text:
\section{Introduction}
\label{sec:intro}

\begin{figure}[h!]
 \centering
   \subfigure[\label{fig:drone}]
   {\includegraphics[width=0.6\textwidth]{figures/husky_axes.png} \label{fig:husky}}

   \subfigure[\label{fig:tool}]
   {\includegraphics[width=0.258\textwidth]{figures/arm.pdf} \label{fig:arm}}
	\vspace{-5pt}
\caption{\footnotesize{We demonstrate our approach for learning (i) reactive obstacle avoidance policies for a differential drive ground vehicle model equipped with a depth sensor, and (ii) neural network-based grasping policies for a manipulator model equipped with an RGB-D sensor.
Our approach provides strong guarantees on the performance of the learned policies on novel environments even with a relatively small number of training environments (e.g., a guaranteed expected collision-free traversal rate of $87.9\%$ using 1000 training environments for the obstacle avoidance example and a guaranteed expected success rate of $70.6\%$ for the grasping example using 2000 training objects).} \label{fig:examples}}
\vspace*{-5pt}
\end{figure}

\begin{figure}[h!]
 \centering
   \subfigure[\label{fig:lab_swing}]{
    \includegraphics[width=0.4\columnwidth]{figures/lab_swing.jpeg}}
   \subfigure[\label{fig:swingb}]{
    \includegraphics[width=0.428\columnwidth]{figures/netted_area_labelled.png}}
   \vspace{-5pt}
\caption{\footnotesize{Pictured in (a) is a Parrot Swing drone --- a quadrotor/fixed-wing hybrid vehicle. We demonstrate our approach for learning reactive obstacle avoidance policies for the Swing given a simulated depth sensor. Our approach provides a guaranteed expected collision-free traversal rate of 88.6$\%$ on novel environments using 1000 simulated training environments. When testing on unseen environments within the netted area pictured in (b), the Swing succeeds in 18/20 trials. Videos of representative trials can be found at https://youtu.be/p5CjcSsojg8.}\label{fig:swing}}
\vspace{-10pt}
\end{figure}

Imagine an unmanned aerial vehicle that successfully navigates a thousand different obstacle environments or a robotic manipulator that successfully grasps a million objects in our dataset. How likely are these systems to succeed on a novel (i.e., previously unseen) environment or object? How can we explicitly learn control policies that provably generalize well to environments or objects that our robot has not previously encountered? Current approaches for designing control policies for robotic systems either do not provide such guarantees on generalization or provide guarantees only under very restrictive assumptions (e.g., strong assumptions on the geometry of a novel environment \citep{Schouwenaars04, Fraichard07, Althoff15, Majumdar17}).

The goal of this paper is to develop an approach for learning control policies for robotic systems that provably generalize well with high probability to novel environments given a dataset of example environments.

 The key conceptual idea for enabling this is to establish a precise analogy between generalization of policies to novel environments and generalization in supervised learning. This analogy allows us to translate techniques for learning hypotheses with generalization guarantees in the supervised learning setting into techniques for learning control policies for robot tasks with performance guarantees on novel environments.

In order to obtain more insight into this analogy, suppose we have a dataset of $N$ objects. A simple approach to learning a grasping policy is to synthesize one that achieves the best possible performance on these $N$ objects. However, such a strategy might result in an overly complex policy that overfits to the specific objects at hand. This is a particularly important challenge for robotics applications since datasets are generally relatively small (e.g., as compared to training sets for image classification tasks).  In order to learn a policy that generalizes well to novel environments, we may need to add a ``regularizer" that penalizes the ``complexity" of the policy. This raises the following questions: (1) what form should this regularizer take?; and (2) can we provide a formal guarantee on the performance of the resulting policy on novel environments?

The analogous questions for \emph{supervised} learning algorithms have been extensively studied in the literature on \emph{generalization theory} in machine learning. Here we leverage PAC-Bayes theory (Probably Approximately Correct Bayes) \citep{McAllester99}, which provides some of the tightest known generalization bounds for classical supervised learning approaches \citep{Langford03, Seeger02, Germain09}. Very recently, PAC-Bayes analysis has also been used to train deep neural networks with guarantees on generalization performance \citep{Dziugaite17, Neyshabur17, Neyshabur17a}. As we will see, we can leverage PAC-Bayes theory to provide precise answers to both questions posed above; it will allow us to specify a regularizer for designing (stochastic) control policies that provably generalize well (with high probability) to novel environments.

\subsection{Statement of Contributions}
\label{sec:contributions}

To our knowledge, the results in this paper constitute the first attempt to provide generalization guarantees on learning-based control policies for robotic systems with continuous state and action spaces, complicated (e.g., nonlinear) dynamics, and rich sensory inputs (e.g., depth images).  To this end, this paper makes four primary contributions. First, we provide a \emph{reduction} that allows us to translate generalization bounds for supervised learning problems to generalization bounds for control policies. We apply this reduction to translate PAC-Bayes bounds to the control setting we consider here (Section \ref{sec:pac bayes control}). Second, we propose learning algorithms that minimize the regularized cost functions specified by PAC-Bayes theory in order to synthesize control policies with generalization guarantees (Section \ref{sec:computing pac-bayes controllers}). In the setting where we are optimizing over a finite policy space (Section \ref{sec:finite policy space}), the corresponding optimization problem can be solved using \emph{convex} optimization techniques (\emph{Relative Entropy Programs (REPs)} in particular). In the more general setting of continuously-parameterized policies (Section \ref{sec:continuous policy space}), we rely on stochastic gradient descent to perform the optimization. Third, in Section \ref{sec:domain shift} we present an extension of our basic approach that allows us to learn policies that are distributionally robust (i.e., handle settings where test environments are drawn from a different distribution than training environments). Fourth, we demonstrate our approach in simulation for learning (i) depth sensor-based reactive obstacle avoidance policies for the ground robot model shown in Figure \ref{fig:husky} (Section \ref{sec:reactive obstacle avoidance control}), and (ii) neural network-based grasping policies for the manipulator model shown in Figure \ref{fig:arm} (Section \ref{sec:grasping}). Finally, we also present hardware results for reactive obstacle avoidance control with the Parrot Swing drone shown in Figure \ref{fig:lab_swing} (Section \ref{sec:hardware}).
Our simulation and hardware results demonstrate that we are able to obtain strong generalization guarantees even with a relatively small number of training environments. We compare the bounds obtained from PAC-Bayes theory with exhaustive sampling to illustrate the tightness of the bounds.

A preliminary version of this work \citep{Majumdar18} was presented at the Conference on Robot Learning (CoRL) 2018. In this significantly revised and extended version, we additionally present: (i) an extension of our basic approach for providing generalization guarantees in settings where test environments are drawn from a different distribution to training environments (Section \ref{sec:domain shift}), (ii) an application of our framework for learning neural-network based grasping policies (Section \ref{sec:grasping}), (iii) a method for handling stochastic dynamics (Section \ref{sec:stochastic rollouts}), (iv) hardware implementation of the depth sensor-based reactive obstacle avoidance policies (Section \ref{sec:hardware}), and (v) a more thorough discussion of challenges associated with our approach and promising future directions (Section \ref{sec:conclusions}).

\subsection{Related Work}
\label{sec:related work}

One approach for synthesizing control policies with guaranteed performance is to leverage robust control techniques (e.g., H-infinity control \citep{Francis87} or chance-constrained programming \citep{Charnes59, Blackmore06, Vitus11, Ono15}). However, such techniques typically require an explicit description of the uncertainty affecting the system. While uncertainty models for the robot's dynamics or measurements can often be obtained via system identification, assuming an uncertainty model for the environment (e.g., a distribution over all possible environment geometries) is unrealistic. One way to address this is to assume that a novel environment satisfies conditions that allow a real-time planner to \emph{always} succeed. For example, in the context of navigation, this constraint could be satisfied by hand-coding emergency maneuvers (e.g., stopping maneuvers or loiter circles) that are always guaranteed to succeed \citep{Schouwenaars04, Fraichard07, Althoff15}. However, requiring the existence of such emergency maneuvers can lead to extremely conservative behavior. Another approach is to assume that the environment satisfies certain geometric conditions (e.g., large separation between obstacles) that allow for safe navigation \citep{Majumdar17}. However, such conditions are rarely satisfied by real-world environments. Moreover, such conditions are domain specific; it is not clear how one would specify such constraints for problems other than navigation (e.g., grasping).

