monthyeardate[], affil0affil0affiliationtext: Mini-course given at University
of Oxford

#  Introduction to SPDEs from Probability and PDE

A. Mayorcas

###### Contents

  1. 1  Introduction and Preliminaries
    1. 1.0.1 Conventions, Notation and Terminology
    2. 1.1 Randomness in PDE
      1. 1.1.1 Passive Scalar in a Turbulent Vector Field
      2. 1.1.2 A Worked Example of the Variational Method
    3. 1.2 Bounded Operators on Infinite Dimensional Spaces
      1. 1.2.1 Trace Class and Hilbert–Schmidt Operators
      2. 1.2.2 Pseudo Inverses of Linear Operators
    4. 1.3 Lebesgue and Bochner Integration
    5. 1.4 Probability and Random Variables
      1. 1.4.1 Gaussian Measures on Hilbert Spaces
      2. 1.4.2 Generalised Gaussian Random Variables
    6. 1.5 Stochastic Analysis in Infinite Dimensions
      1. 1.5.1 Banach Space Valued Martingales
      2. 1.5.2 Hilbert Space Valued Wiener Processes
      3. 1.5.3 Generalised Wiener Processes
      4. 1.5.4 Stochastic Integration Against $Q$-Wiener Processes
      5. 1.5.5 Stochastic Integration Against Generalised Wiener Processes
  2. 2 Variational Approach to PDE
    1. 2.1 Unbounded Operators Between Banach Spaces
    2. 2.2 Linear and Coercive PDE
    3. 2.3 Monotonicity and Weak Continuity
    4. 2.4 A Compactness Method
  3. 3 Variational Approach to Monotone SPDE
    1. 3.1 Semi-Martingales and Itô’s Formula
    2. 3.2 Well-Posedness of Monotone and Coercive SPDE
    3. 3.3 Examples
  4. 4 Pathwise Approach to SPDE
    1. 4.1 Regularity of Stochastic Processes
    2. 4.2 Pathwise Approach to SPDE: Stochastic Burger’s Equation

##  Chapter 1  Introduction and Preliminaries

This mini-course aims to serve as a brief introduction to the study of
stochastic partial differential equations. The main goal of these notes is to
provide a useful resource for both PDE and stochastic analysts. We aim to
demonstrate both how some standard PDE techniques can be extended to the
stochastic setting and how the tools of stochastic analysis can be extended to
infinite dimensional settings, common to the analysis of PDE. The main focus
of this course is on the variational method for coercive-monotone SPDE. The
tools we introduce along the way should be familiar to researchers in both
areas and hopefully this provides a sufficiently broad base of tools for
applications to problems beyond the scope of this course. In Chapter 4 we give
a brief introduction to the pathwise approach to SPDE, which has been a topic
of significant research in recent years, sparked by the development of novel
approaches to singular SPDE, [Hai09, GIP15].

The variational method for SPDE stems from early work by E. Pardoux, [Par72],
which extended the variational approach to PDE developed by J. Lions, [Lio11].
The variational approach to SPDE has been developed in a number of directions
and we refer to [LR17, PR07] for a more comprehensive treatment of the
literature and material. In particular, however, we mention the early works by
I. Gyongy & N. Krylov, [GK81a, GK81b, Gyö82] in which the authors extend the
results presented here to the case of general semi-martingale drivers.
Alongside the variational approach there are broadly two other, non-pathwise,
frameworks for solving SPDE; the perspective of martingale problems, [MV88,
Par07] and the semi-group approach, [DPZ14, Hai09]. We also wish to mention
the book by F. Flandoli, [Fla11] that focuses on stochastic perturbations of
fluid equations, the collected work, [DKM+09], that contains an overview of a
number of different problems in the field of SPDE and the book by L. Zambotti
on stochastic obstacle problems, [Zam17].

These notes were written to accompany a $6$ hour mini-course delivered at the
University of Oxford in April 2021. The four chapters broadly follow the
material presented across four lectures, with more background, references and
details to some arguments. In the remainder of this chapter we first list some
conventions, notation and terminology that we use throughout, Subsection
1.0.1. Then in Section 1.1 we give some background on various motivations for
studying SPDE problems, focussing in particular on a presentation of a
simplified version of the Kraichnan model for a passive scalar in a turbulent
fluid. In the second half of that section, Subsection 1.1.2, we give a worked
example of the variational method that is presented abstractly in Chapter 3.
The remainder of the first chapter is dedicated to recalling some necessary
background material and developing the theory of stochastic integration in
Hilbert spaces. In Chapter 2 we return to deterministic problems and present a
summary of J. Lions variational approach to coercive PDE. Chapter 3 contains
the main focus of the course, the variational approach to coercive-monotone
SPDE. We present the abstract method along with an extension of the worked
example given in Subsection 1.1.2. Finally in Chapter 4 we give a short
introduction to the pathwise approach to non-singular SPDE, using a stochastic
Burger’s equation with white noise forcing as an example.

##### Acknowledgements

A.M. would like to thank the Centres for Doctoral Training in PDEs and The
Mathematics Random Systems at the University of Oxford through which this
lecture course was originally given. I would also like to thank M. Veraar and
Y. Han for comments, corrections and suggestions on improvements to the
manuscript.

####  1.0.1 Conventions, Notation and Terminology

Throughout these notes we use the conventional notation for stochastic
equations, that is we use the shorthand

$\mathop{}\\!\mathrm{d}u_{t}=A(u_{t})\mathop{}\\!\mathrm{d}t+B(u_{t})\mathop{}\\!\mathrm{d}W_{t},\quad\iff\quad
u_{t}=u_{0}+\int_{0}^{t}A(u_{s})\mathop{}\\!\mathrm{d}s+\int_{0}^{t}B(u_{s})\mathop{}\\!\mathrm{d}W_{s}.$
(1.0.1)

Furthermore, when we refer to weak solutions for problems such as (1.0.1), we
mean a weak solution in the PDE sense, that is we ask (1.0.1) to hold in the
sense of distributions. This is not to be confused with the notion of weak
solution in stochastic analysis, which typically means a solution not adapted
to a given filtration. The solutions we consider will always be weak PDE
solutions but strong stochastic solutions. We adopt a slightly unusual
convention and write $[X]_{t}$ for the quadratic variation of a stochastic
process, instead of the usual angle brackets. This is to avoid confusion with
the standard notation for inner products on Hilbert spaces and duality
pairings between Banach spaces.

Unless otherwise specified, all Banach and Hilbert spaces are separable and
whenever we refer to a basis $(e_{k})_{k\geq 1}$ of a Hilbert space we mean a
complete, orthonormal system. For a Banach space $E$ and a Hilbert space $U$
we typically write $\|\,\cdot\,\|_{E}$ for the norm on $E$ and
$|\,\cdot\,|_{U},\,\langle\,\cdot\,,\,\cdot\,\rangle_{U}$ for the norm and
inner product on $U$. If such an expression appears without an explicit label
it will always be explained what is meant. We write $E^{\ast}$ for the dual of
$E$, the set of bounded, linear functionals $\varphi:E\rightarrow\mathbb{R}$.
We say that a sequence $(u_{n})_{n\geq 1}\subset E$ converges weakly in $E$,
if $(\varphi(u_{n}))_{n\geq 1}\subset\mathbb{R}$ converges in $\mathbb{R}$ for
every $\varphi\in E^{\ast}$. We say that a sequence $(\varphi_{n})_{n\geq
1}\subset E^{\ast}$ converges in weak-$*$ if the evaluations
$(\varphi_{n}(u))_{n\geq 1}\subset\mathbb{R}$ are a convergent sequence for
every $u\in E$. Through the Riesz representation theorem we identify the dual
of a Hilbert space with itself, so that we say a sequence $(u_{n})_{n\geq
1}\subset U$ converges weakly if $(\langle h,u_{n}\rangle_{U})_{n\geq
1}\subset\mathbb{R}$ is convergent for every $h\in U$. We say that a map
$[0,\infty)\ni t\mapsto u_{t}\in E$ is continuous if it is continuous with
respect to the strong topology on $E$. Given a $T>0$, we equip the space of
continuous maps $u:[0,T]\rightarrow E$ with the structure of a Banach space,
by defining the norm

$\|u\|_{C_{T}E}:=\sup_{t\in[0,T]}\|u_{t}\|_{E}.$

Given a measure space $(E,\mathcal{B}(E),\mu)$, a measurable space
$(F,\mathcal{B}(F))$ and a measurable map $X:E\rightarrow F$, the push-forward
of $\mu$ by $X$, $X\\#\mu=\mu(X^{-1}(\,\,\cdot\,))$ defines a measure on $F$
and for any integrable function $g:F\rightarrow\mathbb{R}$ and
$A\in\mathcal{B}(F)$, we have the identity

$\int_{A}g(x)\mathop{}\\!\mathrm{d}X\\#\mu(x)=\int_{X^{-1}(A)}g(X(x))\mathop{}\\!\mathrm{d}\mu(x).$

Given a family of sets $\\{A\\}$, we write $\sigma(\\{A\\})$ for the smallest
$\sigma$-algebra containing the collection. Given a measurable space
$(E,\mathcal{B}(E))$ we write $\mathcal{P}(E)$ for the set of probability
measures on $E$. We say that a sequence of probability measures
$(\mu_{n})_{n\geq 1}\subset\mathcal{P}(E)$ converges weakly to
$\mu\in\mathcal{P}(E)$ if
$\int_{E}f\mathop{}\\!\mathrm{d}\mu^{n}\rightarrow\int_{E}f\mathop{}\\!\mathrm{d}\mu$
for all $f:E\rightarrow\mathbb{R}$, continuous and bounded. We say that a
sequence of random variables converges weakly if their laws converge weakly.
The strong topology on $\mathcal{P}(E)$ is the topology induced by the total
variation distance.

For two scalars, $s,\,t\in\mathbb{R}$, we use the notation $s\wedge t$ to
denote their minimum and $s\vee t$ to denote their maximum. Typically we use
this when considering functions $u:[0,\infty)\rightarrow E$, where we write
$u_{t\wedge s}$ to mean the value of $u$ at the minimum of $s,\,t$. For
functions of multiple variables, $u:\mathbb{R}^{d}\rightarrow\mathbb{R}$, we
use standard multi-index notation to denote partial derivatives, i.e for
$k\in\mathbb{N}^{d}$, $\partial_{x}^{k}u$ denotes the $|k|^{\text{th}}$
partial derivative in the coordinates describe by the multi-index
$k=(k_{1},\,\ldots,k_{d})$. If we write $D^{|k|}$ we mean the vector of all
possible $|k|^{\text{th}}$ order derivatives.

###  1.1 Randomness in PDE

In very abstract terms we may think of a typical PDE problem as being
constituted of the following elements; a finite dimensional metric space,
$\Gamma$, a partial differential operator $L(u,Du,\ldots)$, acting on
functions $u:\Gamma\rightarrow\mathbb{R}^{n}$, a non-linear functional, $F$,
and some data $g$ defined on the boundary $\partial\Gamma$. The associated PDE
problem is to find $u$, solving

$\begin{cases}Lu=F(u,Du,D^{2}u,\ldots),&\text{inside }\Gamma,\\\
u|_{\partial\Gamma}=g,\end{cases}$ (1.1.1)

in a suitable way. The distinction between the linear part, $L$, and non-
linear part, $F$, is not necessary but provides a useful framing for the
equations considered in these notes.

In this setting there are essentially four ways in which noise can be included
in a PDE problem; through the linear operator $L$, through the non-linearity
$F$, through the domain, $\Gamma$ or the boundary data
$u|_{\partial\Gamma}=g$. We will not explicitly consider the latter two cases;
some treatment of random data equations can be found in [Fla17, Tzv16] while
some treatment of equation on random domains can be found in [XT06, HPS16]. In
fact we will mostly restrict ourselves to cases where randomness occurs in the
right hand side of the equation, $F$.

Motivations for studying SPDE come from many directions in both mathematics
and the applied sciences. To mention only a few references, they have found
applications in: constructive quantum field theory, [DH87, GH18, Hai14b,
Kup18]; the study of disordered media, [Lab19, AC15]; non-linear filtering,
[Zak69, Par79]; population dynamics, [Daw72, Fle75] and stochastic fluid
dynamics including models of turbulence, [Kra68, Fla11]. In particular we
refer to the introductions of [DPZ14, Par07, DKM+09] for more detailed
summaries of some of these applications along with others not mentioned here.
In order to motivate this course a little better we proceed to describe how an
SPDE model can arise in the study of a passive scalar transported by a
turbulent flow. This model is related to the Kraichnan model of turbulence,
[Kra68], mentioned above.

####  1.1.1 Passive Scalar in a Turbulent Vector Field

We present a simplified version of a model developed by separately by R.
Kraichnan, [Kra68] and A. Kazantsev [Kaz68], both of which were inspired by
the work of G. Batchelor on turbulence, [Bat00, Mof02]. More detailed
presentations and analysis of this model and its variants can be found among
the non-exhaustive references, [Fla11, FL19, FGL21, GY21, MK99, CDG07]. We
will use SPDE related to this model as recurring examples throughout the
course.

Let us imagine that we are interested in the diffusion of a given
concentration in a two dimensional fluid. We may model this situation by the
simple transport equation, on $[0,\infty)\times\mathbb{R}^{d}$, with
$\Gamma_{0}\subset\mathbb{R}^{2}$ closed, bounded, $\partial\Gamma_{0}$
sufficiently smooth, and $(v_{t})_{t\geq 0}$ a family of divergence free
vector fields,

$\begin{cases}\partial_{t}u_{t}-v_{t}\cdot\nabla u_{t}=0,&\text{ in }\Gamma\\\
u|_{t=0}=\frac{1}{|\Gamma_{0}|}\mathds{1}_{\Gamma_{0}}.&\end{cases}$ (1.1.2)

The solution $u_{t}:\Gamma\rightarrow\mathbb{R}$ describes the concentration
at time $t\geq 0$. Formally speaking, due to the divergence free assumption,
for any solution to (1.1.2), one has
$\frac{1}{|\Gamma|}\int_{\Gamma}u_{0}(x)\mathop{}\\!\mathrm{d}x=\frac{1}{|\Gamma|}\int_{\Gamma}u_{t}(x)\mathop{}\\!\mathrm{d}x$.
That is mass cannot be lost during the evolution, but the local density may
vary. For simplicity, let us assume that $\Gamma=\mathbb{T}^{2}$, the two
dimensional torus and as a toy example let us consider the simple vector
fields,

$v_{t}(x,y)=\begin{bmatrix}-y\\\ x\end{bmatrix}\theta_{t},$

for $t\mapsto\theta_{t}\in\mathbb{R}$ a differentiable function. The
characteristics of $\eqref{eq:SimpleTransport}$ are the system of ODEs,

$\begin{cases}\frac{d}{dt}X_{t}=Y_{t}\frac{d}{dt}\theta_{t},\\\
\frac{d}{dt}Y_{t}=-X_{t}\frac{d}{dt}\theta_{t}.\end{cases}$ (1.1.3)

Provided the paths $t\mapsto\theta_{t}\in\mathbb{R}$ are sufficiently regular
(and we ignore the issue of boundary data) we can hope to solve (1.1.3) and
using a flow map then give the solution to (1.1.2) as
$u(t,x,y)=u_{0}(X^{-1}_{t}(x),Y^{-1}_{t}(y))$, where $(X^{-1}_{t},Y^{-1}_{t})$
denotes the backwards flow associated to (1.1.3), see [Eva10, Sec. 3.2].

Now we imagine that there is some uncertainty over the coefficient
$\theta_{t}$. For example the concentration might be a dye diffusing far out
at sea and we only have intermittent readings on the local fluid velocity.
Alternatively, as in [Kra68], we might intentionally introduce some randomness
to model a turbulent velocity field. We can account for both situations by
fixing a probability space $(\Omega,\mathcal{F},\mathbb{P})$ and now
considering random coefficients, given by a measurable map,
$(\omega,t)\mapsto\theta_{t}(\omega)\in\mathbb{R}$. We can also see the map
$\Omega\ni\omega\mapsto\theta(\omega)$ as a random mapping into the space of
paths. If we assume that for $\mathbb{P}$-a.a. $\omega\in\Omega$, we still
have $\theta(\omega)\in C^{1}(\mathbb{R}_{+};\mathbb{R})$, then we may replace
(1.1.3) with the random system of characteristics,

$\begin{cases}\frac{d}{dt}X_{t}(\omega)=Y_{t}(\omega)\frac{d}{dt}\theta_{t}(\omega),\\\
\frac{d}{dt}Y_{t}(\omega)=-X_{t}(\omega)\frac{d}{dt}\theta_{t}(\omega).\end{cases}$
(1.1.4)

Again, up to solving this system, we provide a solution the random PDE,

$\begin{cases}\partial_{t}u_{t}(\omega)-v_{t}(\omega)\cdot\nabla
u_{t}(\omega)=0,&\text{ in }\mathbb{T}^{2}\\\
u|_{t=0}=\frac{1}{|\Gamma_{0}|}\mathds{1}_{\Gamma_{0}}.\end{cases}$ (1.1.5)

By integrating over $\omega\in\Omega$ we can now answer more nuanced questions
regarding the evolution of $u$. For example we could study the expected
density in a certain region, or the expected time to total mixing. The idea is
that if the random coefficients are described by a sufficiently rich
probability space, and of a form suitable to some statistical properties of
the situation at hand, then our model can be made robust to many realised
scenarios.

To this end, we note that the model considered so far is very simple and
unlikely to describe many situations of interest sufficiently well. In order
to enrich the model let us first develop the form of vector fields we
consider. We now define,

$v_{t}(x,y):=\sum_{k=1}^{\infty}\sigma_{k}v_{k}(x,y)\theta^{k}_{t},\quad
v_{k}(x,y)=\begin{bmatrix}-(y-a_{k})\\\
x-b_{k}\end{bmatrix},\quad\theta^{k}_{t}\sim\theta_{t}\quad\text{i.i.d,}$
(1.1.6)

with $(a_{k},b_{k})_{k\geq 1}\subset\Gamma$ given centres of rotation, the
coefficients $\sigma_{k}$ chosen so as to ensure the sum converges suitably,
$\mathbb{P}$-a.s. (note that this is possible since we specified that $\Gamma$
be bounded) and $(\theta^{k}_{t})_{k\geq 1}$ a family of i.i.d random scalar
paths. Without loss of generality we may assume that
$\mathbb{E}[\theta^{k}_{t}]=0$ for all $k,\,t$. This superposition of random,
divergence free, vector fields now gives a lot of scope to accommodate an
expected fluid flow and with randomness built in to account for uncertainty.
In the Kraichnan model the spatial features of the vector field flow is
achieved in a different manner, but the end results are qualitatively similar,
see [CDG07, Sec. III] and [GY21, Sec. 5] for presentations of this more
detailed model.

