping
${t\mapsto S_{\mathbb{C}}(t)}$, where
${S_{\mathbb{C}}(t):=[S(t)]_{\mathbb{C}}}$, extends to a bounded holomorphic
function on an open sector $\Sigma_{\omega}\subseteq\mathbb{C}$ for some angle
$\omega\in(0,\pi)$ (see Definition B.1 in Appendix B);

  2. (iii)

$A_{\mathbb{C}}$ admits a bounded $H^{\infty}$-calculus with
$\omega_{H^{\infty}}(A_{\mathbb{C}})<\tfrac{\pi}{2}$, see Definition B.3;

  3. (iv)

$A$ has a bounded inverse.

Under Assumption 3.1(i), Lemma B.6 allows us to use several results from [31,
37, 61] for $C_{0}$-semigroups and their generators on complex spaces also for
$(S(t))_{t\geq 0}$ and $-A$. For instance, by [31, Theorem II.1.4] and [61,
Chapter 1, Theorem 2.2] the operator $A$ is closed and densely defined, and
the $C_{0}$-semigroup $(S(t))_{t\geq 0}$ satisfies

$\exists
M\in[1,\infty),\;w\in\mathbb{R}:\quad\|S(t)\|_{\mathscr{L}(H)}=\|S_{\mathbb{C}}(t)\|_{\mathscr{L}(H_{\mathbb{C}})}\leq
Me^{-wt}\quad\forall t\geq 0.$ (3.2)

If the conditions (i), (ii) and (iv) are satisfied, then (3.2) holds for some
$w\in(0,\infty)$, see e.g. [61, p. 70]. In this case, $(S(t))_{t\geq 0}$ is
said to be _exponentially stable_. Moreover, we note that Assumption 3.1(ii)
is equivalent to the operator $A_{\mathbb{C}}$ being sectorial with
$\omega(A_{\mathbb{C}})<\tfrac{\pi}{2}$ by Theorem B.2, and that consequently
condition (iii) implies (ii) since
$\omega(A_{\mathbb{C}})\leq\omega_{H^{\infty}}(A_{\mathbb{C}})$ by Remark B.5.
Whenever the conditions (i) and (ii) are satisfied, we have the following
useful estimate (see [37, Proposition 3.4.3]):

$\forall
c\in[0,\infty):\quad\|A^{c}S(t)\|_{\mathscr{L}(H)}=\|A_{\mathbb{C}}^{c}S_{\mathbb{C}}(t)\|_{\mathscr{L}(H_{\mathbb{C}})}\lesssim_{c}t^{-c}\quad\forall
t\in(0,\infty).$ (3.3)

As a first step towards defining the parabolic operator $\mathcal{B}$, we
define the Bochner space counterpart
$\mathcal{A}\colon\mathsf{D}(\mathcal{A})\subseteq L^{2}(0,T;H)\to
L^{2}(0,T;H)$ of $A$ by

$\begin{split}&[\mathcal{A}v](\vartheta):=Av(\vartheta),\quad\;\;v\in\mathsf{D}(\mathcal{A}),\
\text{a.a.\ }\vartheta\in(0,T),\\\
&\mathsf{D}(\mathcal{A})=L^{2}(0,T;\mathsf{D}(A)):=\bigl{\\{}v\in
L^{2}(0,T;H):\|Av(\,\cdot\,)\|_{L^{2}(0,T;H)}<\infty\bigr{\\}}.\end{split}$
(3.4)

The $C_{0}$-semigroup $(S(t))_{t\geq 0}$ on $H$, generated by $-A$, can be
associated to a family of operators $(\mathcal{S}(t))_{t\geq 0}$ on
$L^{2}(0,T;H)$ in a similar way:

$[\mathcal{S}(t)v](\vartheta):=S(t)v(\vartheta),\quad\;\;t\geq 0,\ v\in
L^{2}(0,T;H),\ \text{a.a.\ }\vartheta\in(0,T).$ (3.5)

It turns out that $(\mathcal{S}(t))_{t\geq
0}\subseteq\mathscr{L}(L^{2}(0,T;H))$ is again a $C_{0}$-semigroup, with
infinitesimal generator $-\mathcal{A}$, see Proposition A.3 in Appendix A.

In addition, we consider the family of zero-padded right-translation operators
$(\mathcal{T}(t))_{t\geq 0}$ on $L^{2}(0,T;H)$, defined by

$[\mathcal{T}(t)v](\vartheta):=\widetilde{v}(\vartheta-t),\quad\;\;t\geq 0,\
v\in L^{2}(0,T;H),\ \text{a.a.\ }\vartheta\in(0,T),$ (3.6)

where $\widetilde{v}\in L^{2}(-\infty,T;H)$ denotes the extension of $v$ by
zero to $(-\infty,T)$. As shown in Proposition A.5 in Appendix A, also
$(\mathcal{T}(t))_{t\geq 0}\subseteq\mathscr{L}(L^{2}(0,T;H))$ is a
$C_{0}$-semigroup and its infinitesimal generator is given by $-\partial_{t}$,
where

$\partial_{t}\colon\mathsf{D}(\partial_{t})\subseteq L^{2}(0,T;H)\to
L^{2}(0,T;H),\qquad\mathsf{D}(\partial_{t})=H^{1}_{0,\\{0\\}}(0,T;H),$ (3.7)

denotes the Bochner–Sobolev vector-valued weak derivative. We point out that
the domain $\mathsf{D}(\partial_{t})=H^{1}_{0,\\{0\\}}(0,T;H)$ encodes the
zero initial condition of the SPDE (3.1). Furthermore, note that it readily
follows from the definitions in (3.5) and (3.6) that, for all $t\geq 0$, every
$v\in L^{2}(0,T;H)$, and a.a. $\vartheta\in(0,T)$,

$[\mathcal{S}(t)\mathcal{T}(t)v](\vartheta)=[\mathcal{T}(t)\mathcal{S}(t)v](\vartheta)=S(t)\widetilde{v}(\vartheta-t),$

i.e., the semigroups $(\mathcal{S}(t))_{t\geq 0}$ and $(\mathcal{T}(t))_{t\geq
0}$ commute.

We now define the sum operator
$\partial_{t}+\mathcal{A}\colon\mathsf{D}(\partial_{t}+\mathcal{A})\subseteq
L^{2}(0,T;H)\to L^{2}(0,T;H)$ on its natural domain, that is

$(\partial_{t}+\mathcal{A})v:=\partial_{t}v+\mathcal{A}v,\quad
v\in\mathsf{D}(\partial_{t}+\mathcal{A})=H^{1}_{0,\\{0\\}}(0,T;H)\cap
L^{2}(0,T;\mathsf{D}(A)),$ (3.8)

with $\mathcal{A}$ and $\partial_{t}$ as given in (3.4) and (3.7),
respectively. The next proposition shows that the closure of
$-(\partial_{t}+\mathcal{A})$ again generates a $C_{0}$-semigroup, namely the
product semigroup of $(\mathcal{S}(t))_{t\geq 0}$ and $(\mathcal{T}(t))_{t\geq
0}$.

######  Proposition 3.2.

Let Assumption 3.1(i) be satisfied. The closure
$\mathcal{B}:=\overline{\partial_{t}+\mathcal{A}}$ of the sum operator
$\partial_{t}+\mathcal{A}$ defined in (3.8) exists and $-\mathcal{B}$
generates the $C_{0}$-semigroup $(\mathcal{S}(t)\mathcal{T}(t))_{t\geq 0}$ on
$L^{2}(0,T;H)$, which satisfies

$\|\mathcal{S}(t)\mathcal{T}(t)\|_{\mathscr{L}(L^{2}(0,T;H))}=\|\mathcal{T}(t)\mathcal{S}(t)\|_{\mathscr{L}(L^{2}(0,T;H))}=\begin{cases}\|S(t)\|_{\mathscr{L}(H)}&\text{if
}0\leq t<T,\\\ 0&\text{if }t\geq T,\end{cases}$

where $(\mathcal{S}(t))_{t\geq 0}$ and $(\mathcal{T}(t))_{t\geq 0}$ are
defined as in (3.5) and (3.6), respectively.

###### Proof.

By the commutativity of the semigroups $(\mathcal{S}(t))_{t\geq 0}$ and
$(\mathcal{T}(t))_{t\geq 0}$, we may conclude that
$(\mathcal{T}(t)\mathcal{S}(t))_{t\geq 0}$ is a $C_{0}$-semigroup whose
generator is an extension of $-(\partial_{t}+\mathcal{A})$, and the domain of
the generator contains $H^{1}_{0,\\{0\\}}(0,T;H)\cap L^{2}(0,T;\mathsf{D}(A))$
as a subspace that is dense with respect to the graph norm, see [31, Example
II.2.7]. Subsequently, Lemma A.2 shows that the generator is the closure of
$-(\partial_{t}+\mathcal{A})$.

Fix $t\in[0,T)$. The inequality
$\|\mathcal{T}(t)\mathcal{S}(t)\|_{\mathscr{L}(L^{2}(0,T;H))}\leq\|S(t)\|_{\mathscr{L}(H)}$
follows by the contractivity of $\mathcal{T}(t)$ and the operator norm
isometry from Lemma A.1(a). Now we turn to the reverse inequality. By
definition of the operator norm on $\mathscr{L}(H)$, there exists a normalized
sequence $(x_{n})_{n\in\mathbb{N}}$ in $H$ such that
$\|S(t)x_{n}\|_{H}\geq\|S(t)\|_{\mathscr{L}(H)}-\frac{1}{n}$ holds for all
$n\in\mathbb{N}$. Correspondingly, define the sequence
$(v_{n})_{n\in\mathbb{N}}$ in $L^{2}(0,T;H)$ by
$v_{n}(\vartheta):=(T-t)^{-\nicefrac{{1}}{{2}}}\mathbf{1}_{(0,T-t)}(\vartheta)x_{n}$
for every $\vartheta\in(0,T)$ and all $n\in\mathbb{N}$. Note that
$\|v_{n}\|_{L^{2}(0,T;H)}=1$ for every $n\in\mathbb{N}$, and

$\|\mathcal{T}(t)\mathcal{S}(t)v_{n}\|_{L^{2}(0,T;H)}=\|(T-t)^{-\nicefrac{{1}}{{2}}}\mathbf{1}_{(t,T)}\|_{L^{2}(0,T)}\|S(t)x_{n}\|_{H}\geq\|S(t)\|_{\mathscr{L}(H)}-\tfrac{1}{n}.$

As this holds for all $n\in\mathbb{N}$, we conclude that
$\|\mathcal{T}(t)\mathcal{S}(t)\|_{\mathscr{L}(L^{2}(0,T;H))}\geq\|S(t)\|_{\mathscr{L}(H)}$.
The final assertion for $t\geq T$ follows from the fact that
$\mathcal{T}(t)=0$ for $t\geq T$. ∎

######  Remark 3.3.

The closure $\mathcal{B}=\overline{\partial_{t}+\mathcal{A}}$ appearing in
Proposition 3.2 raises the question of when the sum operator itself is closed.
The answer is intimately related to the subject of maximal $L^{p}$-regularity;
we refer the reader to [29] or [47] for detailed accounts of this theory. In
the Hilbert space setting, the sum turns out to be closed under Assumptions
3.1(i),(ii). Indeed, $[\partial_{t}]_{\mathbb{C}}$ has a bounded
$H^{\infty}$-calculus with
${\omega_{H^{\infty}}([\partial_{t}]_{\mathbb{C}})\leq\tfrac{\pi}{2}}$ since
$(\mathcal{T}(t))_{t\geq 0}$ and $(\mathcal{T}_{\mathbb{C}}(t))_{t\geq 0}$ are
contractive, see Definition B.3 in Appendix B and [42, Theorem 10.2.24]. By
Assumption 3.1(ii) and Theorem B.2, we have
$\omega(A_{\mathbb{C}})<\tfrac{\pi}{2}$, and the same follows for
$\mathcal{A}_{\mathbb{C}}$ by applying Lemma A.1(a) to its resolvent
operators. Thus, we may conclude with [47, Theorem 12.13] that
$[\partial_{t}+\mathcal{A}]_{\mathbb{C}}$ is closed, so that the same holds
for $\partial_{t}+\mathcal{A}$.

We are now in the position to define fractional powers of the parabolic
operator. For $\gamma\in(0,\infty)$ we work with the following representation
(see Appendix B.2.2):

$\mathcal{B}^{-\gamma}:=\frac{1}{\Gamma(\gamma)}\int_{0}^{\infty}s^{\gamma-1}\mathcal{S}(s)\mathcal{T}(s)\mathop{}\\!\mathrm{d}s=\frac{1}{\Gamma(\gamma)}\int_{0}^{T}s^{\gamma-1}\mathcal{S}(s)\mathcal{T}(s)\mathop{}\\!\mathrm{d}s.$
(3.9)

Note that, for any $\gamma\in(0,\infty)$, this definition yields a well-
defined bounded linear operator on $L^{2}(0,T;H)$, since the product semigroup
$(\mathcal{S}(t)\mathcal{T}(t))_{t\geq 0}$ was seen to be exponentially stable
(in fact, eventually zero) in Proposition 3.2.

The next result shows that the pointwise evaluation of
$\mathcal{B}^{-\gamma}f$ at $t\in[0,T]$ is meaningful, provided that
$\gamma>\frac{1}{2}$.

######  Proposition 3.4.

Suppose Assumption 3.1(i) and let
$p\in(1,\infty),\gamma\in(\nicefrac{{1}}{{p}},\infty)$. Then

$f\mapsto\mathscr{B}_{\gamma,p}f,\quad\;\;[\mathscr{B}_{\gamma,p}f](t):=\frac{1}{\Gamma(\gamma)}\int_{0}^{t}(t-s)^{\gamma-1}S(t-s)f(s)\mathop{}\\!\mathrm{d}s\quad\forall
t\in[0,T],$ (3.10)

defines a bounded linear operator, mapping $f\in L^{p}(0,T;H)$ into
$C_{0,\\{0\\}}([0,T];H)$.

In particular, if $\gamma\in(\nicefrac{{1}}{{2}},\infty)$, we have for the
negative fractional parabolic operator ${\mathcal{B}^{-\gamma}}$ defined by
(3.9) when acting on $f\in L^{2}(0,T;H)$ the pointwise formula

$[\mathcal{B}^{-\gamma}f](t)=[\mathscr{B}_{\gamma,2}f](t)=\frac{1}{\Gamma(\gamma)}\int_{0}^{t}(t-s)^{\gamma-1}S(t-s)f(s)\mathop{}\\!\mathrm{d}s\quad\forall
t\in[0,T].$ (3.11)

###### Proof.

By [25, Proposition 5.9], for $p\in(1,\infty)$ and
$\gamma\in(\nicefrac{{1}}{{p}},\infty)$, the operator $\mathscr{B}_{\gamma,p}$
defined by (3.10) maps continuously from $L^{p}(0,T;H)$ to
$C_{0,\\{0\\}}([0,T];H)$.

Next, note that for all $f\in L^{2}(0,T;H)$ and a.a. $t\in[0,T]$, we obtain by
(3.9)

$\displaystyle[\mathcal{B}^{-\gamma}f](t)$
$\displaystyle=\frac{1}{\Gamma(\gamma)}\int_{0}^{\infty}s^{\gamma-1}[\mathcal{S}(s)\mathcal{T}(s)f](t)\mathop{}\\!\mathrm{d}s=\frac{1}{\Gamma(\gamma)}\int_{0}^{t}s^{\gamma-1}S(s)f(t-s)\mathop{}\\!\mathrm{d}s$
$\displaystyle=\frac{1}{\Gamma(\gamma)}\int_{0}^{t}(t-s)^{\gamma-1}S(t-s)f(s)\mathop{}\\!\mathrm{d}s=[\mathscr{B}_{\gamma,2}f](t).$

Thus, by the first part of this proposition, for every
$\gamma\in(\nicefrac{{1}}{{2}},\infty)$, we have that
$\mathsf{R}(\mathcal{B}^{-\gamma})\subseteq C_{0,\\{0\\}}([0,T];H)$ and the
above identities hold pointwise in $t\in[0,T]$. ∎

######  Remark 3.5.

Propositions 3.2 and 3.4 require only Assumption 3.1(i), i.e., that $-A$
generates the $C_{0}$-semigroup $(S(t))_{t\geq 0}$. Exponential stability or
uniform boundedness of $(S(t))_{t\geq 0}$ are not needed, since we consider
linear operators on $L^{2}(0,T;H)$ (instead of $L^{2}(0,\infty;H)$), allowing
us to use uniform boundedness of $(S(t))_{t\geq 0}$ on the compact interval
$[0,T]$ to derive exponential stability of
$(\mathcal{S}(t)\mathcal{T}(t))_{t\geq 0}$.

In what follows, we may also consider the operator
$\mathcal{B}^{-\gamma*}:=(\mathcal{B}^{-\gamma})^{*}$. More specifically, we
will use it in the next section to define a weak solution to the fractional
parabolic SPDE (3.1). The following lemma provides useful results for the
adjoint $\mathcal{B}^{-\gamma*}$ which are analogous to those for
$\mathcal{B}^{-\gamma}$ in Proposition 3.4. For ease of presentation, the
proof has been moved to Subsection A.3 of Appendix A.

######  Lemma 3.6.

Suppose Assumption 3.1(i) and let $\gamma\in(\nicefrac{{1}}{{2}},\infty)$. The
adjoint negative fractional parabolic operator ${\mathcal{B}^{-\gamma*}}$ maps
$g\in L^{2}(0,T;H)$ into $C_{0,\\{T\\}}([0,T];H)$, and

$[\mathcal{B}^{-\gamma*}g](s)=\frac{1}{\Gamma(\gamma)}\int_{s}^{T}(t-s)^{\gamma-1}[S(t-s)]^{*}g(t)\mathop{}\\!\mathrm{d}t\quad\forall
s\in[0,T].$ (3.12)

Finally, we note that $\mathcal{B}^{-\gamma*}=(\mathcal{B}^{*})^{-\gamma}$.
To see that the fractional power on the right-hand side is indeed well-
defined, we use [61, Chapter 1, Corollary 10.6] and conclude that
$-\mathcal{B}^{*}$ is the generator of the $C_{0}$-semigroup
$([\mathcal{S}(t)\mathcal{T}(t)]^{*})_{t\geq 0}$, which clearly inherits the
exponential stability from $(\mathcal{S}(t)\mathcal{T}(t))_{t\geq 0}$ since
their norms are equal. The identity is then obtained as follows,

$\mathcal{B}^{-\gamma*}=\biggl{(}\frac{1}{\Gamma(\gamma)}\int_{0}^{\infty}s^{\gamma-1}\mathcal{S}(s)\mathcal{T}(s)\mathop{}\\!\mathrm{d}s\biggr{)}^{*}=\frac{1}{\Gamma(\gamma)}\int_{0}^{\infty}s^{\gamma-1}[\mathcal{S}(s)\mathcal{T}(s)]^{*}\mathop{}\\!\mathrm{d}s=(\mathcal{B}^{*})^{-\gamma}\\!,$

where the first and last identities are due to (3.9) and the second is a
consequence of the general ability to interchange Bochner integrals and
duality pairings.

###  3.2. Solution concepts, existence and uniqueness

We now turn towards defining solutions to (3.1) for fractional powers
$\gamma\in(0,\infty)$. Recall from Section 2 that
$(\Omega,\mathcal{F},\mathbb{P})$ is a complete probability space equipped
with a normal filtration $(\mathcal{F}_{t})_{t\geq 0}$, and that
$(W^{Q}(t))_{t\geq 0}$ is a cylindrical $Q$-Wiener process on $H$ with respect
to $(\mathcal{F}_{t})_{t\geq 0}$, where $Q\in\mathscr{L}(H)$ is self-adjoint
and strictly positive.

Having defined and investigated the parabolic operator $\mathcal{B}$, its
domain and its fractional powers, we are now in particular able to invert the
fractional parabolic operator $\mathcal{B}^{\gamma}$. Equation (3.11)
suggests the following definition of a fractional stochastic convolution as a
_mild solution_ to (3.1).