Another conceptually appealing approach for synthesizing policies with guaranteed performance on a priori unknown environments is to model the problem as a Partially Observable Markov Decision Process (POMDP) \citep{Kaelbling98}, where the environment is part of the (partially observed) state of the system \citep{Richter15}. Computational considerations aside, such an approach is made infeasible by the need to specify a distribution over environments the robot might encounter. Unfortunately, specifying such a distribution over real-world environments is an extremely challenging endeavor. Thus, many approaches (including ours) assume that we only have \emph{indirect} access to the true underlying distribution over environments in the form of examples. For example, \citep{Richter15, Richter17} propose an approximation to the POMDP framework in the context of navigation by learning to predict future collision probabilities from past data. The work on deep-learning based approaches for control represents another prominent set of techniques where interactions with example environments are used to learn control policies (see, e.g., \citep{Lenz15, Levine16, Agrawal16, Mahler17, Tobin17, Gupta17, Gupta17a, Zhu17, Sunderhauf18}). While the approaches mentioned above have led to impressive empirical demonstrations, it is challenging to guarantee that such methods will perform well on environments that are not part of the training data (especially when a limited number of training examples are available, as is often the case for robotics applications). Our work seeks to address this challenge using ideas from generalization theory.

The primary theoretical framework we utilize in this paper is PAC-Bayes generalization theory \citep{McAllester99}. PAC-Bayes theory provides some of the tightest known generalization bounds for classical supervised learning problems \citep{Langford03, Seeger02, Germain09} and has recently been applied to explain and promote generalization in deep learning \citep{Dziugaite17, Neyshabur17, Neyshabur17a}. PAC-Bayes theory has also been applied to learn control policies for Markov Decision Processes (MDPs) with provable sample complexity bounds  \citep{Fard10, Fard12}. These approaches also exploit the intuition (see Section \ref{sec:intro}) that ``regularizing" policies in an appropriate manner can prevent overfitting and lead to sample efficiency (see also \citep{Neu17, Kearns00, Bagnell01, Bagnell04, Schulman15} for other approaches that exploit this intuition in the reinforcement learning context). However, we note that the focus of our work is quite different from the work on PAC-Bayes MDP bounds (and the more general framework of PAC MDP bounds \citep{Kearns02, Brafman02, Fu14}), which consider the standard reinforcement learning setup where a control policy must be learned through multiple interactions with a given MDP (with unknown transition dynamics and/or rewards).
In contrast, here we focus on \emph{zero-shot} generalization to a novel environment (e.g., obstacle environments or objects). In other words, a policy learned from examples of different environments must immediately perform well on a new one (i.e., without further exploratory interactions with the new environment). We further note that \citep{Fard10} considers finite state and action spaces along with policies that depend on full state feedback while \citep{Fard12} relaxes the assumption on finite state spaces but retains the other modeling assumptions. In contrast, we target systems with continuous state and action spaces and synthesize control policies that rely on rich sensory inputs.

On the algorithmic front, we make significant use of Relative Entropy Programs (REPs) \citep{Chandrasekaran17}. REPs constitute a rich class of \emph{convex} optimization problems that generalize many other problems including linear programs, geometric programs, and second-order cone programs \citep{Boyd04}. REPs are optimization problems in which a linear functional of the decision variables is minimized subject to linear constraints and conic constraints given by a \emph{relative entropy cone}. REPs are amenable to efficient solution techniques (e.g., interior point methods \citep{Nesterov94}) and can be solved using existing software packages (e.g., Mosek \citep{mosek}, SCS \citep{SCS, ODonoghue16}, and ECOS  \citep{ECOS}). We refer the reader to \citep{Chandrasekaran17} for a more thorough introduction to REPs. Importantly for us, REPs can handle constraints of the form $\KL(p \| q) \leq c$, where $p$ and $q$ are decision variables corresponding to probability vectors, $\KL(\cdot \|| \cdot)$ represents the Kullback-Leibler divergence, and $c$ is a scalar decision variable. As we will see, this allows us to use REPs to learn control policies using the PAC-Bayes framework in the setting where we are optimizing over a finite set of policies.

\subsection{Notation}

We use the notation $v[i]$ to refer to the i-th component of a vector $v \in \RR^n$. We use $\RR^n_+$ to denote the set of elementwise nonnegative vectors in $\RR^n$, $\mathbb{Z}_+$ to denote nonnegative integers, and $\odot$ to denote element-wise multiplication.


TITLE: Jacobi symbol: number theory
QUESTION [0 upvotes]: What is Jacobi symbol of $(-2p^2/4a^2 + p^2)^l$ where $p$ is congruent to $3 mod 4$, $a$ is odd and $l$ is even?

REPLY [1 votes]: I guess you mean $$\Big(\frac{-2p^2}{4a^2 + p^2}\Big)^l$$
Since $l$ is even the value is either $1$ or $0$. It is $0$ if
$\gcd(2p^2, 4a^2 + p^2) \ne 1$. Since $4a^2 + p^2$ is odd a non-zero GCD will be a multiple of $p$, i.e. $\gcd(2a, p) = p$. And because $a$ shall be odd you must have  $a=(2n+1)p, \; n \in \mathbb{Z}$.


TITLE: Definability of ordinals in various signatures
QUESTION [6 upvotes]: Recently, I've been studying what the definable subsets of the countable ordinals "look like" from the perspective of bare-bones first order logic (not set theory) equipped with various ways to "access" the structure of the ordinals.
For example, we may have a signature consisting only of a 2-arity relational symbol $S$ which we interpret in a structure $\mathcal{A}$ with underlying set $\omega_1$ as the set of $(\alpha,\beta)$ such that $\beta$ is the successor of $\alpha$. We can then ask questions about which subsets of $\mathcal{A}$ are definable by first-order logic sentences with this signature, where a subset $S\subset\mathcal{A}$ is considered definable if there is a first order logic sentence $\phi(x)$ for which the set of satisfying assignments of $x$ is $S$. In our example, we can define the set of all countable successor ordinals via the formula $\exists y:S(y,x)$.
We can also ask questions like "what is the smallest ordinal $\alpha$ such that $\alpha$ is undefinable in the sense that $\{\alpha\}$ is undefinable" and such. In the example above, it's clear to see that in fact no ordinal is definable, so the smallest undefinable ordinal is zero. I am particularly interested in how the smallest undefinable ordinal grows as we have stronger and stronger signatures. For example, I have been able to convince myself that with the signature $\{<\}$ with the obvious interpretation in $\omega_1$ as the "less than relation", the smallest undefinable ordinal is $\omega^\omega$ (though I haven't written my argument out formally yet).
My question is: has anyone studied questions like these? Is it known what the smallest definable ordinal is for various other signatures, like $\{ADD(x,y,z)\}$ which is true on all $x,y,z$ so that $x+y=z$, or even other signatures with multiplication, exponentiation, veblen functions, or more? Are there any known generalizations of these ideas? Any help or related literature would be appreciated.

REPLY [6 votes]: I do not have enough reputation to add a comment. The following paper may be useful for you. It contains results extending the work of Tarski, Mostowski, and Doner, as well as some very nice historical overview and references.
Buchi, Siefkes - The Complete Extensions of the Monadic Second Order Theory of Countable Ordinals.
Weak monadic second order logic appears already in Ehrenfeucht's original work. Even if you are exclusively interested in first order results, (weak) monadic second order logic can play a role.
For example the first order theory of ordinal addition coincides with the first order theory of ordinal addition inside $\omega^{\omega^{\omega}}$ (by Ehrenfeuct), while $(\omega^{\omega^{\omega}},+)$ is a reduct of a generalised power of $(\omega,+)$ with 'exponent' being the weak monadic second order version of $(\omega^{\omega},<)$ (the Feferman-Vaught theorem is the correct tool to understand this). For more details there is Thomas - Ehrenfeucht, Vaught, and the decidability of the weak monadic theory of successor, the details here are all correct but I think the conclusions have some issues.
There is also more recent work on the automata side such as Cachat - Tree Automata Make Ordinal Theory Easy. I know nothing about the content of this but if you want a comprehensive overview of the area, this is maybe a starting point.