While the spatial properties of the flow (1.1.6) can now be adjusted to a
large extent, the time dependence is constrained by requiring that
$t\mapsto\theta^{k}_{t}$ be at least $C^{1}$. For example, if we choose the
$\sigma_{k}$ so that the series is $\ell^{2}$ convergent, we have,

$\displaystyle\mathbb{E}[|v_{t}(x,y)-v_{s}(x,y)|^{2}]$
$\displaystyle=\sum_{k=1}^{\infty}\sigma_{k}^{2}|v_{k}(x,y)|^{2}\mathbb{E}[|\theta_{t}^{k}-\theta_{s}^{k}|^{2}]$
$\displaystyle\leq\sup_{k\geq
1}\mathbb{E}[|\partial_{t}\theta^{k}_{s}|^{2}]|t-s|^{2}\|\sigma_{\,\cdot\,}v_{\,\cdot\,}(x,y)\|^{2}_{\ell^{2}(\mathbb{N})}.$

That is, the values $v_{t}(x,y)$ and $v_{s}(x,y)$ must be relatively well
correlated. So if we expect the vector fields at a given location to oscillate
faster than linearly in time we will be unable to capture this by our current
model. Following similar considerations, in [Kra68], Kraichnan proposed an
extension, allowing for arbitrarily short correlation times for the vector
flows. In order to achieve this one can replace the $(\theta^{k})_{k\geq 1}$
with less regular trajectories, a common choice being Brownian motions,
$(W^{k})_{k\geq 1}$, where one has,

$\mathbb{E}[|W^{k}_{t}-W^{k}_{s}|^{2}]=|t-s|.$

More pertinently, when we consider the characteristic equations, we replace
$\frac{\mathop{}\\!\mathrm{d}}{\mathop{}\\!\mathrm{d}t}\theta_{t}$ with the
differential $\mathop{}\\!\mathrm{d}W_{t}$ and if we compute the covariation
between $\mathop{}\\!\mathrm{d}W_{t}$ and $\mathop{}\\!\mathrm{d}W_{s}$ we
formally have,

$\mathbb{E}[\mathop{}\\!\mathrm{d}W_{t}\mathop{}\\!\mathrm{d}W_{s}]=\delta(t-s).$

So we see that replacing $(\theta^{k})_{k\geq 1}$ with $(W^{k})_{k\geq 1}$
allows for a fluid flow with arbitrarily small correlation times. While this
extreme may also not be physically well justified it provides more flexibility
than the previous regularity requirements, gives way to a mathematically rich
theory and can be further adapted to more physically relevant settings.

The price to pay for introducing vector fields with this more singular
behaviour is that the ODEs, (1.1.4), are no longer trivially well-posed. In
particular, one has that $\mathbb{P}$-a.s., the paths $t\mapsto W^{k}_{t}$ are
only almost $1/2$-Hölder continuous and so
$t\mapsto\mathop{}\\!\mathrm{d}W^{k}_{t}$ represents a proper distribution. In
order to solve (1.1.4) in this case then we require a definition of integrals
$\int_{0}^{t}X_{s}\mathop{}\\!\mathrm{d}W^{k}_{s}$ for paths $t\mapsto X_{t}$
of no better regularity than $t\mapsto W_{t}$. This is a primary aim of
stochastic analysis, which the remainder of this chapter is dedicated to.
While we refer to (1.1.5) as a random PDE, we refer to,

$\begin{cases}\mathop{}\\!\mathrm{d}u_{t}-\sum_{k=1}^{\infty}\sigma_{k}v_{k}\cdot\nabla
u_{t}\mathop{}\\!\mathrm{d}W_{t}^{k}=0,&\text{ in }\mathbb{T}^{2},\\\
u|_{t=0}=u_{0},&\end{cases}$ (1.1.7)

as a stochastic PDE (SPDE). This is to emphasise the fact that one requires
some additional theory in order to handle the time integral in (1.1.7).

####  1.1.2 A Worked Example of the Variational Method

In the final subsection of this introductory portion of the notes let us
present a model example of the variational method that we will cover in more
detail in the rest of the course. This example serves both as a motivation for
the model described at the end of Chapter 3 and an example walk through of the
abstract method in a relatively simple case. Let
$(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\geq 0},\mathbb{P})$ be a filtered
probability space, carrying $(W_{t})_{t\geq 0}$ a standard, $\mathbb{R}^{d}$
valued Brownian motion. We consider the semi-linear, parabolic SPDE, for
simplicity on the $d$-dimensional torus, $\mathbb{T}^{d}$,

$\begin{cases}\mathop{}\\!\mathrm{d}u=\Delta
u\mathop{}\\!\mathrm{d}t+\sqrt{\sigma}\nabla
u\cdot\mathop{}\\!\mathrm{d}W_{t},&\text{ on
}\mathbb{R}_{+}\times\mathbb{T}^{d},\\\ u|_{t=0}=u_{0}.\end{cases}$ (1.1.8)

######  Remark 1.1.1.

Note that (1.1.8) has the structure of a diffusive equivalent of the transport
equation (1.1.2) with $v_{t}=\mathop{}\\!\mathrm{d}W_{t}$. This can be
generalised to the superposition example discussed in that context
$v_{t}=\sum_{k}\sqrt{\sigma}_{k}v_{k}\mathop{}\\!\mathrm{d}W^{k}_{t}$.

For the sake of presentation we restrict to the one dimensional case here,
although the method applies almost mutatis mutandi in higher dimensions. Let
$(e_{k})_{k\in\mathbb{Z}}$ be the standard Fourier basis of
$L^{2}(\mathbb{T})$. Then defining $u_{k;t}:=\langle u_{t},e_{k}\rangle$, we
derive from (1.1.8) the system of SDEs,

$\mathop{}\\!\mathrm{d}u_{k;t}=-(2\pi
k)^{2}u_{k;t}\mathop{}\\!\mathrm{d}t-2\sqrt{\sigma}\pi
iku_{k;t}\mathop{}\\!\mathrm{d}W_{t},\quad u_{k;0}=\langle
u_{0},e_{k}\rangle.$ (1.1.9)

The linear system, (1.1.9), is seen to be well-posed by a stochastic analogue
of the standard Cauchy–Lipschitz theory, see [Øks13, Thm. 5.2.1]. Granting
that for any $n\geq 1$, we have global, unique solutions, $(u_{k})_{k=1}^{n}$,
we set

$u_{n;t}=\sum_{|k|\leq n}u_{k;t}e_{k},\quad u_{n;0}:=\sum_{|k|\leq
n}u_{k;0}e_{k},$

So, at least formally, $u_{n}$, is a smooth approximation to $u$ solving
(1.1.8). The question is whether we can really take the limit and define a
solution to (1.1.8) as $u:=\lim_{n\rightarrow\infty}u_{n}$. The key is to
obtain a suitable a priori bound on the solutions $u_{n}$, uniform in $n\geq
1$. By the usual Parseval’s identity, it holds that

$\|u_{n;t}\|^{2}_{L^{2}(\mathbb{T})}=\sum_{|k|\leq n}|u_{k;t}|^{2}.$

Applying Itô’s formula (the chain rule for Itô processes, see [Øks13, Thm.
4.1.2 & 4.2.1] and Section 3.1 below), we see that for each $k=1,\ldots n$,
$|u_{k;t}|^{2}$ solves the SDE,

$\displaystyle\mathop{}\\!\mathrm{d}|u_{k;t}|^{2}$
$\displaystyle=2u_{k;t}\mathop{}\\!\mathrm{d}u_{k;t}+\mathop{}\\!\mathrm{d}[u_{k}]_{t}$
$\displaystyle=-2(2\pi
k)^{2}u_{k;t}u_{k;t}\mathop{}\\!\mathrm{d}t+2\sqrt{\sigma}(2\pi
ku_{k;t})u_{k;t}\mathop{}\\!\mathrm{d}W_{t}+\sigma(2\pi
iku_{k;t})^{2}\mathop{}\\!\mathrm{d}t$
$\displaystyle=2\partial_{xx}u_{k;t}u_{k;t}\mathop{}\\!\mathrm{d}t+2\sqrt{\sigma}\partial_{x}u_{k;t}u_{k;t}\mathop{}\\!\mathrm{d}W_{t}+\sigma|\partial_{x}u_{k;t}|^{2}\mathop{}\\!\mathrm{d}t.$

Here $[u_{k}]_{t}$ denotes the quadratic variation of the process $t\mapsto
u_{k;t}$, in our case this is just $4\sigma\pi^{2}|u_{k;t}|^{2}t$, since
$[W]_{t}=t$. For more details see [Øks13, Ex. 2.17] and Section 3.1 below.
Summing these equations up, or equivalently applying Itô’s formula directly to
the PDE, and converting to integral form, we get the identity,

$\|u_{n;t}\|^{2}_{L^{2}}=\|u_{n;0}\|^{2}_{L^{2}}+\int_{0}^{t}\left(2\langle\partial_{xx}u_{n;t},u_{n;t}\rangle+\sigma\langle\partial_{x}u_{n;s},\partial_{x}u_{n;s}\rangle\right)\mathop{}\\!\mathrm{d}s+2\sqrt{\sigma}\int_{0}^{t}\langle
u_{n;s},\partial_{x}u_{n;s}\rangle\mathop{}\\!\mathrm{d}W_{s}.$

The final integral may, at this stage be understood, as a finite dimensional
stochastic integral. Integrating by parts in the first term and applying the
product rule in the final term we write,

$\|u_{n;t}\|^{2}_{L^{2}}=\|u_{n;0}\|^{2}_{L^{2}}-(2-\sigma)\int_{0}^{t}\|\partial_{x}u_{n;s}\|^{2}_{L^{2}}\mathop{}\\!\mathrm{d}s+\sqrt{\sigma}\int_{0}^{t}\int_{\mathbb{T}}\partial_{x}(u_{n;s}(x))^{2}\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}W_{s}.$
(1.1.10)

We see that the last term contains the integral of a derivative on
$\mathbb{T}$ and so it disappears for all $\omega\in\Omega$ such that the
integral is well defined, that is,

$\int_{0}^{t}\int_{\mathbb{T}}\partial_{x}(u_{n;s}(x))^{2}\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}W_{s}(\omega)=0\quad\text{for
}\mathbb{P}\text{-a.a. }\omega\in\Omega.$ (1.1.11)

This is no accident and will also be true in higher dimensions if this term is
of the form $\nabla\cdot(v_{t}u_{t})\mathop{}\\!\mathrm{d}W_{t}$ with
$\nabla\cdot v_{t}=0$. However, in many cases, even if the stochastic integral
does not vanish $\mathbb{P}$-a.s. it will be zero in expectation, so long as
it is well-defined. In this case a stopping time argument can be run under the
expectation.

With this in hand, we now define, for $f\in C^{\infty}(\mathbb{T})$, the norm
$\|f\|^{2}_{H^{1}}:=\|f\|^{2}_{L^{2}}+\|\partial_{x}f\|^{2}_{L^{2}}$, and
rewrite (1.1.10) as,

$\displaystyle\|u_{n;t}\|^{2}_{L^{2}}+(2-\sigma)\int_{0}^{t}\|u_{n;s}\|^{2}_{H^{1}}\mathop{}\\!\mathrm{d}s$
$\displaystyle=\|u_{n;0}\|^{2}_{L^{2}}+(2-\sigma)\int_{0}^{t}\|u_{n;s}\|^{2}_{L^{2}}\mathop{}\\!\mathrm{d}s.$
(1.1.12)

which by (1.1.11) holds for $\mathbb{P}$-a.a. $\omega\in\Omega$. So now, if
$(2-\sigma)>0$ and we fix some $T>0$, we may apply Grönwall’s inequality, to
obtain the estimate,

$\sup_{t\in[0,T]}\|u_{n;t}\|^{2}_{L^{2}}+(2-\sigma)\int_{0}^{T}\|u_{n;s}\|^{2}_{H^{1}}\mathop{}\\!\mathrm{d}s\leq\|u_{n;0}\|^{2}_{L^{2}}\,e^{(2-\sigma)T}$
(1.1.13)

So, assuming $(2-\sigma)>0$, we have shown,

$u_{n}(\omega)\in C([0,T];L^{2}(\mathbb{T}))\cap
L^{2}([0,T];H^{1}(\mathbb{T})),\quad\text{for }\mathbb{P}\text{-a.a.
}\omega\in\Omega.$

Furthermore, the quantity
$\|u_{n}(\omega)\|_{C_{T}L^{2}}+\|u_{n}(\omega)\|_{L^{2}_{T}H^{1}}$ depends
only on $\|u_{n;0}\|^{2}_{L^{2}}$ \- which we assume to be deterministic for
ease. Now if $u_{0}\in L^{2}(\mathbb{T})$ we easily have,

$\|u_{n;0}\|^{2}_{L^{2}}=\sum_{|k|\leq
n}|u_{k;0}|^{2}\leq\sum_{k}|u_{k;0}|^{2}=\|u_{0}\|^{2}_{L^{2}}.$

Therefore the bound (1.1.12), and hence the norm
$\|u_{n}(\omega)\|_{C_{T}L^{2}}+\|u_{n}(\omega)\|_{L^{2}_{T}H^{1}}$ is
controlled independently of $n\geq 1$. Concretely,

$\sup_{n\geq
1}\left(\|u_{n}(\omega)\|_{C_{T}L^{2}}+\|u_{n}(\omega)\|_{L^{2}_{T}H^{1}}\right)\leq\|u_{0}\|_{L^{2}}e^{(2-\sigma)T}.$

This uniform bound is essentially enough to obtain a weak solution to (1.1.8)
as the limit $u_{n}\rightarrow u$. We will detail this final step in the
Chapter 3 since it requires some functional analysis which we present in a
more general setting.

The main takeaway, however, is that the above argument applies only for
$\sqrt{\sigma}<\sqrt{2}$. Philosophically, this strategy is based on the
ellipticity of the operator
$u\mapsto(-\Delta\,\,\mathop{}\\!\mathrm{d}t+\sqrt{\sigma}\nabla\,\,\cdot\,\mathop{}\\!\mathrm{d}W_{t})u$
which we have seen does not hold for $\sqrt{\sigma}\geq\sqrt{2}$. In fact,
using the notion of a Stratonovich integral (a stochastic integral which does
obey the usual chain rule), we can re-write (1.1.8) in the form,

$\begin{cases}\mathop{}\\!\mathrm{d}u_{t}=\left(1-\frac{\sigma}{2}\right)\Delta
u\mathop{}\\!\mathrm{d}t+\sqrt{\sigma}\,\nabla
u\circ\mathop{}\\!\mathrm{d}W_{t}&\text{ on
}\mathbb{R}_{+}\times\mathbb{T}^{d},\\\ u|_{t=0}=u_{0}.\end{cases}$ (1.1.14)

Now if we take $\sigma>2$ we see that the deterministic part of (1.1.14)
becomes a backwards heat equation.

In the second half of this introductory chapter we recall some standard
material and prepare some preliminaries that we need throughout the course.
The majority of this background material is presented without proof and in
general we refer to [Bre11, Eva10, DPZ14, CE15] for details where they are
omitted.

###  1.2 Bounded Operators on Infinite Dimensional Spaces

Given Banach spaces $E,\,F$ we write $L(E;F)$ for the set of bounded, linear
operators $O:E\rightarrow F$. We define the operator norm of $O$ by the
expression

$\|O\|_{\text{Op}(E;F)}:=\sup_{\|x\|_{E}\leq 1}\|Ox\|_{F}=\sup_{x\in
F}\frac{\|Ox\|_{F}}{\|x\|_{E}}.$

$L(E;F)$ is itself a Banach space, although non-separable if both $E,\,F$ are
infinite dimensional.

######  Definition 1.2.1 (Finite Rank and Compact Operators).

Given Banach spaces, $E,\,F$ we say that an operator $O\in L(E;F)$ has finite
rank if $O(E)$ is a finite dimensional subspace of $F$. We say that an
operator $O\in L(E;F)$ is compact if for any bounded set $B\subset E$,
$O(B)\subset E$ is compact.

By definition, any finite rank operator is also compact. The following theorem
says that in fact all compact operators can be approximated by finite rank
operators.

######  Theorem 1.2.2 (Representation of Compact Operators).

An operator $O\in L(E;F)$ is compact if there exists a sequence
$(O_{n})_{n\geq 1}\subset L(E;F)$ of finite rank operators such that
$\|O-O_{n}\|_{\text{Op}(E;F)}\rightarrow 0$. If $U,\,H$ are a Hilbert spaces
and $O\in L(U;H)$ is compact, then there exist orthonormal families,
$(\tilde{e}_{k})_{k\geq 1}$ of $U$ and $(\tilde{f}_{k})_{k\geq 1}$ of $H$, and
a sequence of real numbers, $(\lambda_{k})_{k\geq 1}$, which if they converge
must converge to zero, such that

$Ox=\sum_{k=1}^{\infty}\lambda_{k}\langle
x,\tilde{e}_{k}\rangle_{U}\tilde{f}_{k},\quad\text{for all }x\in U.$ (1.2.1)

###### Proof.

See [Bre11, Ch. 6] in particular Corollary 6.2 and Theorem 6.8. ∎

Note that the families $(\tilde{e}_{k})_{k\geq 1},\,(\tilde{f}_{k})_{k\geq 1}$
need not be complete, in the sense that they need not form bases of $U,\,H$.
Consider $O$ a finite rank operator, for example. If $U,\,H$ are Hilbert
spaces and $O\in L(U;H)$, then there is an operator $O^{\ast}\in L(H;U)$ such
that,

$\langle Ox,h\rangle_{H}=\langle x,O^{\ast}h\rangle_{U},\quad\text{ for all
}x\in U,\,h\in H.$

The operator $O^{\ast}$ is known as the adjoint (or Hermitian adjoint) of $O$.
If $O\in L(H):=L(H;H)$ is such that $O=O^{\ast}$, then $O$ is said to be self-
adjoint. If $O$ is a compact, self-adjoint operator then there exists a
representation of the form (1.2.1) with $(f_{k})_{k\geq 1}=(e_{k})_{k\geq 1}$
an orthonormal basis of $H$. See [Bre11, Thm. 6.11].

An operator $O\in L(H)$ is said to be positive if for any $h\in H$,

$\langle Oh,h\rangle\geq 0.$

The class of non-negative, symmetric operators $O\in L(H)$ will be of
particular importance. To every non-negative operator, $O\in L(H)$, there
exists a second non-negative operator, $O^{1/2},\in L(H)$, known as the square
root of $O$, such that $O=O^{1/2}(O^{1/2})^{\ast}$. See [LR17, Thm. 3.1.28].
If $O\in L(H)$ is symmetric and non-negative, then
$O^{1/2}=(O^{1/2})^{\ast}\in L(H)$.

####  1.2.1 Trace Class and Hilbert–Schmidt Operators

We refer to [DPZ14, App. C] for more details on much of the material in this
subsection.

######  Definition 1.2.3 (Trace Class Operators).

Let $E,\,F$ be Banach spaces and $T\in L(E;F)$. We say that $T$ is a trace
class (or nuclear) operator, if there exist two sequences, $(a_{k})_{k\geq
1}\subset E^{\ast},\,(b_{k})_{k\geq 1}\subset F$, such that

$\sum_{k=1}^{\infty}\|a_{k}\|_{E^{*}}\|b_{k}\|_{F}<\infty,$

and for any $x\in E$,

$Tx=\sum_{k=1}^{\infty}a_{k}(x)b_{k}.$

We write $L_{1}(E;F)$ for the set of trace class operators and equip this
space with the norm,

$\|T\|_{L_{1}}:=\inf\left\\{\sum_{k=1}^{\infty}\|a_{k}\|_{E^{\ast}}\|b_{k}\|_{F}<\infty\,:\,Tx=\sum_{k=1}^{\infty}a_{k}(x)b_{k}\right\\},$
(1.2.2)

under which $L_{1}(E;F)$ becomes a Banach space.

Let $U,\,H$ be Hilbert spaces and $T\in L_{1}(U;H)$, then by making the
identification, $U^{\ast}\cong U$, there exist sequences $(a_{k})_{k\geq
1}\subset U,\,(b_{k})_{k\geq 1}\subset H$ such that, for all $x\in U$,

$Tx=\sum_{k=1}^{\infty}\langle a_{k},x\rangle b_{k}.$

If $T\in L_{1}(H):=L_{1}(H;H)$, and $(e_{k})_{k\geq 1}\subset H$ is a basis of
$H$, then we define the trace of $T$ by the expression,

$\operatorname*{Tr}T:=\sum_{k=1}^{\infty}\langle
Te_{k},e_{k}\rangle\in\mathbb{R}.$ (1.2.3)

It is relatively straightforward to check that the definition of
$\operatorname*{Tr}T$ is independent of the choice of basis $(e_{k})_{k\geq
1}\subset H$, see [DPZ14, Prop. C.1], and that
$|\operatorname*{Tr}T|\leq\|T\|_{L_{1}}$. The following proposition says that
the equality $|\operatorname*{Tr}T|=\|T\|_{L_{1}}$ holds when $T\in L_{1}(H)$
is symmetric and non-negative. In this case, the property of finite trace is
equivalent to being trace class.