######  Definition 3.7.

Let Assumption 3.1(i) hold and define, for $\gamma\in(0,\infty)$, the
stochastic convolution

$\widetilde{Z}_{\gamma}(t):=\frac{1}{\Gamma(\gamma)}\int_{0}^{t}(t-s)^{\gamma-1}S(t-s)\mathop{}\\!\mathrm{d}W^{Q}(s),\quad
t\in[0,T].$ (3.13)

A predictable $H$-valued stochastic process
$Z_{\gamma}:=(Z_{\gamma}(t))_{t\in[0,T]}$ is called a _mild solution_ to (3.1)
if, for all $t\in[0,T]$, it satisfies
$Z_{\gamma}(t)=\widetilde{Z}_{\gamma}(t)$, $\mathbb{P}$-a.s.

We first address existence and mean-square continuity of mild solutions.
Furthermore, we adapt the Da Prato–Kwapień–Zabczyk factorization method (see
[24], [25, Section 5.3]) to establish the existence of a pathwise continuous
modification.

######  Theorem 3.8.

Let Assumption 3.1(i) be satisfied and let $\gamma\in(0,\infty)$ be such that

$\exists\,\delta\in[0,\gamma):\quad\int_{0}^{T}\bigl{\|}t^{\gamma-1-\delta}S(t)Q^{\frac{1}{2}}\bigr{\|}_{\mathscr{L}_{2}(H)}^{2}\mathop{}\\!\mathrm{d}t<\infty.$
(3.14)

The stochastic convolution $\widetilde{Z}_{\gamma}(t)$ in (3.13) belongs to
$L^{2}(\Omega;H)$ for all $t\in[0,T]$ if and only if (3.14) holds with
$\delta=0$. In this case, the mapping $t\mapsto\widetilde{Z}_{\gamma}(t)$ is
an element of $C([0,T];L^{p}(\Omega;H))$ for all $p\in[1,\infty)$; in
particular, there exists a mild solution in the sense of Definition 3.7, and
it is mean-square continuous.

Whenever (3.14) holds for some $\delta\in(0,\gamma)$, then for every
$p\in[1,\infty)$ there exists a modification of $\widetilde{Z}_{\gamma}$ with
continuous sample paths belonging to $L^{p}(\Omega;C([0,T];H))$. In
particular, the mild solution has a modification with continuous sample paths.

###### Proof.

We first consider the case $\delta=0$ in (3.14). By the Itô isometry (see e.g.
[52, Proposition 2.3.5 and p. 32]), we obtain the identity

$\sup_{t\in[0,T]}\|\widetilde{Z}_{\gamma}(t)\|_{L^{2}(\Omega;H)}^{2}=\frac{1}{|\Gamma(\gamma)|^{2}}\int_{0}^{T}\bigl{\|}t^{\gamma-1}S(t)Q^{\frac{1}{2}}\bigr{\|}_{\mathscr{L}_{2}(H)}^{2}\mathop{}\\!\mathrm{d}t.$

Therefore, $\widetilde{Z}_{\gamma}(t)\in L^{2}(\Omega;H)$ holds for all
$t\in[0,T]$ if and only if (3.14) is satisfied with $\delta=0$. The fact that
$t\mapsto\widetilde{Z}_{\gamma}(t)$ belongs to $C([0,T];L^{p}(\Omega;H))$ for
all $p\in[1,\infty)$ will be shown in greater generality in Proposition 3.18,
see Subsection 3.3.3.

Moreover, note that $\widetilde{Z}_{\gamma}\colon[0,T]\times\Omega\to H$ is
measurable and $(\mathcal{F}_{t})_{t\in[0,T]}$-adapted, and that mean-square
continuity implies continuity in probability, so that we may apply [63,
Proposition 3.21] to conclude that there exists a predictable modification
$Z_{\gamma}$ of $\widetilde{Z}_{\gamma}$. Then, $Z_{\gamma}$ is a mild
solution to (3.1) in the sense of Definition 3.7.

Now suppose that (3.14) holds for some $\delta\in(0,\gamma)$ and let
$p\in(\nicefrac{{1}}{{\delta}}\vee 1,\infty)$. By the above considerations,
$\widetilde{Z}_{\gamma-\delta}$ and $\widetilde{Z}_{\gamma}$ exist as elements
of $C([0,T];L^{p}(\Omega;H))$. In particular, we have

$\widetilde{Z}_{\gamma-\delta}\in L^{p}(0,T;L^{p}(\Omega;H))\cong
L^{p}(\Omega;L^{p}(0,T;H)),$

where the latter identification holds by Fubini’s theorem. For this reason,
there exists a set $\Omega_{0}\in\mathcal{F}$ with $\mathbb{P}(\Omega_{0})=0$
such that

$\forall\omega\in\Omega_{0}^{c}=\Omega\setminus\Omega_{0}:\quad\widetilde{Z}_{\gamma-\delta}(\,\cdot\,,\omega)\in
L^{p}(0,T;H).$

We recall the linear operator

$\mathscr{B}_{\delta,p}\colon L^{p}(0,T;H)\to C_{0,\\{0\\}}([0,T];H)$

from (3.10) and claim that the process $\widehat{Z}_{\gamma}$ defined for
$t\in[0,T]$ and $\omega\in\Omega$ by

$\widehat{Z}_{\gamma}(t,\omega):=\begin{cases}\bigl{[}\mathscr{B}_{\delta,p}\widetilde{Z}_{\gamma-\delta}\bigr{]}(t,\omega)&\text{if
}(t,\omega)\in[0,T]\times\Omega_{0}^{c},\\\ 0&\text{if
}(t,\omega)\in[0,T]\times\Omega_{0},\end{cases}$

is the desired continuous modification of $\widetilde{Z}_{\gamma}$. To this
end, firstly note that for every $\omega\in\Omega$ the mapping
$t\mapsto\widehat{Z}_{\gamma}(t,\omega)$ indeed is continuous and
$\widehat{Z}_{\gamma}\in L^{p}(\Omega;C([0,T];H))$; this follows from
Proposition 3.4 since $\delta\in(\nicefrac{{1}}{{p}},\infty)$. In order to
show that $\widehat{Z}_{\gamma}$ is a modification of
$\widetilde{Z}_{\gamma}$, we fix $t\in[0,T]$ and employ formulas (3.10) and
(3.13) along with the semigroup property to obtain that

$\displaystyle\widehat{Z}_{\gamma}(t)=\bigl{[}\mathscr{B}_{\delta,p}\widetilde{Z}_{\gamma-\delta}\bigr{]}(t)=\frac{1}{\Gamma(\delta)}\int_{0}^{t}(t-s)^{\delta-1}S(t-s)\widetilde{Z}_{\gamma-\delta}(s)\mathop{}\\!\mathrm{d}s$
$\displaystyle=\frac{1}{\Gamma(\delta)\Gamma(\gamma-\delta)}\int_{0}^{t}(t-s)^{\delta-1}S(t-s)\biggl{[}\int_{0}^{s}(s-r)^{\gamma-\delta-1}S(s-r)\mathop{}\\!\mathrm{d}W^{Q}(r)\biggr{]}\mathop{}\\!\mathrm{d}s$
$\displaystyle=\frac{1}{\Gamma(\delta)\Gamma(\gamma-\delta)}\int_{0}^{t}\int_{0}^{s}(t-s)^{\delta-1}(s-r)^{\gamma-\delta-1}S(t-r)\mathop{}\\!\mathrm{d}W^{Q}(r)\mathop{}\\!\mathrm{d}s,\quad\mathbb{P}\text{-a.s.}$
(3.15)

We set $\widetilde{M}_{T}:=\sup_{t\in[0,T]}\|S(t)\|_{\mathscr{L}(H)}$,
$K_{T}:=\int_{0}^{T}\bigl{\|}t^{\gamma-1-\delta}S(t)Q^{\frac{1}{2}}\bigr{\|}_{\mathscr{L}_{2}(H)}^{2}\mathop{}\\!\mathrm{d}t$
and find

$\displaystyle\int_{0}^{t}\biggl{[}\int_{0}^{s}\bigl{\|}(t-s)^{\delta-1}(s-r)^{\gamma-\delta-1}S(t-r)Q^{\frac{1}{2}}\bigr{\|}_{\mathscr{L}_{2}(H)}^{2}\mathop{}\\!\mathrm{d}r\biggr{]}^{\nicefrac{{1}}{{2}}}\mathop{}\\!\mathrm{d}s$
$\displaystyle\quad\leq\widetilde{M}_{T}\int_{0}^{t}(t-s)^{\delta-1}\biggl{[}\int_{0}^{s}\bigl{\|}(s-r)^{\gamma-\delta-1}S(s-r)Q^{\frac{1}{2}}\bigr{\|}_{\mathscr{L}_{2}(H)}^{2}\mathop{}\\!\mathrm{d}r\biggr{]}^{\nicefrac{{1}}{{2}}}\mathop{}\\!\mathrm{d}s$
$\displaystyle\quad=\widetilde{M}_{T}\int_{0}^{t}(t-s)^{\delta-1}\biggl{[}\int_{0}^{s}\bigl{\|}r^{\gamma-1-\delta}S(r)Q^{\frac{1}{2}}\bigr{\|}_{\mathscr{L}_{2}(H)}^{2}\mathop{}\\!\mathrm{d}r\biggr{]}^{\nicefrac{{1}}{{2}}}\mathop{}\\!\mathrm{d}s\leq\frac{\widetilde{M}_{T}T^{\delta}\sqrt{K_{T}}}{\delta}<\infty.$

This estimate shows that

$s\mapsto\mathbf{1}_{(0,t)}(s)\mathbf{1}_{(0,s)}(\,\cdot\,)(t-s)^{\delta-1}(s-\,\cdot\,)^{\gamma-\delta-1}S(t-\,\cdot\,)Q^{\frac{1}{2}}\in
L^{1}(0,T;L^{2}(0,T;\mathscr{L}_{2}(H))),$

and the stochastic Fubini theorem [63, Theorem 8.14] may be used in (3.15),
yielding

$\widehat{Z}_{\gamma}(t)=\frac{1}{\Gamma(\delta)\Gamma(\gamma-\delta)}\int_{0}^{t}\biggl{[}\int_{r}^{t}(t-s)^{\delta-1}(s-r)^{\gamma-\delta-1}\mathop{}\\!\mathrm{d}s\biggr{]}S(t-r)\mathop{}\\!\mathrm{d}W^{Q}(r),\quad\mathbb{P}\text{-a.s.}$

Using the change of variables $u(s):=\frac{s-r}{t-r}$ and [59, Formula
5.12.1], we derive that

$\displaystyle(t-r)^{1-\gamma}\int_{r}^{t}(t-s)^{\delta-1}(s-r)^{\gamma-\delta-1}\mathop{}\\!\mathrm{d}s=\int_{0}^{1}(1-u)^{\delta-1}u^{\gamma-\delta-1}\mathop{}\\!\mathrm{d}u=\frac{\Gamma(\gamma-\delta)\Gamma(\delta)}{\Gamma(\gamma)},$

which shows that $\widehat{Z}_{\gamma}(t)=\widetilde{Z}_{\gamma}(t)$ holds
$\mathbb{P}$-a.s. Since $t\in[0,T]$ was arbitrary this implies that
$\widehat{Z}_{\gamma}$ is a modification of $\widetilde{Z}_{\gamma}$ and
completes the proof for $p\in(\nicefrac{{1}}{{\delta}}\vee 1,\infty)$.
Finally, the case $p\in[1,\nicefrac{{1}}{{\delta}}\vee 1]$ follows from the
nestedness of the $L^{p}(\Omega;C([0,T];H))$ spaces. ∎

In order to provide a more rigorous justification for the Definition 3.7 of a
mild solution to (3.1), we proceed as follows: We seek a further suitable
solution concept of a _weak solution_ , which follows “naturally” from (3.1)
using $L^{2}(0,T;H)$ inner products, and show that weak and mild solutions are
equivalent. For this, we first define the _weak stochastic Itô integral_ for
$f\colon(0,T)\to\mathscr{L}(H)$ and $g\colon(0,T)\to H$ by

$\int_{0}^{t}\bigl{(}f(s)\mathop{}\\!\mathrm{d}W^{Q}(s),g(s)\bigr{)}_{H}:=\int_{0}^{t}\widetilde{f}_{g}(s)\mathop{}\\!\mathrm{d}W^{Q}(s),\quad
t\in[0,T],$

where
$\int_{0}^{T}\bigl{\|}Q^{\frac{1}{2}}[f(s)]^{*}g(s)\bigr{\|}_{H}^{2}\mathop{}\\!\mathrm{d}s<\infty$
and $\widetilde{f}_{g}\colon(0,T)\to\mathscr{L}(H;\mathbb{R})$ is defined by

$\widetilde{f}_{g}(s)x:=(f(s)x,g(s))_{H}\quad\forall x\in H,\quad\forall
s\in(0,T),$

cf. [52, Lemma 2.4.2].

######  Definition 3.9.

Let Assumption 3.1(i) hold and let $\gamma\in(0,\infty)$. A predictable
$H$-valued stochastic process $Y_{\gamma}:=(Y_{\gamma}(t))_{t\in[0,T]}$ is
called a _weak solution_ to (3.1) if it is mean-square continuous and, in
addition,

$\forall\psi\in\mathsf{D}(\mathcal{B}^{\gamma*}):\quad(Y_{\gamma},\mathcal{B}^{\gamma*}\psi)_{L^{2}(0,T;H)}=\int_{0}^{T}\bigl{(}\mathop{}\\!\mathrm{d}W^{Q}(t),\psi(t)\bigr{)}_{H},\quad\mathbb{P}\text{-a.s.}$
(3.16)

######  Remark 3.10.

For $\gamma=1$, a natural weak solution concept is the formulation given in
[63, Definition 9.11]: A predictable $H$-valued process
$(Y_{1}(t))_{t\in[0,T]}$ is a weak solution to (3.1) if
$\sup_{t\in[0,T]}\|Y_{1}(t)\|_{L^{2}(\Omega;H)}<\infty$ and, for all
$t\in[0,T]$ and $y\in\mathsf{D}(A^{*})$,

$(Y_{1}(t),y)_{H}=-\int_{0}^{t}(Y_{1}(s),A^{*}y)_{H}\mathop{}\\!\mathrm{d}s+\bigl{(}W^{Q}(t),y\bigr{)}_{H},\quad\text{$\mathbb{P}$-a.s.}$

Provided that Assumption 3.1(i) and (3.14) are satisfied, by [63, Theorem
9.15] an $H$-valued stochastic process is a weak solution in this sense if and
only if it is a mild solution in the sense of Definition 3.7 with $\gamma=1$.

In the next proposition we generalize this result to an arbitrary fractional
power $\gamma$ and show that, under the same conditions, the mild solution in
the sense of Definition 3.7 is equivalent to the weak solution in the sense of
Definition 3.9.

######  Proposition 3.11.

Suppose that Assumption 3.1(i) holds and let $\gamma\in(0,\infty)$ be such
that (3.14) is satisfied. Then, a stochastic process is a mild solution in the
sense of Definition 3.7 if and only if it is a weak solution in the sense of
Definition 3.9. Moreover, mild and weak solutions are unique up to
modification. If one requires continuity of the sample paths, mild and weak
solutions are unique up to indistinguishability.

###### Proof.

First, we show that a mild solution $Z_{\gamma}$ is a weak solution. Note that
mean-square continuity follows from Theorem 3.8. Fix an arbitrary
$\psi\in\mathsf{D}(\mathcal{B}^{\gamma*})$. Then,

$\displaystyle(Z_{\gamma}$
$\displaystyle,\mathcal{B}^{\gamma*}\psi)_{L^{2}(0,T;H)}=\frac{1}{\Gamma(\gamma)}\int_{0}^{T}\biggl{(}\int_{0}^{t}(t-s)^{\gamma-1}S(t-s)\mathop{}\\!\mathrm{d}W^{Q}(s),[\mathcal{B}^{\gamma*}\psi](t)\biggr{)}_{H}\mathop{}\\!\mathrm{d}t$
$\displaystyle=\frac{1}{\Gamma(\gamma)}\int_{0}^{T}\int_{0}^{T}\bigl{(}\mathbf{1}_{(0,t)}(s)(t-s)^{\gamma-1}S(t-s)\mathop{}\\!\mathrm{d}W^{Q}(s),[\mathcal{B}^{\gamma*}\psi](t)\bigr{)}_{H}\mathop{}\\!\mathrm{d}t$
(3.17)

holds $\mathbb{P}$-a.s. Here, we used that
$\bigl{(}\int_{0}^{T}f(s)\mathop{}\\!\mathrm{d}W^{Q}(s),x\bigr{)}_{H}=\int_{0}^{T}(f(s)\mathop{}\\!\mathrm{d}W^{Q}(s),x)_{H}$
for all $f\colon(0,T)\to\mathscr{L}(H)$ and $x\in H$, which readily is derived
from the definition of the weak stochastic integral and the continuity of
inner products. We now would like to apply the stochastic Fubini theorem, see
e.g. [63, Theorem 8.14], in order to interchange the inner weak stochastic
integral and the outer deterministic integral. Again by the definition of the
weak stochastic integral we have, for a.a. $t\in(0,T)$,

$\int_{0}^{T}\bigl{(}\mathbf{1}_{(0,t)}(s)(t-s)^{\gamma-1}S(t-s)\mathop{}\\!\mathrm{d}W^{Q}(s),[\mathcal{B}^{\gamma*}\psi](t)\bigr{)}_{H}=\int_{0}^{T}\Psi(s,t)\mathop{}\\!\mathrm{d}W^{Q}(s),\quad\mathbb{P}\text{-a.s.},$

where the integrand $\Psi(s,t)\colon H\to\mathbb{R}$ is deterministic and, for
$s,t\in(0,T)$, defined by

$\Psi(s,t)x:=\bigl{(}\mathbf{1}_{(0,t)}(s)(t-s)^{\gamma-1}S(t-s)x,[\mathcal{B}^{\gamma*}\psi](t)\bigr{)}_{H}\quad\forall
x\in H.$ (3.18)

Thus, the usage of the stochastic Fubini theorem is justified if
$t\mapsto\Psi(\,\cdot\,,t)Q^{\frac{1}{2}}$ is in
$L^{1}(0,T;L^{2}(0,T;\mathscr{L}_{2}(H;\mathbb{R})))$. Given an orthonormal
basis $(g_{j})_{j\in\mathbb{N}}$ for $H$, we obtain