TITLE: Projectives in the category of discrete G-modules
QUESTION [5 upvotes]: If $G$ is a profinite group, then the category $Mod(G)$ of discrete $G$-modules has sufficiently many injectives (Neukirch, Schmidt, Wingberg: Cohomology of Number Fields, 2.6.5).
Since the cited book says nothing on projectives within this category, I guess $Mod(G)$ doesn't have sufficiently many projectives. Can you give me an example of a discrete $G$-module $M$ such that there is no epimorphism  $P \to M$ with $P$ projective ?  If there are little projectives, is it even possible to classify them ?
Added: Note that a discrete $G$-module $M$ is an abelian group (with the discrete topology) with a continuous $G$-action $G \times M \to M$ (N-S-W, 1.1.5). For example $\mathbb{Z}$ with trivial $G$-action is a discrete $G$-module.

REPLY [2 votes]: Here is a link a proof that the category $\text{Mod}(G)$ does not have enough projectives, when $G$ is an infinite profinite group. I think the proof is fairly elementary. There might be a few details missing though - sorry about that.


TITLE: Why the arccos of adjacent leg is not equal to arcsin to the oposit leg for the right triangle
QUESTION [0 upvotes]: Silly question. I have a right triangle with  adjacent leg $A = 81.67$ and an opposite leg $B = -1225.53$. So hypotenuse $C$ will be $\sqrt{(A^2 + B^2)} = 1228.25$.
$\arccos(A / C) = 1.50$ and $\arcsin(B / C) = 3.07$. Why are they not the same, what is the reason of difference?
( I have used python for calculations)
P.S.
Negative size means that leg goes from origin to the negative X direction

REPLY [1 votes]: You did your computation wrong; the arcsin of $1225.53/1228.25$ is approximately $1.504$ radians.
The arcsin will never produce a value outside the range $-\pi/2 \le u \le \pi/2$, which is approximately $-1.57 \le u \le 1.57$.
(The original question had a negative-sign before the $1225.53$; in that case, the arcsin is $-1.504$ radians, again, not the same as $3.07$.)


text:
\begin{document}
\title{Brownian Sheet and Quasi-Sure Analysis}
\author{Davar Khoshnevisan}
\address{Department of Mathematics, University of Utah,
   Salt Lake City, UT 84112--0090}
   \thanks{Research supported in part by a grant from
   the NSF}
\subjclass{Primary. 60-Hxx, 60-02;  Secondary. 58J65, 60-06}
\email{davar@math.utah.edu}
\keywords{Quasi-Sure analysis, Brownian sheet,
   Ornstein--Uhlenbeck process on classical Wiener space, capacity}
\date{August 30, 2002}

\dedicatory{To Professor Miklos Cs\"org\H{o}
   on the occasion of his 70th birthday.}

\begin{abstract}
   We present a self-contained and
   modern survey of some
   existing quasi-sure results
   via the connection to the Brownian sheet.
   Among other things, we prove that
   quasi-every continuous function:
   (i) satisfies the local law of the iterated
   logarithm; (ii) has L\'evy's modulus of
   continuity for Brownian motion; (iii)
   is nowhere differentiable; and (iv)
   has a nontrivial quadratic variation.
   We also present
   a hint of how to extend (iii) to
   obtain a quasi-sure refinement of the
   M.\@ Cs\"org\H{o}--P.\@ R\'ev\'esz
   modulus of continuity for almost every continuous
   function along the lines suggested by M.\@ Fukushima.
\end{abstract}

\maketitle

\section{Introduction}

Throughout, we let $\Omega$ denote the space of all continuous
functions $f:[0,1]\to\mathbb{R}$. As usual,
$\Omega$ is endowed with the compact-open topology
[i.e., the topology of uniform convergence] and its corresponding
Borel sigma-algebra $\mathcal{B}(\Omega)$.
Then a number of classical, as well as
modern, theorems of probability theory can be interpreted
a saying something about the ``typical'' function in $\Omega$
in the following sense:\footnote{
The Baire category theorem provides us with another
notion of ``a typical function $f\in\Omega$,''
with a rich and colorful history dating back
to the works of Baire, Borel, and Lebesgue.
Among many gems of thought,
J.-P.\@ Kahane's article (\cite[\S3]{kahane}) contains a
delightful discussion of the history of this subject.}
If we endow
$(\Omega,\mathcal{B}(\Omega))$ with the standard
Wiener measure we then obtain the
\emph{classical Wiener space}, and various
``probabilistic'' results hold for almost
every $f\in\Omega$. Here and throughout,
``almost every''
is tacitly understood to hold with respect
to Wiener's measure
on $(\Omega,\mathcal{B}(\Omega))$.  As two notable
examples we can consider the following, although
frequently one
thinks of these as statements about the Brownian motion:
\begin{enumerate}
   \item \label{lil}
      (Khintchine~\cite{khintchine}).
      \emph{Almost every $f\in\Omega$
      satisfies the local law of the iterated
      logarithm; i.e.,}
      \begin{equation}
         \limsup_{x\to 0} \frac{f(x)}{\sqrt{2x\ln
         \ln \left(\frac 1x\right)}}=-
         \liminf_{x\to 0} \frac{f(x)}{\sqrt{2x\ln
         \ln \left(\frac 1x\right)}}=1.
      \end{equation}
   \item \label{pwz}
      (Paley, Wiener, and Zygmund~\cite{paley-et-al}).
      \emph{Almost every
      $f\in \Omega$ is nowhere-differentiable.}
\end{enumerate}
Other examples abound.

One might ask for a
more restrictive notion of what it means for
$f\in\Omega$ to be ``typical.'' In the
appendix to \cite{meyer}, D.\@ Williams
has proposed this problem, and has shown us an
interesting, less restrictive, class of
``typical functions'' that is motivated
by infinite-dimensional diffusion theory,
and in particular the work of P.\@ Malliavin
in the said area (\cite{malliavin}).

Let $W$ denote the
two-parameter Brownian sheet, based on which
we can construct the
\emph{Ornstein--Uhlenbeck Brownian sheet},
\begin{equation}\label{U}
   U(s,t) := e^{-s/2} W(e^s,t),
   \qquad\forall s,t\ge 0.
\end{equation}
One can think of this two-parameter process
as the ``evaluations'' of the
following infinite-dimensional
(in fact, $\Omega$-valued) stochastic process that
is called the \emph{Ornstein--Uhlenbeck
process in Wiener space}:
\begin{equation}\label{Y}
   Y_s := U(s,\bullet),\qquad\forall s\ge 0.
\end{equation}
It is not difficult to see
that $Y$ is an $\Omega$-valued diffusion;
this follows at once from Theorem~\ref{th:SMP}
below, for instance. Moreover, since
$Y_0 =W(1,\bullet)$ is a standard Brownian motion,
it follows that the process $Y$ is a stationary
diffusion on the space of continuous function,
and the invariant measure of $Y$ is Wiener's
measure.

Next, consider the hitting probabilities $\text{\rm Cap}(\bullet)$
of the diffusion $Y$ killed at an independent mean-one
exponential random variable; i.e., for any
Borel set $G\subset\Omega$,
\begin{equation}
   \text{\rm Cap}(G):= \int_0^\infty
   e^{-s} \IP \left\{
   \exists s\ge 0:\ Y_s\in G\right\}
   \, ds.
\end{equation}

Following D.\@ Williams, we then say
that a Borel measurable set $G\subseteq\Omega$ holds
\emph{quasi-surely} if $\text{\rm Cap}(G^\mathsf{c})=0$.

It is not difficult to see that the set function
$\text{\rm Cap}$ is a natural capacity in the sense
of G.~Choquet. From this it follows that $G$ holds
quasi-surely if and only
if its complement is a capacity-zero set;
i.e., it is almost-surely never visited by
the Ornstein--Uhlenbeck process on $\Omega$.
Equivalently---and this requires only a
moment of reflection---$G$ holds quasi-surely if and
only if
\begin{equation}
   \IP\{ \forall s\ge 0:\ U(s,\bullet)\in G\}=1.
\end{equation}
Thanks to (\ref{U}), the quasi-sure analysis of
subsets of $\Omega$ can be related to the Brownian sheet.