######  Proposition 1.2.4 (Symmetric Non-negative Trace Class Operators).

Let $T\in L(H)$, be symmetric and non-negative. Then $T\in L_{1}(H)$ if and
only if, for some (equivalently any) basis $(e_{k})_{k\geq 1}\subset H$,

$|\operatorname*{Tr}T|=\operatorname*{Tr}T=\sum_{k=1}^{\infty}\langle
Te_{k},e_{k}\rangle<\infty.$

Furthermore, in this case, $\operatorname*{Tr}T=\|T\|_{L_{1}}$.

###### Proof.

Let $T^{1/2}$, denote the non-negative square root of $T$, and $(e_{k})_{k\geq
1}$ be a basis of $H$, so that for any $x\in H$,

$T^{1/2}x=\sum_{k=1}^{\infty}\langle T^{1/2}x,e_{k}\rangle e_{k}.$

Therefore, for any $x\in H$ and $N\geq 1$,

$\displaystyle\left\|T^{1/2}x-\sum_{k=1}^{N}\langle T^{1/2}x,e_{k}\rangle
e_{k}\right\|_{H}^{2}=\sum_{k=N+1}^{\infty}|\langle
T^{1/2}x,e_{k}\rangle|^{2}$
$\displaystyle\leq|x|^{2}\sum_{k=N+1}^{\infty}\|T^{1/2}e_{k}\|^{2}_{H}$
$\displaystyle=|x|^{2}\sum_{k=N+1}^{\infty}\langle Te_{k},e_{k}\rangle$
$\displaystyle\leq|x|^{2}\operatorname*{Tr}T.$

It follows that $T^{1/2}$ is the limit in operator norm of a sequence of
finite rank operators and so $T^{1/2}$ is a compact operator and hence
$T=T^{1/2}T^{1/2}$ is also compact. Therefore, by Theorem 1.2.2, there exists
a basis $(f_{k})_{k\geq 1}\subset H$ and a sequence of non-negative, real
numbers, $(\lambda_{k})_{k\geq 1}\subset\mathbb{R}_{+}$, accumulating to $0$,
such that

$Tx=\sum_{k=1}^{\infty}\lambda_{k}\langle x,f_{k}\rangle f_{k}.$ (1.2.4)

Using this formula, we have that $\langle
Te_{k},e_{k}\rangle=\sum_{k=1}^{\infty}\lambda_{k}\langle
e_{k},f_{k}\rangle^{2}$ and so,

$\operatorname*{Tr}T=\sum_{k=1}^{\infty}\langle
Te_{k},e_{k}\rangle=\sum_{k=1}^{\infty}\sum_{l=1}^{\infty}\lambda_{k}|\langle
e_{k},f_{l}\rangle|^{2}=\sum_{k=1}^{\infty}\lambda_{k}<\infty.$

Thus we have shown that $T\in L_{1}(H)$, with sequences $(a_{k})_{k\geq
1}=(f_{k})_{k\geq 1}$ and $(b_{k})_{k\geq 1}=(\lambda_{k}f_{k})_{k\geq 1}$,
and that $\operatorname*{Tr}T\geq\|T\|_{L_{1}}$. Hence
$\operatorname*{Tr}T=\|T\|_{L_{1}}$. ∎

######  Definition 1.2.5 (Hilbert–Schmidt Operator).

Let $U,\,H$ be Hilbert spaces with bases $(e_{k})_{k\geq 1},\,(f_{k})_{k\geq
1}$ respectively. Then $T\in L(U;H)$ is said to be a Hilbert–Schmidt operator,
if,

$\|T\|^{2}_{L_{2}}:=\sum_{k=1}^{\infty}\|Te_{k}\|^{2}_{H}=\sum_{k=1}^{\infty}\sum_{l=1}^{\infty}\langle
Te_{k},f_{l}\rangle_{H}^{2}=\sum_{l=1}^{\infty}\|T^{\ast}f_{l}\|_{U}^{2}\,<\,\infty.$
(1.2.5)

######  Remark 1.2.6.

The space $L_{2}(U;H)$ is a separable Hilbert space when equipped with the
norm $\|T\|_{L_{2}}$ and scalar product,

$\langle T,S\rangle=\sum_{k=1}^{\infty}\langle
Te_{k},Se_{k}\rangle_{H},\quad\text{ for }T,\,S\in L_{2}(U;H),$

and the sequence $(e_{k}\otimes f_{k})_{k\geq 1}\subset U\otimes H$ provides a
complete orthonormal basis for $L_{2}(U;H)$. It follows that $L(U;H)$ is
densely embedded in $L_{2}(U;H)$.

######  Remark 1.2.7.

It follows from (1.2.5) that the definition of $\|T\|_{L_{2}}$ is independent
of the choice of bases for $U,\,H$ and that
$\|T^{\ast}\|_{L_{2}}=\|T\|_{L_{2}}$.

We see that $T\in L(H)$ is trace class if and only if
$|T|^{1/2}:=(TT^{\ast})^{1/2}\in L_{2}(H)$. Furthermore, since both trace
class and Hilbert–Schmidt operators are seen to be compact, from Theorem
1.2.2, for any $T\in L_{2}(U;H)$ and $TT^{\ast}\in L_{1}(H)$, there exist
bases $(e_{k})_{k\geq 1}\subset U$ and $(f_{k})_{k\geq 1}\subset H$ and a
sequence of positive real numbers, $(\lambda_{k})_{k\geq 1}$ such that, for
all $x\in U$, $h\in H$,

$\displaystyle Tx$ $\displaystyle=\sum_{k\geq 1}\sqrt{\lambda_{k}}\langle
x,e_{k}\rangle_{U}f_{k},$ (1.2.6) $\displaystyle(TT^{\ast})h$
$\displaystyle=\sum_{k\geq 1}\lambda_{k}\langle h,f_{k}\rangle_{H}f_{k}.$

####  1.2.2 Pseudo Inverses of Linear Operators

In the following sections we turn to the study of Gaussian random variables
and Wiener processes taking values in infinite dimensional spaces. Instead of
being described by a covariance matrix these random variables are described by
covariance operators. In developing the theory of stochastic integration with
respect to generalised Wiener processes, Subsection 1.5.5, we will be
particularly concerned with the pre-images of these operators. It will
therefore be useful to recall some facts regarding the images and pre-imagines
of linear operators and their pseudo-inverses. Throughout we let $U,\,H$ be
two Hilbert spaces and recall that $L(U;H)$ is the set of bounded linear
operators from $U$ to $H$.

######  Lemma 1.2.8.

Let $O\in L(U;H)$ and $R\geq 0$. Then,

  * •

the set $O\left(\overline{B_{U}(0,R)}\right):=\\{Ox\in H\,:\,x\in
U,\,\|x\|_{U}\,\leq\,R\\}$ is convex and closed,

  * •

the set $O^{-1}\left(\overline{B_{H}(0,R)}\right):=\\{x\in
U\,:\,\|Ox\|_{H}\,\leq\,R\\}$ is convex and closed.

###### Proof.

Convexity of both sets follows from linearity of $O$. We only show that
$O\left(\overline{B_{U}(0,R)}\right)$ is closed, since it follows directly by
continuity of $O$ that $O^{-1}\left(\overline{B_{H}(0,R)}\right)$, as the pre-
image of a closed set, is closed. Recall that a convex subset of a Hilbert
space is closed if and only if it is weakly compact. By the Banach–Alaoglou
theorem, any bounded set in $U$ is weakly compact, and since $O$ is strongly,
and therefore also weakly, continuous, the image of a weakly compact set under
$O$ is also weakly compact in $H$. It therefore follows from convexity that
$O\left(\overline{B_{U}(0,R)}\right)$ is also closed. ∎

For $h\in H\setminus\\{0\\}$, let us define the set,

$O^{-1}(h):=\\{x\in U\,:\,Ox=h\\}.$

It follows from Lemma 1.2.8 that $O^{-1}(h)$ is convex and closed. By
continuity of $O$ it also does not contain $0\in U$. Therefore, there is a
unique element of $O^{-1}(h)$ that minimizes the distance to $0\in U$. This
leads to the following definition.

######  Definition 1.2.9 (Pseudo-inverse).

Let $O\in L(U;H)$ and the map $H\setminus\\{0\\}\ni h\mapsto O^{-1}h$ be
defined by setting,

$O^{-1}h:=\left\\{y\in O^{-1}(h)\,:\,\|y\|_{U}\leq\|x\|_{U},\,\forall\,x\in
O^{-1}(h)\right\\}.$

Then $O^{-1}\in L(H\setminus\\{0\\};U)$ is the pseudo-inverse of $O$.

Recall that for a linear operator $O\in L(U;H)$, we define
$\text{Ker}(O):=\\{x\in U\,:\,Ox=0\in H\\}$ and $\text{Ker}(O)^{\perp}=\\{y\in
U\,:\,\langle x,y\rangle_{U}=0,\forall\,x\in\text{Ker}(O)\\}=O^{\ast}(H)$.

######  Proposition 1.2.10.

Let $O\in L(U;H)$ and $O^{-1}$ be its pseudo-inverse.

  1. i)

The space $O(U)$ is a Hilbert space when equipped with the inner product,

$\langle h,g\rangle_{O(U)}:=\langle O^{-1}h,O^{-1}h\rangle_{U},\quad\text{ for
all }h,\,g\in O(U).$

  2. ii)

Let $(e_{k})_{k\geq 1}$ be a basis of $\text{Ker}(O)^{\perp}$. Then,
$(Oe_{k})_{k\geq 1}$ is a basis of
$(O(U),\langle\,\cdot\,,\,\cdot\,\rangle_{O(U)})$.

###### Proof.

Both conclusions are a consequence of the fact that
$O:\text{Ker}(O)^{\perp}\rightarrow O(U)$ is an isometry when
$\text{Ker}(O)^{\perp}$ is equipped with the inner product of $U$ and $O(U)$
is equipped with the inner product $\langle\,\cdot\,,\,\cdot\,\rangle_{O(U)}$
defined above. ∎

######  Proposition 1.2.11.

Let
$(U_{1},\langle\,\cdot\,,\,\cdot\,\rangle_{1}),\,(U_{2},\langle\,\cdot\,,\,\cdot\,\rangle_{2})$
be two Hilbert spaces and $O_{1}\in L(U_{1},H)$, $O_{2}\in L(U_{2},H)$. Then
the following both hold,

  1. i)

If there exists a $c\geq 0$ such that for all $h\in H$,
$\|O^{\ast}_{1}h\|_{1}\leq c\|O^{\ast}_{2}h\|_{2}$ then

$\\{O_{1}x\,:\,x\in U_{1},\,\|x\|_{1}\leq 1\\}\subseteq\\{O_{2}y\,:\,y\in
U_{2},\,\|x\|_{2}\leq 1\\}.$

In particular, $\text{Im}O_{1}\subseteq\text{Im}O_{2}$.

  2. ii)

If for all $h\in H$, $\|O_{1}^{\ast}h\|_{1}=\|O_{2}^{\ast}h\|_{2}$, then
$\text{Im}O_{1}=\text{Im}(O_{2})$ and for all $h\in\text{Im}O_{2}$,
$\|O^{-1}_{1}h\|_{1}=\|O^{-1}_{2}h\|_{2}$.

###### Proof.

See the proof of [PR07, Prop. C.0.5] ∎

######  Corollary 1.2.12.

Let $O\in L(U;H)$ and set $Q=OO^{\ast}\in L(H)$. Then it holds that,

$\text{Im}Q^{1/2}=\text{Im}O,\quad\text{and}\quad\|Q^{-1/2}h\|_{H}=\|O^{-1}h\|_{U},\quad\text{for
all}\quad h\in\text{Im}O,$

where $Q^{-1/2}$ is the pseudo-inverse of $Q^{1/2}$.

###### Proof.

First note that $Q^{1/2}$ is well defined since $OO^{\ast}$ is always a non-
negative operator. Furthermore, since $Q^{1/2}$ is symmetric, for all $h\in
H$, we have that

$\|(Q^{1/2})^{\ast}h\|^{2}_{H}=\|Q^{1/2}h\|^{2}_{H}=\langle
Qh,h\rangle_{H}=\langle OO^{\ast}h,h\rangle_{H}=\|O^{\ast}h\|^{2}_{U}.$

Therefore, the conclusion follows by Item ii) of Proposition 1.2.11. ∎

###  1.3 Lebesgue and Bochner Integration

We very briefly recall the construction of the Lebesgue and Bochner integrals.
We will build the stochastic integral in a similar manner as we build the
Lebesgue integral below. The expectation of infinite dimensional random
variables gives a recurring example of a Bochner integral in our analysis.

Let $(E,\mathcal{B}(E),\mu)$ be a topological, measure space, equipped with
its Borel sigma algebra, $\mathcal{B}(E)$. We define the set of simple
functions, $\mathcal{E}$, to be all functions, $g:E\rightarrow\mathbb{R}$ of
the form

$g=\sum_{k=1}^{N}c_{k}\mathds{1}_{A_{k}},\quad
c_{k}\in\mathbb{R},\,\,A_{k}\in\mathcal{B}(E),\,N\in\mathbb{N}.$

Define the integral of $g\in\mathcal{E}$ by the expression

$\int_{E}g(x)\mathop{}\\!\mathrm{d}\mu(x)=\sum_{k=1}^{N}c_{k}\mu(A_{k})\,\in\,\mathbb{R},$

and then note that for any $p\in[1,\infty]$, the space $\mathcal{E}$ equipped
with the norm,

$\|g\|_{L^{p}}:=\begin{cases}\displaystyle\sum_{k=1}^{N}|c_{k}|^{p}|A_{k},&p\in[1,\infty)\\\
\displaystyle\sup_{k=1,\ldots,N}|c_{k}|,&p=\infty,\end{cases}$

defines a complete, normed, vector space. We set $L^{p}(\mu)$ to be the
completion of $\mathcal{E}$ under the norm $\|\,\cdot\,\|_{L^{p}}$ and the
Lebesgue integral of a measurable function $f:E\rightarrow\mathbb{R}$ as the
equivalence class of limit points of all simple approximating sequences
$g_{n}\rightarrow f\in L^{p}(\mu)$. Extending to vector valued functions can
be done component by component.

A similar notion of integral can be defined for maps taking values in infinite
dimensional spaces, this is the Bochner integral. Let $(V,\|\,\cdot\,\|_{V})$
be a Banach space and define by $\mathcal{E}_{V}$ the set of $V$ valued simple
maps $g:E\rightarrow V$ of the form

$g=\sum_{k=1}^{N}c_{k}\mathds{1}_{A_{k}},\quad c_{k}\in
V,A_{k}\in\mathcal{B}(E),\,N\in\mathbb{N},$

and we now set

$\int_{E}g(x)\mathop{}\\!\mathrm{d}\mu(x)=\sum_{k=1}^{N}c_{k}\mu(A_{k})\,\in
V.$

This is mutatis mutandi the same construction as for the Lebesgue integral.
However, in order to extend to general measurable maps $f:E\rightarrow V$ we
now say that $f$ is Bochner integrable if there exists a sequence of simple
function $(g_{n})_{n\geq 1}\subset\mathcal{E}_{V}$ such that

$\lim_{n\rightarrow\infty}\int_{E}\|f(x)-g_{n}(x)\|_{V}\mathop{}\\!\mathrm{d}\mu(x)=0,$

where each integral is understood as the Lebesgue integral of the map $E\ni
x\mapsto\|f(x)-g_{n}(x)\|_{V}\,\in\mathbb{R}$.

A version of the dominated convergence theorem holds for the Bochner integral.

######  Theorem 1.3.1 (DCT for Bochner Integrals).

Let $(E,\mathcal{B}(E),\mu)$ be a topological measure space and $V$ be a
Banach space, $f:E\rightarrow V$ be Bochner integrable and $(f_{n})_{n\geq 1}$
a sequence of Bochner integral maps, such that for $\mu$-a.e $x\in E$,
$f(x)=\lim_{n\rightarrow\infty}f_{n}(x)$. Then, if there exists a real,
integrable function $g\in L^{1}(\mu;\mathbb{R})$ such that for $\mu$-a.e $x\in
E$, $\|f_{n}(x)\|_{V}\leq g(x)$, it holds that

$\lim_{n\rightarrow\infty}\int_{E}\|f-f_{n}\|_{V}\,\mathop{}\\!\mathrm{d}\mu=0,$

and so

$\int_{E}f_{n}\mathop{}\\!\mathrm{d}\mu\rightarrow\int_{E}f\mathop{}\\!\mathrm{d}\mu.$

###### Proof.

The first assertion is a consequence of the DCT for Lebesgue integrals, since
by assumption $\|f_{n}(x)\|_{V}\rightarrow\|f(x)\|_{V}\in\mathbb{R}$ for
$\mu$-a.e $x\in E$. The second assertion follows from the first and the
discussion above regarding construction of the Bochner integral. ∎

###  1.4 Probability and Random Variables

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space, consisting of an
abstract measurable space $(\Omega,\mathcal{F})$, equipped with $\mathbb{P}$ a
probability. We also assume we are given, $(E,\mathcal{B}(E))$ a measurable,
separable, Banach space, with $\mathcal{B}(E)$ the Borel sigma algebra induced
by the norm $\|\,\cdot\,\|_{E}$. We will assume both these to be in place
without further comment. We say that a measurable map $X:\Omega\rightarrow E$
is an $E$ valued random variable. We write
$\mathcal{L}(X):=X_{\\#}\mathbb{P}\in\mathcal{P}(E)$ that is, for any
$A\in\mathcal{B}(E)$,

$\mathcal{L}(X)(A)=\mathbb{P}[\omega\in\Omega\,:\,X(\omega)\in A].$

We say that a property holds $\mathbb{P}$-a.s. if the $\mathbb{P}$ measure of
the set where the property does not hold is zero. The following proposition
says that the Borel $\sigma$-algebra, $\mathcal{B}(E)$ is generated by the
cylinder sets of $E^{\ast}$. We recall that given a family of sets $A\in
2^{E}$, the notation $\sigma(A)$ denotes the smallest $\sigma$-algebra
containing $A$.

######  Proposition 1.4.1.

Let $(E,\mathcal{B}(E))$ be a separable, Banach space equipped with its Borel
$\sigma$-algebra. Firstly, there exists a sequence $(\varphi_{n})_{n\geq
1}\subset E^{\ast}$ such that,

$\|x\|_{E}=\sup_{n\geq 1}|\varphi_{n}(x)|,\quad\text{for any}\quad x\in E.$
(1.4.1)

As a result,

$\mathcal{B}(E)=\sigma(\\{x\in E\,:\,\varphi(x)\leq
R,\,R\in\mathbb{R},\,\varphi\in E^{\ast}\\}),$ (1.4.2)

and so $X$ is an $E$ valued random variable, if and only if $\varphi(X)$ is a
real random variable for all $\varphi\in E^{\ast}$.

###### Proof.

We show (1.4.1) first. Let $(x_{n})_{n\geq 1}\subset E$ be a dense set. By the
Hahn–Banach theorem, [Bre11, Thm. 1.1], for all $n\geq 1$ there exists a
$\varphi_{n}\in E^{\ast}$ such that $\varphi_{n}(x_{n})=\|x_{n}\|_{E}$ and
$\|\varphi_{n}\|_{E^{\ast}}=1$. We show that the sequence
$(\varphi_{n})_{n\geq 1}$ has the property (1.4.1). Fix $x\in E$ and since
$\|\varphi_{n}\|_{E^{\ast}}\leq 1$ it holds that
$|\varphi_{n}(x)|\leq\|x\|_{E}$ for each $n\geq 1$ and therefore $\sup_{n\geq
1}|\varphi_{n}(x)|\leq\|x\|_{E}$. On the other hand, for any $\varepsilon>0$,
there exists an $x_{n}$, such that $\|x-x_{n}\|_{E}<\varepsilon$ and so
$|\varphi_{n}(x)-\|x_{n}\|_{E}|=|\varphi_{n}(x-x_{n})|\leq\varepsilon$.
Therefore, $|\varphi_{n}(x)|>\|x\|_{E}-2\varepsilon$, and since $\varepsilon$
was arbitrary we conclude that (1.4.1) holds.