$\displaystyle\bigl{\|}\Psi(s,t)Q^{\frac{1}{2}}\bigr{\|}_{\mathscr{L}_{2}(H;\mathbb{R})}^{2}$
$\displaystyle=\sum_{j=1}^{\infty}\bigl{|}\bigl{(}\mathbf{1}_{(0,t)}(s)(t-s)^{\gamma-1}S(t-s)Q^{\frac{1}{2}}g_{j},[\mathcal{B}^{\gamma*}\psi](t)\bigr{)}_{H}\bigr{|}^{2}$
$\displaystyle\leq\bigl{\|}\mathbf{1}_{(0,t)}(s)(t-s)^{\gamma-1}S(t-s)Q^{\frac{1}{2}}\bigr{\|}_{\mathscr{L}_{2}(H)}^{2}\bigl{\|}[\mathcal{B}^{\gamma*}\psi](t)\bigr{\|}_{H}^{2}$

by the Cauchy–Schwarz inequality on $H$. From this, it follows that

$\displaystyle\bigl{\|}t\mapsto\Psi(\,\cdot\,,t)$ $\displaystyle
Q^{\frac{1}{2}}\bigr{\|}_{L^{1}(0,T;L^{2}(0,T;\mathscr{L}_{2}(H;\mathbb{R})))}=\int_{0}^{T}\biggl{(}\int_{0}^{T}\bigl{\|}\Psi(s,t)Q^{\frac{1}{2}}\bigr{\|}_{\mathscr{L}_{2}(H;\mathbb{R})}^{2}\mathop{}\\!\mathrm{d}s\biggr{)}^{\nicefrac{{1}}{{2}}}\mathop{}\\!\mathrm{d}t$
$\displaystyle\leq\int_{0}^{T}\biggl{(}\int_{0}^{t}\bigl{\|}(t-s)^{\gamma-1}S(t-s)Q^{\frac{1}{2}}\bigr{\|}_{\mathscr{L}_{2}(H)}^{2}\bigl{\|}[\mathcal{B}^{\gamma*}\psi](t)\bigr{\|}_{H}^{2}\mathop{}\\!\mathrm{d}s\biggr{)}^{\nicefrac{{1}}{{2}}}\mathop{}\\!\mathrm{d}t$
$\displaystyle=\int_{0}^{T}\biggl{(}\int_{0}^{t}\bigl{\|}s^{\gamma-1}S(s)Q^{\frac{1}{2}}\bigr{\|}_{\mathscr{L}_{2}(H)}^{2}\mathop{}\\!\mathrm{d}s\biggr{)}^{\nicefrac{{1}}{{2}}}\bigl{\|}[\mathcal{B}^{\gamma*}\psi](t)\bigr{\|}_{H}\mathop{}\\!\mathrm{d}t$
$\displaystyle\leq
T^{\nicefrac{{1}}{{2}}}\|\mathcal{B}^{\gamma*}\psi\|_{L^{2}(0,T;H)}\biggl{(}\int_{0}^{T}\bigl{\|}s^{\gamma-1}S(s)Q^{\frac{1}{2}}\bigr{\|}_{\mathscr{L}_{2}(H)}^{2}\mathop{}\\!\mathrm{d}s\biggr{)}^{\nicefrac{{1}}{{2}}}<\infty,$

where we used the Cauchy–Schwarz inequality on $L^{2}(0,T)$ in the last step.
Owing to (3.14), the integral in the final expression is finite. Applying the
stochastic Fubini theorem to (3.17), taking adjoints in (3.18) and using the
continuity of $(\,\cdot\,,\,\cdot\,)_{H}$ gives

$\displaystyle(Z_{\gamma},\mathcal{B}^{\gamma*}\psi$
$\displaystyle)_{L^{2}(0,T;H)}=\frac{1}{\Gamma(\gamma)}\int_{0}^{T}\int_{0}^{T}\Psi(s,t)\mathop{}\\!\mathrm{d}t\mathop{}\\!\mathrm{d}W^{Q}(s)$
$\displaystyle=\int_{0}^{T}\biggl{(}\mathop{}\\!\mathrm{d}W^{Q}(s),\frac{1}{\Gamma(\gamma)}\int_{s}^{T}(t-s)^{\gamma-1}[S(t-s)]^{*}[\mathcal{B}^{\gamma*}\psi](t)\mathop{}\\!\mathrm{d}t\biggr{)}_{H}$
$\displaystyle=\int_{0}^{T}\bigl{(}\mathop{}\\!\mathrm{d}W^{Q}(s),[\mathcal{B}^{-\gamma*}\mathcal{B}^{\gamma*}\psi](s)\bigr{)}_{H}=\int_{0}^{T}\bigl{(}\mathop{}\\!\mathrm{d}W^{Q}(s),\psi(s)\bigr{)}_{H},\qquad\mathbb{P}\text{-a.s.},$

where we used (3.12) in the third line. Therefore, $Z_{\gamma}$ is a weak
solution.

Conversely, suppose that $Y_{\gamma}$ is a weak solution, let an arbitrary
$\phi\in L^{2}(0,T;H)$ be given and set
$\psi:=\mathcal{B}^{-\gamma*}\phi\in\mathsf{D}(\mathcal{B}^{\gamma*})$.
Substituting this into (3.16) gives

$(Y_{\gamma},\phi)_{L^{2}(0,T;H)}=\int_{0}^{T}\bigl{(}\mathop{}\\!\mathrm{d}W^{Q}(t),[\mathcal{B}^{-\gamma*}\phi](t)\bigr{)}_{H},\quad\mathbb{P}\text{-a.s.}$

Let $(\widetilde{Z}_{\gamma}(t))_{t\in[0,T]}$ be the stochastic convolution in
(3.13). Since the condition for the stochastic Fubini theorem still holds
after replacing $\mathcal{B}^{\gamma*}\psi$ by $\phi$ in (3.18), the proof of
the previous implication can be read backwards to see that

$\forall\phi\in
L^{2}(0,T;H):\quad\mathbb{P}\bigl{(}(Y_{\gamma},\phi)_{L^{2}(0,T;H)}=(\widetilde{Z}_{\gamma},\phi)_{L^{2}(0,T;H)}\bigr{)}=1.$

By separability of $H$, also $\mathbb{P}(Y_{\gamma}=\widetilde{Z}_{\gamma}$ in
$L^{2}(0,T;H))=1$ holds so that by Fubini $Y_{\gamma}=\widetilde{Z}_{\gamma}$
in $L^{2}(0,T;L^{2}(\Omega;H))$ follows. Since both $Y_{\gamma}$ and
$\widetilde{Z}_{\gamma}$ are mean-square continuous, this shows that, for all
$t\in[0,T]$, $Y_{\gamma}(t)=\widetilde{Z}_{\gamma}(t)$ in $L^{2}(\Omega;H)$.
Therefore, for all $t\in[0,T]$, we have that
$Y_{\gamma}(t)=\widetilde{Z}_{\gamma}(t)$, $\mathbb{P}$-a.s., i.e.,
$Y_{\gamma}$ is a mild solution.

It thus suffices to prove uniqueness only for mild solutions. By Definition
3.7, mild solutions are modifications of the stochastic convolution
$\widetilde{Z}_{\gamma}$ in (3.13), hence of each other. If two mild solutions
are moreover known to have continuous sample paths, then they are
indistinguishable by [63, Proposition 3.17]. ∎

###  3.3. Spatiotemporal regularity of solutions

We now investigate spatiotemporal regularity of the mild solution $Z_{\gamma}$
in Definition 3.7. We start by stating our main results, Theorem 3.12 and
Corollary 3.13, in Subsection 3.3.1. In Subsection 3.3.2 we derive a
simplified condition for spatiotemporal regularity, which is easier to check
in applications and sufficient whenever $A$ satisfies Assumptions
3.1(i),(iii),(iv), see Proposition 3.14. In addition, we explicitly discuss
the setting of a Gelfand triple $V\hookrightarrow H\cong H^{*}\hookrightarrow
V^{*}$ in the case that the operator $A$ is induced by a (not necessarily
symmetric) bilinear form $\mathfrak{a}\colon V\times V\to\mathbb{R}$ which is
continuous and satisfies a Gårding inequality. Subsection 3.3.3 is devoted to
the proof of Theorem 3.12.

####  3.3.1. Main results

In Theorem 3.12 below, temporal regularity is measured by the
differentiability $n\in\mathbb{N}_{0}$ as well as the Hölder exponent
$\tau\in[0,1)$. Spatial regularity is expressed by means of vector spaces
which are defined in terms of fractional powers of $A$ (see Subsection B.2.2
in Appendix B) as follows:

$\dot{H}_{A}^{\sigma}:=\mathsf{D}\bigl{(}A^{\nicefrac{{\sigma}}{{2}}}\bigr{)},\qquad(x,y)_{\dot{H}_{A}^{\sigma}}:=\bigl{(}A^{\nicefrac{{\sigma}}{{2}}}x,A^{\nicefrac{{\sigma}}{{2}}}y\bigr{)}_{H},\quad\sigma\in[0,\infty).$

For $\sigma\in(0,\infty)$, $\dot{H}_{A}^{\sigma}$ is a Hilbert space provided
that Assumptions 3.1(i),(ii),(iv) are satisfied. In this case, we have the
embeddings
$\dot{H}^{\sigma^{\prime}}_{A}\hookrightarrow\dot{H}^{\sigma}_{A}\hookrightarrow
H$ for all $\sigma^{\prime}\geq\sigma\geq 0$. Note, in particular, that we do
not need to assume that $A$ is self-adjoint.

######  Theorem 3.12.

Suppose that Assumptions 3.1(i),(ii) are satisfied and let
$n\in\mathbb{N}_{0}$, $\sigma\in[0,\infty)$ and
$\gamma\in\bigl{(}\frac{\sigma-r}{2}+n,\infty\bigr{)}$, where $r\in[0,\sigma]$
is such that $Q^{\frac{1}{2}}\in\mathscr{L}(H;\dot{H}_{A}^{r})$. In the case
that $\sigma\in(0,\infty)$, suppose furthermore that Assumption 3.1(iv) is
fulfilled. Under the condition

$\int_{0}^{T}\bigl{\|}t^{\gamma-1-n}S(t)Q^{\frac{1}{2}}\bigr{\|}_{\mathscr{L}_{2}(H;\dot{H}^{\sigma}_{A})}^{2}\mathop{}\\!\mathrm{d}t<\infty,$
(3.19)

the mild solution $Z_{\gamma}$ (or, equivalently, the weak solution
$Y_{\gamma}$) in the sense of Definition 3.7 (or 3.9) belongs to
$C^{n,0}([0,T];L^{p}(\Omega;\dot{H}^{\sigma}_{A}))$ for every
$p\in[1,\infty)$.

If additionally $\gamma\geq n+\tau+\frac{1}{2}$ and
$A^{n+\tau+\frac{1}{2}-\gamma}Q^{\frac{1}{2}}\in\mathscr{L}_{2}(H;\dot{H}_{A}^{\sigma})$
hold for some $\tau\in(0,1)$, then we have $Z_{\gamma}\in
C^{n,\tau}([0,T];L^{p}(\Omega;\dot{H}^{\sigma}_{A}))$ for every
$p\in[1,\infty)$.

An application of the Kolmogorov–Chentsov continuity theorem, see e.g. [20,
Theorem 3.9], allows us to (partially) transport the temporal regularity
result of Theorem 3.12 to the pathwise setting, as seen in the next corollary.

######  Corollary 3.13.

Suppose that Assumptions 3.1(i),(ii) are satisfied. Let $\sigma\in[0,\infty)$,
$r\in[0,\sigma]$, $\gamma\in\bigl{(}\frac{\sigma-r}{2},\infty\bigr{)}$ and
$\tau\in(0,1)$ be such that $Q^{\frac{1}{2}}\in\mathscr{L}(H;\dot{H}_{A}^{r})$
and $\gamma\geq\tau+\frac{1}{2}$. If ${\sigma\in(0,\infty)}$, suppose also
that Assumption 3.1(iv) holds. If the condition

$\bigl{\|}A^{\tau+\frac{1}{2}-\gamma}Q^{\frac{1}{2}}\bigr{\|}_{\mathscr{L}_{2}(H;\dot{H}_{A}^{\sigma})}+\int_{0}^{T}\bigl{\|}t^{\gamma-1}S(t)Q^{\frac{1}{2}}\bigr{\|}_{\mathscr{L}_{2}(H;\dot{H}^{\sigma}_{A})}^{2}\mathop{}\\!\mathrm{d}t<\infty$

is satisfied, then for all $p\in[1,\infty)$ and every
$\tau^{\prime}\in[0,\tau)$ there exists a modification $\widehat{Z}_{\gamma}$
of the mild solution $Z_{\gamma}$ (or, equivalently, of the weak solution
$Y_{\gamma}$) in the sense of Definition 3.7 (or 3.9) such that
$\widehat{Z}_{\gamma}$ has $\tau^{\prime}$-Hölder continuous sample paths and
belongs to
$L^{p}\bigl{(}\Omega;C^{0,\tau^{\prime}}([0,T];\dot{H}^{\sigma}_{A})\bigr{)}$.

###### Proof.

We first invoke Theorem 3.12 with $n=0$ and $\tau\in(0,1)$ to establish that
$Z_{\gamma}$ belongs to $C^{0,\tau}([0,T];L^{q}(\Omega;\dot{H}_{A}^{\sigma}))$
for every $q\in[1,\infty)$. The result then follows by choosing $q\geq 1$
sufficiently large, applying the Kolmogorov–Chentsov continuity theorem (see
e.g. [20, Theorem 3.9]), and using nestedness of the $L^{p}$ spaces. ∎

####  3.3.2. A simplified condition and its application to the Gårding
inequality case

Whenever also Assumption 3.1(iii) holds, it is possible to replace the
condition in (3.19) by one which is simpler to check in practice. In this
case, the operator $A$ satisfies _square function estimates_ (see Subsection
B.2.3 in Appendix B), one of which is used to prove the next result.

######  Proposition 3.14.

Suppose that Assumptions 3.1(i),(iii),(iv) are satisfied. Let
$\sigma,\delta\in[0,\infty)$ and
$\gamma\in\bigl{(}\frac{1}{2}+\delta,\infty\bigr{)}\cap\bigl{[}\frac{1}{2}+\delta+\frac{\sigma-r}{2},\infty\bigr{)}$,
where $r\in[0,\sigma]$ is taken such that
$Q^{\frac{1}{2}}\in\mathscr{L}(H;\dot{H}_{A}^{r})$. Then,

$\int_{0}^{\infty}\bigl{\|}t^{\gamma-1-\delta}S(t)Q^{\frac{1}{2}}\bigr{\|}_{\mathscr{L}_{2}(H;\dot{H}^{\sigma}_{A})}^{2}\mathop{}\\!\mathrm{d}t\eqsim_{(\gamma,\delta)}\bigl{\|}A^{\delta+\frac{1}{2}-\gamma}Q^{\frac{1}{2}}\bigr{\|}_{\mathscr{L}_{2}(H;\dot{H}^{\sigma}_{A})}^{2}.$

###### Proof.

Applying Lemma B.7, see Appendix B, with
$a:=\gamma-\delta-\frac{1}{2}\in(0,\infty)$ and
$x:=A^{\frac{\sigma}{2}+\delta+\frac{1}{2}-\gamma}Q^{\frac{1}{2}}y\in H$ for
$y\in H$ shows that

$\int_{0}^{\infty}\bigl{\|}t^{\gamma-1-\delta}A^{\gamma-\delta-\frac{1}{2}}S(t)A^{\delta+\frac{1}{2}-\gamma}Q^{\frac{1}{2}}y\bigr{\|}_{\dot{H}^{\sigma}_{A}}^{2}\mathop{}\\!\mathrm{d}t\eqsim_{(\gamma,\delta)}\bigl{\|}A^{\delta+\frac{1}{2}-\gamma}Q^{\frac{1}{2}}y\bigr{\|}_{\dot{H}^{\sigma}_{A}}^{2}\quad\forall
y\in H.$

Summing both sides over an orthonormal basis for $H$ and using the
Fubini–Tonelli theorem to interchange integration and summation on the left-
hand side yields the desired conclusion. ∎

######  Remark 3.15.

Proposition 3.14 shows that under the additional assumption that
$A_{\mathbb{C}}$ admits a bounded $H^{\infty}$-calculus with
$\omega_{H^{\infty}}(A_{\mathbb{C}})<\tfrac{\pi}{2}$, which e.g. is satisfied
whenever $A$ is self-adjoint and strictly positive, it suffices to check that
$\gamma>n+\frac{(\sigma-r)\vee 1}{2}$, $\gamma\geq
n+\frac{1+(\sigma-r)\vee(2\tau)}{2}$ and that the Hilbert–Schmidt norm
$\|A^{n+\tau+\frac{1}{2}-\gamma}Q^{\frac{1}{2}}\|_{\mathscr{L}_{2}(H;\dot{H}^{\sigma}_{A})}$
is bounded to conclude the regularity results of Theorem 3.12. This condition
coincides with the one imposed in [46, Section 4, Theorem 6] to derive
regularity in the non-fractional case $\gamma=1$ for $p=2$, $\sigma=0$, $n=0$
and $\tau\in[0,\nicefrac{{1}}{{2}}]$.

######  Corollary 3.16.

Let $\delta\in[0,\infty)$ and
$\gamma\in\bigl{(}\frac{1}{2}+\delta,\infty\bigr{)}$. Suppose that $A$
satisfies Assumption 3.1(i) and that there exists a constant
$\eta\in[0,\infty)$ such that $\widehat{A}:=A+\eta I$ satisfies Assumptions
3.1(i),(iii),(iv) and
$\widehat{A}^{\delta+\frac{1}{2}-\gamma}Q^{\frac{1}{2}}\in\mathscr{L}_{2}(H)$.
Then, the mild solution $Z_{\gamma}$ in the sense of Definition 3.7 exists and
belongs to $C([0,T];L^{p}(\Omega;H))$ for every $p\in[1,\infty)$. If
$\delta>0$, then for every $p\in[1,\infty)$ there exists a modification of
$Z_{\gamma}$ in $L^{p}(\Omega;C([0,T];H))$ which has continuous sample paths.

###### Proof.

Note that $S(t)=e^{\eta t}\widehat{S}(t)$ holds for every $t\geq 0$, where
$(\widehat{S}(t))_{t\geq 0}$ denotes the $C_{0}$-semigroup generated by
$-\widehat{A}$. Hence, by Proposition 3.14 we find that

$\displaystyle\int_{0}^{T}\bigl{\|}t^{\gamma-1-\delta}S(t)Q^{\frac{1}{2}}\bigr{\|}_{\mathscr{L}_{2}(H)}^{2}\mathop{}\\!\mathrm{d}t$
$\displaystyle\leq e^{2\eta
T}\int_{0}^{T}\bigl{\|}t^{\gamma-1-\delta}\widehat{S}(t)Q^{\frac{1}{2}}\bigr{\|}_{\mathscr{L}_{2}(H)}^{2}\mathop{}\\!\mathrm{d}t$
$\displaystyle\lesssim_{(\gamma,\delta)}e^{2\eta
T}\bigl{\|}\widehat{A}^{\delta+\frac{1}{2}-\gamma}Q^{\frac{1}{2}}\bigr{\|}_{\mathscr{L}_{2}(H)}^{2}<\infty.$

The claim now follows from Theorem 3.8. ∎

We illustrate the utility of Corollary 3.16 in the following example. It is
concerned with the case that the operator $A$ is induced by a continuous
bilinear form $\mathfrak{a}\colon V\times V\to\mathbb{R}$, where
$V\hookrightarrow H$ is dense in $H$, and $\mathfrak{a}$ is not necessarily
coercive on $V$; see also [37, Section 7.3.2]. We note that this setting
applies to a variety of important applications, including symmetric and non-
symmetric differential operators of even orders.