An alternative, more direct, approach was proposed by
M.\@ Fukushima (\cite{fukushima}) who used the properties
of the Dirichlet form associated with the infinite-dimensional
process $Y$ to produce interesting quasi-sure theorems.
This was an exciting new development on the intersection of
probability and infinite-dimensional analysis,
and has led to a rich body of works;
cf.~\cite{CKS:00,CKS:99,CR,DM,DW1,DW2,
fitzsimmons,fukushima,KD,KRS,KS,
Komatsu-Takashima:84a,Komatsu-Takashima:84b,kono:84,Lyons:86,
meyer:82,mountford:92,mountford:89,OP,penrose:90,penrose:89,
shigekawa:84,takeda,walsh:86,walsh:84,zimmerman}. (Not all
of these references employ the quasi-sure notation
in their presentation.)

The said connection to the Brownian sheet makes it
clear that whenever $G\subseteq\Omega$ holds quasi-surely,
then $G$ holds almost-surely as well. For a converse, it
has been noted in~\cite[p. 165]{fukushima}
that there are events that hold almost-surely and not
quasi-surely. For instance, consider $G$ to be
the collection of all continuous functions
$f:[0,1]\to\mathbb{R}$ such that $f(1)\neq 0$.
It is clear then that $G$ holds almost surely;
equivalently, with probability one a Brownian motion $B$
satisfies $B(1)\neq 0$. On the other hand, $G$ does
\emph{not} hold quasi-surely. [This is equivalent to the statement
that the Brownian sheet $W$ satisfies
$W(t,1)=0$ for some $t\ge 1$, which happens with
probability one
since $t\mapsto W(t,1)$ is a Brownian motion,
and hence is point-recurrent.]

I will say a few things in the final section
of this paper about the
aforementioned analytical methods and their
potential-theoretic connections in turn.
However, this paper
is chiefly concerned with the aspects of quasi-sure analysis
that are close in spirit to what I believe may be
the general theme of this volume; namely, methods
that are based on finite-dimensional processes,
concentration, and
Gaussian inequalities.

On a few occasions, $\Omega$ will denote the
space of continuous functions $f:[0,1]\to\mathbb{R}^d$, and
$W$ will denote $d$-dimensional two-parameter
Brownian sheet, where $d\ge 1$. However, this should not cause
any confusion.

\section{A Strong Markov Property}\label{sec:SMP}
The following is an infinite-dimensional
strong Markov property of
the Brownian sheet. It is not a particularly
difficult result, but it is useful.
In addition, this is a
natural place to start our discussion.

Let $\mathfrak{F}:= \{ \mathfrak{F}(s);\, s\ge 0\}$
denote the filtration of $\sigma$-algebras defined
as follows:
For any $s\ge 0$, we first define
$\mathfrak{F}_{_{00}}(s)$ to be the $\sigma$-algebra
generated by the random variables
$\{W(r,t);\ r\in[0,s],\ t\ge 0\}$.
To each $\mathfrak{F}_{_{00}}(s)$
we can add all the $\IP$-null sets
and call the resulting $\sigma$-algebra
$\mathfrak{F}_{_0}(s)$. Finally, we make this
completed filtration right-continuous
in the usual way; namely, we let
$\mathfrak{F}(s):= \cap_{u>s}\mathfrak{F}_{_0}(u)$.

\begin{theorem}[A Strong Markov Property]
   \label{th:SMP}
   If $S$ is a finite $\mathfrak{F}$-stopping time,
   then the process
   $t\mapsto W(S,t)$ is measurable
   with respect to $\mathfrak{F}(S)$. Moreover,
   the infinite-dimensional process $s\mapsto
   W(S+s,\bullet)-W(S,\bullet)$
   is totally independent of $\mathfrak{F}(S)$,
   and has the same law as $W$.
\end{theorem}

\begin{remark}
   This is a simple consequence of J.\@ B.\@ Walsh's
   strong Markov property with respect to weak
   stopping points; cf.~\cite[Theorem 3.6]{walsh:86}
   or~\cite[Theorem 1.6]{walsh:84} for details.
\end{remark}

\begin{proof}
   Let $I(j;n)$ denote the half-open
   interval
   $[ j2^{-n},(j+1)2^{-n})$, and
   for any fixed real number $r>0$, define
   \begin{equation}
      S_{n,r}:= \sum_{j=0}^{\lfloor
      2^n r \rfloor} j2^{-n}
      \mathbf{1}_{ I(j;n) }(S).
   \end{equation}
   Since $S$ is an $\mathfrak{F}$-stopping time,
   so is $S_{n,r}$ for any fixed $n,r$;
   moreover, we have $S_{n,r}\le S\wedge r$, and
   as $n\uparrow+\infty$, then
   $S_{n,r}\uparrow S\mathbf{1}_{\{ S\le r\}}$.
   Now
   \begin{equation}
      W(S_{n,r},t)=\sum_{j=0}^{\lfloor
      2^n r \rfloor} W(j2^{-n},t)
      \mathbf{1}_{ I(j;n) }(S).
   \end{equation}
   In particular, for any $t_1,\ldots,t_k\ge 0$,
   the vector $(W(S_{n,r},t_i))_{1\le i\le k}$
   is $\mathfrak{F} (S_{n,r})$-measurable,
   which is another way to say that $W(S_{n,r},\bullet)$
   is $\mathfrak{F} (S_{n,r})$-measurable. Since
   $\mathfrak{F} (S_{n,r})\subseteq\mathfrak{F}(S)$,
   this shows that $W(S_{n,r},\bullet)$ is
   $\mathfrak{F}(S)$-measurable. Let $n\uparrow\infty$
   and $r\uparrow\infty$ (along rationals), and use
   the path continuity of $W$ to see that
   $W(S,\bullet)$ is $\mathfrak{F}(S)$-measurable,
   as asserted. Suppose $\Phi$ is the random function
   \begin{equation}
        \Phi(u):=
        \prod_{i=1}^m \phi_i \Big(
         W(u+s_i,t_i) - W(u,t_i)
         \Big),
   \end{equation}
   where $\phi_1,\ldots,\phi_m$ are
   bounded continuous
   functions, and $t_1,\ldots,t_m\ge 0$. Then,
   for any bounded $\mathfrak{F}(S_{n,r})$-measurable
   random variable $\xi$,
   \begin{equation}
      \IE \left\{ \Phi ( S_{n,r} )  \cdot \xi \right\}
      \quad= \sum_{j=0}^{\lfloor 2^n r\rfloor}
      \IE  \Big\{ \Phi ( j2^{-n} )  \cdot \xi
      \mathbf{1}_{
      I(n;j)}(S) \Big\}.
   \end{equation}
   The term ``$\xi$ times the indicator function'' is
   $\mathfrak{F}(j2^{-n})$-measurable since
   $S$ is a stopping time. Therefore, the stationary
   independent-increments property of Brownian sheet
   implies that
   \begin{equation}
      \IE  \left\{ \Phi ( S_{n,r} )  \cdot \xi \right\}
      = \IE \{\Phi(0)\}\cdot
      \sum_{j=0}^{\lfloor 2^n r\rfloor}
      \IE  \left\{  \xi \mathbf{1}_{
      I(n;j)}(S) \right\}=\IE \{ \Phi(0)\}\cdot \IE \{\xi\}.
   \end{equation}
   This shows that a.s.,
   \begin{equation}
      \IE  \left\{\Phi(S_{n,r})\, \Big|\,\mathfrak{F}(S_{n,r})
      \right\}
      = \IE \{ \Phi(0)\},
   \end{equation}
   which is the desired
   strong Markov property in the case that
   $S\equiv S_{n,r}$. For the general case,
   we let $n,r\uparrow\infty$ along rationals,
   and use the fact
   that $\mathfrak{F}(S_{n,r})\uparrow \mathfrak{F}(S)$,
   and that $\Phi(S_{n,r})\to \Phi(S)$ boundedly,
   together with H.\@ F\"ollmer's multiparameter
   version
   of Hunt's lemma (\cite[Lemma 2.3]{follmer}), to see that
   $\IE \{\Phi(S)\,|\,\mathfrak{F}(S)\}= \IE \{\Phi(0)\}$. This
   completes our proof.
\end{proof}