We use (1.4.1) to prove (1.4.2). It follows from (1.4.1), that for $a\in E$,
$R>0$,

$\overline{B}_{E}(a,R)=\\{x\in E\,:\,\|x-a\|_{E}\leq R\\}=\bigcap_{n\geq
1}\\{x\in E\,:\,|\varphi_{n}(x)-\varphi_{n}(a)|\leq R\\}.$

Therefore, the smallest $\sigma$-algebra containing all sets of the form
$\\{x\in E\,:\,\varphi(x)\leq R,\,R\in\mathbb{R},\,\varphi\in E^{\ast}\\}$
contains the closed balls of $E$ and thus contains $\mathcal{B}(E)$. On the
other hand, every $\varphi\in E^{\ast}$ is a continuous map from
$(E,\mathcal{B}(E))\rightarrow(\mathbb{R},\mathcal{B}(\mathbb{R}))$ and so is
measurable. Therefore, the reverse inclusion holds from which (1.4.2) follows.
Since a map $X:\Omega\rightarrow E$ is an $E$ valued random variable if and
only if it is measurable with respect to $\mathcal{F}$ and $\mathcal{B}(E)$,
the final assertion follows since (1.4.2) is equivalent to the statement
$\mathcal{B}(E)=\sigma(\\{\varphi\in E^{\ast}\\})$. ∎

The following corollary of Proposition 1.4.1 allows us to turn some questions
regarding $E$ valued random variables into questions regarding real valued
random variables through the dual mapping $\omega\mapsto\varphi(X(\omega))$
for $\varphi\in E^{\ast}$.

######  Corollary 1.4.2.

Let $(E,\mathcal{B}(E))$ be a separable Banach space equipped with its Borel
$\sigma$-algebra. Then the following all hold,

  1. i)

The set of $E$ valued random variables is a vector space. That is, if $X,\,Y$
are $E$ valued random variables, then for any $\alpha,\,\beta\in\mathbb{R}$,
$\alpha X+\beta Y$ is an $E$ valued random variable.

  2. ii)

If $(X_{n})_{n\geq 1}$ is a sequence of $E$ valued random variables such that
$X_{n}(\omega)\rightharpoonup X(\omega)\in E$, $\mathbb{P}$-a.s. then
$X:\Omega\rightarrow E$ is an $E$ valued random variable.

  3. iii)

If $X$ is an $E$ valued random variable, then
$\Omega\ni\omega\mapsto\|X(\omega)\|_{E}$ is a real random variable.

  4. iv)

If $X,\,Y$ are two $E$ valued random variables. Then $X=Y$, $\mathbb{P}$-a.s.
if and only if $\varphi(X)=\varphi(Y)$ $\mathbb{P}$-a.s. for all $\varphi\in
E^{\ast}$.

###### Proof.

Items i) and ii) follow from Proposition 1.4.1 and the fact that both
properties hold for real random variables. Item iii) follows from the fact
that by definition $\|\,\cdot\,\|_{E}\in E^{\ast}$. In order to show item iv),
by linearity of the space $E$ it suffices to show that if
$\varphi(X)=0\in\mathbb{R}$, $\mathbb{P}$-a.s. for all $\varphi\in E^{\ast}$,
then $X=0\in E$, $\mathbb{P}$-a.s.. Let, $(\varphi_{n})_{n\geq 1}$ be the
sequence from (1.4.1), then if by assumption $\varphi_{n}(X)=0$,
$\mathbb{P}$-a.s. it holds that $\sup_{n\geq 1}|\varphi_{n}(X)|=\|X\|=0$,
$\mathbb{P}$-a.s., which implies the conclusion. ∎

We say that an $E$ valued random variable, $X$, is simple if there exists an
$N\geq 1$ and sequences $(x_{k})_{k=1}^{N}\subset E$,
$(A_{k})_{k=1}^{N}\subset\mathcal{F}$, such that,

$X(\omega)=\sum_{k=1}^{N}x_{k}\mathds{1}_{A_{k}}(\omega),\quad\text{for}\quad\mathbb{P}\text{-a.a.
}\omega\in\Omega.$ (1.4.3)

It follows from, Corollary 1.4.2, that for any sequence of simple random
variables $(X_{n})_{n\geq 1}$ converging, weakly or strongly, to $X\in E$,
defines an $E$ valued random variable. Conversely, it also holds that to any
$E$ valued random variable, there exists a sequence of simple random variables
converging strongly to it. We give this result in the more general setting of
separable metric spaces.

######  Proposition 1.4.3.

Let $(E,\rho_{E})$ be a separable metric space and $X$ be an $E$ valued random
variable. Then there exists a sequence of simple random variables
$(X_{n})_{n\geq 1}$, of the form, (1.4.3), such that
$\lim_{n\rightarrow\infty}\rho_{E}(X(\omega),X_{n}(\omega))\rightarrow 0$,
$\mathbb{P}$-a.s..

###### Proof.

Let $(x_{k})_{k\geq 1}\subset E$ be a countable dense set. Then for every
$n\geq 1$ and $\mathbb{P}$-a.a. $\omega\in\Omega$, we define

$\displaystyle\rho_{n}(\omega)$
$\displaystyle=\min\\{\rho(X(\omega),x_{k}),\,k=1,\ldots,n\\},$ $\displaystyle
k_{n}(\omega)$ $\displaystyle=\min\\{k\leq
n\,:\,\rho_{n}(\omega)=\rho(X(\omega),x_{k})\\},$ $\displaystyle
X_{n}(\omega)$ $\displaystyle=x_{k_{n}(\omega)}.$

The random variables $X_{n}$ are simple, since each $X_{n}$ takes values only
in the set $\\{x_{1},\ldots,x_{n}\\}$. Furthermore, since the sequence
$(x_{k})_{k\geq 1}$ is dense, one has that
$\lim_{n\rightarrow\infty}\rho_{n}(\omega)=0$, $\mathbb{P}$-a.s. Since
$\rho_{n}(\omega)=\rho_{E}(X(\omega),X_{n}(\omega))$, the conclusion follows.
∎

For a simple random variable we define the Bochner integrals,

$\mathbb{E}[X]:=\int_{\Omega}X(\omega)\mathop{}\\!\mathrm{d}\mathbb{P}(\omega)=\sum_{k=1}^{N}\mathbb{E}[x_{k}]\mathbb{P}[A_{k}],$

and

$\mathbb{E}[\|X\|_{E}]=\sum_{k=1}^{N}\mathbb{E}[\|x_{k}\|_{E}]\mathbb{P}[A_{k}].$

By completeness of $(E,\,\|\,\cdot\,\|_{E})$, an $E$ valued random variable is
Bochner integrable if and only if

$\int_{\Omega}\|X(\omega)\|_{E}\mathop{}\\!\mathrm{d}\mathbb{P}(\omega)<\infty.$

Therefore, using Proposition 1.4.3 we may extend the Bochner integral,
$\mathbb{E}[X]$, to all $E$ valued random variables, as the limit of the
Bochner integral along a sequence of approximating simple random variables.
For $p\in[1,\infty]$ we define the spaces $L^{p}(\Omega;E)$ as the completions
of the simple random variables under the norms,

$\|X\|_{p}:=\begin{cases}\mathbb{E}\left[\|X\|_{E}^{p}\right]^{\frac{1}{p}},&\text{
if }p\in[1,\infty),\\\
\displaystyle\operatorname*{ess\,sup}_{\omega\in\Omega}\|X(\omega)\|_{E},&\text{
if }p=\infty.\end{cases}$

Note that $\mathbb{E}[X]\in E$ while $\mathbb{E}[\|X\|_{E}^{p}]\in\mathbb{R}$.
For concision and where it will not cause confusion we sometimes write the
Bochner integral as

$\int_{\Omega}X\mathop{}\\!\mathrm{d}\mathbb{P}:=\int_{\Omega}X(\omega)\mathop{}\\!\mathrm{d}\mathbb{P}(\omega).$

The Bochner integral of a random variable obeys analogues of the dominated
convergence and monotone convergence theorems for the Lebesgue integral. Let
$(F,\|\,\cdot\,\|_{E})$ be a second separable Banach space and consider a
linear operator $A:D(A)\subseteq E\rightarrow F$. Recall that we say $A$ is
closed if the set,

$\text{graph}(A):=\\{(x,y)\in E\times F\,:\,x\in D(A),\,y=Ax\\},$

is closed in the product space $E\times F$. We equip $D(A)$ with the graph
norm, $\|x\|_{D(A)}:=\|x\|_{E}+\|Ax\|_{F}$. We have the following result
regarding $D(A)\subseteq E$ valued random variables.

######  Theorem 1.4.4.

Let $E,\,F$ be separable Banach spaces and $A:D(A)\subseteq E\rightarrow F$ be
a closed operator such that $D(A)\in\mathcal{B}(E)$. If $X:\Omega\rightarrow
E$ is $\mathbb{P}$-a.s. $D(A)$ valued, then $AX$ is an $F$ valued random
variable and $X$ is a $D(A)$ valued random variable (seen as a Banach space
equipped with the graph norm). If $X$ is Bochner integrable and

$\int_{\Omega}\|AX(\omega)\|_{E}\mathop{}\\!\mathrm{d}\mathbb{P}(\omega)<\infty,$

then

$A\int_{\Omega}X(\omega)\mathop{}\\!\mathrm{d}\mathbb{P}(\omega)=\int_{\Omega}AX(\omega)\mathop{}\\!\mathrm{d}\mathbb{P}(\omega).$
(1.4.4)

###### Proof.

The first statements essentially follow from the definitions, using the fact
that $(D(A),\|\,\cdot\,\|_{D(A)})$ is itself a Banach space. To prove (1.4.4)
consider a sequence of simple random variables $(X_{n})_{n\geq 1}\subset D(A)$
approximating $X$ in $D(A)$, that is

$\|X-X_{n}\|_{D(A)}:=\|X-X_{n}\|_{E}+\|AX-AX_{n}\|_{F}\rightarrow 0.$

It follows directly that we have

$\int_{\Omega}\|X(\omega)-X_{n}(\omega)\|_{D(A)}\mathop{}\\!\mathrm{d}\mathbb{P}(\omega)\rightarrow
0.$

Thus, by definition of the Bochner integral, we have that both

$\int_{\Omega}X_{n}(\omega)\mathop{}\\!\mathrm{d}\mathbb{P}(\omega)\rightarrow\int_{\Omega}X(\omega)\mathop{}\\!\mathrm{d}\mathbb{P}(\omega),\qquad\int_{\Omega}AX_{n}(\omega)\mathop{}\\!\mathrm{d}\mathbb{P}(\omega)\rightarrow\int_{\Omega}AX(\omega)\mathop{}\\!\mathrm{d}\mathbb{P}(\omega).$

However, it also follows from the definition of the Bochner integral of simple
random variables that

$\int_{\Omega}AX_{n}(\omega)\mathop{}\\!\mathrm{d}\mathbb{P}(\omega)=A\int_{\Omega}X_{n}(\omega)\mathop{}\\!\mathrm{d}\mathbb{P}(\omega),$

therefore (1.4.4) follows from the closedness of $A$. ∎

######  Remark 1.4.5.

It follows from 1.4.4 that $\varphi\in L(E,F)$ and if $X$ is $E$ valued random
variable that is Bochner integrable then,

$\varphi\left(\int_{\Omega}X(\omega)\mathop{}\\!\mathrm{d}\mathbb{P}(\omega)\right)=\int_{\Omega}\varphi(X(\omega))\mathop{}\\!\mathrm{d}\mathbb{P}(\omega).$

That is $\varphi(\mathbb{E}[X])=\mathbb{E}[\varphi(X)]$. In fact, the
expectation is characterised as the unique element $x\in E$ such that,

$\varphi(x)=\mathbb{E}[\varphi(X)],\quad\text{ for all }\varphi\in E^{\ast}.$

In the sequel we employ the notion of a conditional expectation, in particular
when studying martingales taking values in Hilbert spaces. Existence and
uniqueness of the conditional expectation for $E$ valued random variables
follows in a similar manner as in the finite dimensional case.

######  Theorem 1.4.6.

Let $X$ be a Bochner integrable, $E$ valued random variable and let
$\mathcal{G}\subseteq\mathcal{F}$ be a sub sigma-algebra of $\mathcal{F}$.
Then there exists a unique, up to a $\mathbb{P}$ measure zero set, integrable,
$E$ valued random variable, $Z$, that is $\mathcal{G}$ measurable and such
that

$\int_{A}X\mathop{}\\!\mathrm{d}\mathbb{P}=\int_{A}Z\mathop{}\\!\mathrm{d}\mathbb{P},\quad\text{
for all }A\in\mathcal{G}.$ (1.4.5)

We write $\mathbb{E}[X|\mathcal{G}]$ for the random variable $Z$ and refer to
it as the conditional expectation of $X$ with respect to $\mathcal{G}$.

######  Remark 1.4.7.

Often, we wish to talk about the conditional expectation of a random variable,
$X$, with respect to another random variable $Y$. In this case we use the
abuse of notation $\mathbb{E}[X|Y]$ by which we really mean
$\mathbb{E}[X|\sigma(Y)]$, where $\sigma(Y)$ is the $\sigma$-algebra generated
by $Y$.

###### Proof of Theorem 1.4.6.

We begin by showing uniqueness. Assume that there exists two random variables
$Z,\,\tilde{Z}$ such that both equal $\mathbb{E}[X|\mathcal{G}]$. Therefore,
using Remark 1.4.5 it holds that for any $\varphi\in E^{\ast}$,

$\int_{A}\varphi(Z)\mathop{}\\!\mathrm{d}\mathbb{P}=\varphi\left(\int_{A}Z\mathop{}\\!\mathrm{d}\mathbb{P}\right)=\int_{A}\varphi(X)\mathop{}\\!\mathrm{d}\mathbb{P},\quad\text{
for all }A\in\mathcal{G}.$

Similarly,
$\int_{A}\varphi(\tilde{Z})\mathop{}\\!\mathrm{d}\mathbb{P}=\int_{A}\varphi(X)\mathop{}\\!\mathrm{d}\mathbb{P}$.
Therefore, for all $\varphi\in E^{\ast}$, $\varphi(Z),\,\varphi(\tilde{Z})$
both equal the real, conditional expectation
$\mathbb{E}[\varphi(X)|\mathcal{G}]$. Since uniqueness holds for the finite
dimensional conditional expectation we have $\varphi(Z)=\varphi(\tilde{Z})$
for all $\varphi\in E^{\ast}$ and so by Item iv) of Corollary 1.4.2 we have
$Z=\tilde{Z}$ $\mathbb{P}$-a.s.

To prove existence, let $X=\sum_{k=1}^{N}x_{k}\mathds{1}_{A_{k}}$ be a simple
random variable, then we define,

$Z=\sum_{k=1}^{N}x_{k}\mathbb{P}[A_{k}|\mathcal{G}],$

where $\mathbb{P}[A_{k}|\mathcal{G}]$ is the usual conditional probability of
$A_{k}$ with respect to $\mathcal{G}$. It is clear that $Z$ satisfies (1.4.5)
and by the law of total expectation we have

$\mathbb{E}[\|Z\|_{E}]\leq\sum_{k=1}^{N}\mathbb{E}[\|x_{k}\|_{E}]\mathbb{P}[A_{k}]=\mathbb{E}[\|X\|_{E}].$
(1.4.6)

Now let $(X_{n})_{n\geq 1}$ be a sequence of simple random variables,
approximating $X$ and $Z_{n}=\mathbb{E}[X_{n}|\mathcal{G}]$. So then for
$n,\,m\in\mathbb{N}$, by (1.4.6), we have

$\mathbb{E}[\|Z_{n}-Z_{m}\|_{E}]\leq\mathbb{E}[\|X_{n}-X_{m}\|_{E}]\rightarrow
0,\quad\text{ as }n,\,m\rightarrow\infty.$

Therefore $Z_{n}$ is a Cauchy sequence in
$L^{1}(\Omega,\mathcal{G},\mathbb{P};E)$ and thus there exists a
$\mathcal{G}$-measurable random variable $Z:=\lim_{n\rightarrow\infty}Z_{n}$
which by continuity of the Bochner integral also satisfies (1.4.5). ∎

####  1.4.1 Gaussian Measures on Hilbert Spaces

Recall that a real random variable $X:\Omega\rightarrow\mathbb{R}$ is called
Gaussian if there exist $m\in\mathbb{R}$, $q\in[0,\infty)$ such that for any
$A\in\mathcal{B}(\mathbb{R})$,

$\mathcal{L}(X)(A):=\mathbb{P}[X\in
A]=\frac{1}{\sqrt{4\pi\sigma}}\int_{A}e^{-\frac{|x-m|^{2}}{2q}}\mathop{}\\!\mathrm{d}x.$

In the multivariate case we say that an $\mathbb{R}^{d}$ valued random
variable is Gaussian if there exist $m\in\mathbb{R}^{d}$ and
$Q\in\mathbb{R}^{d\times d}$, a non-negative, symmetric matrix such that for
any $A\in\mathcal{B}(\mathbb{R}^{d})$,

$\mathcal{L}(X)(A):=\mathbb{P}[X\in
A]=\frac{1}{(4\pi)^{d/2}\sqrt{\det{Q}}}\int_{A}e^{-\frac{1}{2}\langle
Q^{-1}(x-m),x-m\rangle}\mathop{}\\!\mathrm{d}x$

In both cases, if the variance is zero we interpret the measure,
$\mathcal{L}(X)$, as a dirac mass centred at $m$. Both definitions can also be
concisely given in terms of characteristic functions (Fourier transforms) of
measures. A measure $\mu\in\mathcal{P}(\mathbb{R}^{d})$ is a Gaussian measure
if and only if its characteristic function
$\mathbb{R}^{d}\ni\lambda\mapsto\hat{\mu}(\lambda):=\mathbb{E}[e^{i\langle\lambda,X\rangle}]$
is given by the expression

$\hat{\mu}(\lambda)=e^{i\langle m,\lambda\rangle-\frac{1}{2}\langle
Q\lambda,\lambda\rangle}\quad\text{ for all }\lambda\in\mathbb{R}^{d},$
(1.4.7)

where $m\in\mathbb{R}^{d}$ and $Q\in\mathbb{R}^{d\times d}$ is a symmetric,
non-negative matrix. We say that an $\mathbb{R}^{d}$ valued random variable is
Gaussian if its law, $\mathcal{L}(X)$ is a Gaussian measure in this sense.

These definitions motivate the notion of Hilbert space valued random variables
in possibly infinite dimensions. We fix a separable, Hilbert space $U$ and
identify $U^{\ast}$ with $U$ from now on. As a result we change notation from
the previous section, instead of writing $g(h)$ for $g\in U^{\ast},\,f\in U$,
we write $\langle f,g\rangle$, where the inner product is understood as the
inner product on $U$.

######  Definition 1.4.8 ($U$-Valued Gaussian Random Variable).

Let $X$ be a $U$-valued random variable. Then we say that $X$ is Gaussian if
and only if $\langle X,g\rangle$ is a real Gaussian random variable, for every
$g\in U$. We say that a measure $\mu\in\mathcal{P}(U)$ is Gaussian if
$\mu(A)=\mathbb{P}[X\in A]$ for a Gaussian random variable $X$.