######  Example 3.17.

Let $(V,(\,\cdot\,,\,\cdot\,)_{V})$ be a Hilbert space which is densely and
continuously embedded in $H$. Suppose that $A\colon\mathsf{D}(A)\subseteq H\to
H$ is induced by a bilinear form $\mathfrak{a}\colon V\times V\to\mathbb{R}$
which is bounded and satisfies a Gårding inequality, i.e., there exist
constants $\alpha_{0},\alpha_{1}\in(0,\infty)$ and $\eta\in[0,\infty)$ such
that

$\displaystyle\qquad\quad|\mathfrak{a}(u,v)|$
$\displaystyle\leq\alpha_{1}\|u\|_{V}\|v\|_{V}$ $\displaystyle\forall u,v\in
V,\qquad\quad$ (3.20) $\displaystyle\qquad\quad\mathfrak{a}(u,u)$
$\displaystyle\geq\alpha_{0}\|u\|_{V}^{2}-\eta\|u\|_{H}^{2}$
$\displaystyle\forall u\in V.\qquad\quad$ (3.21)

The Gårding inequality (3.21) can be interpreted as coercivity of the bilinear
form $\hat{\mathfrak{a}}(u,v):=\mathfrak{a}(u,v)+\eta(u,v)_{H}$ on $V$,
associated with $\widehat{A}=A+\eta I$, while (3.20) implies that
$\hat{\mathfrak{a}}$ is bounded. The complexified sesquilinear form
$\hat{\mathfrak{a}}_{\mathbb{C}}\colon V_{\mathbb{C}}\times
V_{\mathbb{C}}\to\mathbb{C}$, which is defined analogously to (B.2) and
induces the operator $\widehat{A}_{\mathbb{C}}$, inherits the boundedness and
coercivity from $\hat{\mathfrak{a}}$. Thus, there exist
$\widehat{\alpha}_{0},\widehat{\alpha}_{1}\in(0,\infty)$ such that

$\displaystyle\qquad\quad|\hat{\mathfrak{a}}_{\mathbb{C}}(u,v)|$
$\displaystyle\leq\widehat{\alpha}_{1}\|u\|_{V_{\mathbb{C}}}\|v\|_{V_{\mathbb{C}}}$
$\displaystyle\forall u,v\in V_{\mathbb{C}},\qquad\quad$
$\displaystyle\qquad\quad\operatorname{Re}{\hat{\mathfrak{a}}_{\mathbb{C}}(u,u)}$
$\displaystyle\geq\widehat{\alpha}_{0}\|u\|_{V_{\mathbb{C}}}^{2}$
$\displaystyle\forall u\in V_{\mathbb{C}}.\qquad\quad$

Therefore,
$\widehat{\alpha}_{0}\|u\|_{V_{\mathbb{C}}}^{2}\leq\operatorname{Re}{\hat{\mathfrak{a}}_{\mathbb{C}}(u,u)}\leq|\hat{\mathfrak{a}}_{\mathbb{C}}(u,u)|\leq\widehat{\alpha}_{1}\|u\|_{V_{\mathbb{C}}}^{2}\leq\tfrac{\widehat{\alpha}_{1}}{\widehat{\alpha}_{0}}\operatorname{Re}{\hat{\mathfrak{a}}_{\mathbb{C}}(u,u)}$
follows for every $u\in V_{\mathbb{C}}$. If $V_{\mathbb{C}}\neq\\{0\\}$, these
estimates imply that $\widehat{\alpha}_{0}\leq\widehat{\alpha}_{1}$ and

$|\operatorname{Im}{\hat{\mathfrak{a}}_{\mathbb{C}}}(u,u)|=\sqrt{|\hat{\mathfrak{a}}_{\mathbb{C}}(u,u)|^{2}-|\operatorname{Re}{\hat{\mathfrak{a}}_{\mathbb{C}}}(u,u)|^{2}}\leq\Bigl{(}\tfrac{\widehat{\alpha}_{1}^{2}}{\widehat{\alpha}_{0}^{2}}-1\Bigr{)}^{\nicefrac{{1}}{{2}}}\operatorname{Re}{\hat{\mathfrak{a}}_{\mathbb{C}}}(u,u)\quad\forall
u\in V_{\mathbb{C}}.$

This shows that $-\widehat{A}_{\mathbb{C}}$ generates a bounded analytic
$C_{0}$-semigroup $(\widehat{S}_{\mathbb{C}}(t))_{t\geq 0}$ of contractions on
$H_{\mathbb{C}}$, cf. [60, Theorem 1.54], where we used that
$(-\infty,0)\subseteq\rho(\widehat{A}_{\mathbb{C}})$ by [60, Proposition
1.22]. Applying [42, Theorems 10.2.24 and 10.4.21] and using that
$\omega(\widehat{A}_{\mathbb{C}})\in\bigl{[}0,\tfrac{\pi}{2}\bigr{)}$, since
$(\widehat{S}_{\mathbb{C}}(t))_{t\geq 0}$ is bounded analytic (see Theorem
B.2), we find that $\widehat{A}_{\mathbb{C}}$ admits a bounded
$H^{\infty}$-calculus of angle
$\omega_{H^{\infty}}(\widehat{A}_{\mathbb{C}})=\omega(\widehat{A}_{\mathbb{C}})\in\bigl{[}0,\tfrac{\pi}{2}\bigr{)}$.
Thus, we are in the setting of Corollary 3.16. In particular, the existence of
a mean-square continuous mild solution to (3.1) for $\gamma>\frac{1}{2}$
follows if
$\|\widehat{A}^{\frac{1}{2}-\gamma}Q^{\frac{1}{2}}\|_{\mathscr{L}_{2}(H)}<\infty$.

####  3.3.3. The proof of Theorem 3.12

We split the proof of Theorem 3.12 into several intermediate results. Before
stating and proving these, we introduce the following function, which
generalizes the integrand in (3.13) used to define mild solutions. Given
$a\in\mathbb{R}$, $b\in[0,\infty)$ and $\sigma\in[0,\infty)$, define
$\Phi_{a,b}\colon(0,\infty)\to\mathscr{L}(H;\dot{H}_{A}^{\sigma})$ by

$\Phi_{a,b}(t):=t^{a}A^{b}S(t)Q^{\frac{1}{2}},\qquad t\in(0,\infty).$ (3.22)

Note that a mild solution $Z_{\gamma}$ in the sense of Definition 3.7
satisfies the relation

$\forall t\in[0,T]:\quad
Z_{\gamma}(t)=\frac{1}{\Gamma(\gamma)}\int_{0}^{t}\Phi_{\gamma-1,0}(t-s)\mathop{}\\!\mathrm{d}\widehat{W}(s),\quad\text{$\mathbb{P}$-a.s.},$

where $\widehat{W}(t):=Q^{-\frac{1}{2}}W^{Q}(t)$, $t\geq 0$, is a cylindrical
Wiener process.

The first result quantifies spatial regularity of the continuous-in-time
stochastic convolution with $\Phi_{a,b}$ in
$L^{p}(\Omega;\dot{H}^{\sigma}_{A})$-sense. Recall from Section 2 that
$(W(t))_{t\geq 0}$ denotes an (arbitrary) $H$-valued cylindrical Wiener
process with respect to $(\mathcal{F}_{t})_{t\geq 0}$.

######  Proposition 3.18.

Let Assumption 3.1(i) hold, and let $a\in\mathbb{R}$, $b,\sigma\in[0,\infty)$
and $T\in(0,\infty)$ be given. If $\sigma\neq 0$, then suppose moreover that
Assumptions 3.1(ii),(iv) are satisfied. If the function $\Phi_{a,b}$ defined
in (3.22) belongs to $L^{2}(0,T;\mathscr{L}_{2}(H;\dot{H}^{\sigma}_{A}))$,
i.e.,

$\int_{0}^{T}\|\Phi_{a,b}(t)\|_{\mathscr{L}_{2}(H;\dot{H}_{A}^{\sigma})}^{2}\mathop{}\\!\mathrm{d}t<\infty,$

then $t\mapsto\int_{0}^{t}\Phi_{a,b}(t-s)\mathop{}\\!\mathrm{d}W(s)$ belongs
to $C([0,T];L^{p}(\Omega;\dot{H}_{A}^{\sigma}))$ for all $p\in[1,\infty)$.

###### Proof.

We first note that the assumption $\Phi_{a,b}\in
L^{2}(0,T;\mathscr{L}_{2}(H;\dot{H}^{\sigma}_{A}))$, combined with the
Burkholder–Davis–Gundy inequality (see [52, Theorem 6.1.2]) and the continuous
embedding

$L^{2}(\Omega;\dot{H}^{\sigma}_{A})\hookrightarrow
L^{p}(\Omega;\dot{H}^{\sigma}_{A}),\qquad p\in[1,2),\;\sigma\in[0,\infty),$
(3.23)

imply that $\int_{0}^{t}\Phi_{a,b}(t-s)\mathop{}\\!\mathrm{d}W(s)$ indeed is a
well-defined element of $L^{p}(\Omega;\dot{H}_{A}^{\sigma})$ for all
$t\in[0,T]$ and every $p\in[1,\infty)$.

It remains to check the $L^{p}(\Omega;\dot{H}^{\sigma}_{A})$-continuity of
$t\mapsto\int_{0}^{t}\Phi_{a,b}(t-s)\mathop{}\\!\mathrm{d}W(s)$. For fixed
$t\in[0,T)$ and $h\in(0,T-t]$, we split the stochastic integrals as follows:

$\displaystyle\int_{0}^{t+h}\Phi_{a,b}(t+h-s)\mathop{}\\!\mathrm{d}W(s)-\int_{0}^{t}\Phi_{a,b}(t-s)\mathop{}\\!\mathrm{d}W(s)$
$\displaystyle\quad=\int_{t}^{t+h}\Phi_{a,b}(t+h-s)\mathop{}\\!\mathrm{d}W(s)+\int_{0}^{t}[\Phi_{a,b}(t+h-s)-\Phi_{a,b}(t-s)]\mathop{}\\!\mathrm{d}W(s).$

For $p\in[2,\infty)$, the Burkholder–Davis–Gundy inequality yields

$\displaystyle\biggl{\|}\int_{t}^{t+h}\Phi_{a,b}(t+h-s)\mathop{}\\!\mathrm{d}W(s)+\int_{0}^{t}[\Phi_{a,b}(t+h-s)-\Phi_{a,b}(t-s)]\mathop{}\\!\mathrm{d}W(s)\biggr{\|}_{L^{p}(\Omega;\dot{H}^{\sigma}_{A})}$
$\displaystyle\lesssim_{p}\biggl{[}\int_{t}^{t+h}\|\Phi_{a,b}(t+h-s)\|_{\mathscr{L}_{2}(H;\dot{H}_{A}^{\sigma})}^{2}\mathop{}\\!\mathrm{d}s\biggr{]}^{\nicefrac{{1}}{{2}}}$
$\displaystyle\qquad+\biggl{[}\int_{0}^{t}\|\Phi_{a,b}(t+h-s)-\Phi_{a,b}(t-s)\|_{\mathscr{L}_{2}(H;\dot{H}_{A}^{\sigma})}^{2}\mathop{}\\!\mathrm{d}s\biggr{]}^{\nicefrac{{1}}{{2}}}$
$\displaystyle=\biggl{[}\int_{0}^{h}\|\Phi_{a,b}(u)\|_{\mathscr{L}_{2}(H;\dot{H}_{A}^{\sigma})}^{2}\mathop{}\\!\mathrm{d}u\biggr{]}^{\nicefrac{{1}}{{2}}}+\biggl{[}\int_{0}^{t}\|\Phi_{a,b}(r+h)-\Phi_{a,b}(r)\|_{\mathscr{L}_{2}(H;\dot{H}_{A}^{\sigma})}^{2}\mathop{}\\!\mathrm{d}r\biggr{]}^{\nicefrac{{1}}{{2}}},$

where $u:=t+h-s$ and $r:=t-s$. Since $\Phi_{a,b}\in
L^{2}(0,T;\mathscr{L}_{2}(H;\dot{H}_{A}^{\sigma}))$ the first integral tends
to zero as $h\downarrow 0$ by dominated convergence. The second term tends to
zero by Lemma A.4, see Appendix A.

For $t\in(0,T]$ and $h\in[-t,0)$, the difference of stochastic integrals can
be rewritten using $\int_{0}^{t}=\int_{0}^{t+h}+\int_{t+h}^{t}$. Thus, we
obtain, for every $p\in[2,\infty)$, the bound

$\displaystyle\biggl{\|}\int_{0}^{t+h}\Phi_{a,b}(t+h-s)\mathop{}\\!\mathrm{d}W(s)-\int_{0}^{t}\Phi_{a,b}(t-s)\mathop{}\\!\mathrm{d}W(s)\biggr{\|}_{L^{p}(\Omega;\dot{H}^{\sigma}_{A})}$
$\displaystyle\lesssim_{p}\biggl{[}\int_{0}^{-h}\|\Phi_{a,b}(r)\|_{\mathscr{L}_{2}(H;\dot{H}_{A}^{\sigma})}^{2}\mathop{}\\!\mathrm{d}r\biggr{]}^{\nicefrac{{1}}{{2}}}+\biggl{[}\int_{-h}^{t}\|\Phi_{a,b}(r+h)-\Phi_{a,b}(r)\|_{\mathscr{L}_{2}(H;\dot{H}_{A}^{\sigma})}^{2}\mathop{}\\!\mathrm{d}r\biggr{]}^{\nicefrac{{1}}{{2}}},$

where we again used the change of variables $r:=t-s$. Both terms on the last
line tend to zero, again by dominated convergence and Lemma A.4, respectively.

Finally, we note that the result for $p=2$ implies that for $p\in[1,2)$ by
(3.23). ∎

Furthermore, we obtain the following result regarding the temporal Hölder
continuity of the stochastic convolution with the function $\Phi_{a,b}$ in
(3.22).

######  Proposition 3.19.

Let Assumptions 3.1(i),(ii) hold, let $T\in(0,\infty)$,
${a\in\bigl{(}-\frac{1}{2},\infty\bigr{)}}$, ${b,\sigma\in[0,\infty)}$ and
$\tau\in\bigl{(}0,a+\frac{1}{2}\bigr{]}\cap(0,1)$. If $\sigma\neq 0$, then
suppose also that Assumption 3.1(iv) holds. If
$A^{-a-\frac{1}{2}+b+\tau}Q^{\frac{1}{2}}\in\mathscr{L}_{2}(H;\dot{H}_{A}^{\sigma})$
and $\Phi_{a,b}$ is defined by (3.22), then
$t\mapsto\int_{0}^{t}\Phi_{a,b}(t-s)\mathop{}\\!\mathrm{d}W(s)$ belongs to
$C^{0,\tau}([0,T];L^{p}(\Omega;\dot{H}_{A}^{\sigma}))$ for all
$p\in[1,\infty)$.

###### Proof.

For $t\in[0,T)$ and $h\in(0,T-t]$, we obtain

$\displaystyle\biggl{\|}$
$\displaystyle\int_{0}^{t+h}\Phi_{a,b}(t+h-s)\mathop{}\\!\mathrm{d}W(s)-\int_{0}^{t}\Phi_{a,b}(t-s)\mathop{}\\!\mathrm{d}W(s)\biggr{\|}_{L^{p}(\Omega;\dot{H}^{\sigma}_{A})}$
$\displaystyle\leq\biggl{\|}\int_{0}^{t}\bigl{[}\Phi_{a,b}(t+h-s)-\Phi_{a,b}(t-s)\bigr{]}\mathop{}\\!\mathrm{d}W(s)\biggr{\|}_{L^{p}(\Omega;\dot{H}^{\sigma}_{A})}$
$\displaystyle\quad+\biggl{\|}\int_{t}^{t+h}\Phi_{a,b}(t+h-s)\mathop{}\\!\mathrm{d}W(s)\biggr{\|}_{L^{p}(\Omega;\dot{H}^{\sigma}_{A})}\lesssim_{(p,a,\tau)}h^{\tau}\bigl{\|}A^{-a-\frac{1}{2}+b+\tau}Q^{\frac{1}{2}}\bigr{\|}_{\mathscr{L}_{2}(H;\dot{H}^{\sigma}_{A})}$

by Lemmas A.6 and A.7, see Appendix A. The analogous result for the case that
$t\in(0,T]$ and $h\in[-t,0)$ follows upon splitting
$\int_{0}^{t}=\int_{0}^{t+h}+\int_{t+h}^{t}$ and applying the lemmas with
$\bar{t}:=t+h\in[0,T)$ and $\bar{h}:=-h\in(0,T-\bar{t}\,]$. ∎

We now investigate temporal mean-square differentiability. To this end, we
need the following estimate which is implied by (3.3): For all
$a\in\mathbb{R}$, $b\in[0,\infty)$, we have

$\forall
c\in[0,\infty):\quad\|\Phi_{a,b}(t)x\|_{H}\lesssim_{c}t^{a-c}\bigl{\|}A^{b-c}Q^{\frac{1}{2}}x\bigr{\|}_{H}\quad\forall
x\in\mathsf{D}\bigl{(}A^{b-c}Q^{\frac{1}{2}}\bigr{)}.$ (3.24)

The next lemma records some information about the derivatives of $\Phi_{a,b}$
in (3.22).

######  Lemma 3.20.

Let Assumptions 3.1(i),(ii) be satisfied, and let $a\in\mathbb{R}$,
$b,\sigma\in[0,\infty)$. If $\sigma\in(0,\infty)$, suppose furthermore that
Assumption 3.1(iv) holds. Then, the function $\Phi_{a,b}$ defined by (3.22)
belongs to $C^{\infty}((0,\infty);\mathscr{L}(H;\dot{H}_{A}^{\sigma}))$ with
$k$th derivative

$\frac{\mathrm{d}^{k}}{\mathrm{d}t^{k}}\,\Phi_{a,b}(t)=\sum_{j=0}^{k}C_{a,j,k}t^{a-(k-j)}A^{b+j}S(t)Q^{\frac{1}{2}}=\sum_{j=0}^{k}C_{a,j,k}\Phi_{a-(k-j),b+j}(t),$
(3.25)

where $C_{a,j,k}:=(-1)^{j}\binom{k}{j}\prod_{i=1}^{k-j}(a-(k-j)+i)$ for
$a\in\mathbb{R}$, $j,k\in\mathbb{N}_{0}$, $j\leq k$.

Moreover, if $r\in[0,2b+\sigma]$ is such that
$Q^{\frac{1}{2}}\in\mathscr{L}(H;\dot{H}_{A}^{r})$ and $n\in\mathbb{N}_{0}$
satisfies $n<a-b-\tfrac{\sigma-r}{2}$, then $\Phi_{a,b}$ has a continuous
extension in $C^{n}([0,\infty);\mathscr{L}(H;\dot{H}^{\sigma}_{A}))$ with all
$n$ derivatives vanishing at zero.

###### Proof.

Since $(S(t))_{t\geq 0}$ is assumed to be analytic, $S(\,\cdot\,)$ is
infinitely differentiable from $(0,\infty)$ to $\mathscr{L}(H)$, with $j$th
derivative $(-A)^{j}S(\,\cdot\,)$ and, for $t\in(0,\infty)$,
$\varepsilon:=\frac{t}{2}$,

$\displaystyle\bigl{[}A^{b+\frac{\sigma}{2}}S(\,\cdot\,)\bigr{]}^{(j)}(t)$
$\displaystyle=\bigl{[}S(\,\cdot\,-\varepsilon)A^{b+\frac{\sigma}{2}}S(\varepsilon)\bigr{]}^{(j)}(t)$
$\displaystyle=(-A)^{j}S(t-\varepsilon)A^{b+\frac{\sigma}{2}}S(\varepsilon)=(-1)^{j}A^{j+b+\frac{\sigma}{2}}S(t).$

Here, the limits for the derivatives are taken in the $\mathscr{L}(H)$ norm.
This is equivalent to $[A^{b}S(\,\cdot\,)]^{(j)}(t)=(-1)^{j}A^{j+b}S(t)$ with
respect to the $\mathscr{L}(H;\dot{H}_{A}^{\sigma})$ norm. The expression for
the $k$th derivative of $\Phi_{a,b}$ thus follows from the Leibniz rule.