The above immediately yields a 0--1
law of the following form:

\begin{corollary}[A Zero-One Law]
   \label{co:0-1}
   If $S$ is a finite $\mathfrak{F}$-stopping time,
   then the following $\sigma$-algebra is trivial:
   \begin{equation}
      \mathfrak{A}(S) := \bigcap_{s\in\mathbb{Q}_+}
      \sigma \Big\{ W(S+s,\bullet)-W(S,\bullet)\Big\}.
   \end{equation}
\end{corollary}

\begin{proof}
   This follows from Theorem~\ref{th:SMP}
   and R.\@ M.\@ Blumenthal's $0$-$1$ law (cf.~\cite{blumenthal}).
   However, we include an argument that we will need
   later on, but do not wish to repeat.
   Consider the infinite-dimensional process
   $X(s;\delta):= W(S+s,\bullet)-W(S+\delta,\bullet)$,
   as $s$ varies over $(\delta,\infty)$ and $\delta>0$
   is a fixed number.
   Thanks to Theorem~\ref{th:SMP} applied to
   the $\mathfrak{F}$-stopping times of the form
   $r+T$ (where $r$ is nonrandom),
   for any fixed $\delta>0$, all
   $s_1,s_2,\ldots,s_n> \delta$, and
   for all bounded continuous functionals
   $\phi_1,\ldots,\phi_m$,
   \begin{equation}
      \IE \left.\left\{ \prod_{j=1}^m \phi_j
      \left( X(s_j;\delta) \right)
      \, \right|\, \mathfrak{X}(\delta)
      \right\} =
      \IE  \left\{ \prod_{j=1}^m \phi_j
      \left( X(s_j;\delta) \right)\right\},\qquad\text{a.s.},
   \end{equation}
   where $\mathfrak{X}(\delta)$
   is the $\sigma$-algebra generated by
   $\{W(S+u,\bullet)-W(S,\bullet);\, u\in [0,\delta)\}$. Since
   $\cap_{\delta\in\mathbb{Q}_+}\mathfrak{X}(\delta)
   =\mathfrak{A}(S)$, we can let $\delta\downarrow 0$
   and apply Hunt's lemma
   (cf. C.\@ Dellacherie and
   P.-A.\@  Meyer~\cite[Chapter V, p. 25]{DM:80}) to deduce that
   \begin{equation}\label{eq:trivial}
         \IE \left.\left\{ \prod_{j=1}^m \phi_j
         \left( X(s_j;0) \right)
         \, \right|\, \mathfrak{A}(S)
         \right\} =
         \IE  \left\{ \prod_{j=1}^m \phi_j
         \left( X(s_j;0) \right)\right\},\qquad\text{a.s.}
   \end{equation}
   The restriction $s_i>\delta$ has been removed
   since $\delta$ has been allowed to go to zero
   while keeping the $s_i$'s fixed. Thus,
   (\ref{eq:trivial}) holds for all bounded continuous
   functionals $\phi_1,\ldots,\phi_m$ and all
   $s_1,\ldots,s_m>0$. A monotone class argument
   shows that $\mathfrak{A}(S)$ is independent of
   itself, and is trivial as a result.
\end{proof}

\section{A Law of the Iterated Logarithm}
\label{sec:zimmerman}

\begin{theorem}[G.~J.~Zimmerman~{\cite[Theorem 3]{zimmerman}}]
    \label{th:zimmerman}
    Quasi-every $f\in\Omega$
    satisfies the law of the iterated logarithm.
\end{theorem}

\begin{remark}
   For an analytic proof see
   \cite[Theorem 4]{fukushima}.
\end{remark}

Equivalently, Zimmerman's LIL
states that with probability one,
\begin{equation}
    \limsup_{t \downarrow 0}\frac{ U(s,t) }{
    \sqrt{2 t \log \log \left( \frac 1t \right)}} =1,
    \qquad\forall
    s\in[0,1].
\end{equation}
Of course, the point is that the null set in question
does not depend upon $s\in[0,1]$ (or for that
matter upon $s\ge 0$ by scaling).
It is easy to see that the preceding equation can be translated
to the following statement about the Brownian
sheet: With probability one,
\begin{equation}\label{eq:zimmerman}
    \limsup_{t \downarrow 0}\frac{ W(s,t) }{
    \sqrt{2st \log \log \left( \frac 1t \right)}} =1,
    \qquad\forall
    s\in[1,e].
\end{equation}
In the next two subsections we will prove this particular
reformulation of Theorem \ref{th:zimmerman}.
Before getting on with proofs, I would like
to mention---without proof---the following
theorem of \cite{mountford:92}.
Recall that a function $G:\mathbb{R}_+\to\mathbb{R}$
is an \emph{upper function}
for a function $g:\mathbb{R}_+\to\mathbb{R}$
if there exists $x_0>0$
such that for all $x\in[0,x_0]$, $g(x)\le G(x)$.

\begin{theorem}[T.\@ S.\@ Mountford~\cite{mountford:92}]
    \label{th:mountford}
    An increasing function
    $t\mapsto\sqrt{t}\phi \left( \frac 1t\right)$ is in the upper
    class of quasi-every function in $\Omega$ if
    and only if
    \begin{equation}\label{eq:mountford}
        \int_4^\infty \phi^3(t) e^{-\phi^2(t)/2} \,
        \frac{dt}{t}<+\infty.
    \end{equation}
\end{theorem}
On the other hand, the upper class for almost
all continuous paths has a different characterization
that is described in the
classic paper \cite{erdos}; it states the following:

\begin{theorem}[P.\@ Erd\H{o}s~\cite{erdos}]
    \label{th:erdos}
    An increasing function
    $t\mapsto\sqrt{t}\phi\left( \frac 1t\right)$ is in the upper
    class of almost every function in $\Omega$ if
    and only if
    \begin{equation}\label{eq:erdos}
        \int_4^\infty \phi (t) e^{-\phi^2(t)/2} \,
        \frac{dt}{t}<+\infty.
    \end{equation}
\end{theorem}

To illustrate, for any $\alpha>0$ define
\begin{equation}
    \phi_\alpha(t) := \sqrt{2 \log\log \left(
    \frac 1t\right)
    +\alpha \log\log\log \left(
    \frac 1t\right) },\qquad\forall t>4,
\end{equation}
and note that when $\alpha\in(3,5]$,
$\phi_\alpha$ is an upper
function for almost all continuous paths,
but it is \emph{not} an upper function
for quasi-all of them.

We conclude this section by describing our
proof of Theorem \ref{th:zimmerman}.

\subsection{Upper Bound}\label{sec:zimmermanUB}
As in Khintchine's classical proof of the
law of iterated logarithm, we begin by verifying
the following half of (\ref{eq:zimmerman}):
With probability one,
\begin{equation}\label{eq:zimmerman-UB}
    \limsup_{t \downarrow 0}
    \frac{ W(s,t) }{
    \sqrt{2st \log \log \left( \frac 1t \right)}}
    \le 1,\qquad\forall
    s\in[1,e].
\end{equation}

To prove this, we will need an infinite-dimensional
reflection principle that we state in the following
abstract form.

\begin{lemma}[The Reflection Principle]\label{le:reflection}
    If $B$ is a continuous Brownian motion in a separable
    Banach space $\mathbb{B}$, and if $\mathcal{N}$ is any
    seminorm on $\mathbb{B}$ that is compatible with the
    topology of
    $\mathbb{B}$, then for all $T,\lambda>0$,
    \begin{equation}
        \IP \left\{ \sup_{t\in[0,T]}
        \mathcal{N}(B(t)) \ge \lambda \right\}
        \le 2 \IP \left\{ \mathcal{N}(B(T))
        \ge\lambda \right\}.
    \end{equation}
\end{lemma}

\begin{proof}
    We follow the original ideas of
    D.\@ Andr\'e and P.\@ L\'evy that were developed
    for 1-dimensional Brownian motion.