It follows from the Riesz representation theorem, that given a Gaussian
measure $\mu\in\mathcal{P}(U)$, there exists $m\in U$ and $Q:U\rightarrow U$,
such that for all $h,\,g\in U$,

$\displaystyle\langle m,h\rangle$ $\displaystyle=\int_{U}\langle
x,h\rangle\mathop{}\\!\mathrm{d}\mu(x)=\mathbb{E}[\langle X,h\rangle],$
(1.4.8) $\displaystyle\langle Qh,g\rangle$ $\displaystyle=\int_{U}\langle
x,h\rangle\langle x,g\rangle\mathop{}\\!\mathrm{d}\mu(x)=\mathbb{E}[\langle
X,h\rangle\langle X,g\rangle].$

From now on we only deal with centred Gaussian measures, that is measures for
which $m=0$. By linearity one may always translate a centred measure to a non-
centred measure. It is immediate that $Q$ is linear, symmetric and non-
negative. We will shortly demonstrate that in fact $Q$ defines a trace class
operator from $U$ to itself.

At this point we remark that much of the theory presented below for Gaussian
measures on Hilbert spaces extends quite naturally, to the more general
setting of separable, reflexive Banach spaces, satisfying either a
$2$-martingale or uniform martingale difference (UMD) property, see [vNVW07]
for an overview on these topics. A benefit of the theory of stochastic
integration in UMD Banach spaces is that it leads to maximal regularity of the
constructed integrals. On the other hand, the theory presented here can be
implemented to develop a theory for Banach space valued integrals at the price
of a small loss in regularity, see [Hai09, Sec. 5]. For an approach to SPDEs
valued in $L^{\infty}$ based spaces, such as $\mathcal{C}^{\alpha}$,
$W^{k,\infty}$ we refer to the pathwise approach briefly presented in Chapter
4 here.

For a measure, $\mu$, on a Hilbert space $U$, we define the characteristic
function $U\ni h\mapsto\hat{\mu}(h)\in\mathbb{C}$, by the expression,

$\hat{\mu}(h):=\int_{U}e^{i\langle g,h\rangle}\mathop{}\\!\mathrm{d}\mu(g).$

It follows from Definition 1.4.8 and (1.4.8) that for $\mu$ a centred,
Gaussian measure on $U$, one has

$\hat{\mu}(h)=e^{-\frac{1}{2}\langle Qh,h\rangle},\quad\text{ for all }h\in
U.$ (1.4.9)

To see this, use the fact that by definition, there exists a $U$ valued random
variable, $X$, such that $\langle X,h\rangle$ is a one dimensional Gaussian
and the expression (1.4.7) for the Fourier transform of finite dimensional
Gaussian measures. In fact, Theorem 1.4.12 below, shows that any measure with
characteristic function of the form (1.4.9) is a Gaussian measure.

We say that two measures, $\mu,\,\nu\in\mathcal{P}(U)$ are equal, if
$\mu(A)=\nu(A)$ for all $A\in\mathcal{B}(U)$. A useful extension of Item iv)
of Corollary 1.4.2 shows that the characteristic function uniquely determines
the measure. We note that the following sequence of results hold directly on
separable Banach spaces, not only Hilbert spaces provided one appropriately
defines $\hat{\mu}$ in terms of elements of $E^{\ast}$.

######  Proposition 1.4.9.

Let $\mu,\,\nu\in\mathcal{P}(U)$ be two measures on a Hilbert space $U$. Then
$\mu=\nu$ if and only if $\hat{\mu}(g)=\hat{\nu}(g)$ for all $g\in U$.

###### Proof.

By linearity, it suffices to show that if $\hat{\mu}\equiv 0$ then $\mu=0$.
For $\tilde{\mu}\in\mathcal{P}(\mathbb{R})$ and $\varphi\in
C^{\infty}_{c}(\mathbb{R})$ one has

$\int_{\mathbb{R}}\varphi(x)\mathop{}\\!\mathrm{d}\tilde{\mu}(x)=\frac{1}{2\pi}\int_{\mathbb{R}}\int_{\mathbb{R}}\hat{\varphi}(\xi)e^{-i\xi
x}\mathop{}\\!\mathrm{d}\xi\,\tilde{\mu}(\mathop{}\\!\mathrm{d}x)=\frac{1}{2\pi}\int_{\mathbb{R}}\hat{\varphi}(\xi)\hat{\tilde{\mu}}(-\xi)\mathop{}\\!\mathrm{d}\xi.$

Therefore, approximating any continuous and bounded function by smooth
functions we see that the Fourier transform uniquely determines one
dimensional probability measures. The infinite dimensional case then follows
from Item iv) of Corollary 1.4.2. ∎

So far we have not in fact shown that measures satisfying Definition 1.4.8 in
fact exist. In order to do so we establish some a priori properties of
Gaussian measures. Recall that given an angle $\theta\in[0,2\pi)$ a vector in
$U^{2}$ is rotated under the map
$R_{\theta}(h,g):=(h\cos\theta-g\sin\theta,h\sin\theta+g\cos\theta)$. By
extension we define the rotation of $\mu\in\mathcal{P}(U)$ by $\theta$, as the
image of $\mu\otimes\mu$ under the map,

$\displaystyle\mathfrak{R}_{\theta}:\mathcal{P}(U)\otimes\mathcal{P}(U)$
$\displaystyle\rightarrow\mathcal{P}(U)\otimes\mathcal{P}(U)$
$\displaystyle\mu\otimes\nu$ $\displaystyle\mapsto
R_{\theta}\\#(\mu\otimes\nu)$

######  Lemma 1.4.10.

Let $\mu$ be a Gaussian measure on $U$. Then for any $\theta\in[0,2\pi)$, one
has $\mathfrak{R}_{\theta}(\mu\times\mu)=\mu\otimes\mu$.

###### Proof.

It follows from Proposition 1.4.9 that is enough to check that the identity,
$\left(\widehat{\mu\otimes\mu}\right)\circ
R^{-1}_{\theta}=\widehat{\mu\otimes\mu}:U^{2}\rightarrow\mathbb{C}^{2}$, holds
for the characteristic functions. This can be checked using the explicit
expression (1.4.9). ∎

A consequence of this rotation invariance is a bound on exponential moments of
Gaussian measures known as Fernique’s theorem. Note that the theorem holds for
all measures that are invariant by rotations of $\pi/4$ degrees, not just
Gaussian measures.

######  Theorem 1.4.11 (Fernique’s Theorem).

Let $\mu\in\mathcal{P}(U)$ be a probability measure such that
$\mathfrak{R}_{\pi/4}(\mu\otimes\mu)=\mu\otimes\mu$. Then there exists an
$\alpha>0$ such that

$\int_{U}e^{\alpha\|h\|^{2}_{U}}\mathop{}\\!\mathrm{d}\mu(h)<\infty.$

###### Proof.

Let $\tau,\,\sigma>0$. Then by $\pi/4$-rotation invariance we have that

$\displaystyle\mu(\|h\|_{U}\leq\tau)\mu(\|g\|_{U}>\sigma)$
$\displaystyle=\mu\left(\left\|\frac{h-g}{\sqrt{2}}\right\|_{U}\leq\tau\right)\mu\left(\frac{\|h+g\|_{U}}{\sqrt{2}}>\sigma\right)$
$\displaystyle=\int_{\left\|\frac{h-g}{\sqrt{2}}\right\|_{U}\leq\tau}\int_{\left\|\frac{h+g}{\sqrt{2}}\right\|_{U}>\sigma}\mathop{}\\!\mathrm{d}\mu(h)\mathop{}\\!\mathrm{d}\mu(g)$

By the triangle inequality we see that

$\min\\{\|h\|_{U},\,\|g\|_{U}\\}\geq\frac{1}{2}\left(\|h+g\|_{U}-\|h-g\|_{U}\right),$

and so it follows that together, the conditions, $\|h-g\|_{U}\leq\sqrt{2}\tau$
and $\|h+g\|_{U}>\sqrt{2}\sigma$, that
$\min\\{\|h\|_{U},\,\|g\|_{U}\\}\geq\frac{\sigma-\tau}{\sqrt{2}}$. So using
the above equality, we have that

$\mu(\|h\|_{U}\leq\tau)\mu(\|g\|_{U}>\sigma)\leq\int_{\|h\|_{U}>\frac{\sigma-\tau}{\sqrt{2}}}\int_{\|g\|_{U}>\frac{\sigma-\tau}{\sqrt{2}}}\mathop{}\\!\mathrm{d}\mu(h)\mathop{}\\!\mathrm{d}\mu(g)=\mu\left(\|h\|_{U}>\frac{\tau-\sigma}{\sqrt{2}}\right)^{2}.$

Since $\|h\|_{U}<\infty$ $\mu$-a.s., there exists some $\sigma>0$ such that
$\mu(\|h\|_{U}\leq\sigma)\geq\frac{3}{4}$. So now we set $t_{0}=\sigma$ to be
the minimal value such that this holds and for $n>0$ we define,
$t_{n}=\frac{t_{n+1}-\sigma}{\sqrt{2}}$. So it follows that

$\mu(\|h\|_{U}>t_{n+1})\leq\frac{\mu\left(\|h\|_{U}>\frac{t_{n+1}-\sigma}{\sqrt{2}}\right)}{\mu(\|h\|_{U}\leq\sigma)}\leq\frac{4}{3}\mu(\|h\|_{U}>t_{n})^{2}.$

Therefore, defining the shorthand, $k_{n}:=\frac{4}{3}\mu(\|h\|_{U}>t_{n})$,
we have shown the recursion $k_{n+1}\leq k_{n}^{2}$ with
$k_{0}\leq\frac{1}{3}$. Repeatedly applying this inequality gives the bound
$\mu(\|h\|_{U}>t_{n})\leq 3^{-2n}$. On the other hand, one can check
explicitly that $t_{n}=\frac{\sqrt{2}^{n+1}-1}{\sqrt{2}-1}\sigma\leq
2^{n/2}(2+\sqrt{2})\sigma$ so that in particular $t_{n+1}\leq 2^{n/5}5\sigma$.
Therefore in combination we have

$\mu(\|h\|_{U}>t_{n})\leq 3^{-\frac{t^{2}_{n+1}}{25\sigma^{2}}}.$

So it follows that there exists a universal constant $\tilde{\alpha}>0$ such
that $\mu(\|h\|_{U}>t)\leq e^{-\frac{2\tilde{\alpha}t^{2}}{\sigma^{2}}}$,
holds for every $t>\sigma$. So integrating by parts we have that

$\displaystyle\int_{U}\exp\left(\frac{\tilde{\alpha}\|h\|^{2}_{U}}{\sigma^{2}}\right)\mathop{}\\!\mathrm{d}\mu(h)$
$\displaystyle\leq
e^{\tilde{\alpha}}+\frac{2\tilde{\alpha}}{\sigma^{2}}\int_{\sigma}^{\infty}te^{\frac{\tilde{\alpha}t^{2}}{\sigma^{2}}}\mu(\|h\|_{U}>t)\mathop{}\\!\mathrm{d}t$
$\displaystyle\leq
e^{\tilde{\alpha}}+2\tilde{\alpha}\int_{1}^{\infty}te^{-\tilde{\alpha}t^{2}}\mathop{}\\!\mathrm{d}t$
$\displaystyle<\infty,$

which is the stated result with $\alpha=\tilde{\alpha}/\sigma$. ∎

It follows from Fernique’s theorem that Gaussian measures have finite moments
of all orders. In fact one can show an even stronger result, namely that the
second moment of a Gaussian measure controls all higher moments. We postpone
this proof to Chapter 4 since we do not make use of it here. We employ Theorem
1.4.11 here to give the following characterisation of Gaussian measures on a
Hilbert space.

######  Theorem 1.4.12.

Let $U$ be a Hilbert space. A measure $\mu\in\mathcal{P}(U)$ is Gaussian if
and only if there exist $m\in U$ and $Q\in L_{1}(U)$ such that

$\hat{\mu}(h)=e^{i\langle m,h\rangle-\frac{1}{2}\langle
Qh,h\rangle},\quad\text{ for all }h\in U.$ (1.4.10)

Furthermore one has the identity,

$\int_{U}\|h\|^{2}_{U}\mathop{}\\!\mathrm{d}\mu(h)=\operatorname*{Tr}Q.$
(1.4.11)

Finally, if $\mu=\mathcal{L}(X)$ then we say that $X\sim\mathcal{N}(m,Q)$.

###### Proof.

As above, without loss of generality we may prove the result only for centred
Gaussian measures, so we set $m=0$. Furthermore, we have already seen in
(1.4.9) that given a Gaussian measure there exists a linear, non-negative,
symmetric operator $Q:U\rightarrow U$ such that (1.4.10) holds. Therefore, in
one direction it suffices to show that this operator $Q$ is in fact trace
class and in turn that the expression (1.4.11) is finite.

It follows from Theorem 1.4.11 that given a Gaussian measure, a priori one has
the bound

$\int_{U}\|h\|^{2}_{U}\mathop{}\\!\mathrm{d}\mu(h)\leq\frac{1}{\alpha}\int_{U}e^{\alpha\|h\|^{2}_{U}}\mathop{}\\!\mathrm{d}\mu(h)<\infty.$

Now let $(e_{n})_{n\geq 1}\subset U$ be a basis, so that applying Lebesgue’s
dominated convergence in the first step, we have,

$\int_{U}\|h\|^{2}_{U}\mathop{}\\!\mathrm{d}\mu(h)=\sum_{k=1}^{\infty}\int_{U}\langle
h,e_{k}\rangle^{2}\mathop{}\\!\mathrm{d}\mu(h)=\sum_{k=1}^{\infty}\langle
Qe_{k},e_{k}\rangle=\operatorname*{Tr}Q.$

That is, if a measure, $\mu\in\mathcal{P}(U)$, satisfies (1.4.8), then $Q\in
L_{1}(U)$.

To prove the converse, let $Q\in L_{1}(U)$ so that by (1.2.4) there exists a
basis $(e_{n})_{n\geq 1}\subset U$ such that $Qe_{n}=\lambda_{n}e_{n}$ for a
summable sequence $(\lambda_{n})_{n\geq 1}\subset\mathbb{R}_{\geq 0}$. Then,
let $(\beta_{n})_{n\geq 1}$ be a family of i.i.d, real valued standard normal
random variables, which exist by Kolmogorov’s extensions theorem. By
definition, $\sum_{n\geq
1}\lambda_{n}\mathbb{E}[\beta_{n}^{2}]=\operatorname*{Tr}Q$ and so the series
$\sum_{n\geq 1}\sqrt{\lambda_{n}}\beta_{n}e_{n}$ is finite in mean square and
so there exists a subsequence of partial sums converging $\mathbb{P}$-a.s. in
$U$. It follows that $X:=\sum_{n\geq
1}^{\infty}\sqrt{\lambda_{n}}\beta_{n}e_{n}$ defines a $U$ valued Gaussian
random variable in the sense of Definition 1.4.8.

In both cases the form of the characteristic function, (1.4.10), can be
computed explicitly. ∎

In the case of Banach space valued Gaussian random variables, one can instead
show that the covariance operators is a compact operator. See [Hai09, Ex.
3.16].

####  1.4.2 Generalised Gaussian Random Variables

One may wonder if we can allow for more general covariance operators. An
example of particular interest is the white-noise, otherwise known as a
cylindrical Gaussian random variable. Let $U$ be as above, $(e_{k})_{k\geq 1}$
be any orthonormal basis and $(\beta_{k})_{k\geq 1}$ be i.i.d, real, standard
normal, random variables. Then consider the random variable defined by the
formal sum,

$W:=\sum_{k=1}\beta_{k}e_{k}.$ (1.4.12)

This corresponds to formally setting $Q=\text{Id}_{U}$, the identity map on
$U$. Since the identity map is not trace-class, $W$, cannot be a $U$ valued
Gaussian random variable. However, if we consider $W$ as a random variable
taking values in a suitable space larger than $U$, it turns out to be
perfectly well defined Gaussian random. For example, consider the
Hilbert–Schmidt extension, $U_{\text{ex}}$, of $U$, defined to be the
completion of $U$ under the norm,

$\|x\|^{2}_{U_{\text{ex}}}:=\sum_{k=1}^{\infty}\frac{1}{k^{2}}\langle
x,e_{k}\rangle^{2}.$

The inclusion map $\iota:U\rightarrow U_{\text{ex}}$ is Hilbert–Schmidt and
since $U\subset U_{\text{ex}}$, for any $h\in U$, the action $\langle
W,h\rangle_{U_{\text{ex}}}$ is well defined. Interpreting all inner products
in the only admissible sense, we then have, for $h,\,g\in E$,

$\mathbb{E}[\langle W,h\rangle]=0,\quad\mathbb{E}[\langle W,h\rangle\langle
W,g\rangle]=\langle h,g\rangle.$

Thus we may treat $W$ as defining a Gaussian random variable on $U$, which
takes values almost surely in $U_{\text{ex}}$. This leads us to the following
definition.

######  Definition 1.4.13 (Generalised Gaussian Random Variable).

Given a Hilbert space, $U$, we say that a linear map $h\mapsto X_{h}$ defines
a generalised Gaussian random variable, if $X_{h}$ is a real Gaussian random
variable for every $h\in U$ and if $h_{n}\rightarrow h\in U$,

$\mathbb{E}\left[|X_{h}-X_{h_{n}}|^{2}\right]\rightarrow 0.$

Consider the example of the white noise, $W$, above. The map $h\mapsto\langle
W,h\rangle=:W_{h}$ is linear and defines a family of real, Gaussian random
variables. Furthermore, for $(h_{n})_{n\geq 1}\subset U$ converging to $h\in
U$ we have,

$\mathbb{E}\left[|W_{h}-W_{h_{n}}|^{2}\right]=\|h_{n}-h\|^{2}_{U}\rightarrow
0.$

It follows from the definition of a generalised Gaussian random variable, that
there exists a bi-linear form, $K:E\times E\rightarrow\mathbb{R}$ and a
symmetric, non-negative operator $Q:E\rightarrow E$ such that,

$\mathbb{E}\left[X_{h}X_{g}\right]=K(h,g)=\langle Qh,g\rangle,\quad\text{ for
all }h,\,g\in U.$

As before, we call $Q$ the covariance operator of $X$. In the case of the
white noise, $W$, the covariance operator is the identity on $U$. In fact, all
generalised Gaussian random variables can be written formally as

$X=\sum_{k=1}\beta_{k}Q^{1/2}e_{k},$ (1.4.13)

where $(\beta_{k})_{k\geq 1}$ are a family of i.i.d, real, standard normals,
$(e_{k})_{k\geq 1}$ is an orthonormal basis of $U$ and $Q\in L(U)$ is a
symmetric and non-negative operator. Note that $Q$ need not be trace class,
however, since it is symmetric and non-negative it admits an invertible,
square root. From this we define the reproducing kernel space of $X$.

######  Definition 1.4.14 (Reproducing Kernel Space).

Let $X$ be a generalised Gaussian random variable on a Hilbert space $U$, with
covariance $Q$. Then, we define the reproducing kernel of $X$ to be the space
$U_{0}:=Q^{1/2}(U)$. Note that $U_{0}$ is itself a Hilbert space, with inner
product $\langle h,g\rangle_{U_{0}}=\langle Q^{-1/2}h,Q^{-1/2}g\rangle_{U}$.

This definition allows us to give proper meaning to the generalised Gaussian
random variables as defined in (1.4.13).