Now let $r\in[0,2b+\sigma]$, $n\in\mathbb{N}_{0}$ be such that
$n<a-b-\frac{\sigma-r}{2}$ and
$Q^{\frac{1}{2}}\in\mathscr{L}(H;\dot{H}_{A}^{r})$. To prove the second claim,
we derive that for all $k\in\\{0,1,\dots,n\\}$ and $t\in(0,\infty)$

$\displaystyle\biggl{\|}\frac{\mathrm{d}^{k}}{\mathrm{d}t^{k}}\,\Phi_{a,b}(t)\biggr{\|}_{\mathscr{L}(H;\dot{H}^{\sigma}_{A})}$
$\displaystyle=\biggl{\|}\sum_{j=0}^{k}C_{a,j,k}t^{a-(k-j)}A^{b+j+\frac{\sigma-r}{2}}S(t)A^{\frac{r}{2}}Q^{\frac{1}{2}}\biggr{\|}_{\mathscr{L}(H)}$
$\displaystyle\lesssim_{(a,b,k,r,\sigma)}t^{a-k-b-\frac{\sigma-r}{2}}\bigl{\|}Q^{\frac{1}{2}}\bigr{\|}_{\mathscr{L}(H;\dot{H}_{A}^{r})}$

by applying (3.24) to each summand with $c:=b+j+\frac{\sigma-r}{2}\geq 0$.
Furthermore, since $a-k-b-\frac{\sigma-r}{2}\geq a-n-b-\frac{\sigma-r}{2}>0$,
the above quantity tends to zero as $t\downarrow 0$. Hence, extending
$t\mapsto\frac{\mathrm{d}^{k}}{\mathrm{d}t^{k}}\,\Phi_{a,b}(t)$ by zero at
$t=0$ gives a function in $C([0,\infty);\mathscr{L}(H;\dot{H}^{\sigma}_{A}))$
for all $k\in\\{0,1,\dots,n\\}$. Inductively it follows then that the $k$th
derivative of the zero extension is the zero extension of the original $k$th
derivative. ∎

######  Proposition 3.21.

Let $\sigma\in[0,\infty)$, and whenever ${\sigma\in(0,\infty)}$ require
additionally Assumptions 3.1(i),(ii),(iv). Suppose that $\Psi\in
H^{1}_{0,\\{0\\}}(0,T;\mathscr{L}_{2}(H;\dot{H}^{\sigma}_{A}))$ and let
$\Psi^{\prime}$ denote its weak derivative. Then, for every $p\in[1,\infty)$,
the stochastic convolution
$t\mapsto\int_{0}^{t}\Psi(t-s)\mathop{}\\!\mathrm{d}W(s)$ is differentiable
from $[0,T]$ to $L^{p}(\Omega;\dot{H}^{\sigma}_{A})$, with derivative

$\displaystyle\frac{\mathop{}\\!\mathrm{d}}{\mathop{}\\!\mathrm{d}t}\int_{0}^{t}\Psi(t-s)\mathop{}\\!\mathrm{d}W(s)=\int_{0}^{t}\Psi^{\prime}(t-s)\mathop{}\\!\mathrm{d}W(s)\quad\forall\,t\in[0,T].$
(3.26)

###### Proof.

For $t\in[0,T)$ and $h\in(0,T-t]$, we can write

$\displaystyle\frac{1}{h}\biggl{[}\int_{0}^{t+h}\Psi(t+h-s)\mathop{}\\!\mathrm{d}W(s)-\int_{0}^{t}\Psi(t-s)\mathop{}\\!\mathrm{d}W(s)\biggr{]}-\int_{0}^{t}\Psi^{\prime}(t-s)\mathop{}\\!\mathrm{d}W(s)$
$\displaystyle=\int_{0}^{t}\biggl{[}\frac{\Psi(t+h-s)-\Psi(t-s)}{h}-\Psi^{\prime}(t-s)\biggr{]}\mathop{}\\!\mathrm{d}W(s)+\frac{1}{h}\int_{t}^{t+h}\Psi(t+h-s)\mathop{}\\!\mathrm{d}W(s)$
$\displaystyle=:I^{h^{+}}_{1}+I^{h^{+}}_{2}\\!.$

For $t\in(0,T]$ and $h\in[-t,0)$, we instead have

$\displaystyle\frac{1}{h}\biggl{[}$
$\displaystyle\int_{0}^{t+h}\Psi(t+h-s)\mathop{}\\!\mathrm{d}W(s)-\int_{0}^{t}\Psi(t-s)\mathop{}\\!\mathrm{d}W(s)\biggr{]}-\int_{0}^{t}\Psi^{\prime}(t-s)\mathop{}\\!\mathrm{d}W(s)$
$\displaystyle=\int_{0}^{t+h}\biggl{[}\frac{\Psi(t+h-s)-\Psi(t-s)}{h}-\Psi^{\prime}(t-s)\biggr{]}\mathop{}\\!\mathrm{d}W(s)$
$\displaystyle\quad-\frac{1}{h}\int_{t+h}^{t}\Psi(t-s)\mathop{}\\!\mathrm{d}W(s)-\int_{t+h}^{t}\Psi^{\prime}(t-s)\mathop{}\\!\mathrm{d}W(s)=:I_{1}^{h^{-}}+I_{2}^{h^{-}}+I_{3}^{h^{-}}\\!.$

We first deal with the terms $I_{2}^{h^{\pm}}$. Note that $\Psi\in
H^{1}_{0,\\{0\\}}(0,T;\mathscr{L}_{2}(H;\dot{H}^{\sigma}_{A}))$ implies
$\Psi(u)=\int_{0}^{u}\Psi^{\prime}(r)\mathop{}\\!\mathrm{d}r$ for all
$u\in(0,|h|)$, see [32, §5.9.2, Theorem 2]. In conjunction with the
Burkholder–Davis–Gundy inequality (combined with the embedding (3.23) if
$p\in[1,2)$) and the Cauchy–Schwarz inequality, this leads to

$\displaystyle\bigl{\|}I_{2}^{h^{\pm}}$
$\displaystyle\bigr{\|}_{L^{p}(\Omega;\dot{H}^{\sigma}_{A})}\lesssim_{p}\frac{1}{|h|}\biggl{[}\int_{0}^{|h|}\|\Psi(u)\|_{\mathscr{L}_{2}(H;\dot{H}^{\sigma}_{A})}^{2}\mathop{}\\!\mathrm{d}u\biggr{]}^{\nicefrac{{1}}{{2}}}$
$\displaystyle\leq\frac{1}{|h|}\biggl{[}\int_{0}^{|h|}\biggl{(}\int_{0}^{u}\|\Psi^{\prime}(r)\|_{\mathscr{L}_{2}(H;\dot{H}^{\sigma}_{A})}\mathop{}\\!\mathrm{d}r\biggr{)}^{2}\mathop{}\\!\mathrm{d}u\biggr{]}^{\nicefrac{{1}}{{2}}}\leq\|\Psi^{\prime}\|_{L^{2}(0,|h|;\mathscr{L}_{2}(H;\dot{H}^{\sigma}_{A}))}.$

Moreover, we find that

$\displaystyle\bigl{\|}I_{3}^{h^{-}}\bigr{\|}_{L^{p}(\Omega;\dot{H}^{\sigma}_{A})}$
$\displaystyle\lesssim_{p}\biggl{[}\int_{t+h}^{t}\|\Psi^{\prime}(t-s)\|_{\mathscr{L}_{2}(H;\dot{H}^{\sigma}_{A})}^{2}\mathop{}\\!\mathrm{d}s\biggr{]}^{\nicefrac{{1}}{{2}}}$
$\displaystyle=\biggl{[}\int_{0}^{|h|}\|\Psi^{\prime}(u)\|_{\mathscr{L}_{2}(H;\dot{H}^{\sigma}_{A})}^{2}\mathop{}\\!\mathrm{d}u\biggr{]}^{\nicefrac{{1}}{{2}}}=\|\Psi^{\prime}\|_{L^{2}(0,|h|;\mathscr{L}_{2}(H;\dot{H}^{\sigma}_{A}))}.$

Since $\Psi^{\prime}\in L^{2}(0,T;\mathscr{L}_{2}(H;\dot{H}^{\sigma}_{A}))$,
we have that
$\|\Psi^{\prime}\|_{L^{2}(0,|h|;\mathscr{L}_{2}(H;\dot{H}^{\sigma}_{A}))}\to
0$ as $h\to 0$ by dominated convergence. Thus, it remains to deal with the
$I_{1}^{h^{\pm}}$ terms. For the case of positive $h$, we find using the
definition of the difference quotient $D_{h}$ (see Equation (A.6) in
Subsection A.4 of Appendix A) that

$\displaystyle\bigl{\|}I_{1}^{h^{+}}\bigr{\|}_{L^{p}(\Omega;\dot{H}^{\sigma}_{A})}$
$\displaystyle\lesssim_{p}\biggl{[}\int_{0}^{t}\biggl{\|}\frac{\Psi(t+h-s)-\Psi(t-s)}{h}-\Psi^{\prime}(t-s)\biggr{\|}_{\mathscr{L}_{2}(H;\dot{H}^{\sigma}_{A})}^{2}\mathop{}\\!\mathrm{d}s\biggr{]}^{\nicefrac{{1}}{{2}}}$
$\displaystyle=\biggl{[}\int_{0}^{t}\biggl{\|}\frac{\Psi(u+h)-\Psi(u)}{h}-\Psi^{\prime}(u)\biggr{\|}_{\mathscr{L}_{2}(H;\dot{H}^{\sigma}_{A})}^{2}\mathop{}\\!\mathrm{d}u\biggr{]}^{\nicefrac{{1}}{{2}}}$
$\displaystyle=\|D_{h}\Psi-\Psi^{\prime}\|_{L^{2}(0,t;\mathscr{L}_{2}(H;\dot{H}^{\sigma}_{A}))}.$

For the case of negative $h$, we arrive at

$\displaystyle\bigl{\|}I_{1}^{h^{-}}\bigr{\|}_{L^{p}(\Omega;\dot{H}^{\sigma}_{A})}$
$\displaystyle\lesssim_{p}\biggl{[}\int_{0}^{t+h}\biggl{\|}\frac{\Psi(t+h-s)-\Psi(t-s)}{h}-\Psi^{\prime}(t-s)\biggr{\|}_{\mathscr{L}_{2}(H;\dot{H}^{\sigma}_{A})}^{2}\mathop{}\\!\mathrm{d}s\biggr{]}^{\nicefrac{{1}}{{2}}}$
$\displaystyle=\biggl{[}\int_{-h}^{t}\biggl{\|}\frac{\Psi(u+h)-\Psi(u)}{h}-\Psi^{\prime}(u)\biggr{\|}_{\mathscr{L}_{2}(H;\dot{H}^{\sigma}_{A})}^{2}\mathop{}\\!\mathrm{d}u\biggr{]}^{\nicefrac{{1}}{{2}}}$
$\displaystyle=\|D_{h}\Psi-\Psi^{\prime}\|_{L^{2}(-h,t;\mathscr{L}_{2}(H;\dot{H}^{\sigma}_{A}))}.$

The convergence $\lim_{h\to
0}\|I_{1}^{h^{\pm}}\|_{L^{p}(\Omega;\dot{H}^{\sigma}_{A})}=0$ follows then
from Proposition A.8. ∎

We are now ready to prove Theorem 3.12.

###### Proof of Theorem 3.12.

We first claim that the mild solution, interpreted as a mapping
$Z_{\gamma}\colon[0,T]\to L^{p}(\Omega;\dot{H}^{\sigma}_{A})$, is $n$ times
differentiable and that, for every $k\in\\{0,1,\dots,n\\}$ and all
$t\in[0,T]$, its $k$th derivative satisfies

$Z_{\gamma}^{(k)}(t)=\frac{1}{\Gamma(\gamma)}\int_{0}^{t}\Phi_{\gamma-1,0}^{(k)}(t-s)\mathop{}\\!\mathrm{d}\widehat{W}(s),\quad\text{$\mathbb{P}$-a.s.},$
(3.27)

where $\Phi_{\gamma-1,0}^{(k)}$ is the $k$th derivative of $\Phi_{\gamma-1,0}$
given by (3.25), and $\widehat{W}$ is the cylindrical Wiener process
$\widehat{W}(t):=Q^{-\frac{1}{2}}W^{Q}(t)$, $t\geq 0$. We prove this by
induction with respect to $k$. For $k=0$, the identity (3.27) follows from
Definition 3.7 and (3.22). Now let ${k\in\\{0,1,\dots,n-1\\}}$ and suppose
that $Z_{\gamma}$ is $k$ times differentiable and (3.27) holds. Then, the
induction hypothesis and Lemma 3.20 show that, for all $t\in[0,T]$,

$\displaystyle\frac{\mathrm{d}^{k+1}}{\mathop{}\\!\mathrm{d}t^{k+1}}\,Z_{\gamma}(t)$
$\displaystyle=\frac{\mathrm{d}}{\mathop{}\\!\mathrm{d}t}\,Z_{\gamma}^{(k)}(t)=\frac{\mathop{}\\!\mathrm{d}}{\mathop{}\\!\mathrm{d}t}\biggl{[}\frac{1}{\Gamma(\gamma)}\int_{0}^{t}\Phi^{(k)}_{\gamma-1,0}(t-s)\mathop{}\\!\mathrm{d}\widehat{W}(s)\biggr{]}$
$\displaystyle=\frac{1}{\Gamma(\gamma)}\frac{\mathop{}\\!\mathrm{d}}{\mathop{}\\!\mathrm{d}t}\int_{0}^{t}\sum_{j=0}^{k}C_{\gamma-1,j,k}\Phi_{\gamma-1-(k-j),j}(t-s)\mathop{}\\!\mathrm{d}\widehat{W}(s),\quad\text{$\mathbb{P}$-a.s.}$

Fixing an arbitrary $j\in\\{0,1,\dots,k\\}$, it suffices to verify that
$\Psi:=\Phi_{\gamma-1-(k-j),j}$ satisfies the conditions of Proposition 3.21,
so that (3.26) holds for the cylindrical Wiener process $\widehat{W}$. Indeed,
having proved this for an arbitrary $j$, by linearity

$\displaystyle\frac{\mathrm{d}^{k+1}}{\mathop{}\\!\mathrm{d}t^{k+1}}\,Z_{\gamma}(t)$
$\displaystyle=\frac{1}{\Gamma(\gamma)}\int_{0}^{t}\sum_{j=0}^{k}C_{\gamma-1,j,k}\Phi_{\gamma-1-(k-j),j}^{\prime}(t-s)\mathop{}\\!\mathrm{d}\widehat{W}(s)$
$\displaystyle=\frac{1}{\Gamma(\gamma)}\int_{0}^{t}\Phi_{\gamma-1,0}^{(k+1)}(t-s)\mathop{}\\!\mathrm{d}\widehat{W}(s),\quad\text{$\mathbb{P}$-a.s.},$

follows, where the latter identity is an equality of the operator-valued
integrands.

Using (3.3) with $c:=b$, the identity
$A^{\frac{\sigma}{2}}\Phi_{a,b}(t)=2^{a}(t/2)^{a}A^{b}S(t/2)A^{\frac{\sigma}{2}}S(t/2)Q^{\frac{1}{2}}$
and a change of variables $u:=t/2$, we observe that

$\begin{split}\|\Phi_{a,b}&\|_{L^{2}(0,T;\mathscr{L}_{2}(H;\dot{H}_{A}^{\sigma}))}\lesssim_{(a,b)}\biggl{[}\int_{0}^{T}(t/2)^{2(a-b)}\bigl{\|}A^{\frac{\sigma}{2}}S(t/2)Q^{\frac{1}{2}}\bigr{\|}_{\mathscr{L}_{2}(H)}^{2}\frac{\mathop{}\\!\mathrm{d}t}{2}\biggr{]}^{\nicefrac{{1}}{{2}}}\\\
&=\biggl{[}\int_{0}^{T}\|\Phi_{a-b,0}(t/2)\|_{\mathscr{L}_{2}(H;\dot{H}_{A}^{\sigma})}^{2}\frac{\mathop{}\\!\mathrm{d}t}{2}\biggr{]}^{\nicefrac{{1}}{{2}}}\leq\|\Phi_{a-b,0}\|_{L^{2}(0,T;\mathscr{L}_{2}(H;\dot{H}_{A}^{\sigma}))}\end{split}$
(3.28)

holds for all $a\in\mathbb{R}$ and $b\in[0,\infty)$. For
$\Psi=\Phi_{\gamma-1-(k-j),j}$ we use (3.28) to obtain

$\|\Psi\|_{L^{2}(0,T;\mathscr{L}_{2}(H;\dot{H}_{A}^{\sigma}))}\lesssim_{(\gamma,k,j)}\|\Phi_{\gamma-1-k,0}\|_{L^{2}(0,T;\mathscr{L}_{2}(H;\dot{H}_{A}^{\sigma}))}.$

The norm on the right-hand side is finite by (3.19), since $k\leq n-1<n$.
Next, noting that
$\gamma-1-k-\frac{\sigma-r}{2}\geq\gamma-n-\frac{\sigma-r}{2}>0$, the second
assertion of Lemma 3.20 implies that $t\mapsto\Psi(t)$ has a continuous
extension in $C_{0,\\{0\\}}([0,T];\mathscr{L}(H;\dot{H}_{A}^{\sigma}))$.
Furthermore, also by Lemma 3.20, $\Psi$ is differentiable from $(0,T)$ to
$\mathscr{L}(H;\dot{H}^{\sigma}_{A})$, with derivative

$\Psi^{\prime}=(\gamma-1-(k-j))\Phi_{\gamma-1-(k-j)-1,j}-\Phi_{\gamma-1-(k-j),j+1}.$

Applying the triangle inequality and (3.28) then shows that

$\|\Psi^{\prime}\|_{L^{2}(0,T;\mathscr{L}_{2}(H;\dot{H}_{A}^{\sigma}))}\lesssim_{(\gamma,k,j)}\|\Phi_{\gamma-1-(k+1),0}\|_{L^{2}(0,T;\mathscr{L}_{2}(H;\dot{H}_{A}^{\sigma}))},$

where the norm on the right-hand side is finite by (3.19), as ${k+1\leq n}$.
Since
$\mathscr{L}_{2}(H;\dot{H}^{\sigma}_{A})\hookrightarrow\mathscr{L}(H;\dot{H}^{\sigma}_{A})$,
Lemma A.9 implies that $\Psi\in
H^{1}_{0,\\{0\\}}(0,T;\mathscr{L}_{2}(H;\dot{H}^{\sigma}_{A}))$. Thus, we may
indeed use Proposition 3.21, and the differentiability follows.

It remains to show that the $n$th derivative $Z_{\gamma}^{(n)}$ is (Hölder)
continuous, i.e., $Z_{\gamma}^{(n)}\in
C^{0,\tau}([0,T];L^{p}(\Omega;\dot{H}_{A}^{\sigma}))$. To this end, we use
(3.27) and (3.25), and write

$\forall t\in[0,T]:\quad
Z_{\gamma}^{(n)}(t)=\frac{1}{\Gamma(\gamma)}\sum_{j=0}^{n}C_{\gamma-1,j,n}\int_{0}^{t}\Phi_{\gamma-1-(n-j),j}(t-s)\mathop{}\\!\mathrm{d}\widehat{W}(s),\quad\mathbb{P}\text{-a.s.}$

The case $\tau=0$ (i.e., continuity) follows after applying, for all
$j\in\\{0,1,\ldots,n\\}$, Proposition 3.18 with $a=\gamma-1-(n-j)$ and $b=j$.
Note that $\Phi_{\gamma-1-(n-j),j}$ indeed is an element of
$L^{2}(0,T;\mathscr{L}_{2}(H;\dot{H}_{A}^{\sigma}))$ for all
$j\in\\{0,\ldots,n\\}$ by (3.19) and (3.28). For
$\tau\in\bigl{(}0,\gamma-n-\frac{1}{2}\bigr{]}\cap(0,1)$, the Hölder
continuity of $Z_{\gamma}^{(n)}$ follows from Proposition 3.19 which we may
apply, for all $j\in\\{0,1,\ldots,n\\}$, with $a=\gamma-1-(n-j)$ and $b=j$,
since
$A^{n+\tau+\frac{1}{2}-\gamma}Q^{\frac{1}{2}}\in\mathscr{L}_{2}(H;\dot{H}_{A}^{\sigma})$
is assumed. ∎

##  4\. Covariance structure

In this section, we study the covariance structure of solutions to (3.1). More
specifically, we consider the mild solution process
$(Z_{\gamma}(t))_{t\in[0,T]}$ from Definition 3.7. The covariance structure of
$Z_{\gamma}$ will be expressed in terms of the family of _covariance
operators_ $(Q_{Z_{\gamma}}(s,t))_{s,t\in[0,T]}\subseteq\mathscr{L}(H)$ which
satisfies, for all $s,t\in[0,T]$, that

$(Q_{Z_{\gamma}}(s,t)x,y)_{H}=\mathbb{E}[(Z_{\gamma}(s)-\mathbb{E}[Z_{\gamma}(s)],x)_{H}(Z_{\gamma}(t)-\mathbb{E}[Z_{\gamma}(t)],y)_{H}]\quad\forall
x,y\in H.$

Note that this family is well-defined whenever $Z_{\gamma}$ is square-
integrable, e.g., under the assumptions made in Theorem 3.8. Note also that
$\mathbb{E}[Z_{\gamma}(t)]=0$ for all $t\in[0,T]$.