    Consider the stopping time
    \begin{equation}
        \sigma :=\inf\left\{ t>0:\,
        \mathcal{N}(B(t))\ge\lambda
        \right\}.
    \end{equation}
    Since $\mathcal{N}$ is compatible with the topology of
    $\mathbb{B}$, and since $W$ is continuous,
    $\mathcal{N}(B(\sigma))=\lambda$ on $\{\omega:\,
    \sigma(\omega)<+\infty\}$. Now
    \begin{eqnarray}
        \lefteqn{\IP\left\{ \sup_{t\in[0,T]}
            \mathcal{N}(B(t))\ge\lambda \right\} =
            \IP\left\{ \mathcal{N}(B(T))
            \ge\lambda\right\}+
            \IP\left\{ \sigma<T ~,~
            \mathcal{N}(B(T))<\lambda\right\}}
            \nonumber\\
        & =& \IP\left\{ \mathcal{N}(B(T))
            \ge\lambda\right\}+
            \IP\left\{ \sigma<T ~,~
            \mathcal{N}(B(T)-B(\sigma)+B(\sigma))<
            \lambda\right\}\\%\qquad\qquad\qquad\quad\\
        & =& \IP\left\{ \mathcal{N}(B(T))
            \ge\lambda\right\}+
            \IP\left\{ \sigma<T ~,~
            \mathcal{N}(-B(T) +2B(\sigma))<\lambda\right\},
            \nonumber
    \end{eqnarray}
    thanks to the symmetry and independent-increments (i.e.,
    the strong Markov; cf. Theorem~\ref{th:SMP})
    properties of $W$. But the seminorm
    property of $\mathcal{N}$ insures us of its subadditivity.
    Thus, on $\{\sigma<+\infty\}$ we have
    \begin{equation}
        \mathcal{N}(-B(T)+2B(\sigma))\ge
        2\mathcal{N}(B(\sigma)) - \mathcal{N}(B(T))
        = 2\lambda - \mathcal{N}(B(T)).
    \end{equation}
    The previous two displays, used in conjunction,
    prove the result.
\end{proof}

\begin{proof}[Proof of (\ref{eq:zimmerman-UB})]
    Fix $c,\theta>1$, and consider the measurable events
    \begin{equation}
        \mathsf{F}_n := \left\{\omega:\
        \exists s\in[1,e],\,
        \sup_{0\le t\le \theta^{-n}} W(s,t) \ge
        \sqrt{2cs \theta^{-n}\log\log \theta^n}
        \right\}.
    \end{equation}
    We can rewrite $\mathsf{F}_n$ as follows.
    \begin{equation}
        \mathsf{F}_n  = \left\{\omega:\
        \sup_{0\le t\le \theta^{-n}}
        \mathcal{N}(B(t)) \ge \sqrt{2c \theta^{-n}
        \log\log \theta^n}  \right\},
    \end{equation}
    where for all $f\in\Omega$,
    $\mathcal{N}$ is the seminorm
    \begin{equation}
        \mathcal{N}(f)  = \sup_{1\le s\le e} \frac{
            f(s)}{\sqrt{s}},
    \end{equation}
    and $t\mapsto B(t)$ is the following Brownian motion
    in the Banach space $\mathbb{B}$ of continuous
    functions on $[1,e]$ endowed with its compact-open
    topology:
    \begin{equation}
        B(t) (s)  = \frac{W(s,t)}{\sqrt{s}}.
    \end{equation}
    It follows readily that
    all of the assumptions of the reflection principle
    are verified in the present context; cf. Lemma
    \ref{le:reflection}. Thus, the latter lemma implies
    that
    \begin{equation}\begin{split}
        \IP \left( \mathsf{F}_n \right)
            & \le 2 \IP\left\{ \sup_{1\le s\le e}
            \frac{W(s,\theta^{-n})}{\sqrt{s}}
            \ge\sqrt{2c \theta^{-n}
            \log\log \theta^n} \right\}\\
        & = 2 \IP \left\{ \sup_{0\le s\le 1}
            O(s) \ge \sqrt{2c \log\log \theta^n} \right\},
    \end{split}\end{equation}
    where $O$ denotes a one-parameter Ornstein--Uhlenbeck
    process. C.\@ Borell's inequality (\cite{borell})
    shows that as $n\to\infty$, we have the estimate
    $\IP \left\{ \mathsf{F}_n \right\}
    \le n^{-c+o(1)}.$
    Since $c>1$, $n\mapsto \IP\{ \mathsf{F}_n\}$
    forms a summable
    sequence in $n$; thus, by the Borel--Cantelli
    lemma, with probability one,
    eventually $\mathsf{F}_n$ does not occur.
    Equivalently, with probability one,
    \begin{equation}
        \limsup_{n\to\infty}
        \frac{\sup_{0\le t\le \theta^{-n}}
        W(s,t)}{\sqrt{2 s \theta^{-n}
        \log\log \theta^n}}\le \sqrt{c},
        \qquad\forall s\in[1,e].
    \end{equation}
    Since $c,\theta>1$  are arbitrary, monotonicity arguments
    yield (\ref{eq:zimmerman-UB}).
\end{proof}

\subsection{Lower Bound}
Theorem \ref{th:zimmerman} now follows once we show that
\begin{equation}
    \label{eq:zimmerman-LB}
    \limsup_{t\downarrow 0}
    \frac{ W(s,t) }{
    \sqrt{2st \log \log \left( \frac 1t \right)}}
    \ge 1,\qquad\forall
    s\in[1,e].
\end{equation}
\begin{proof}[Proof of (\ref{eq:zimmerman-LB})]
   Fix four constants $\varepsilon,c\in(0,1)$, $\tau>0$,
   and $\theta>1$,
   and consider the events
   \begin{equation}
       \mathsf{E}_n
       := \left\{ \omega:\
       \forall s\in[\tau, \tau(1+\varepsilon) ],\
       \frac{W(s,\theta^{-n})-
       W(s,\theta^{-n-1})}{ \sqrt{2cs
       (\theta^{-n}-\theta^{-n-1})
       \log\log \theta^n}}\ge 1 \right\}.
   \end{equation}
   Evidently, independently of $\tau>0$,
   \begin{equation}
      \IP ( \mathsf{E}_n )
      = \IP \left\{ \inf_{1\le s\le 1+\varepsilon}
      \frac{B(s)}{\sqrt{s}} \ge
      \sqrt{2c \log\log \theta^n}
      \right\},
   \end{equation}
   where $B$ is a Brownian motion. Trivially, for any
   given $c'\in(c,1)$, and as $n\to\infty$,
   \begin{equation}\begin{split}
      \IP  &\left( \mathsf{E}_n \right)
         \ge \IP\left\{ \inf_{
         1\le s\le 1+\varepsilon}
         B(s) \ge \sqrt{2c (1+\varepsilon)
         \log\log \theta^n}
         \right\}\hskip1in\\
      & \ge \IP \left\{ \sup_{
         1\le s\le 1+\varepsilon}
         \left| B(s)-B(1) \right| \le 1
         \right\} \IP \left\{ B(1) \ge
         \sqrt{2c'(1+\varepsilon)
         \log\log \theta^n }\right\}  \\
      &= n^{-c'(1+\varepsilon) + o(1) }.
   \end{split}\label{eq:lower}\end{equation}
   Now if we also insist that $c(1+\varepsilon)<1$,
   then we can arrange things so that
   $c'(1+\varepsilon)<1$. In this case, the independence
   of $\mathsf{E}_1,\mathsf{E}_2,\ldots$, used in conjunction
   with (\ref{eq:lower}) and the Borel--Cantelli lemma,
   shows that infinitely many $\mathsf{E}_n$'s occur
   with probability one. Consequently, as long
   as $c(1+\varepsilon)<1$, then outside one null
   set, the following holds simultaneously for all
   $s\in[\tau,\tau(1+\varepsilon)]$:
   \begin{equation}\label{eq:19}\begin{split}
      &\limsup_{n\to\infty} \frac{W(s,\theta^{-n} )}{
         \sqrt{2cs(\theta^{-n}-\theta^{-n-1})
         \log\log \theta^n}}
         \hskip1.9in\\
      &\ \ge 1 - \limsup_{n\to\infty} \frac{
         |W(s,\theta^{-n-1} ) |}{
         \sqrt{2cs(\theta^{-n}-\theta^{-n-1})\log\log
         \theta^n}}.\hskip1.5in
   \end{split}\end{equation}
   The already-proven upper bound (cf.
   \S\ref{sec:zimmermanUB})
   implies that a.s., and simultaneously for all
   $s\in[\tau,\tau(1+\varepsilon)]$,
   \begin{equation}\label{eq:20}\begin{split}
      &\limsup_{n\to\infty} \frac{
         |W( s,\theta^{-n-1} ) |}{
         \sqrt{2cs(\theta^{-n}-\theta^{-n-1})\log\log
         \theta^n}}\hskip1.9in\\
      &\ = \frac{1}{\sqrt{\theta-1}}
         \limsup_{n\to\infty} \frac{
         |W( s,\theta^{-n-1} ) |}{
         \sqrt{2cs \theta^{-n-1} \log\log
         \theta^{n+1}}} \le \frac{1}{\sqrt{c(\theta-1)}}.
   \end{split}\end{equation}
   Thus, by (\ref{eq:19}) and (\ref{eq:20}),
   a.s., and simultaneously for all
   $s\in[\tau,\tau(1+\varepsilon)]$,
   \begin{equation}\label{eq:21}\begin{split}
      &\limsup_{t\to 0} \frac{W(s,t)}{
         \sqrt{2cs t\log\log \left( \frac 1t \right)}}
         \ge \limsup_{n\to\infty} \frac{W( s,\theta^{-n})}{
         \sqrt{2c s \theta^{-n} \log\log
         \theta^n}}\hskip1in\\
      &\ \ge \sqrt{ 1-\frac{1}{\theta} }\cdot
         \limsup_{n\to\infty}
         \frac{W( s,\theta^{-n} ) }{
         \sqrt{2c s (\theta^{-n}-\theta^{-n-1})
         \log\log \theta^n}}
         \ge \sqrt{1 - \frac 1\theta} -
         \sqrt{\frac {1}{c\theta}}.
   \end{split}\end{equation}
   Since $\theta>1$ is arbitrary, we can let
   $\theta\uparrow+\infty$
   along a rational sequence to deduce that if
   $c(1+\varepsilon)>1$, then almost surely,
   \begin{equation}
      \limsup_{t\to 0} \frac{W(s,t)}{
      \sqrt{2cs t\log\log \left( \frac 1t \right)}} \ge 1,
      \qquad \forall s\in[\tau,\tau(1+\varepsilon)],\
      \forall\tau\in\mathbb{Q}_+.
   \end{equation}
   Let $c\uparrow (1+\varepsilon)^{-1}$ along a rational
   sequence to see that
   \begin{equation}
      \limsup_{t\to 0} \frac{W(s,t)}{
      \sqrt{2 s t\log\log \left( \frac 1t \right)}} \ge \sqrt{
      \frac{1}{1+\varepsilon}},
      \qquad \forall s\in[\tau,\tau(1+\varepsilon)],\
      \forall\tau\in\mathbb{Q}_+.
   \end{equation}
   Equation (\ref{eq:zimmerman-LB}) follows from this
   readily.
\end{proof}