######  Proposition 1.4.15.

Let $U,\,U_{1}$ be Hilbert spaces and $Q\in L(U)$ be such that
$Q^{1/2}(U)=:U_{0}\subseteq U_{1}$ with Hilbert–Schmidt embedding $\iota
Q^{1/2}:U\rightarrow U_{1}$. Then the sum on the right hand side of (1.4.13)
defines a generalised Gaussian random variable on $U$ in the sense of
Definition 1.4.13. Furthermore, the definition of $X$ as a generalised
Gaussian random variable on $U$ is independent of the choice of $U_{1}$ and
$\iota$. Finally, defining $Q_{1}:=\iota Q\iota^{\ast}$, one has that $\iota
Q^{1/2}:U\rightarrow Q^{1/2}_{1}(U_{1})$ defines an isomorphism

###### Proof.

Almost sure convergence of the sum $\eqref{eq:GenGaussianSum}$ in $U_{1}$
follows from the bound,

$\displaystyle\sum_{k\geq 1}\mathbb{E}[\beta^{2}_{k}]\langle
Q^{1/2}e_{k},Q^{1/2}e_{k}\rangle_{U_{1}}$ $\displaystyle=\sum_{k\geq
1}\langle\iota Q^{1/2}e_{k},\iota Q^{1/2}e_{k}\rangle_{U_{1}}$
$\displaystyle=\sum_{k\geq 1}\langle\iota Q\iota^{\ast}e_{k},e_{k}\rangle$
$\displaystyle=\operatorname*{Tr}\iota Q\iota^{\ast}<\infty.$

Where the last bracket is understood as the duality bracket between $U$ and
$U_{1}$. Recall that since $\iota Q^{1/2}$ is Hilbert–Schmidt, $\iota
Q\iota^{\ast}$ is trace class by definition. It therefore follows that
$X=\sum_{k\geq 1}\beta_{k}Q^{1/2}e_{k}$ defines a $U_{1}$ valued Gaussian
random variable, on $U$ in the sense of Definition 1.4.8. Furthermore, it is
readily checked that for $X$ defined in this way, the map,

$U\ni h\mapsto X_{h}:=\sum_{k=1}\beta_{k}\langle
Q^{1/2}e_{k},e_{k}\rangle\langle h,e_{k}\rangle,$

defines a generalised Gaussian random variable, with zero mean, in the sense
of Definition 1.4.13. Furthermore, since the choice of $\iota,\,U_{1}$ does
not play a role in the action $h\mapsto X_{h}$ they do not affect the
definition, provided the embedding $\iota Q^{1/2}:U\rightarrow U_{1}$ is
Hilbert–Schmidt.

To see that $\iota Q^{1/2}$ defines an isomorphism between $U$ and
$Q_{1}^{1/2}(U_{1})$, we first note that since $Q_{1}=(\iota Q)(\iota
Q^{1/2})^{\ast}$, by Corollary 1.2.12, we have
$Q^{1/2}_{1}(U_{1})=\iota(U_{0})$. Furthermore, from the same result we have
that $\|Q^{-1/2}_{1}h_{1}\|_{U_{1}}=\|\iota^{-1}u_{1}\|_{U_{0}}$ for all
$h_{1}\in\iota(U_{0})$. However, since the mapping $\iota:U_{0}\rightarrow
U_{1}$ is surjective, we may replace $h_{1}$ by $\iota h_{0}$ for any
$h_{0}\in U_{0}$ and the final claim is shown. ∎

In the case of the white noise, $W$, defined above, the Hilbert–Schmidt
extension of $U$ provides an example of $U_{1}$. However, due to the final
assertion of Proposition 1.4.15, the definition of the white noise is
independent of the choice of $U_{1}$, provided the embedding
$\iota:U_{0}\rightarrow U_{1}$ is Hilbert–Schmidt.

######  Example 1.4.16.

Let $U=L^{2}([0,T])$ and $W$ be a white noise defined on $U$. Then, define the
real process, $t\mapsto W_{t}:=\langle W,\mathds{1}_{[0,t]}\rangle$. For any
$t,\,s\in[0,T]$ we have,

$\mathbb{E}[W_{t}]=0,\quad\mathbb{E}[W_{t}W_{s}]=t\wedge s.$

The mapping $[0,T]\ni t\mapsto W_{t}\in\mathbb{R}$ is by definition
$\mathbb{P}$-a.s. continuous. Finally, if we consider the filtration
$(\mathcal{F}_{t})_{t\in[0,T]}$ generated by $(W_{t})_{t\in[0,T]}$, then by
linearity, for any $t>s$ we have

$\mathbb{E}[W_{t}|\mathcal{F}_{s}]=\mathbb{E}[W_{s}+W_{t-s}|\mathcal{F}_{s}]=W_{s}.$

So by Levy’s characterisation, $t\mapsto W_{t}$ defines a real Brownian
motion.

###  1.5 Stochastic Analysis in Infinite Dimensions

The main purpose of this chapter is to build up a theory of stochastic
analysis in infinite dimensional Hilbert spaces. In Chapter 3 we will use this
theory to understand a certain class of SPDE as infinite dimensional
stochastic evolution equations. We first develop some theory of abstract
martingales in Banach spaces, after which we turn to the specific case of
Gaussian processes in Hilbert spaces and conclude by developing a theory of
stochastic integration for these processes.

######  Definition 1.5.1 (Stochastic Processes).

Given a Banach space $E$, equipped with its Borel $\sigma$-algebra and
$(\Omega,\mathcal{F},\mathbb{P})$ an abstract probability space, we say that a
measurable map $X:[0,\infty)\times\Omega\rightarrow E$ is an $E$ valued
stochastic process.

We typically suppress the dependence of $X$ on $\omega\in\Omega$ and use the
notations $(X_{t})_{t\in[0,\infty)}$ and $[0,\infty)\ni t\mapsto X_{t}$ to
denote these processes.

######  Definition 1.5.2 (Modifications of Stochastic Processes).

Let $X,\,Y:[0,\infty)\times\Omega\rightarrow E$ be two stochastic processes.
Then we say that $Y$ is a modification of $X$ if for all $t\in[0,\infty)$, one
has

$\mathbb{P}[X_{t}=Y_{t}]=1.$

In what follows we often deal with stochastic processes restricted to an
interval $[0,T]$ for $T\in(0,\infty)$. In this case these definition only
change up to modifying the interval of definition.

####  1.5.1 Banach Space Valued Martingales

As in finite dimensional stochastic analysis, the class of Martingale
processes will play a central role. We introduce the concept of a filtered
probability space. We say that a family $(\mathcal{F}_{t})_{t\geq 0}$, of
$\sigma$-algebras is non-decreasing if
$\mathcal{F}_{s}\subseteq\mathcal{F}_{t}$ for all $0\leq s<t<\infty$. We also
introduce the notations,

$\mathcal{F}_{t+}=\bigcap_{s>t}\mathcal{F}_{s},\quad\mathcal{F}_{t-}=\bigcap_{s<t}\mathcal{F}_{s}.$

######  Definition 1.5.3 (Filtered Probability space).

We say that a tuple, $(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\geq
0},\mathbb{P})$, consisting of the the usual triple; state space $\Omega$,
$\sigma$-algebra, $\mathcal{F}$, and probability measure, $\mathbb{P}$, and in
addition a non-decreasing family of sub-$\sigma$-algebras,
$(\mathcal{F}_{t})_{t\geq 0}$ is a filtered probability space. We say that a
filtration is normal if $\mathcal{F}_{0}$ contains all $\mathbb{P}$-null sets
of $\mathcal{F}$ and if

$\mathcal{F}_{t}=\mathcal{F}_{t+}\quad\text{ for all }t\geq 0.$

Given a filtered probability space, we define the notion of an
$(\mathcal{F}_{t})_{t\geq 0}$-martingale.

######  Definition 1.5.4 (Martingales).

Let $(M_{t})_{t\geq 0}$ be a stochastic process, taking values in $E$ and
$(\mathcal{F}_{t})_{t\geq 0}$ be a filtration on
$(\Omega,\mathcal{F},\mathbb{P})$. Then we say that $(M_{t})_{t\geq 0}$ is an
$(\mathcal{F}_{t})_{t\geq 0}$-martingale if,

  1. 1.

$\mathbb{E}[\|M_{t}\|_{E}]<\infty$ for all $t\geq 0$,

  2. 2.

$M_{t}$ is $\mathcal{F}_{t}$ measurable for all $t\geq 0$,

  3. 3.

$\mathbb{E}[M_{t}\,|\,\mathcal{F}_{s}]=M_{s}$ $\mathbb{P}$-a.s. for all $0\leq
s\leq t<\infty$.

We say that a stochastic process is a sub-martingale if the final property is
replaced with the inequality $\mathbb{E}[M_{t}|\mathcal{F}_{s}]\geq M_{s}$ and
a super-martingale if it is replaced by the inequality
$\mathbb{E}[M_{t}|\mathcal{F}_{s}]\leq M_{s}$.

Given a filtered probability space
$(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\geq 0},\mathbb{P})$, we recall the
notion of a stopping time.

######  Definition 1.5.5 (Stopping Times).

We say that a random variable
$\Omega\ni\omega\mapsto\tau(\omega)\in\mathbb{R}_{+}$ is an
$(\mathcal{F}_{t})_{t\geq 0}$ stopping time, if for any $t\geq 0$,

$\\{\omega\in\Omega\,:\,\tau(\omega)\leq t\\}\in\mathcal{F}_{t}.$

######  Remark 1.5.6.

If $(\mathcal{F}_{t})_{t\geq 0}$ is normal, in the sense of Definition 1.5.3,
then any deterministic time $T\geq 0$ is an $(\mathcal{F}_{t})_{t\geq 0}$
stopping time.

######  Remark 1.5.7.

We will mostly be concerned with so called hitting times, that is given an $E$
valued process, $(X_{t})_{t\geq 0}$, for $R\geq 0$ and $\varphi\in E^{\ast}$,
we define

$\tau_{R}:=\inf\\{t\geq 0\,:|\varphi(X_{t})|>R\\}.$

If $t\mapsto X_{t}$ is almost surely continuous and $(\mathcal{F}_{t})_{t\geq
0}$ is normal then any random time of this form is an
$(\mathcal{F}_{t})_{t\geq 0}$ stopping time.

The following example shows that in some cases the requirements of being a
true martingale in the sense of Definition 1.5.4 are too stringent.

######  Example 1.5.8.

Let $(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\geq 0},\mathbb{P})$ be a
filtered probability space carrying two random variables, $\xi$ which is
$\mathcal{F}_{1}$ measurable, takes values in $\mathbb{R}_{+}$ and such that
$\mathbb{E}[\xi]=\infty$ and $\eta$ which is $\mathcal{F}_{2}$ measurable,
independent of $\mathcal{F}_{2-}$ and takes values $\pm 1$ each with
probability $1/2$. We define the process $[0,\infty)\ni t\mapsto
X_{t}:=\mathds{1}_{t\geq 2}\eta\xi$. It follows immediately that
$\mathbb{E}[|X_{2}|]=\mathbb{E}[\xi]=\infty$ so that $X$ cannot be a
martingale. However, we can define the sequence of stopping times, for $R\geq
0$,

$\tau_{R}:=\begin{cases}R&\text{if }\xi\leq R,\\\ 1&\text{if
}\xi>R.\end{cases}$

Since $\xi(\omega)<\infty$ $\mathbb{P}$-a.s., $\tau_{R}\nearrow\infty$
$\mathbb{P}$-a.s. and for every $n\geq 0$,
$X^{\tau_{R}}_{\,\cdot\,}:=X_{\tau_{R}\wedge\,\cdot\,}$ is a bounded
martingale.

This leads us to the definition local martingales, which will be recurring
objects in our analysis. In particular the stochastic integrals we define in
Subsections 1.5.4 & 1.5.5 will only be local martingales for the integrands we
typically consider.

######  Definition 1.5.9 (Local Martingale).

We say that an $E$ valued stochastic process $(M_{t})_{t\geq 0}$ is a local
martingale, if there exists a sequence of $(\mathcal{F}_{t})_{t\geq 0}$
stopping times $(\tau_{n})_{n\geq 0}$ such that $\tau_{n}\rightarrow\infty$,
$\mathbb{P}$-a.s. and for any $n\geq 0$, the stopped process
$(M^{\tau_{n}}_{t})_{t\geq 0}:=(M_{\tau_{n}\wedge t})_{t\geq 0}$ is a
martingale in the sense of Definition 1.5.4. Local sub/super-martingales are
defined analogously. The sequence $(\tau_{n})_{n\geq 0}$ is called a
localising sequence for the local martingale $M$.

As in the previous section we make a connection to real martingales.

######  Proposition 1.5.10.

Let $(M_{t})_{t\geq 0}$ be an $E$ valued stochastic process such that
$\mathbb{E}[\|M_{t}\|_{E}]<\infty$ for all $t\geq 0$ and
$(\mathcal{F}_{t})_{t\geq 0}$ be a filtration as in Definition 1.5.4. Then
$(M_{t})_{t\geq 0}$ is an $E$ valued martingale if and only if $(\langle
M_{t},h\rangle)_{t\geq 0}$ is a real, $(\mathcal{F}_{t})_{t\geq 0}$-martingale
for all $h\in E$. Furthermore, if $(M_{t})_{t\geq 0}$ is an
$(\mathcal{F}_{t})_{t\geq 0}$-sub-martingale, then $(\|M_{t}\|^{p}_{E})_{t\geq
0}$ is a real $(\mathcal{F}_{t})_{t\geq 0}$-sub-martingale for any
$p\in[1,\infty)$.

###### Proof.

To prove the equivalence, it is clear that if $(M_{t})_{t\geq 0}$ is an $E$
valued $(\mathcal{F}_{t})_{t\geq 0}$-martingale then $(\langle
M_{t},h\rangle)_{t\geq 0}$ is a real $(\mathcal{F}_{t})_{t\geq 0}$-martingale
for all $h\in E$. On the other hand, if $\langle M_{t},h\rangle$ is
$\mathcal{F}_{t}$ measurable for all $t\geq 0$ and $h\in E$ then the element
$M_{t}\in E$ must be $\mathcal{F}_{t}$ measurable for every $t\geq 0$. To show
the martingale property it suffices to recall the construction of the
conditional expectation in Hilbert spaces from Theorem 1.4.6.

To show that the norm is a real martingale, recall that by Hahn–Banach there
exists a sequence of elements $(h_{k})_{k\geq 1}$ such that
$\|M_{t}\|_{E}=\sup_{k\geq 1}\langle M_{t},h_{k}\rangle$. Therefore, from the
first equivalence, $\mathbb{P}$-a.s., we have that

$\displaystyle\mathbb{E}[\|M_{t}\|_{E}|\mathcal{F}_{s}]\geq\sup_{k\geq
1}\mathbb{E}[\langle M_{t},h_{k}\rangle|\mathcal{F}_{s}]=\sup_{k\geq 1}\langle
M_{s},h_{k}\rangle=\|M_{t}\|_{E}.$

This proves the claim for $p=1$, to show the same for $p>1$ it suffices to
apply Jensen in the first inequality. ∎

We have the following version of Doob’s maximal inequality.

######  Theorem 1.5.11.

Let $(M_{t})_{t\in[0,T]}$ be a right continuous, $E$ valued
$(\mathcal{F}_{t})_{t\in[0,T]}$-martingale. Then for any $p>1$,

$\mathbb{E}\left[\sup_{t\in[0,T]}\|M_{t}\|^{p}_{E}\right]^{\frac{1}{p}}\leq\frac{p}{p-1}\mathbb{E}[\|M_{T}\|^{p}_{E}]^{\frac{1}{p}}.$

###### Proof.

The result follows from Proposition 1.5.10 and the same result for real
valued, non-negative sub-martingales. See [CE15, Thm. 4.5.6] ∎

From now on we will focus in particular on the square integrable, continuous,
martingales, i.e $E$ valued, continuous,
$(\mathcal{F}_{t})_{t\in[0,T]}$-martingales such that

$\|M\|_{\mathcal{M}^{2}_{T}}:=\sup_{t\in[0,T]}\mathbb{E}\left[\|M_{t}\|^{2}_{E}\right]=\mathbb{E}\left[\|M_{T}\|^{2}_{E}\right]<\infty.$
(1.5.1)

Identifying $\mathcal{M}^{2}_{T}(E)$ as a space as a subspace of the Banach
space $L^{2}(\Omega;C([0,T];E))$ it follows that $\mathcal{M}^{2}_{T}(E)$ is
itself a Banach space if it can be shown to be closed. However, by
construction of the conditional expectation, it is easy to show that even
convergence in $L^{1}(\Omega;C([0,T];E))$ preserves the martingale property
and so $\mathcal{M}^{2}_{T}(E)$ is a Banach space, with norm given by (1.5.1).
Note that finiteness of (1.5.1) is not enough to ensure that a local
martingale, $M$, is a true martingale, so that we may also speak of
continuous, square integrable, local martingales and we write
$\mathcal{M}^{2}_{T;\text{loc}}(E)$ for this space.

Our ultimate goal in this chapter is to build the stochastic integral with
respect to a given element of $\mathcal{M}^{2}_{T}(E)$, concretely a
$Q$-Wiener process. In order to do so we proceed in analogy with the
construction of the Lebesgue integral. We first identify a suitable space of
simple processes, for which the integral is naturally defined. We then show
that this integral is an isometry from the simple processes, equipped with a
suitable norm, to the space $\mathcal{M}_{T}^{2}$. We then define the
stochastic integral as the limit under approximation by simple processes. From
now on we keep as implicit a filtered probability space
$(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\in[0,T]};\mathbb{P})$ and assume
that all martingales are with respect to $(\mathcal{F}_{t})_{t\in[0,T]}$. We
also assume without further comment that all martingales are zero at $t=0$.

####  1.5.2 Hilbert Space Valued Wiener Processes

In the context of stochastic evolution equations we will wish to consider
noise terms given by evolving processes of random variables. That is a Hilbert
space valued stochastic process. Given a spatial domain,
$\Gamma\subset\mathbb{R}^{d}$, we typically think of $U$ in the previous
section, being a Hilbert space of functions
$f:\Gamma\rightarrow\mathbb{R}^{m}$ and the noise as being a sequence of
random elements in $U$ (or some larger space) indexed continuously by $t\geq
0$. We begin by defining the class of $Q$-Wiener processes.

######  Definition 1.5.12.

Given a Hilbert space $U$ and a trace class operator $Q\in L_{1}(U)$, we say
that $(W_{t})_{t\geq 0}$ is a standard $Q$-Wiener process if,

  1. 1.

$W_{0}=0\in U$,

  2. 2.

the map $\mathbb{R}_{+}\ni t\mapsto W_{t}\in U$ is $\mathbb{P}$-a.s.
continuous,

  3. 3.

for any $0\leq s<t<\infty$, one has

$W_{t}-W_{s}\sim\mathcal{N}(0,(t-s)Q)\quad\text{and}\quad W_{t}-W_{s}\perp
W_{s}.$

The qualification standard here refers to the condition $W_{0}=0$, which
entails by the martingale property (see below) that $\mathbb{E}[\langle
W_{t},h\rangle]=0$ for all $t\geq 0$. Setting $W_{0}=m$ for a deterministic
element $m\in E$ gives a non-centred $Q$-Wiener process. One can also consider
random initial data, provided the independent increments assumption still
holds. This can be useful for example when studying invariant behaviour of
stochastic systems, as one may wish to start the system already from its
invariant distribution. From now on we only treat standard $Q$-Wiener
processes and remove the qualifier from all subsequent statements.

Given Theorem 1.4.12 we can give a characterisation of the $Q$-Wiener
processes.

######  Theorem 1.5.13.

Let $Q\in L_{1}(U)$ and $(e_{k})_{k\geq 1},\,(\lambda_{k})_{k\geq 1}$ be an
orthonormal basis of $U$ generated by $Q$ and associated eigenvalues. Then
$(W_{t})_{t\geq 0}$ is a $Q$-Wiener process if and only if,

$W_{t}=\sum_{k\geq 1}\sqrt{\lambda_{k}}\beta^{k}_{t}e_{k},$ (1.5.2)

where $(\beta^{k})_{k\geq 1}$ are independent, real, Brownian motions.

###### Proof.

This result essentially follows from the characterisation of Gaussian random
variables given by Theorem 1.4.12. It follows directly that the random
variables $(W_{t})_{t\geq 0}$ defined in this way are centred, Gaussian,
$U$-valued random variables. Since $Q$ is trace class, the right hand side of
(1.5.2) is square summable in expectation and so by the dominated convergence
theorem we may exchange summation and the expectation. Therefore, independence
of the increments, $W_{t}-W_{s}$, follows from independence of the increments
$\beta^{k}_{t}-\beta_{s}^{k}$. finally, almost sure continuity of the paths
follows also from almost sure continuity of the maps $t\mapsto\beta^{k}_{t}$,
although in Chapter 4 we show this more directly by a generalisation of the
Kolmogorov continuity criterion. ∎

For the remainder of this chapter, unless otherwise mentioned $Q\in L_{1}(U)$.
Given a filtered probability space we define the notion of a $Q$-Wiener
process with respect to a filtration.