We present three results on the covariance operators of the mild solution
$Z_{\gamma}$. The most general result is Proposition 4.1, which provides an
explicit integral representation of $Q_{Z_{\gamma}}(s,t)$. Corollary 4.2 is
concerned with the asymptotic behavior of the covariance operator
$Q_{Z_{\gamma}}(t,t)$ as $t\to\infty$. Subsequently, in Corollary 4.3 we
consider a situation in which the covariance is separable in time and space,
and prove that the temporal part is asymptotically of Matérn type.

######  Proposition 4.1.

Let Assumption 3.1(i) be satisfied and $\gamma\in(0,\infty)$ be such that
(3.14) holds. The covariance operators $(Q_{Z_{\gamma}}(s,t))_{s,t\in[0,T]}$
of $Z_{\gamma}$ admit the representation

$Q_{Z_{\gamma}}(s,t)=\frac{1}{\Gamma(\gamma)^{2}}\int_{0}^{s\wedge
t}[(s-r)(t-r)]^{\gamma-1}S(t-r)Q[S(s-r)]^{*}\mathop{}\\!\mathrm{d}r.$ (4.1)

###### Proof.

Square-integrability of $Z_{\gamma}$ is a consequence of Theorem 3.8 and
(3.14). In order to prove the integral representation (4.1), for $s\in[0,T]$,
$r\in(0,s)$ and $x\in H$, we define $f(s,r;x)\in\mathscr{L}(H;\mathbb{R})$ by

$f(s,r;x)z:=[\Gamma(\gamma)]^{-1}(z,(s-r)^{\gamma-1}[S(s-r)]^{*}x)_{H},\quad
z\in H.$

We proceed similarly as in [45, Lemma 3.10] and obtain (4.1) from the Itô
isometry combined with the polarization identity:

$\displaystyle\mathbb{E}[(Z_{\gamma}(s),x)_{H}$
$\displaystyle(Z_{\gamma}(t),y)_{H}]=\mathbb{E}\left[\int_{0}^{s}f(s,r;x)\mathop{}\\!\mathrm{d}W^{Q}(r)\int_{0}^{t}f(t,\tau;y)\mathop{}\\!\mathrm{d}W^{Q}(\tau)\right]$
$\displaystyle=\int_{0}^{s\wedge
t}\bigl{(}f(s,r;x)Q^{\frac{1}{2}},f(t,r;y)Q^{\frac{1}{2}}\bigr{)}_{\mathscr{L}_{2}(H;\mathbb{R})}\mathop{}\\!\mathrm{d}r$
$\displaystyle=\frac{1}{\Gamma(\gamma)^{2}}\int_{0}^{s\wedge
t}[(s-r)(t-r)]^{\gamma-1}(S(t-r)Q[S(s-r)]^{*}x,y)_{H}\mathop{}\\!\mathrm{d}r.$

Then, (4.1) follows from exchanging the order of integration and taking the
inner product, which is justified since $(0,s\wedge t)\ni
r\mapsto[(s-r)(t-r)]^{\gamma-1}S(t-r)Q[S(s-r)]^{*}x$ is integrable by (3.14).
∎

By imposing more assumptions on the operator $A$, one can obtain explicit
representations of the asymptotic covariance structure of $Z_{\gamma}$ as
$t\to\infty$, as the next two corollaries show. Note that, if (3.14) holds for
$\delta=0$ and $T=\infty$, in Definition 3.7 the stochastic convolution
$\widetilde{Z}_{\gamma}$ and the mild solution $Z_{\gamma}$ are well-defined
on the infinite time interval $[0,\infty)$. It is thus meaningful to consider
the asymptotic behavior.

######  Corollary 4.2.

Let Assumptions 3.1(i),(ii),(iv) be satisfied and let
${\gamma\in(\nicefrac{{1}}{{2}},\infty)}$. Suppose that (3.14) holds for
$\delta=0$ and $T=\infty$. If for every $t\in[0,\infty)$ the operator $S(t)$
is self-adjoint and commutes with the covariance operator $Q$ of $W^{Q}$, we
have

$\lim_{t\to\infty}Q_{Z_{\gamma}}(t,t)=\Gamma(\gamma-\nicefrac{{1}}{{2}})\bigl{[}2\sqrt{\pi}\Gamma(\gamma)\bigr{]}^{-1}A^{1-2\gamma}Q\quad\text{in
}\mathscr{L}(H).$

###### Proof.

Starting from the identity (4.1) for a fixed $t=s\in[0,\infty)$, we recall
self-adjointness of the operators $(S(t))_{t\geq 0}$ and the commutativity
with $Q$ to obtain that

$\displaystyle Q_{Z_{\gamma}}(t,t)$
$\displaystyle=\frac{1}{\Gamma(\gamma)^{2}}\int_{0}^{t}(t-r)^{2(\gamma-1)}S(t-r)QS(t-r)\mathop{}\\!\mathrm{d}r$
$\displaystyle=\frac{1}{\Gamma(\gamma)^{2}}\int_{0}^{t}(t-r)^{2\gamma-2}QS(2t-2r)\mathop{}\\!\mathrm{d}r=\frac{2^{1-2\gamma}}{\Gamma(\gamma)^{2}}\int_{0}^{2t}u^{2\gamma-2}QS(u)\mathop{}\\!\mathrm{d}u,$

where we also used the semigroup property and the change of variables
${u:=2(t-r)}$. Now we interchange the bounded linear operator $Q$ with the
integral, and pass to the limit $t\to\infty$ in $\mathscr{L}(H)$, which by
(B.3) with $\alpha:=2\gamma-1\in(0,\infty)$ gives

$\lim_{t\to\infty}Q_{Z_{\gamma}}(t,t)=2^{1-2\gamma}\Gamma(2\gamma-1)[\Gamma(\gamma)]^{-2}A^{1-2\gamma}Q=\Gamma(\gamma-\nicefrac{{1}}{{2}})\bigl{[}2\sqrt{\pi}\Gamma(\gamma)\bigr{]}^{-1}A^{1-2\gamma}Q.$

The last equality follows by applying the Legendre duplication formula for the
gamma function (see e.g. [59, Formula 5.5.5]) to
$\Gamma(2\gamma-1)=\Gamma(2[\gamma-\nicefrac{{1}}{{2}}])$. ∎

######  Corollary 4.3.

Suppose the setting of Corollary 4.2 and let $A:=\kappa I$ for
${\kappa\in(0,\infty)}$. Then the covariance function of $Z_{\gamma}$ is
separable and its temporal part is asymptotically of Matérn type, i.e., there
is a function
$\varrho_{Z_{\gamma}}\colon[0,\infty)\times[0,\infty)\to\mathbb{R}$ such that

$\displaystyle\forall s,t\in[0,\infty):$ $\displaystyle
Q_{Z_{\gamma}}(s,t)=\varrho_{Z_{\gamma}}(s,t)\,Q,$ $\displaystyle\forall
h\in\mathbb{R}\setminus\\{0\\}:$
$\displaystyle\lim_{t\to\infty}\varrho_{Z_{\gamma}}(t,t+h)=\frac{2^{\frac{1}{2}-\gamma}\kappa^{1-2\gamma}}{\sqrt{\pi}\Gamma(\gamma)}(\kappa|h|)^{\gamma-\frac{1}{2}}K_{\gamma-\frac{1}{2}}(\kappa|h|).$
(4.2)

######  Remark 4.4.

On the right-hand side of (4.2), one recognizes the Matérn covariance function
(1.1) with smoothness parameter $\nu=\gamma-\nicefrac{{1}}{{2}}$, correlation
length parameter $\kappa$ and variance
$\sigma^{2}=\kappa^{1-2\gamma}\Gamma(\gamma-\nicefrac{{1}}{{2}})\bigl{[}2\sqrt{\pi}\Gamma(\gamma)\bigr{]}^{-1}$.

###### Proof of Corollary 4.3.

For $s,t\geq 0$, the integral representation (4.1) yields

$Q_{Z_{\gamma}}(s,t)=\frac{1}{\Gamma(\gamma)^{2}}\int_{0}^{s\wedge
t}[(s-r)(t-r)]^{\gamma-1}e^{-\kappa(s+t-2r)}\mathop{}\\!\mathrm{d}r\,Q=\varrho_{Z_{\gamma}}(s,t)\,Q,$

where we moved the bounded operator $Q\in\mathscr{L}(H)$ out of the integral.
Next, we fix $h\in(0,\infty)$, let $t\in[0,\infty)$ and perform the change of
variables ${u:=h+2(t-r)}$,

$\varrho_{Z_{\gamma}}(t,t+h)=\varrho_{Z_{\gamma}}(t+h,t)=\frac{2^{1-2\gamma}}{\Gamma(\gamma)^{2}}\int_{h}^{2t+h}[(u+h)(u-h)]^{\gamma-1}e^{-\kappa
u}\mathop{}\\!\mathrm{d}u.$

Thus, by passing to the limit $t\to\infty$, we obtain

$\displaystyle\lim_{t\to\infty}\varrho_{Z_{\gamma}}(t,t+h)=\frac{2^{1-2\gamma}}{\Gamma(\gamma)^{2}}\int_{h}^{\infty}\bigl{(}u^{2}-h^{2}\bigr{)}^{\gamma-1}e^{-\kappa
u}\mathop{}\\!\mathrm{d}u$
$\displaystyle\quad=\frac{2^{1-2\gamma}}{\Gamma(\gamma)^{2}}\mathcal{L}\bigl{[}u\mapsto\bigl{(}u^{2}-h^{2}\bigr{)}^{\gamma-1}\mathbf{1}_{(h,\infty)}(u)\bigr{]}(\kappa)=\frac{2^{1-2\gamma}}{\Gamma(\gamma)^{2}}\frac{(2h)^{\gamma-\frac{1}{2}}\Gamma(\gamma)}{\sqrt{\pi}\kappa^{\gamma-\frac{1}{2}}}K_{\gamma-\frac{1}{2}}(\kappa
h),$

where $\mathcal{L}[f](\kappa)$ denotes the Laplace transform of the function
$f\colon[0,\infty)\to\mathbb{R}$ evaluated at $\kappa$, and the last identity
follows from [58, Chapter I, Formula 3.13]. ∎

##  5\. Spatiotemporal Whittle–Matérn fields

In this section, we demonstrate how the results of the previous Sections 3 and
4 can be related to the widely used statistical models involving generalized
Whittle–Matérn operators (1.3) on ${H=L^{2}(\mathcal{X})}$, where
$\mathcal{X}=\mathcal{D}\subsetneq\mathbb{R}^{d}$ is a bounded domain in the
Euclidean space (see Subsection 5.1) or a surface $\mathcal{X}=\mathcal{M}$
(see Subsection 5.2).

###  5.1. Bounded Euclidean domains

Throughout this subsection, let
${\emptyset\neq\mathcal{D}\subsetneq\mathbb{R}^{d}}$ be a bounded, connected
and open domain. In order to rigorously define the symmetric, strongly
elliptic second-order differential operator $L$, formally given by (1.3), as a
linear operator on $L^{2}(\mathcal{D})$, we make the following assumptions on
its coefficients $\kappa\colon\mathcal{D}\to\mathbb{R}$ and
$a\colon\mathcal{D}\to\mathbb{R}^{d\times d}_{\rm sym}$, as well as on the
spatial domain $\mathcal{D}\subsetneq\mathbb{R}^{d}$.

######  Assumption 5.1 (Euclidean domain—minimal conditions).

  1. (i)

$\mathcal{D}$ has a Lipschitz continuous boundary $\partial\mathcal{D}$;

  2. (ii)

$a\in L^{\infty}\bigl{(}\mathcal{D};\mathbb{R}^{d\times d}_{\rm sym}\bigr{)}$
is strongly elliptic, i.e.,

$\exists\,\theta>0:\qquad\operatornamewithlimits{ess\,inf}_{x\in\mathcal{D}}\,\xi^{\top}a(x)\xi\geq\theta\|\xi\|_{\mathbb{R}^{d}}^{2}\qquad\forall\xi\in\mathbb{R}^{d};$

  3. (iii)

$\kappa\in L^{\infty}(\mathcal{D})$.

Under these assumptions, we introduce the bilinear form

$\mathfrak{a}_{L}\colon H_{0}^{1}(\mathcal{D})\times
H_{0}^{1}(\mathcal{D})\to\mathbb{R},\qquad\mathfrak{a}_{L}(u,v):=(a\nabla
u,\nabla v)_{L^{2}(\mathcal{D})}+(\kappa^{2}u,v)_{L^{2}(\mathcal{D})},$

which is symmetric, continuous and coercive. We say that $u\in
H^{1}_{0}(\mathcal{D})$ belongs to the domain $\mathsf{D}(L)$ of the
differential operator $L$ if and only if
$|\mathfrak{a}_{L}(u,v)|\lesssim_{u}\|v\|_{L^{2}(\mathcal{D})}$ holds for all
$v\in H^{1}_{0}(\mathcal{D})$. In this case, we define $Lu$ as the unique
element of $L^{2}(\mathcal{D})$ which satisfies the relation
$\mathfrak{a}_{L}(u,v)=(Lu,v)_{L^{2}(\mathcal{D})}$ for all $v\in
H^{1}_{0}(\mathcal{D})$.

By the Lax–Milgram theorem the inverse
$L^{-1}\in\mathscr{L}(L^{2}(\mathcal{D});H_{0}^{1}(\mathcal{D}))$ exists and
can be extended to
$L^{-1}\in\mathscr{L}(H_{0}^{1}(\mathcal{D})^{*};H_{0}^{1}(\mathcal{D}))$.
Moreover, it is a consequence of the Rellich–Kondrachov theorem (see [2,
Theorem 6.3]) that $L^{-1}$ is compact on $L^{2}(\mathcal{D})$. For this
reason, the spectral theorem for self-adjoint compact operators is applicable
and shows that there exist an orthonormal basis $(e_{j})_{j\in\mathbb{N}}$ for
$L^{2}(\mathcal{D})$ and a non-decreasing sequence
$(\lambda_{j})_{j\in\mathbb{N}}$ of positive real numbers accumulating only at
infinity such that $Le_{j}=\lambda_{j}e_{j}$ holds for all $j\in\mathbb{N}$.
Furthermore, the eigenvalues of $L$ satisfy the following asymptotic behavior,
known as Weyl’s law [27, Theorem 6.3.1]:

$\lambda_{j}\eqsim j^{\nicefrac{{2}}{{d}}}\quad\forall j\in\mathbb{N}.$ (5.1)

In this setting, for two differential operators $L$ and $\widetilde{L}$ on
$L^{2}(\mathcal{D})$ with coefficients $a,\kappa$ and
$\widetilde{a},\widetilde{\kappa}$, respectively, we obtain the following
corollary from the regularity results in Section 3 for spatiotemporal
Whittle–Matérn fields, where $A:=L^{\beta}$ and $Q:=\widetilde{L}^{-\alpha}$.

######  Corollary 5.2.

Let $\alpha,\beta,\sigma\in[0,\infty)$, set
$r:=\frac{\alpha}{\beta}\wedge\sigma$ if $\beta>0$ and $r:=\sigma$ if
$\beta=0$, and suppose that $n\in\mathbb{N}_{0}$, $\tau\in[0,1)$ and
$\gamma\in\bigl{(}n+\frac{(\sigma-r)\vee 1}{2},\infty\bigr{)}$ are such that

$\gamma\geq
n+\tfrac{1+(\sigma-r)\vee(2\tau)}{2}\quad\text{and}\quad\beta\gamma>\tfrac{d}{4}-\tfrac{\alpha}{2}+\beta\bigl{(}n+\tau+\tfrac{1+\sigma}{2}\bigr{)}.$
(5.2)

Let $L\colon\mathsf{D}(L)\subseteq H^{1}_{0}(\mathcal{D})\to
L^{2}(\mathcal{D})$ and
${\widetilde{L}\colon\mathsf{D}(\widetilde{L})\subseteq
H^{1}_{0}(\mathcal{D})\to L^{2}(\mathcal{D})}$ be symmetric, strongly elliptic
second-order differential operators as defined above, cf. (1.3). Suppose that
Assumption 5.1(i) holds for $\mathcal{D}\subsetneq\mathbb{R}^{d}$, and that
the coefficients $a,\kappa$ of $L$ and $\widetilde{a},\widetilde{\kappa}$ of
$\widetilde{L}$ satisfy Assumptions 5.1(ii),(iii). Assume further that $L$ and
$\widetilde{L}$ diagonalize with respect to the same orthonormal basis
$(e_{j})_{j\in\mathbb{N}}$ for $L^{2}(\mathcal{D})$, i.e., there exist non-
decreasing sequences $(\lambda_{j})_{j\in\mathbb{N}}$,
$(\widetilde{\lambda}_{j})_{j\in\mathbb{N}}$ of positive real numbers such
that ${Le_{j}=\lambda_{j}e_{j}}$ and
$\widetilde{L}e_{j}=\widetilde{\lambda}_{j}e_{j}$ for all $j\in\mathbb{N}$.

Then, setting $A:=L^{\beta}$ and $Q:=\widetilde{L}^{-\alpha}$, the mild
solution $Z_{\gamma}$ to (3.1) in the sense of Definition 3.7, see also (1.4),
belongs to $C^{n,\tau}([0,T];L^{p}(\Omega;\dot{H}^{\sigma}_{A}))$ for all
${p\in[1,\infty)}$. If the above conditions hold with $n=0$ and
$\tau\in(0,1)$, then for every $p\in[1,\infty)$ and all
$\tau^{\prime}\in[0,\tau)$ the mild solution $Z_{\gamma}$ has a modification
$\widehat{Z}_{\gamma}\in
L^{p}\bigl{(}\Omega;C^{0,\tau^{\prime}}([0,T];\dot{H}^{\sigma}_{A})\bigr{)}$.

###### Proof.

By the spectral mapping theorem for fractional powers of operators, see e.g.
[53, Section 5.3], we obtain that
$Ae_{j}=L^{\beta}e_{j}=\lambda_{j}^{\beta}e_{j}$ and
${Qe_{j}=\widetilde{L}^{-\alpha}e_{j}=\widetilde{\lambda}_{j}^{-\alpha}e_{j}}$.
In particular, it follows that $A$ inherits the self-adjointness and strict
positive-definiteness from $L$. This readily implies that $0\in\rho(A)$. By
[42, Proposition 10.2.23] we see that $A_{\mathbb{C}}$ admits a bounded
$H^{\infty}$-calculus of angle $\omega_{H^{\infty}}(A_{\mathbb{C}})=0$,
showing that Assumptions 3.1(i)–(iv) are satisfied for $A$.

Furthermore, we note that, for every $\sigma,s\in[0,\infty)$, we have that
$\dot{H}_{A}^{\sigma}=\dot{H}_{L}^{\sigma\beta}$ and the spaces
$\dot{H}_{L}^{s}$ and $\dot{H}_{\widetilde{L}}^{s}$ are isomorphic. The latter
fact follows from the asymptotic behavior (5.1) of the eigenvalues
$(\lambda_{j})_{j\in\mathbb{N}}$ and
$(\widetilde{\lambda}_{j})_{j\in\mathbb{N}}$, since $L$ and $\widetilde{L}$
have the same eigenfunctions. Thus, we obtain that
$Q^{\frac{1}{2}}=\widetilde{L}^{-\frac{\alpha}{2}}\in\mathscr{L}(H;\dot{H}_{L}^{\alpha})\subseteq\mathscr{L}(H;\dot{H}_{A}^{r})$.