\section{J.\@ B.\@ Walsh's Proof of Theorem~\ref{th:zimmerman}}

The argument that was used to derive
Theorem~\ref{th:zimmerman} (essentially
due to G.\@ J.\@ Zimmerman) is quite natural,
and has other uses in quasi-sure analysis
as we shall see in the next section.
I now wish to present a different
derivation of Theorem~\ref{th:zimmerman}---due to
J.\@ B.\@ Walsh---that is elegant and short.
It also has striking
consequences on the ``propagation of
singularities'' along the Brownian
sheet.
The main ingredient of Walsh's proof is
the celebrated ``section theorem'' of
\cite{meyer} that,
borrowing from the words of M.\@ Sharpe,
``is one of the prime achievements
of [stochastic analysis].'' See
\cite[p. 388]{sharpe}.

\subsection{P.-A.\@  Meyer's Section Theorem}

In order to describe a version of Meyer's section
theorem that is suitable for our needs, we need
to recall a few notions from the general theory
of processes.

Let $(\Omega,\mathfrak{G},\IP)$ denote a
filtered probability space, where the filtration
$\mathfrak{G}
:=(\mathfrak{G}_t)_{t\ge 0}$ satisfies the
``usual conditions'' of stochastic analysis,
i.e., $\mathfrak{G}_0$ contains all the
$\IP$-null sets, and $\mathfrak{G}_t=\cap_{r>t}\mathfrak{G}_r$.
A stochastic process $\{X_t\}_{t\ge 0}$
is said to be
\emph{optional} if: (i) For all $t\ge 0$,
$X_t$ is $\mathfrak{G}_t$-measurable; and
(ii) $t\mapsto X_t(\omega)$ is right-continuous
with left-limits for each $\omega\in\Omega$.
The \emph{optional}
$\sigma$-algebra $\mathfrak{O}$
is the smallest $\sigma$-algebra of
subsets of $[0,\infty)\times\Omega$ that renders
optional processes measurable; i.e.,
$\mathfrak{O}$ is the $\sigma$-algebra generated
by all sets of the form
$\{ (t,\omega)\in[0,\infty)\times\Omega:\
X_t(\omega)\in A \}$ where
$A\subseteq\mathbb{R}$ is measurable
and $X$ is an optional process.
Finally, a stochastic set
$\Gamma \subseteq [0,\infty)\times\Omega$
is \emph{optional} if it is measurable
with respect to $\mathfrak{O}$.

\begin{theorem}[P.-A.\@  Meyer~{\cite[Chapter IV, pp.
   84--85]{meyer}}]
   \label{th:section}
   If $F$ is an optional set, then for
   every $\varepsilon>0$, there exists a stopping
   time $T_\varepsilon:\Omega\to[0,\infty)$ such that
   $\IP\{ T_\varepsilon<+\infty\} \ge \IP\{\Pi (F)
   \}-\varepsilon$,
   where $\Pi$ is the natural projection
   from $[0,\infty)\times\Omega$ onto $\Omega$.
\end{theorem}

\subsection{J.\@ B.\@ Walsh's Proof.}
For all $s\in[0,1]$ define
\begin{equation}
   L_s := \limsup_{t\to 0^+}\frac{W(s,t)}{
   \sqrt{2 t\log \log \left( \frac 1t\right)}}.
\end{equation}
The standard law of the iterated logarithm implies
that for each fixed $s>0$,
$L_s=\sqrt{s}$, a.s. In particular, thanks
to Fubini's theorem,
\begin{equation}\label{zero-leb}
   \text{Leb}\left\{ s>0:\
   L_s \neq \sqrt{s}\right\} =0,\qquad
   \text{a.s.},
\end{equation}
where $\text{Leb}$ denotes Lebesgue's
measure on $\mathbb{R}$.
Our goal is
to show that $\IP\{\forall s>0:\ L_s=\sqrt{s}\}=1$.
Suppose to the contrary
that $\IP\{\exists s>0:\ L_s\neq
\sqrt{s}\}>0$.
We will use the section theorem to obtain
a contradiction. To do so, we need to meet
the conditions of Theorem~\ref{th:section}.
Let $\mathfrak{G}_t:= \mathfrak{F}(t)$ (the
filtration of  \S\ref{sec:SMP}), that
we recall satisfies the usual conditions.
Let
\begin{equation}
   F:= \left\{ (t,\omega):\
   L_t(\omega) \neq \sqrt{t}
   \right\}.
\end{equation}
Since $s\mapsto W(s,\bullet)$ is continuous
(in the space of continuous functions on
$[0,1]$ endowed with compact-open topology),
and since $\mathfrak{F}$ is generated by the
latter process, the stochastic set $F$
is optional. By the section
theorem (Theorem~\ref{th:section}),
there would then exist a finite
$\mathfrak{F}$-stopping time $S$,
such that $\IP\{ L_S \neq \sqrt{S}\}>0$.
(Technical remark: The process $s\mapsto W(s,\bullet)$
is not real-valued. However, by considering
all processes of the form $\int W(s,t)\, \mu(dt)$
where $\mu$ is a linear combination of
point masses, we can see that
$\mathfrak{F}$ is generated by continuous
real-valued processes, so that the section theorem
can be applied as stated.)
Without loss of generality, we can assume
that there exists a $\delta>0$ (fixed and
nonrandom), such that
$\IP\{ L_S <\sqrt{S}-\delta \}>0$.
(If for some $\delta>0$,
$\IP\{ L_S >\sqrt{S} + \delta \}>0$, then a
similar argument can be invoked to
get a contradiction.)