######  Definition 1.5.14.

Let $(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\geq 0},\mathbb{P})$ be a normal,
filtered probability space and $(W_{t})_{t\geq 0}$ be a $Q$-Wiener process. We
say that $(W_{t})_{t\geq 0}$ is a $Q$-Wiener process with respect to
$(\mathcal{F}_{t})_{t\geq 0}$ if,

  1. 1.

$W_{t}$ is $\mathcal{F}_{t}$ measurable for every $t\geq 0$,

  2. 2.

$W_{t}-W_{s}$ is independent of $\mathcal{F}_{s}$ for all $0\leq s<t<\infty$.

Every $Q$-Wiener process naturally induces a normal filtration, defined by
setting, for all $t\geq 0$,

$\tilde{\mathcal{F}}_{t}:=\sigma(W_{t}\,:\,s\leq
t),\quad\tilde{\mathcal{F}}^{0}_{t}:=\sigma(\tilde{\mathcal{F}}_{t}\cup\\{A\in\mathcal{F}\,:\,\mathbb{P}[A]=0\\}),\quad\mathcal{F}_{t}:=\bigcap_{s>t}\tilde{\mathcal{F}}^{0}_{s}.$

From now on we will always consider filtrations generated in this way and so
without further comment we will assume that any $Q$-Wiener process is adapted
to the given filtration. A probabilistically strong solution to our SPDE will
be a stochastic process $(u_{t})_{t\geq 0}$ also adapted to the given
filtration.

As in the finite dimensional case, the $Q$-Wiener processes are martingales.
Let us a fix a $T>0$ and adapt the definitions above accordingly. Then we have
the following lemma.

######  Lemma 1.5.15.

Let $(W)_{t\in[0,T]}$ be a $Q$-Wiener process and
$(\mathcal{F}_{t})_{t\in[0,T]}$ be its natural filtration. Then
$(W_{t})_{t\in[0,T]}\in\mathcal{M}^{2}_{T}(U)$.

###### Proof.

The fact that $(W_{t})_{t\in[0,T]}\in\mathcal{M}^{2}_{T}(U)$ follows from the
definition of $(\mathcal{F}_{t})_{t\in[0,T]}$ and the assumption of
independent increments of $(W_{t})_{t\in[0,T]}$. If we equip $U$ with the
orthonormal basis generated by $Q$, then we have

$\displaystyle\sup_{t\in[0,T]}\mathbb{E}[\|W_{t}\|^{2}_{U}]=\sup_{t\in[0,T]}\mathbb{E}\left[|\langle
W_{t},W_{t}\rangle|\right]$
$\displaystyle=\sup_{t\in[0,T]}\mathbb{E}\left[\sum_{k=1}^{\infty}|\langle
W_{t},e_{k}\rangle|^{2}\right]$ $\displaystyle=T\sum_{k=1}^{\infty}\langle
Qe_{k},e_{k}\rangle$ $\displaystyle=\operatorname*{Tr}Q\,T<\infty.$

∎

####  1.5.3 Generalised Wiener Processes

As in subsection 1.4.2, it is natural to ask if we can extend the definition
of a Wiener process beyond that of Definition 1.5.12. This leads to the
definition of generalised Wiener processes, which includes the white noise (or
cylindrical Wiener) process.

######  Definition 1.5.16 (Generalised Wiener Process).

Given a Hilbert space $U$, we say that a family of linear maps $[0,T]\times
U\ni(t,h)\mapsto W_{h;t}\in\mathbb{R}$ defines a generalised Wiener process if
for each $h\in U$, $t\mapsto W_{h;t}$ defines a real Wiener process and if for
any sequence $(h_{n})_{n\geq 1}\subset U$ converging to $h\in U$ we have,

$\mathbb{E}\left[|W_{h;t}-W_{h_{n};t}|^{2}\right]\rightarrow 0,\quad\text{ for
all }t\geq 0.$

We employ the same ideas as in Subsection 1.4.2 to understand
$(W_{h;t})_{t\geq 0}$ as a Wiener process taking values in a space slightly
larger than $U$. From the definition of a generalised Wiener process it
follows that there exists a bi-linear form $K:U\times U\rightarrow\mathbb{R}$
and a symmetric, non-negative operator $Q\in L(U)$ such that for all
$s,\,t\geq 0$ and $h,g\in U$,

$\mathbb{E}[W_{h;t}W_{g;s}]=t\wedge sK(h,g)=t\wedge s\langle Qh,g\rangle.$

We write $W_{t}$ as a formal sum,

$W_{t}:=\sum_{k\geq 1}\beta^{k}_{t}Qe_{k},$

where $(\beta^{k})_{k\geq 1}$ is a family of i.i.d, standard, real, Brownian
motions and $(e_{k})_{k\geq 1}$ is an orthonormal basis of $U$. As in
Subsection 1.4.2 we define the reproducing kernel of $W$, as the space
$U_{0}:=Q^{1/2}(U)$ and we recall that for any $t\geq 0$, $h\mapsto W_{t;h}$
defines a generalised Gaussian random variable on $U$ and it takes values in
any space $U_{1}\hookleftarrow U_{0}$ such that the embedding
$\iota:U_{0}\rightarrow U_{1}$ is Hilbert–Schmidt. Thus, we may consider
$(W_{h;t})_{t\in[0,T],h\in U}$ as a $U_{1}$ valued $Q$-Wiener process on $U$
and its definition is independent of the choice of $U_{1}$. By definition, the
generalised Wiener process is an element of $\mathcal{M}^{2}_{T}(U_{1})$.

######  Example 1.5.17 (Space-Time White Noise).

A common example of a generalised Wiener process is the space-time white
noise, sometimes called a cylindrical Wiener process. It is the natural
infinite dimensional analogue of the finite dimensional white noise familiar
from classical stochastic analysis. Let $\Gamma\subset\mathbb{R}^{d}$ be a
bounded, domain and $U:=L^{2}(\Gamma)$. Then the space-time white noise on
$[0,T]\times\Gamma$ is the generalised Wiener process $[0,T]\times
L^{2}(\Gamma)\ni(t,h)\mapsto W_{h;t}$ such that, for all $h,\,g\in U$ and
$s,\,t\in[0,T]$,

$\mathbb{E}[W_{t;h}]=0,\quad\mathbb{E}[W_{t;h}W_{s;g}]=t\wedge s\langle
h,g\rangle_{L^{2}(\Gamma)}.$

One can check that for given a family of i.i.d, standard, real Brownian
motions, $(\beta^{k})_{k\geq 1}$ and a orthonormal basis of $L^{2}(\Gamma)$,
$(e_{k})_{k\geq 1}$, the formal sum

$W_{t}:=\sum_{k\geq 1}\beta^{k}_{t}e_{k},$

defines a generalised Gaussian process in the above sense, known as the
cylindrical Gaussian process, and that the sum converges in any Hilbert space
$U_{1}$ such that the embedding $\iota:L^{2}(\Gamma)\rightarrow U_{1}$ is
Hilbert–Schmidt. The formal derivative,
$\mathop{}\\!\mathrm{d}W_{t}=\sum_{k\geq
1}\mathop{}\\!\mathrm{d}\beta^{k}_{t}e_{k}$ is referred to as the space-time
white-noise.

An alternative approach to defining the space-time white noise is to return to
Subsection 1.4.2 and consider the Hilbert space
$U:=L_{0}^{2}([0,T]\times\Gamma)$ of square integrable space-time functions,
zero at zero. Then it is an easy exercise to check that the white noise on
$L_{0}^{2}([0,T]\times\Gamma)$ coincides with the space-time white noise
defined above.

####  1.5.4 Stochastic Integration Against $Q$-Wiener Processes

We conclude this chapter by giving meaning to integrals of the form

$[0,T]\ni t\mapsto\int_{0}^{t}\varphi_{s}\mathop{}\\!\mathrm{d}W_{s},$ (1.5.3)

where $(\varphi_{t})_{t\in[0,T]}$ is an adapted, suitably regular, operator
valued process. We first treat the case when $(W_{t})_{t\in[0,T]}$ is a
$Q$-Wiener process from a Hilbert space to itself and treat this case in the
most detail. We follow this by defining the stochastic integral for
$(W_{t})_{t\in[0,T]}$ a generalised Wiener process, of which the space-time
white noise will be our canonical example.

We fix a Hilbert space, $U$ and assume that $W$ is an $U$ valued $Q$-Wiener
process. We recall the definition of its reproducing kernel, the Hilbert space
$U_{0}:=Q^{1/2}(U)\subseteq U$ equipped with the inner product $\langle
Q^{-1/2}f,Q^{-1/2}g\rangle_{U}$. We also fix a second Hilbert space, $H$; we
will subsequently assume $(\varphi_{t})_{t\in[0,T]}$ takes values in a
subspace of $L(U_{0};H)$, so that the stochastic integral, (1.5.3), takes
values in $H$.

As in the finite dimensional case we first define the stochastic integral for
a suitable class of simple processes, before extending it to a larger space of
integrands.

######  Definition 1.5.18 (Simple Processes in $L(U;H)$).

We say that a process $[0,T]\ni t\mapsto\varphi_{t}\in L(U;H)$ is simple if,
there exist a sequence of deterministic times,
$0=t_{0}<t_{1}<\cdots<t_{n-1}=T$, and a set of $L(U;H)$ valued,
$(\mathcal{F}_{t_{m}})^{n-1}_{m=0}$-measurable, random variables,
$(\varphi_{m})_{m=1}^{n-1}$, taking only finitely many values, such that,

$\varphi_{t}=\varphi_{0}\mathds{1}_{\\{0\\}}(t)+\sum_{m=0}^{n-1}\varphi_{m}\mathds{1}_{(t_{m},t_{m+1}]}(t).$

We write $\mathcal{E}_{T}(U;H)$ for the set of $L(U;H)$ valued simple
processes on $[0,T]$.

We define the space of Hilbert–Schmidt operators $L^{0}_{2}(U_{0};H)$ equipped
with the norm,

$\|\varphi\|^{2}_{L^{0}_{2}}:=\operatorname*{Tr}\left[(Q^{1/2}\varphi)(Q^{1/2}\varphi)^{\ast}\right].$

Note that since $L^{0}_{2}$ is the space of Hilbert–Schmidt operators defined
on the image of a Hilbert–Schmidt operator $Q^{1/2}:U\rightarrow U$, it
follows that that $L(U;H)\subset L^{0}_{2}(U,H)$ and so example the simple
process $\mathcal{E}_{T}(U;H)$ are measurable, $L^{0}_{2}(U_{0},H)$, valued
processes.

This allows us to define the set of processes that are square integrable with
respect to a $Q$-Wiener process. For a measurable process $[0,T]\ni
t\mapsto\varphi_{t}\in L^{0}_{2}(U_{0},H)$, we set,

$\|\varphi\|_{\mathscr{H}_{T}^{2}}:=\mathbb{E}\left[\int_{0}^{T}\|\varphi_{s}\|^{2}_{L^{0}_{2}}\mathop{}\\!\mathrm{d}s\right]^{\frac{1}{2}}=\mathbb{E}\left[\int_{0}^{T}\operatorname*{Tr}\left[(Q^{1/2}\varphi_{s})(Q^{1/2}\varphi_{s})^{\ast}\right]\mathop{}\\!\mathrm{d}s\right]^{\frac{1}{2}}.$

If $\varphi\in\mathcal{E}_{T}$ then, since we specified that the random
variables $(\varphi_{m})_{m=1}^{n-1}$ all take only finitely many values in
$L(U;H)$, it follows directly that $\|\varphi\|_{\mathscr{H}^{2}_{T}}<\infty$
for simple processes.

We are now able to define the stochastic integral for simple process. We will
then show that the map sending $\varphi\in\mathcal{E}_{T}(U;H)$ to its
stochastic integral, (1.5.3), is an isometry between $\mathcal{E}_{T}(U;H)$
and $L^{2}(\Omega;H)$. Given $(W_{t})_{t\in[0,T]}$ we define the linear map,

$\displaystyle\mathcal{E}_{T}(U;H)$
$\displaystyle\longrightarrow\mathcal{M}^{2}_{T}(H)$ (1.5.4)
$\displaystyle\varphi$
$\displaystyle\mapsto\varphi\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}W_{\,\cdot\,}:=\int_{0}^{\,\cdot\,}\varphi_{s}\mathop{}\\!\mathrm{d}W_{s}:=\sum_{k=1}^{n-1}\varphi_{k}(W_{t_{k+1}\wedge\,\cdot\,}-W_{t_{k}\wedge\,\cdot\,}).$

We show in the next proposition that this map does indeed define an isometry
from $(\mathcal{E}_{T},\|\,\cdot\,\|_{\mathscr{H}^{2}_{T}})$ to the space of
square integrable martingales,
$(\mathcal{M}_{T}^{2}(H),\,\|\,\cdot\,\|_{\mathcal{M}^{2}_{T}})$, where the
norm is defined in (1.5.1).

######  Proposition 1.5.19.

Let $\varphi\in\mathcal{E}_{T}(U;H)$. Then,
$(\varphi\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}W_{t})_{t\in[0,T]}$,
defined in (1.5.4), is a continuous, square integrable, $H$-valued martingale,
adapted to the natural filtration of $(W_{t})_{t\in[0,T]}$. Furthermore, one
has that

$\|\varphi\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}W\|_{\mathcal{M}^{2}_{T}}=\sup_{t\in[0,T]}\mathbb{E}[\|\varphi\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}W_{t}\|_{H}^{2}]=\|\varphi\|_{\mathscr{H}^{2}_{T}}.$
(1.5.5)

###### Proof.

The fact that
$t\mapsto\varphi\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}W_{t}$
defines a continuous, $H$-valued martingale is clear from the definitions.
Once can check the martingale property in detail using optional stopping, see
proof of [PR07, Prop. 2.3.2]. We check square integrabillity by proving
(1.5.5) directly. Note that it suffices to show (1.5.5) for $t=t_{k}\in[0,T]$.
Let $\zeta_{k}:=W_{t_{k+1}}-W_{t_{k}}$ so that we have,

$\displaystyle\mathbb{E}[\|(\varphi\cdot W)_{t}\|^{2}_{H}]$
$\displaystyle=\mathbb{E}\left[\left\|\sum_{k=0}^{n-1}\varphi_{k}\zeta_{k}\right\|_{E}^{2}\right]=\mathbb{E}\left[\sum_{k=0}^{n-1}\|\varphi_{k}\zeta_{k}\|_{H}^{2}\right]+2\mathbb{E}\left[\sum_{\begin{subarray}{c}k_{2}=1\\\
k_{1}<k_{2}\end{subarray}}^{n-1}\langle\varphi_{k_{1}}\zeta_{k_{1}},\varphi_{k_{2}}\zeta_{k_{2}}\rangle\right].$

Considering the first term, taking each summand individually and using the
definition of a $Q$-Wiener process, and $(e_{k})_{k\geq 1}$ the basis of
$U_{0}$ associated to the Hilbert–Schmidt operator $\varphi_{k}\in
L_{2}^{0}(U_{0},H)$, and defining $\varphi^{\ast}_{k}\in L_{2}(H;U_{0})$ to be
its adjoint, we have

$\displaystyle\mathbb{E}[\|\varphi_{k}\zeta_{k}\|_{E}^{2}]=\sum_{m=1}^{\infty}\mathbb{E}[|\langle\varphi_{k}\zeta_{k},e_{m}\rangle|^{2}]$
$\displaystyle=\sum_{m=1}^{\infty}\mathbb{E}[|\langle\zeta_{k},\varphi_{k}^{\ast}e_{m}\rangle|^{2}]$
$\displaystyle=(t_{k+1}-t_{k})\sum_{m=1}^{\infty}|\langle
Q\varphi_{k}^{\ast}e_{m},\varphi_{k}^{\ast}e_{m}\rangle|$
$\displaystyle=(t_{k+1}-t_{k})\sum_{m=1}^{\infty}\|Q^{1/2}\varphi_{k}^{\ast}e_{m}\|^{2}_{E}$
$\displaystyle=(t_{k+1}-t_{k})\|\varphi_{k}\|_{L^{0}_{2}}.$

Similarly, by the independence of increments, one has

$\mathbb{E}[\langle\varphi_{k_{1}}\zeta_{k_{1}},\varphi_{k_{2}}\zeta_{k_{2}}\rangle]=0,\quad
k_{1}\neq k_{2}.$

Putting these two results together and summing over $k$ gives equality
(1.5.5). ∎

We now wish to extend the definition of
$\varphi\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}W$
to the set of all $\mathbb{P}$-a.s. $L^{0}_{2}$-predictable processes. Before
doing so however, we introduce the predictable $\sigma$-algebra and the notion
of a predictable process. This is necessary since we will want our stochastic
integration to preserve the martingale property and the notion of a
predictable process ensures it does not look into the future. Note that for
simple processes this was not a problem, since almost by definition their
integrals were martingales.

######  Definition 1.5.20.

[Predictable $\sigma$-algebra] Let $(\mathcal{F}_{t})_{t\in[0,T]}$ be a given
filtration. Then we define, $\mathscr{P}_{T}$, to be the $\sigma$-algebra of
subsets of $[0,T]\times\Omega$ given by,

$\mathscr{P}_{T}:=\sigma\left(\\{(s,t]\times F\,:\,0\leq s<t\leq
T,\,F\in\mathcal{F}_{s}\\}\cup\\{\\{0\\}\times
F\,:\,F\in\mathcal{F}_{0}\\}\right).$

Given a Hilbert space, $\mathcal{X}$, we say that a process
$\varphi:[0,T]\times\Omega\rightarrow\mathcal{X}$ is predictable if it is
$\mathscr{P}_{T}$-measurable.

Let us define the measure space $(\Omega_{T},\mathcal{P}_{T},\mathbb{P}_{T})$
where $\Omega_{T}=[0,T]\times\Omega$, $\mathcal{P}_{T}$ is as above and
$\mathbb{P}_{T}=\lambda_{T}\otimes\mathbb{P}$ with $\lambda_{T}$ the Lebesgue
measure on $[0,T]$. Note that the norm $\|\,\cdot\,\|_{\mathscr{H}^{2}_{T}}$
defined above corresponds to the Bochner integral,

$\|\varphi\|^{2}_{\mathscr{H}^{2}_{T}}=\int_{\Omega_{T}}\|\varphi_{t}(\omega)\|^{2}_{L^{0}_{2}}\mathop{}\\!\mathrm{d}\mathbb{P}_{T}(t,\omega).$

Recall that since $L^{0}_{2}$ is a separable Banach space, this Bochner
integral is well defined, see Section 1.3.

With $W$ fixed, let us define the space of square integrable stochastic
integrands - the fact that they are suitable stochastic integrands will be
proved in Theorem 1.5.23.

######  Definition 1.5.21.

[Stochastic Integrands] We say that
$[0,T]\times\Omega\ni(t,\omega)\mapsto\varphi_{t}(\omega)\in
L^{0}_{2}(U_{0},H)$ is stochastically integrable with respect to $W$ if it is
$\mathscr{P}_{T}$ measurable with respect to the natural filtration of $W$ and
is such that $\|\varphi\|_{\mathscr{H}^{2}_{T}}<\infty$. We denote this space
by $\mathscr{H}^{2}_{T}(W)$ and observe that it forms a Hilbert space with
inner product
$\langle\varphi,\tilde{\varphi}\rangle_{\mathscr{H}^{2}_{T}}=\int_{0}^{T}\langle\varphi_{t},\tilde{\varphi}_{t}\rangle_{L^{0}_{2}}\,\mathop{}\\!\mathrm{d}t$

The following proposition allows us to approximate predictable elements of
$\mathscr{H}^{2}_{T}(W)$ by simple processes in
$\mathscr{H}^{2}_{T}(W)\cap\mathcal{E}_{T}$.