Since
$\gamma\in\bigl{(}\frac{1}{2}+n,\infty\bigr{)}\cap\bigl{[}\frac{1}{2}+n+\frac{\sigma-r}{2},\infty\bigr{)}$
is assumed, by Proposition 3.14 (see also Remark 3.15) the condition (3.19) of
Theorem 3.12 is equivalent to requiring that
$A^{n+\frac{1}{2}-\gamma}Q^{\frac{1}{2}}\in\mathscr{L}_{2}(H;\dot{H}_{A}^{\sigma})$.
Since also
$\gamma\in\bigl{(}\frac{\sigma-r}{2}+n,\infty\bigr{)}\cap\bigl{[}n+\tau+\frac{1}{2},\infty\bigr{)}$,
we therefore conclude with Theorem 3.12 that it suffices to check that the
quantity

$\begin{split}\bigl{\|}&A^{\frac{\sigma}{2}+n+\tau+\frac{1}{2}-\gamma}Q^{\frac{1}{2}}\bigr{\|}_{\mathscr{L}_{2}(H)}^{2}=\bigl{\|}L^{\beta\left(\frac{\sigma}{2}+n+\tau+\frac{1}{2}-\gamma\right)}\widetilde{L}^{-\frac{\alpha}{2}}\bigr{\|}_{\mathscr{L}_{2}(H)}^{2}\\\
&\qquad=\sum_{j=1}^{\infty}\bigl{\|}L^{\beta\left(\frac{\sigma}{2}+n+\tau+\frac{1}{2}-\gamma\right)}\widetilde{L}^{-\frac{\alpha}{2}}e_{j}\bigr{\|}_{H}^{2}=\sum_{j=1}^{\infty}\lambda_{j}^{2\beta\left(\frac{\sigma}{2}+n+\tau+\frac{1}{2}-\gamma\right)}\widetilde{\lambda}_{j}^{-\alpha}\end{split}$
(5.3)

is finite. Indeed, applying Weyl’s law (5.1) to both $L$ and $\widetilde{L}$,
it follows that

$\sum_{j=1}^{\infty}\lambda_{j}^{2\beta\left(\frac{\sigma}{2}+n+\tau+\frac{1}{2}-\gamma\right)}\widetilde{\lambda}_{j}^{-\alpha}\eqsim_{\text{(}\alpha,\beta,\gamma,\sigma,n,\tau)}\sum_{j=1}^{\infty}j^{\frac{4}{d}\left[\beta\left(n+\tau+\frac{1+\sigma}{2}\right)-\beta\gamma-\frac{\alpha}{2}\right]},$

so that (5.3) is finite if and only if (5.2) holds, as we assume. Then, for
all $p\in[1,\infty)$, Theorem 3.8, Theorem 3.12 and Proposition 3.14 yield the
existence of a mild solution $Z_{\gamma}\in
C^{n,\tau}([0,T];L^{p}(\Omega;\dot{H}^{\sigma}_{A}))$, which is unique up to
modification. The last assertion for $n=0$ and $\tau\in(0,1)$ follows from
Corollary 3.13. ∎

The spatial regularity obtained in Corollary 5.2 is measured using the spaces
${\dot{H}^{\sigma}_{A}=\dot{H}_{L}^{\beta\sigma}}$. It would be more
practical to express this in terms of fractional-order Sobolev spaces
$H^{s}(\mathcal{D})$, $s\geq 0$. This raises the question of how
$\dot{H}_{L}^{s}$ and $H^{s}(\mathcal{D})$ relate. The answer to this question
depends on the smoothness of the coefficients $a,\kappa$ and of the boundary
$\partial\mathcal{D}$. We therefore introduce two additional sets of
assumptions: Assumption 5.3 is only slightly more restrictive than the minimal
conditions of Assumption 5.1, whereas Assumption 5.4 requires a high degree of
smoothness.

######  Assumption 5.3 (Euclidean domain—$H^{2}(\mathcal{D})$-regular
setting).

  1. (i)

$\mathcal{D}$ is convex.

  2. (ii)

$a\colon\overline{\mathcal{D}}\to\mathbb{R}_{\text{sym}}^{d\times d}$ is
Lipschitz continuous, i.e.,

$|a_{ij}(x)-a_{ij}(y)|\lesssim\|x-y\|_{\mathbb{R}^{d}}\quad\forall
x,y\in\overline{\mathcal{D}},\quad\forall i,j\in\\{1,\dots,d\\}.$

######  Assumption 5.4 (Euclidean domain—smooth setting).

  1. (i)

The boundary $\partial\mathcal{D}$ is of class $C^{\infty}$;

  2. (ii)

$a_{ij}\in C^{\infty}(\overline{\mathcal{D}})$ holds for all
$i,j\in\\{1,\ldots,d\\}$, i.e., for all entries of $a$;

  3. (iii)

$\kappa\in C^{\infty}(\overline{\mathcal{D}})$.

The results of the next lemma are taken from [21, Lemma 2] and [12, Lemma
3.4].

######  Lemma 5.5.

Let $L\colon\mathsf{D}(L)\subseteq H^{1}_{0}(\mathcal{D})\to
L^{2}(\mathcal{D})$ be a symmetric second-order differential operator as
defined as above, cf. (1.3). Then, the following assertions hold:

  1. (a)

If Assumption 5.1 is satisfied, then $\dot{H}_{L}^{s}\hookrightarrow
H^{s}(\mathcal{D})$ for all ${s\in[0,1]}$. Moreover, the norms
$\|\,\cdot\,\|_{\dot{H}_{L}^{s}}$ and $\|\,\cdot\,\|_{H^{s}(\mathcal{D})}$ are
equivalent on $\dot{H}_{L}^{s}$ for
$s\in[0,1]\setminus\\{\nicefrac{{1}}{{2}}\\}$;

  2. (b)

If Assumptions 5.1 and 5.3 are fulfilled, then

$\bigl{(}\dot{H}_{L}^{s},\|\,\cdot\,\|_{\dot{H}_{L}^{s}}\bigr{)}\cong\bigl{(}H^{s}(\mathcal{D})\cap
H_{0}^{1}(\mathcal{D}),\|\,\cdot\,\|_{H^{s}(\mathcal{D})}\bigr{)}\quad\forall
s\in[1,2];$

  3. (c)

If Assumptions 5.1 and 5.4 are satisfied, then we have
$\dot{H}_{L}^{s}\hookrightarrow H^{s}(\mathcal{D})$ for all $s\in[0,\infty)$,
and the norms $\|\,\cdot\,\|_{\dot{H}_{L}^{s}}$,
$\|\,\cdot\,\|_{H^{s}(\mathcal{D})}$ are equivalent on $\dot{H}_{L}^{s}$ for
every $s\in[0,\infty)\setminus\mathfrak{E}$, where
$\mathfrak{E}:=\\{2k+\nicefrac{{1}}{{2}}:k\in\mathbb{N}_{0}\\}$ is called the
_exclusion set_.

Combining Lemma 5.5 with the results of Corollary 5.2 shows that the mild
solution $Z_{\gamma}$ is an element of
$C^{n,\tau}([0,T];L^{p}(\Omega;H^{\beta\sigma}(\mathcal{D})))$, provided that
$\sigma\beta\in[0,s^{\prime}]$, where $s^{\prime}\in[1,\infty)$ is prescribed
by the smoothness of the coefficients $a,\kappa$ and the boundary
$\partial\mathcal{D}$ via Lemma 5.5(a), (b) or (c). Note that we do not have
to take the exclusion set $\mathfrak{E}$ into account, as we only need the
embedding $\dot{H}_{L}^{s}\hookrightarrow H^{s}(\mathcal{D})$.

Lastly, we consider the covariance structure of the mild solution, as treated
in the abstract setting in Section 4. The most illustrative results are the
asymptotic formulas presented in Corollaries 4.2 and 4.3, which we translate
to the current setting in Corollary 5.6. We see that (Whittle–)Matérn
operators are recovered as marginal spatial or temporal covariance operators.

######  Corollary 5.6.

Consider the setting of Corollary 5.2 with $L=\widetilde{L}$, i.e.,
${Q:=L^{-\alpha}}$. Let $\alpha,\beta\in[0,\infty)$ and
$\gamma\in(\nicefrac{{1}}{{2}},\infty)$ be such that
$\beta\gamma>\frac{1}{2}\bigl{(}\frac{d}{2}-\alpha+\beta\bigr{)}$, and let
$Z_{\gamma}$ be the mild solution in the sense of Definition 3.7. Then the
asymptotic marginal spatial covariance of $Z_{\gamma}$ satisfies

$\lim_{t\to\infty}Q_{Z_{\gamma}}(t,t)=\Gamma(\gamma-\nicefrac{{1}}{{2}})\bigl{[}2\sqrt{\pi}\Gamma(\gamma)\bigr{]}^{-1}L^{\beta(1-2\gamma)-\alpha}\quad\text{in}\;\;\mathscr{L}(L^{2}(\mathcal{D})).$

For $\beta=0$, the covariance of $Z_{\gamma}$ is separable in the sense that
there exists a function
$\varrho_{Z_{\gamma}}\colon[0,\infty)\times[0,\infty)\to\mathbb{R}$ such that

$Q_{Z_{\gamma}}(s,t)=\varrho_{Z_{\gamma}}(s,t)\,L^{-\alpha}\quad\forall
s,t\in[0,\infty),$

and for all $h\in\mathbb{R}\setminus\\{0\\}$ we have

$\lim_{t\to\infty}Q_{Z_{\gamma}}(t,t+h)=2^{\frac{1}{2}-\gamma}\bigl{[}\sqrt{\pi}\Gamma(\gamma)\bigr{]}^{-1}|h|^{\gamma-\frac{1}{2}}K_{\gamma-\frac{1}{2}}(|h|)\,L^{-\alpha}\quad\text{in}\;\;\mathscr{L}(L^{2}(\mathcal{D})).$

###### Proof.

Existence and uniqueness of the mild solution $Z_{\gamma}$ follow from
Corollary 5.2 with $L=\widetilde{L}$ and $n=\tau=\sigma=0$. Recall from its
proof that $A$ satisfies Assumptions 3.1(i)–(iv). Note also that $A=L^{\beta}$
is self-adjoint and $Q=L^{-\alpha}\in\mathscr{L}(L^{2}(\mathcal{D}))$ commutes
with $A$, so that it also commutes with $S(t)$ for all $t\in[0,\infty)$, cf.
[37, Theorem 1.3.2(a)]. All assertions follow thus from Corollaries 4.2 and
4.3. ∎

######  Remark 5.7.

The asymptotic results obtained in Corollary 5.6 are in accordance with the
marginal spatial and temporal covariance functions derived in [49, Section 3,
Proposition 1 and Corollary 1] for the case of the differential operator
$L=\gamma_{s}^{2}-\Delta$ acting on functions defined on all of
$\mathbb{R}^{2}$, where $\gamma_{s}\in(0,\infty)$. Note that, in order to
exploit Fourier techniques, in [49] the “time” variable $t$ is an element of
the whole real axis, $t\in\mathbb{R}$, instead of only its non-negative part.

######  Remark 5.8.

Corollaries 5.2 and 5.6 explain and justify the roles of the parameters
$\alpha$, $\beta$ and $\gamma$. They control three important properties of
spatiotemporal Whittle–Matérn fields. Besides the temporal and spatial
smoothness, measured respectively by the quantities $n+\tau$ and $\sigma$, we
identify a third degree of freedom: The _degree of separability_ , expressed
by the ratio $\frac{\alpha}{\beta}\in[0,\infty]$. Indeed, if
$\frac{\alpha}{\beta}=\infty$, i.e. $\beta=0$, we observe that the covariance
of the field is separable and that its temporal and spatial behavior are
exclusively governed by the parameters $\gamma$ and $\alpha$, respectively. In
contrast, if $\frac{\alpha}{\beta}=0$, i.e. $\alpha=0$, the SPDE is driven by
spatiotemporal Gaussian white noise and the “coloring” of its solution is
fully determined by the fractional parabolic differential operator
$\bigl{(}\partial_{t}+L^{\beta}\bigr{)}^{\gamma}$.

###  5.2. Surfaces

In this subsection, we provide a brief demonstration of how the above results
can be extended to spatiotemporal Whittle–Matérn fields on more general
spatial domains. More precisely, we consider a smooth, closed, connected,
orientable and compact $2$-surface $\mathcal{M}$ immersed in $\mathbb{R}^{3}$
and endowed with the positive surface measure $\nu_{\mathcal{M}}$ on
$\mathcal{B}(\mathcal{M})$, induced by the first fundamental form. An
important example of such a surface is given by the $2$-sphere,
$\mathcal{M}=\mathbb{S}^{2}$.

On $H:=L^{2}(\mathcal{M})$, we consider the following analog of the symmetric,
strongly elliptic second-order differential operator from Subsection 5.1,
formally given by

$Lu:=-\nabla_{\mathcal{M}}\cdot(a\nabla_{\mathcal{M}}u)+\kappa^{2}u,\qquad
u\in\mathsf{D}(L)\subseteq L^{2}(\mathcal{M}),$

where $\nabla_{\mathcal{M}}\,\cdot\,$ and $\nabla_{\mathcal{M}}$ denote the
surface divergence and the surface gradient, respectively. We record the
precise conditions on the surface $\mathcal{M}$ and on the coefficients
$a,\kappa$ in Assumption 5.9 below; with regard to smoothness, they are
analogous to the setting of Assumption 5.4 in the case of a bounded Euclidean
domain.

######  Assumption 5.9 (Surface—smooth setting).

  1. (i)

$a$ is a symmetric tensor field, i.e., $a(x)\colon T_{x}\mathcal{M}\to
T_{x}\mathcal{M}$ is linear and symmetric for all $x\in\mathcal{M}$, where
$T_{x}\mathcal{M}$ denotes the tangent space of $x$. Moreover, $a$ is smooth
and strongly elliptic in the following sense:

$\exists\,\theta>0:\quad\forall x\in\mathcal{M},\;\forall\xi\in
T_{x}\mathcal{M}:\quad\xi^{\top}a(x)\xi\geq\theta\|\xi\|_{\mathbb{R}^{3}}^{2}.$

  2. (ii)

The coefficient $\kappa\colon\mathcal{M}\to\mathbb{R}$ is smooth and bounded
away from zero, i.e., there exists $\kappa_{0}\in(0,\infty)$ such that
$|\kappa(x)|\geq\kappa_{0}$ for all $x\in\mathcal{M}$.

The conditions in Assumption 5.9 are sufficient to ensure that
$L\colon\dot{H}_{L}^{1}\to(\dot{H}_{L}^{1})^{*}$ is boundedly invertible, and
has a compact inverse on $L^{2}(\mathcal{M})$. This allows us to find an
orthonormal basis $(e_{j})_{j\in\mathbb{N}}$ for $L^{2}(\mathcal{M})$ and a
non-decreasing sequence of positive real eigenvalues
$(\lambda_{j})_{j\in\mathbb{N}}$ of $L$ accumulating only at infinity, as in
Subsection 5.1. Moreover, fractional powers $L^{\beta}$ are well-defined for
all $\beta\in\mathbb{R}$, the sequence of eigenvalues still satisfies Weyl’s
law (5.1) (with $d=2$), and a spectral mapping theorem holds, cf. [72,
Theorems XII.1.3 and XII.2.1]. These facts are sufficient to repeat the proofs
of Corollaries 5.2 and 5.6 yielding the analogous results, with $d=2$ and
other obvious modifications to the conditions. In particular, the analog of
Corollary 5.2 on the surface $\mathcal{M}$ implies regularity of the solution
process in the space
$C^{n,\tau}([0,T];L^{p}(\Omega;H^{\beta\sigma}(\mathcal{M})))$.

An important difference from the (smooth) Euclidean setting of Assumption 5.4
is that under Assumption 5.9, the Sobolev space $H^{s}(\mathcal{M})$ and
$\dot{H}_{L}^{s}$ are isomorphic for every $s\in[0,\infty)$, see [72, Example
XII.2.1]. In other words, the absence of a boundary $\partial\mathcal{M}$
implies that one does not need to exclude the exception set $\mathfrak{E}$
from the admissible exponents $s$ in the analog of Lemma 5.5(c).

##  Appendix A Auxiliary results

Throughout this section, $H$ denotes a separable Hilbert space which, if not
specified otherwise, is considered over the real scalar field $\mathbb{R}$.

###  A.1. Bochner counterparts

The first auxiliary result records relations between a (possibly unbounded)
linear operator $A\colon\mathsf{D}(A)\subseteq H\to H$ and its Bochner space
counterpart $\mathcal{A}$ which is defined on a subspace of $L^{2}(0,T;H)$,
where $T\in(0,\infty)$.

######  Lemma A.1.

Let $T\in(0,\infty)$ and $A\colon\mathsf{D}(A)\subseteq H\to H$ be a linear
operator on a real or complex Hilbert space $H$. Consider the associated
operator $\mathcal{A}$ on $L^{2}(0,T;H)$ as defined in (3.4). Then, the
following hold:

  1. (a)

$\mathcal{A}$ is bounded if and only if $A$ is bounded, and in that case we
have

$\|\mathcal{A}\|_{\mathscr{L}(L^{2}(0,T;H))}=\|A\|_{\mathscr{L}(H)};$

  2. (b)

$\mathcal{A}$ is closed if and only if $A$ is.

###### Proof.

If $A$ is bounded, then the inequality
$\|\mathcal{A}\|_{\mathscr{L}(L^{2}(0,T;H))}\leq\|A\|_{\mathscr{L}(H)}$ is
easily verified. Now suppose that $\mathcal{A}$ is bounded. Then for all $x\in
H$ we have

$\displaystyle\|Ax\|_{H}$
$\displaystyle=\|T^{-\nicefrac{{1}}{{2}}}\mathbf{1}_{(0,T)}\otimes
Ax\|_{L^{2}(0,T;H)}=\|\mathcal{A}(T^{-\nicefrac{{1}}{{2}}}\mathbf{1}_{(0,T)}\otimes
x)\|_{L^{2}(0,T;H)}$
$\displaystyle\leq\|\mathcal{A}\|_{\mathscr{L}(L^{2}(0,T;H))}\|T^{-\nicefrac{{1}}{{2}}}\mathbf{1}_{(0,T)}\otimes
x\|_{L^{2}(0,T;H)}=\|\mathcal{A}\|_{\mathscr{L}(L^{2}(0,T;H))}\|x\|_{H}.$

Here, given $f\colon(0,T)\to\mathbb{R}$ and $x\in H$, the function $f\otimes
x\colon(0,T)\to H$ is defined by $[f\otimes x](t):=f(t)x$ for all $t\in(0,T)$.
We thus find that $A$ is bounded with operator norm
$\|A\|_{\mathscr{L}(H)}\leq\|\mathcal{A}\|_{\mathscr{L}(L^{2}(0,T;H))}$, which
finishes the proof of (a).

To prove part (b), first let $A$ be closed and let the sequence
$(v_{n})_{n\in\mathbb{N}}$ in $\mathsf{D}(\mathcal{A})$ be such that $v_{n}\to
v$ and $\mathcal{A}v_{n}\to y$ in $L^{2}(0,T;H)$. We need to prove that
$v\in\mathsf{D}(\mathcal{A})$ and $y=\mathcal{A}v$. Let
$(v_{n_{k}})_{k\in\mathbb{N}}$ be a subsequence such that $v_{n_{k}}\to v$ and
$Av_{n_{k}}\to y$ in $H$, a.e. in $(0,T)$, so that by the closedness of $A$ it
follows that $v(\vartheta)\in\mathsf{D}(A)$ and $y(\vartheta)=Av(\vartheta)$
for a.a. $\vartheta\in(0,T)$. From the latter we obtain that $y=\mathcal{A}v$,
which is meaningful since $v,y\in L^{2}(0,T;H)$ yields that
$v\in\mathsf{D}(\mathcal{A})$.

Now let $\mathcal{A}$ be closed and let $(x_{n})_{n\in\mathbb{N}}$ in
$\mathsf{D}(A)$ be such that $x_{n}\to x$ and $Ax_{n}\to y$ in $H$. This
implies the following convergences in $L^{2}(0,T;H)$:

$\displaystyle\mathbf{1}_{(0,T)}\otimes x_{n}\to\mathbf{1}_{(0,T)}\otimes x,$
$\displaystyle\mathcal{A}(\mathbf{1}_{(0,T)}\otimes
x_{n})=\mathbf{1}_{(0,T)}\otimes Ax_{n}\to\mathbf{1}_{(0,T)}\otimes y.$

Since $\mathcal{A}$ is closed, we deduce that $\mathbf{1}_{(0,T)}\otimes
x\in\mathsf{D}(\mathcal{A})$ and $\mathbf{1}_{(0,T)}\otimes
y=\mathcal{A}(\mathbf{1}_{(0,T)}\otimes x)$, from which we may conclude
$x\in\mathsf{D}(A)$ and $y=Ax$. Hence $A$ is closed. ∎

The following lemma is generally useful for determining the domain of a
generator of a given $C_{0}$-semigroup, and it will subsequently be used to
show that the Bochner space counterpart of a $C_{0}$-semigroup is again a
$C_{0}$-semigroup, see Proposition A.3.