######  Proposition 1.5.22.

The following statements both hold:

  1. i)

If a mapping $\varphi:[0,T]\times\Omega\rightarrow L(U;H)$ is
$L(U;H)$-predictable, then it is also $L^{0}_{2}$-predictable.

  2. ii)

If $\varphi\in\mathscr{H}^{2}_{T}(W)$ is $L^{0}_{2}$-predictable then there
exists a sequence of simple processes $(\varphi^{n})_{n\geq 1}$ such that
$\displaystyle\lim_{n\rightarrow\infty}\|\varphi-\varphi^{n}\|_{\mathscr{H}^{2}_{T}}=0$.

###### Proof.

To prove i), first recall from Remark 1.2.6 that given bases $(e_{k})_{k\geq
1},\,(f_{k})_{k\geq 1}$ of $U$ and $H$ respectively, the family $(e_{k}\otimes
f_{k})_{k\geq 1}$ is a basis of $L_{2}(U;H)$. So it follows that $L(U;H)$ is a
dense, linear subspace of $L_{2}(U;H)$. Furthermore, since
$Q^{1/2}(U)=U_{0}\subset U$ has continuous image in $U$, it follows that
$L(U;H)\subset L_{2}(U_{0};H)=L^{0}_{2}$ with continuous embedding. By an
adaptation of Theorem 1.4.4 it follows that any $L(U;H)$ valued process is
also an $L^{0}_{2}$ valued process, with continuous image. Since
predictability is preserved by continuous transformations the first claim
follows.

As in the proof of i) we use the fact that $L(U;H)$ is densely embedded in
$L^{0}_{2}$ and Proposition 1.4.3 that for $\mathbb{P}_{T}$-a.e
$(t,\omega)\in\Omega_{T}$ there exists a sequence of simple $L(U;H)$ valued
random variables, $(\varphi_{n})_{n\geq 1}\subset\mathcal{E}_{T}$ such that

$\|\varphi_{t}(\omega)-\varphi_{n;t}(\omega)\|_{L^{2}_{0}}\rightarrow
0,\quad\mathbb{P}_{T}\,{\text{-a.s.}}.$

It follows from the assumptions, that therefore
$\|\varphi-\varphi_{n}\|_{\mathscr{H}^{2}_{T}}\rightarrow 0$. Note, however,
that we have not yet treated predictability. So to conclude it suffices to
show that any predictable set $A\in\mathscr{P}_{T}$ can be approximated
arbitrarily by predictable events in $\mathbb{P}_{T}$. However, this follows
from Dynkin’s lemma, since the predictable sets form a $\pi$-system which
generates $\mathcal{P}_{T}$. See the last paragraph in the proof of [DPZ14,
Prop. 4.22] for details. ∎

Therefore, for fixed $(W_{t})_{t\in[0,T]}$, we have defined an isometry,
$\mathscr{H}^{2}_{T}(W)\ni\varphi\mapsto(\varphi\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}W)\in\mathcal{M}^{2}_{T}(U)$,
on a linear, dense subspace. This allows us to extend the map, first to all of
$\mathscr{H}^{2}_{T}(W)$ and then to all $L^{0}_{2}$-predictable processes,
$\varphi$ such that

$\mathbb{P}\left[\int_{0}^{T}\|\varphi_{s}\|^{2}_{L^{0}_{2}}\mathop{}\\!\mathrm{d}s<\infty\right]=1.$
(1.5.6)

We denote this, the full space of integrands, by $\mathscr{H}_{T}(W)$.

######  Theorem 1.5.23.

Given $\varphi\in\mathscr{H}^{2}_{T}(W)$, the stochastic integral,

$[0,T]\ni
t\mapsto\int_{0}^{t}\varphi_{s}\mathop{}\\!\mathrm{d}W_{s}=:\varphi\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}W_{t},$
(1.5.7)

defines a continuous, square integrable, $H$-valued martingale. Furthermore,
if instead $\varphi\in\mathscr{H}_{T}(W)$ then (1.5.7) defines a continuous,
local martingale.

###### Proof.

Combining Proposition 1.5.22 and the isometry property, (1.5.5), for the Itô
integral defined for simple processes, allows us to extend the integral to all
of $\mathscr{H}^{2}_{T}(W)$ by density. Since the martingale property and
continuity of paths are preserved under convergence in $L^{2}(\Omega)$, it
follows that the integral takes values in $\mathcal{M}^{2}_{T}(H)$.

To extend the integral to all of $\mathscr{H}_{T}(W)$, we first claim that for
$\tau$ an $(\mathcal{F}_{t})_{t\in[0,T]}$ stopping time and
$\varphi\in\mathscr{H}^{2}_{T}(W)$, it holds that,

$\int_{0}^{T}\mathds{1}_{[0,\tau]}(s)\varphi_{s}\mathop{}\\!\mathrm{d}W_{s}=\varphi\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}W_{t\wedge\tau},\quad\text{
for all }t\in[0,T],\,\,\mathbb{P}\text{-a.s.}$ (1.5.8)

In other words, stopping the integrand at $\tau$ amounts to stopping the
integral. Identity (1.5.8) can be shown by approximating $\tau$ by simple
stopping times (stopping times taking only finitely many values) and $\varphi$
by simple processes and then passing to the limits. For details see [DPZ14,
Lem. 4.24]. So then for $\varphi\in\mathscr{H}_{T}(W)$, i.e such that (1.5.6)
holds, let us define,

$\tau_{n}:=\inf\left\\{t\in[0,T]\,:\,\int_{0}^{t}\|\varphi_{s}\|^{2}_{L^{0}_{2}}\mathop{}\\!\mathrm{d}s\geq
n\right\\},\quad n\geq 0,$ (1.5.9)

with the convention that $\tau_{n}=T$ if the set is empty. Therefore, by
(1.5.8), the stochastic integrals
$\mathds{1}_{[0,\tau_{n}]}\varphi\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}W=\varphi\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}W_{\,\cdot\,\wedge\tau_{n}}$
are well defined. Furthermore, due to (1.5.6), the sequence $(\tau_{n})_{n\geq
0}$ is $\mathbb{P}$-a.s. increasing so for any $t\in[0,T]$ and
$\mathbb{P}$-a.a. $\omega\in\Omega$ there exists an $n(\omega)\geq 0$ such
that $t\leq\tau_{n}(\omega)$. Therefore, the integral
$t\mapsto\varphi\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}W_{t}=\mathds{1}_{[0,\tau_{n}]}(t)\varphi\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}W_{t}$,
with $t\leq\tau_{n}$, is $\mathbb{P}$-a.s. well defined for all
$\varphi\in\mathscr{H}_{T}(W)$ and the definition is independent of the chosen
sequence $(\tau_{n})_{n\geq 1}$ satisfying (1.5.9). Since $\mathscr{H}_{T}(W)$
contains elements which are not stochastically square integrable on $[0,T]$,
the resulting stochastic integral is only a local martingale, with the
localisation given by a sequence of the form (1.5.9). ∎

####  1.5.5 Stochastic Integration Against Generalised Wiener Processes

We recall that given a, non-negative, symmetric, bounded, linear operator
$Q\in L(U)$, there exists a generalised Wiener process $(W_{t})_{t\in[0,T]}$
on $U$, with covariance $Q$. Furthermore, there exists a pair of Hilbert
spaces $U_{0}:=Q^{1/2}(U),\,U_{1}$ such that $U_{0}\hookrightarrow U_{1}$ and
the embedding $\iota U_{0}\rightarrow U_{1}$ is Hilbert–Schmidt. As a result,
the generalised Wiener process, $[0,T]\times U\ni(t,h)\mapsto W_{h;t}$, can be
seen as taking values in $U_{1}$, although the particular choice of $U_{1}$ is
not important.

As such, we may almost immediately define the stochastic integral with respect
to a generalised Wiener process for all predictable, processes,
$(\varphi_{t})_{t\in[0,T]}\in L_{2}(Q^{1/2}(U_{1});H)$ such that,

$\mathbb{P}\left[\int_{0}^{T}\|\varphi_{s}\|^{2}_{L_{2}(Q^{1/2}(U_{1});H)}\mathop{}\\!\mathrm{d}s<\infty\right]=1.$

It suffices to replace $U_{0}$ with $Q^{1/2}(U_{1})$ in the previous section.
However, this is unsatisfactory, since we know that the definition of the
generalised Wiener process is independent of the choice $U_{1}$ and so it
would be natural for the integral to retain this property. It turns out that
we can retain the same space of integrands as in Subsection 1.5.4. Note that
$U_{0}:=Q^{1/2}(U)$ is still well defined in this case, only now it does not
hold that the embedding $U_{0}\hookrightarrow U$ is Hilbert–Schmidt.

######  Proposition 1.5.24.

Let $(e_{k})_{k\geq 1}$ be an orthonormal basis of $U$, $(\beta^{k})_{k\geq
1}$ be a family of i.i.d, standard, real Brownian motions, $Q\in L(U)$ be non-
negative and symmetric and $\iota:Q^{1/2}(U)=:U_{0}\rightarrow U_{1}$ define a
Hilbert–Schmidt embedding as above. Then, defining the non-negative, symmetric
and trace class operator, $Q_{1}:=\iota Q\iota^{\ast}\in L_{1}(U_{1})$, it
follows that,

$[0,T]\ni t\mapsto W_{t}=\sum_{k=1}^{\infty}\beta^{k}_{t}\iota Q^{1/2}e_{k},$
(1.5.10)

defines $Q_{1}$-Wiener process in $\mathcal{M}^{2}_{T}(U_{1})$. Moreover, one
has that $Q_{1}^{1/2}(U_{1})=\iota(U_{0})\subset U_{1}$ and for all $h_{0}\in
U_{0}$,

$\|h_{0}\|_{U_{0}}=\|Q_{1}^{-1/2}\iota h_{0}\|_{U_{1}}=\|\iota
h_{0}\|_{Q^{1/2}_{1}U_{1}},$

where $Q^{-1/2}_{1}$ is the pseudo-inverse of $Q_{1}^{1/2}$, see Definition
1.2.9. In other words, $\iota:U_{0}\rightarrow Q^{1/2}_{1}U_{1}$ is an
isometry.

###### Proof.

By assumption, $\iota Q^{1/2}$, defines a trace class operator from $U$ to
$U_{1}$ and so it follows that the right hand side of (1.5.10) defines a
$U_{1}$ valued Wiener process. To see that $W$ defines a generalised Wiener
process on $U$ in the sense of Definition 1.5.16 we may apply the same
arguments as in the proof of Proposition 1.4.15. The independence of the
definition on $U_{1},\,\iota$ also follows from the same arguments and we
recall, using, Corollary 1.2.12, that $\iota:U_{0}\rightarrow
Q^{1/2}_{1}U_{1}$ is an isometry. ∎

######  Theorem 1.5.25.

Let $(W_{t})_{t\in[0,T]}$ be a generalised Wiener process on $U$, $[0,T]\ni
t\mapsto\varphi_{t}\in L_{2}(U_{0};H)=:L^{0}_{2}$ be a stochastic process,
such that,

$\mathbb{P}\left[\int_{0}^{T}\|\varphi_{t}\|^{2}_{L^{0}_{2}}\mathop{}\\!\mathrm{d}t<\infty\right]=1.$

Then, letting $U_{1}$, $\iota:U_{0}\rightarrow U_{1}$ be as above, the
stochastic integral defined by setting

$[0,T]\ni
t\mapsto\varphi\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\scriptscriptstyle\bullet$}}}}}W_{t}:=\int_{0}^{t}(\varphi_{s}\circ\iota^{-1})\mathop{}\\!\mathrm{d}W_{s},$
(1.5.11)

is a continuous, $H$-valued, square integrable, local martingale.

###### Proof.

By Proposition 1.5.24, for $Q_{1}=\iota\iota^{\ast}\in N(U_{1})$, we have
$Q^{1/2}_{1}(U_{1})=\iota(U_{0})$ and by the polarisation identity, for any
$\varphi_{0},\psi_{0}\in U_{0}$,

$\langle\iota\varphi_{0},\iota\psi_{0}\rangle_{Q^{1/2}_{1}U_{1}}=\langle
Q^{-1/2}_{1}\iota\varphi_{0},Q^{-1/2}_{1}\iota\psi_{0}\rangle_{U_{1}}=\langle\varphi_{0},\psi_{0}\rangle_{U_{0}}.$

In particular, given $(e_{k})_{k\geq 1}$ a basis of $U_{0}$, it follows that
$(\iota e_{k})_{k\geq 1}$ is a basis of $Q^{1/2}_{1}(U_{1})$. Hence we have
the equivalence,

$\varphi\in L_{2}(U_{0};H)\quad\iff\quad\varphi\circ\iota^{-1}\in
L_{2}(Q_{1}^{1/2}(U_{1});H),$

and so the stochastic integral in (1.5.11) is well defined as a continuous,
$H$-valued, square integrable, local martingale, by the results of Subsection
1.5.4. The independence on the choice of $U_{1}$ and $\iota$ is a consequence
of the same result for the generalised Wiener process. ∎

For another approach to the construction of stochastic integrals against
generalised Wiener processes, we refer to [DPZ14, Subsec. 4.2.1]. For more
details on this approach, see [PR07].

##  Chapter 2 Variational Approach to PDE

In this chapter we present an introduction to the variational approach to
partial differential evolution equations. The method relies on viewing time
dependent PDE in spirit as ordinary differential equations in infinite
dimensional spaces. We use this is our starting point for viewing SPDE as
infinite dimensional stochastic differential equations in the next chapter. A
useful outcome of this approach to PDE and SPDE is a natural approach to
numerical analysis of such problems through the finite element method. This
approach is also sometimes known as the Galerkin method.

###  2.1 Unbounded Operators Between Banach Spaces

We begin by recalling the notion of unbounded operators on Banach spaces. We
refer to [Bre11, Sec. 2.6] for more details.

######  Definition 2.1.1 (Unbounded Linear Operators).

Given Banach spaces $E,\,F$, an unbounded, linear operator $A$ from $E$ to $F$
is a pair $(A,D(A))$, where $D(A)\subseteq E$ is a linear subspace and
$A:D(A)\rightarrow F$ is a linear map. We write write $\mathscr{L}(E;F)$ for
the set of unbounded linear operators from $X$ to $Y$, leaving the dependence
on defining a domain inside $X$ implicit in the notation.

We say that $A$ is densely defined if $D(A)$ is a dense subset of $E$. If $A$
is densely defined from $E$ to $F$ and continuous on its domain then there
exists a unique, continuous extension of $A$ defined on all of $E\rightarrow
F$.

We define the graph of $A$, $G_{A}:=\\{(x,y)\in E\oplus F\,:\,y=Ax\\}$ and say
that $A$ is a closed operator if $G_{A}\subset E\oplus F$ is a closed set.
Explicitly, $A$ is closed if and only if for any sequence
$\\{(x_{n},Ax_{n})\\}_{n\geq 0}\subset E\oplus F$ such that $x_{n}\rightarrow
x$ and $Ax_{n}\rightarrow y$, we have $x\in D(A)$ and $y=Tx$.

######  Example 2.1.2.

Let $C([0,1];\mathbb{R})$ be the set of continuous, real valued functions on
the unit interval, equipped with the norm
$\|f\|_{\infty}:=\sup_{x\in[0,1]}|f(x)|$. Then, for $f\in
C([0,1];\mathbb{R})$, define $Af:=\frac{d}{dx}f$ and

$D\left(\frac{d}{dx}\right):=C^{1}([0,1];\mathbb{R})\subset
C([0,1];\mathbb{R}),$

the set of continuously differentiable maps $f:[0,1]\rightarrow\mathbb{R}$
equipped with the norm
$\|f\|_{C^{1}}:=\|f\|_{\infty}+\|\frac{d}{dx}f\|_{\infty}$. It follows that
$(\frac{d}{dx},C^{1}([0,1];\mathbb{R}))$ is an unbounded, linear operator from
$C([0,1];\mathbb{R})$ to itself in the sense of Definition 2.1.1.

By Stone–Weierstrass, $\frac{d}{dx}$ is densely defined, however, it is also
unbounded on its domain and so cannot be continuously extended to all of
$C([0,1];\mathbb{R})$. To see this, define the sequence $f_{n}(x):=\sin(nx)$,
we have $\|f_{n}\|_{\infty}\equiv 1$ but
$\|\frac{d}{dx}f_{n}\|_{\infty}=n\rightarrow\infty$. On the other hand, it is
easy to see that $\frac{d}{dx}$ is a closed operator, since if
$\|f-f_{n}\|_{C^{1}}\rightarrow 0$, then by definition
$\|\frac{d}{dx}(f-f_{n})\|_{\infty}\rightarrow 0$.

######  Example 2.1.3 (Dirichlet Laplacian).

Let $\Gamma\subset\mathbb{R}^{d}$ be a smooth, bounded domain and define the
second order differential operator, for mappings
$f:\Gamma\rightarrow\mathbb{R}$,

$\Delta f:=\sum_{i=1}^{d}\partial^{2}_{ii}f.$

We consider $\Delta$ as an unbounded operator on the set,

$\mathcal{H}^{1}_{0}(\Gamma):=\left\\{f:\Gamma\rightarrow\mathbb{R}\,:\,\|f\|_{L^{2}(\Gamma)}+\sup_{i=\\{1,\ldots,d\\}}\|\partial_{i}f\|_{L^{2}(\Gamma)}<\infty,\,f|_{\partial\Gamma}=0\right\\},$

with domain of definition, $D(\Delta)$,

$C^{2}_{c}(\Gamma):=\left\\{f:\Gamma\rightarrow\mathbb{R}\,:\,\sup_{i,j=\\{1,\,\ldots,d\\}}\|\partial_{ij}f\|_{\infty}<\infty,\,\text{supp}(f)\text{
is compact in }\Gamma\right\\}.$

The lack of boundedness in Example 2.1.2 arose because we chose too small a
target space for the derivative map. In order that $\Delta$ remain bounded on
its domain in this case, we define
$\mathcal{H}^{-1}(\Gamma):=(\mathcal{H}(\Gamma))^{\ast}$, which can be
explicitly represented as,

$\mathcal{H}^{-1}(\Gamma):=\left\\{h=f+\sum_{i=1}^{d}\partial_{i}g,\,f,\,g\in
L^{2}(\Gamma;\mathbb{R})\right\\}.$ (2.1.1)

Then we consider the unbounded operator,
$\Delta:C^{2}_{c}(\Gamma)\subset\mathcal{H}^{1}_{0}(\Gamma)\rightarrow\mathcal{H}^{-1}(\Gamma)$.
It follows from the density of $C^{2}_{c}(\Gamma)$ in
$\mathcal{H}^{1}_{0}(\Gamma)$ and the fact that $\Delta$ is continuous as a
map from $C^{2}_{c}(\Gamma)$ to $\mathcal{H}^{-1}(\Gamma)$ that there exists a
unique extension of
$\Delta:\mathcal{H}^{1}_{0}(\Gamma)\rightarrow\mathcal{H}^{-1}(\Gamma)$.

Note that by setting $g=0$ in (2.1.1), we see that
$L^{2}(\Gamma)\subset\mathcal{H}^{-1}(\Gamma)$. Furthermore, it is clear that
$\mathcal{H}^{1}_{0}(\Gamma)\subset L^{2}(\Gamma)$ and so we have the
ordering,

$\mathcal{H}^{1}_{0}(\Gamma)\subset
L^{2}(\Gamma)\subset\mathcal{H}^{-1}(\Gamma)=(\mathcal{H}^{1}_{0}(\Gamma))^{\ast}.$

This is an example of a Gelfand triple, and
$\Delta:\mathcal{H}^{1}_{0}(\Gamma)\rightarrow\mathcal{H}^{-1}(\Gamma)$
defines a continuous, coercive operator. We explore this setting in more
general abstraction in the next section.

###  2.2 Linear and Coercive PDE

Let $V$ be a separable, reflexive, Banach space with $V\subset H$ a separable