######  Lemma A.2.

Let $(S(t))_{t\geq 0}$ be a $C_{0}$-semigroup on $H$ with infinitesimal
generator $\widetilde{A}\colon\mathsf{D}(\widetilde{A})\subseteq H\to H$. If
$A\colon\mathsf{D}(A)\subseteq H\to H$ is a linear operator satisfying
$A\subseteq\widetilde{A}$ and $\mathsf{D}(A)$ is dense in
$\mathsf{D}(\widetilde{A})$ with respect to the graph norm
$\|\cdot\|_{\mathsf{D}(\widetilde{A})}$, then $\widetilde{A}=\overline{A}$.

###### Proof.

Let $(x,\widetilde{A}x)\in\mathsf{G}(\widetilde{A})$ and choose a sequence
$(x_{n})_{n\in\mathbb{N}}$ in $\mathsf{D}(A)$ such that ${x_{n}\to x}$ in
$\mathsf{D}(\widetilde{A})$. Using $A\subseteq\widetilde{A}$, we have
$(x_{n},Ax_{n})=(x_{n},\widetilde{A}x_{n})\to(x,\widetilde{A}x)$ with respect
to the product norm on $H\times H$, which shows that
$(x,\widetilde{A}x)\in\overline{\mathsf{G}(A)}$. Conversely, for any
$(x,y)\in\overline{\mathsf{G}(A)}$ there exists a sequence
$(x_{n})_{n\in\mathbb{N}}\subseteq\mathsf{D}(A)$ such that
$(x_{n},\widetilde{A}x_{n})=(x_{n},Ax_{n})\to(x,y)$ in $H\times H$. Since
$\widetilde{A}$ is closed as the generator of a $C_{0}$-semigroup, see [31,
Theorem II.1.4], we find that $(x,y)\in\mathsf{G}(\widetilde{A})$. This proves
$\overline{\mathsf{G}(A)}=\mathsf{G}(\widetilde{A})$. ∎

######  Proposition A.3.

Let $T\in(0,\infty)$ and let Assumption 3.1(i) be satisfied. The family
$(\mathcal{S}(t))_{t\geq 0}$ of operators on $L^{2}(0,T;H)$ given by (3.5) is
a $C_{0}$-semigroup with infinitesimal generator $-\mathcal{A}$, as defined by
(3.4).

###### Proof.

First note that the operators $(\mathcal{S}(t))_{t\geq 0}$ are well-defined in
the sense that they map elements in $L^{2}(0,T;H)$ to $L^{2}(0,T;H)$. In fact,
Lemma A.1(a) shows that
$\|\mathcal{S}(t)\|_{\mathscr{L}(L^{2}(0,T;H))}=\|S(t)\|_{\mathscr{L}(H)}$ for
all $t\geq 0$.

We now check that $(\mathcal{S}(t))_{t\geq 0}$ is a $C_{0}$-semigroup.
Clearly, $\mathcal{S}(0)=I$ and the semigroup property holds. Let $M\geq 1$
and $w\in\mathbb{R}$ be as in (3.2), so that

$\forall h\in[0,1]:\quad\|S(h)\|_{\mathscr{L}(H)}\leq Me^{-wh}\leq
Me^{(-w)\vee 0}=:\widetilde{M}.$

To show strong continuity, let $x\in H$, $h\in(0,1)$ and note that

$\|S(h)x-x\|_{H}^{2}\leq 2\|S(h)x\|_{H}^{2}+2\|x\|_{H}^{2}\leq
2\bigl{(}\widetilde{M}^{2}+1\bigr{)}\|x\|_{H}^{2}.$

By dominated convergence, $\lim_{h\downarrow
0}\|\mathcal{S}(h)v-v\|_{L^{2}(0,T;H)}=0$ for $v\in L^{2}(0,T;H)$.

Next we investigate the infinitesimal generator of $(\mathcal{S}(t))_{t\geq
0}$, which we denote by $-\widetilde{\mathcal{A}}$ for the time being. We wish
to show that $\widetilde{\mathcal{A}}=\mathcal{A}$. Let $x\in\mathsf{D}(A)$
and consider

$\biggl{\|}\frac{1}{h}(S(h)x-x)+Ax\biggr{\|}_{H}^{2}\leq
2\biggl{\|}\frac{1}{h}(S(h)x-x)\biggr{\|}_{H}^{2}+2\|Ax\|_{H}^{2}.$

To bound the first term, we use [61, Chapter 1, Theorem 2.4(d)] and note that,
for every $h\in(0,1)$, we obtain

$\biggl{\|}\frac{1}{h}(S(h)x-x)\biggr{\|}_{H}^{2}=\biggl{\|}\frac{1}{h}\int_{0}^{h}S(s)Ax\mathop{}\\!\mathrm{d}s\biggr{\|}_{H}^{2}\leq\frac{1}{h^{2}}\biggl{|}\int_{0}^{h}\|S(s)Ax\|_{H}\mathop{}\\!\mathrm{d}s\biggr{|}^{2}\leq\widetilde{M}^{2}\|Ax\|_{H}^{2}.$

The two previous displays show that, for $v\in L^{2}(0,T;\mathsf{D}(A))$ and
all $h\in(0,1)$,

$\int_{0}^{T}\biggl{\|}\frac{1}{h}(S(h)v(\vartheta)-v(\vartheta))+Av(\vartheta)\biggr{\|}_{H}^{2}\mathop{}\\!\mathrm{d}\vartheta\leq
2\bigl{(}\widetilde{M}^{2}+1\bigr{)}\|\mathcal{A}v\|_{L^{2}(0,T;H)}^{2}<\infty.$

This justifies the use of the dominated convergence theorem to conclude that

$-\widetilde{\mathcal{A}}v=\lim_{h\downarrow
0}\frac{1}{h}(\mathcal{S}(h)v-v)=-\mathcal{A}v\qquad\text{in}\;\;L^{2}(0,T;H),$

i.e., $-\mathcal{A}\subseteq-\widetilde{\mathcal{A}}$ as
$v\in\mathsf{D}(\mathcal{A})=L^{2}(0,T;\mathsf{D}(A))$ was arbitrary. Since
$\mathsf{D}(\mathcal{A})$ is dense in $L^{2}(0,T;H)$ (by density of
$\mathsf{D}(A)$ in $H$), and $\mathcal{S}(t)$ maps $\mathsf{D}(\mathcal{A})$
to itself for each $t\geq 0$, Proposition II.1.7 of [31] implies that
$\mathsf{D}(\mathcal{A})$ is dense in the domain
$\mathsf{D}(\widetilde{\mathcal{A}})$ of the generator of
$(\mathcal{S}(t))_{t\geq 0}$ with respect to the graph norm
$\|\cdot\|_{\mathsf{D}(\widetilde{\mathcal{A}})}$. Applying Lemma A.2 and
noting that $\mathcal{A}$ is closed by Lemma A.1(b) completes the proof. ∎

###  A.2. Translation operators

######  Lemma A.4.

Let $U$ be a real and separable Hilbert space and let $J:=(0,T)$ for some
$T\in(0,\infty]$. For every $u\in L^{2}(J;U)$ we have that

$\lim_{h\to 0}\|u(\,\cdot\,+h)-u\|_{L^{2}(J_{h};U)}=0.$

Here, we define for each $h\in\mathbb{R}$ the interval $J_{h}:=((-h)\vee
0,T\wedge(T-h))\subseteq J$ and $u(\,\cdot\,+h)\colon J_{h}\to U$ denotes the
function $u$ shifted to the left by an increment $h$.

###### Proof.

Let $v\in C^{\infty}_{c}(J;U)$ and fix an arbitrary
$\varepsilon\in(0,\infty)$. Choose a compact interval $[a,b]\subset[0,\infty)$
such that
$\operatorname{supp}\bigl{(}v(\,\cdot\,+h)-v|_{J_{h}}\bigr{)}\subseteq[a,b]$
for all $h\in[-1,1]$. By the uniform continuity of $v$, there exists a
$\delta\in(0,1)$ such that, for all $h\in(-\delta,\delta)$ and every $t\in
J_{h}$, the estimate $\|v(t+h)-v(t)\|_{U}<\sqrt{\varepsilon/(b-a)}$ holds.
Thus,

$\|v(t+h)-v(t)\|_{L^{2}(J_{h};U)}^{2}<\varepsilon\quad\forall
h\in(-\delta,\delta).$

This shows the desired convergence for functions in the space
$C^{\infty}_{c}(J;U)$, which is dense in $L^{2}(J;U)$; indeed, since the set
of $U$-valued measurable simple functions is dense in $L^{2}(J;U)$ [41, Lemma
1.2.19(1)], it suffices to note that the scalar-valued function space
$C_{c}^{\infty}(J)$ is dense in $L^{2}(J)$ [2, Corollary 2.30]. Combined with
the fact that the translation operator is contractive from $L^{2}(J;U)$ to
$L^{2}(J_{h};U)$ (and thus bounded, uniformly in $h$), the result extends to
$L^{2}(J;U)$. ∎

######  Proposition A.5.

Let $T\in(0,\infty)$. The family $(\mathcal{T}(t))_{t\geq
0}\subseteq\mathscr{L}(L^{2}(0,T;H))$ defined in (3.6) is a $C_{0}$-semigroup
whose infinitesimal generator is given by $-\partial_{t}$, where
$\partial_{t}$ is the Bochner–Sobolev vector-valued weak derivative on
$\mathsf{D}(\partial_{t})=H^{1}_{0,\\{0\\}}(0,T;H)$.

###### Proof.

For each $t\geq 0$, it is clear that $\mathcal{T}(t)$ is a well-defined
contractive linear map on $L^{2}(0,T;H)$. Furthermore, it follows readily from
the definition (3.6) that $\mathcal{T}(0)=I$ and that the semigroup property
is satisfied, since for all $s,t\geq 0$, $v\in L^{2}(0,T;H)$ and a.a.
$\vartheta\in[0,T]$ we have that

$[\mathcal{T}(t)\mathcal{T}(s)v](\vartheta)=\widetilde{[\mathcal{T}(s)v]}(\vartheta-t)=\widetilde{v}(\vartheta-
t-s)=[\mathcal{T}(t+s)v](\vartheta).$

The strong continuity follows from Lemma A.4 for $h\uparrow 0$.

Next, we turn to the generator of $(\mathcal{T}(t))_{t\geq 0}$. To this end,
let an arbitrary ${v\in C_{c}^{\infty}((0,T];H)}$ be given and note that its
extension by zero to $(-\infty,T]$, again denoted by $\widetilde{v}$, is
continuously differentiable with classical (and hence weak) derivative
${\partial_{\vartheta}\widetilde{v}=\widetilde{\partial_{\vartheta}v}}$ by the
compact support of $v$ in $(0,T]$. Fix an arbitrary $\vartheta\in[0,T]$. The
function $t\mapsto\widetilde{v}(\vartheta-t)$ is continuously differentiable
on $[0,\infty)$ with derivative
$t\mapsto-\widetilde{\partial_{\vartheta}v}(\vartheta-t)$ by the chain rule.
Thus, the fundamental theorem of calculus gives

$\mathcal{T}(t)v(\vartheta)-v(\vartheta)=\widetilde{v}(\vartheta-t)-\widetilde{v}(\vartheta)=-\int_{0}^{t}\widetilde{\partial_{\vartheta}v}(\vartheta-s)\mathop{}\\!\mathrm{d}s=-\int_{0}^{t}[\mathcal{T}(s)\partial_{\vartheta}v](\vartheta)\mathop{}\\!\mathrm{d}s$

for every $t\geq 0$. It follows that

$\mathcal{T}(t)v-v=-\int_{0}^{t}\mathcal{T}(s)\partial_{\vartheta}v\mathop{}\\!\mathrm{d}s.$

Furthermore, we know from [61, Chapter 1, Theorem 2.4(b)] that if $R$ denotes
the generator of $(\mathcal{T}(t))_{t\geq 0}$, then we have

$\mathcal{T}(t)v-v=R\int_{0}^{t}\mathcal{T}(s)v\mathop{}\\!\mathrm{d}s,$

hence, combining the previous two displays yields

$R\int_{0}^{t}\mathcal{T}(s)v\mathop{}\\!\mathrm{d}s=-\int_{0}^{t}\mathcal{T}(s)\partial_{\vartheta}v\mathop{}\\!\mathrm{d}s.$
(A.1)

Set $v_{t}:=\frac{1}{t}\int_{0}^{t}\mathcal{T}(s)v\mathop{}\\!\mathrm{d}s$ for
$t\in(0,\infty)$. It follows that $v_{t}\to\mathcal{T}(0)v=v$ in
$L^{2}(0,T;H)$ as $t\downarrow 0$, see e.g. [61, Chapter 1, Theorem 2.4(a)].
Dividing both sides of (A.1) by $t\in(0,\infty)$ and passing to the limit
$t\downarrow 0$, one obtains

$Rv_{t}=R\,\frac{1}{t}\int_{0}^{t}\mathcal{T}(s)v\mathop{}\\!\mathrm{d}s=-\frac{1}{t}\int_{0}^{t}\mathcal{T}(s)\partial_{\vartheta}v\mathop{}\\!\mathrm{d}s\to-\mathcal{T}(0)\partial_{\vartheta}v=-\partial_{\vartheta}v.$

Since $R$ is assumed to be the generator of a $C_{0}$-semigroup, it is in
particular closed by [31, Proposition II.1.4]. Combined with the convergence
$v_{t}\to v$ and $Rv_{t}\to-\partial_{\vartheta}v$ as $t\downarrow 0$, this
yields $v\in\mathsf{D}(R)$ and ${Rv=-\partial_{\vartheta}v}$, hence
$-\partial_{\vartheta}|_{C_{c}^{\infty}((0,T];H)}\subseteq R$.

As $C_{c}^{\infty}((0,T];H)$ is dense in $L^{2}(0,T;H)$ and
$\mathcal{T}(t)C_{c}^{\infty}((0,T];H)\subseteq C_{c}^{\infty}((0,T];H)$ for
all $t\geq 0$, we have that $C_{c}^{\infty}((0,T];H)$ is dense in
$\mathsf{D}(R)$ with respect to the graph norm of $R$ by [31, Proposition
II.1.7]. It is evident from the respective definitions that
$\|\cdot\|_{\mathsf{D}(R)}\eqsim\|\cdot\|_{H^{1}(0,T;H)}$. These observations
together imply

$\mathsf{D}(R)=\overline{C_{c}^{\infty}((0,T];H)}^{\mathsf{D}(R)}=\overline{C_{c}^{\infty}((0,T];H)}^{H^{1}(0,T;H)}=H^{1}_{0,\\{0\\}}(0,T;H).\qed$

###  A.3. The proof of Lemma 3.6

###### Proof of Lemma 3.6.

Analogously to [25, Proposition 5.9] it can be shown that the operator defined
by the right-hand side of (3.12) maps functions in $L^{2}(0,T;H)$ to
$C_{0,\\{T\\}}([0,T];H)$. Now we prove the identity in (3.12). Let ${f,g\in
L^{2}(0,T;H)}$ be arbitrary. By (3.11) and by continuity of the inner product
$(\,\cdot\,,\,\cdot\,)_{H}$, we find that

$\displaystyle(\mathcal{B}^{-\gamma}$ $\displaystyle
f,g)_{L^{2}(0,T;H)}=\int_{0}^{T}\bigl{(}[\mathcal{B}^{-\gamma}f](t),g(t)\bigr{)}_{H}\mathop{}\\!\mathrm{d}t$
$\displaystyle=\int_{0}^{T}\biggl{(}\frac{1}{\Gamma(\gamma)}\int_{0}^{t}(t-s)^{\gamma-1}S(t-s)f(s)\mathop{}\\!\mathrm{d}s,g(t)\biggr{)}_{H}\mathop{}\\!\mathrm{d}t$
$\displaystyle=\frac{1}{\Gamma(\gamma)}\int_{0}^{T}\int_{0}^{T}\bigl{(}\mathbf{1}_{(0,t)}(s)(t-s)^{\gamma-1}S(t-s)f(s),g(t)\bigr{)}_{H}\mathop{}\\!\mathrm{d}s\mathop{}\\!\mathrm{d}t.$
(A.2)

Next, we would like to use Fubini’s theorem to exchange the order of
integration. By (3.2) the semigroup $(S(t))_{t\geq 0}$ is uniformly bounded on
the compact interval $[0,T]$,

$\widetilde{M}_{T}:=\sup\nolimits_{t\in[0,T]}\|S(t)\|_{\mathscr{L}(H)}\leq
Me^{(-wT)\vee 0}<\infty.$

We then use the Cauchy–Schwarz inequality on $H$ and on $L^{2}(0,T)$ as well
as the fact that $\gamma>\tfrac{1}{2}$ to check that

$\displaystyle\int_{0}^{T}\int_{0}^{T}\bigl{|}\bigl{(}\mathbf{1}_{(0,t)}(s)(t-s)^{\gamma-1}S(t-s)f(s),g(t)\bigr{)}_{H}\bigr{|}\mathop{}\\!\mathrm{d}s\mathop{}\\!\mathrm{d}t$
$\displaystyle\quad\leq\widetilde{M}_{T}\int_{0}^{T}\int_{0}^{t}(t-s)^{\gamma-1}\|f(s)\|_{H}\mathop{}\\!\mathrm{d}s\;\|g(t)\|_{H}\mathop{}\\!\mathrm{d}t$
$\displaystyle\quad\leq\widetilde{M}_{T}\,\|f\|_{L^{2}(0,T;H)}\int_{0}^{T}\biggl{(}\int_{0}^{t}(t-s)^{2\gamma-2}\mathop{}\\!\mathrm{d}s\biggr{)}^{\nicefrac{{1}}{{2}}}\|g(t)\|_{H}\mathop{}\\!\mathrm{d}t$
$\displaystyle\quad=\tfrac{\widetilde{M}_{T}}{\sqrt{2\gamma-1}}\,\|f\|_{L^{2}(0,T;H)}\int_{0}^{T}t^{\gamma-\frac{1}{2}}\|g(t)\|_{H}\mathop{}\\!\mathrm{d}t\leq\tfrac{\widetilde{M}_{T}T^{\gamma}}{\sqrt{2\gamma(2\gamma-1)}}\,\|f\|_{L^{2}(0,T;H)}\|g\|_{L^{2}(0,T;H)}$

is finite. This justifies changing the order of integration in (A.2), which
gives

$\displaystyle(\mathcal{B}^{-\gamma}f,g)_{L^{2}(0,T;H)}$
$\displaystyle=\frac{1}{\Gamma(\gamma)}\int_{0}^{T}\int_{0}^{T}\bigl{(}\mathbf{1}_{(s,T)}(t)(t-s)^{\gamma-1}S(t-s)f(s),g(t)\bigr{)}_{H}\mathop{}\\!\mathrm{d}t\mathop{}\\!\mathrm{d}s$
$\displaystyle=\frac{1}{\Gamma(\gamma)}\int_{0}^{T}\int_{0}^{T}\bigl{(}f(s),\mathbf{1}_{(s,T)}(t)(t-s)^{\gamma-1}[S(t-s)]^{*}g(t)\bigr{)}_{H}\mathop{}\\!\mathrm{d}t\mathop{}\\!\mathrm{d}s$
$\displaystyle=\int_{0}^{T}\biggl{(}f(s),\frac{1}{\Gamma(\gamma)}\int_{s}^{T}\mathbf{(}t-s)^{\gamma-1}[S(t-s)]^{*}g(t)\mathop{}\\!\mathrm{d}t\biggr{)}_{H}\mathop{}\\!\mathrm{d}s,$

where we interchanged integrals and inner products as before in the last step.
∎

###  A.4. Hölder continuity and weak derivatives

Recall from Section 2 that $(W(t))_{t\geq 0}$ denotes an $H$-valued