Thanks to the strong
Markov property (Theorem~\ref{th:SMP})
and the usual LIL, for any $s>0$, there exists
a null set off of which,
\begin{equation}
   L_{S+s} \le L_S +
   \limsup_{t\to 0^+} \frac{W(S+s,t)-W(S,t)}{
   \sqrt{2t\log\log \left( \frac{1}{t} \right)}}
   =L_S + \sqrt{s}.
\end{equation}
Therefore, for all $s>0$ sufficiently small,
\begin{equation}
   \IP \left\{L_{S+s} < \sqrt{S+s}
   \right\} \ge \IP  \{ L_{S+s} <\sqrt{S} + \sqrt{s}
   -\delta \}>0.
\end{equation}
We can integrate this $[ds]$
and use Fubini's theorem to deduce
that  with positive
probability,
\begin{equation}
   \text{Leb}
   \left\{  s>0:\ L_{S+s} <\sqrt{S+s} \right\}>0,
\end{equation}
which contradicts (\ref{zero-leb}).\hfill$\square$

\section{Modulus of Continuity}

A well-known result of P.\@ L\'evy (\cite{levy}) states that
almost all $f\in\Omega$ have
the following uniform modulus of continuity:
\begin{equation}\label{eq:modulus}
   \limsup_{\varepsilon\to 0^+} \sup_{
   \scriptstyle u,v\in[0,1]:
   \atop\scriptstyle |u-v|\le\varepsilon}
   \frac{ \left| f(u) - f(v) \right|}{
   \sqrt{2\varepsilon\log\left( \frac{1}{\varepsilon}
   \right)}}=1.
\end{equation}
In an elegant paper that popularized the
subject of quasi-sure analysis, M.\@ Fukushmia
proved the following quasi-sure analogue.

\begin{theorem}[M.\@ Fukushima~{\cite[Theorem 3]{fukushima}}]
   \label{thm:fukushima}
   Quasi-every $f\in\Omega$
   has the uniform modulus of continuity
   described by (\ref{eq:modulus}).
\end{theorem}

The argument of \cite{fukushima} involves
infinite-dimensional analysis and Dirichlet form
estimates. Instead of going that route, we follow a more
classical route that has the advantage of providing
us with a more delicate result. To explain this
extension, we first recall that
in their book~(\cite{CR1}), M.\@ Cs\"org\H{o} and
P.\@ R\'ev\'esz
have shown us that  even if we replace the
$\limsup$ by a proper limit there,
(\ref{eq:modulus}) holds for almost every function.
By adapting their argument, we plan to prove the
following refinement of Theorem
\ref{thm:fukushima}.

\begin{theorem}[M.\@ Fukushima~{\cite[Theorem
   3]{fukushima}}]\label{thm:fukushima'}
   Quasi-every $f\in\Omega$ satisfies
   (\ref{eq:modulus}) with $\limsup$ replaced
   by a proper limit.
\end{theorem}

\begin{proof}
   Our goal is to show that with probability one,
   \begin{equation}
      \lim_{\varepsilon\to 0^+} \sup_{
      \scriptstyle u,v\in[0,1]:
      \atop\scriptstyle |u-v|\le\varepsilon}
      \frac{ \left| U(s,u) - U(s,v) \right|}{
      \sqrt{2\varepsilon
      | \log \varepsilon |}}=1,\qquad\forall s\in[0,1].
   \end{equation}
   Instead, we will prove the following stronger
   result: Almost surely, as $\varepsilon\to 0^+$,
   \begin{equation}\label{unif:goal}
      \sup_{
      \scriptstyle u,v\in[0,1]:
      \atop\scriptstyle |u-v|\le\varepsilon}
      \frac{ \left| U(s,u) - U(s,v) \right|}{
      \sqrt{2\varepsilon
    | \log \varepsilon |}} \longrightarrow 1,
   \end{equation}
   \emph{uniformly} for all $s\in[0,1]$.
   Clearly, this is equivalent to the following statement
   about the Brownian sheet that we propose to derive:
   With probability one,
   \begin{equation}\label{unif:goal'}
      \lim_{\varepsilon\to 0^+}
      \sup_{s\in[1,e]} \left| \sup_{
      \scriptstyle u,v\in[0,1]:
      \atop\scriptstyle |u-v|\le\varepsilon}
      \frac{ \left| W(s,u) - W(s,v) \right|}{
      \sqrt{2s\varepsilon
    | \log \varepsilon |}} -1 \right| = 0.
   \end{equation}

   \subsection{The Upper Bound}
   Fix $0<\theta<1$, and
   define $\delta_n :=n^2\theta^n$,
   and $\Theta_n := \{ j\theta^n:\,
   0\le j\le\theta^{-n}\}$, and notice that
   \begin{equation}
      \IP\left\{ \max_{\scriptstyle
      u,v\in\Theta_n:\atop\scriptstyle
      |u-v|\le\delta_n}
      \left| W(s,u) - W(s,v) \right| \ge
      \sqrt{2s \delta_n \lambda} \right\} \le
      2 n^2 |\Theta_n| e^{-\lambda},
   \end{equation}
   since for every  $v\in\Theta_n$, there are
   no more than $2n^2$ many
   $u\in\Theta_n$ such that $|u-v|\le\delta_n$.
   Let $\lambda := p\log(\delta_n^{-1})$ for a fixed
   $p>1$, and appeal to our abstract form
   of reflection principle
   (Lemma \ref{le:reflection}) to deduce that for
   any $\tau>0$,
   \begin{equation}\begin{split}
      &\IP\left\{ \max_{s\in[0,\tau]}
         \max_{\scriptstyle
         u,v\in\Theta_n:\atop\scriptstyle
         |u-v|\le\delta_n}
         \left| W(s,u) - W(s,v) \right| \ge
         \sqrt{2 p \tau
         \delta_n \log(\delta_n^{-1})} \right\}\\
      &\quad\le 2 \IP\left\{
         \max_{\scriptstyle
         u,v\in\Theta_n:\atop\scriptstyle
         |u-v|\le\delta_n}
         \left| W(\tau,u) - W(\tau,v) \right| \ge
         \sqrt{2 \tau p \delta_n \log(\delta_n^{-1})}
         \right\}\\
      &\quad \le \theta^{n(p-1)+o(n)}.
   \end{split}\end{equation}
   On the other hand, by Kolmogorov's continuity theorem
   (cf. \cite[Chapter 5, Exercise~2.5.1]{MPP}
   for a suitable version
   of the latter theorem), for any integer $k\ge 1$,
   there exists a contant $A_k$ such that for all
   $r\in(0,1)$,
   \begin{equation}\label{eq:kolmogorov}
      \IE \left[ \sup_{s\in[0,e]}
      \sup_{\scriptstyle u,v\in[0,1]:\atop\scriptstyle
      |u-v|\le r} \left| W(s,u)-W(s,v) \right|^k \right]
      \le A_k r^{\frac k2}.
   \end{equation}
   Thus, thanks to the triangle inequality,
   for any fixed $\eta>0$ and $\ell\ge 0$,
   \begin{equation}
      \IP\left\{
      \sup \left| W(s,u)-W(s,v) \right| \ge \psi_n
      \right\} \le \theta^{n(p-1)+o(n)} + 2A_4 n^{-2},
   \end{equation}
   where the supremum is taken over all
   $s\in [1+\ell\eta , 1+(\ell+1)\eta]$
   and $u,v\in[0,1]$ such that $|u-v|\le\delta_n$,
   and $\psi_n:=\sqrt{2p\{ 1+(\ell+1)\eta \}\delta_n\log
   \left( \delta_n^{-1} \right)}+2n\theta^{\frac n2}$.
   The displayed probability is summable in $n$. But
   as $n\to\infty$, we have
   $\delta_n=(1+o(1))\delta_{n+1}$; moreover $p>1$
   is arbitrary. Therefore,
   the Borel--Cantelli lemma and monotonicity
   together show that with probability one
   \begin{equation}
      \limsup_{\varepsilon\to 0^+}\ \sup
      \frac{ \left| W(s,u)-W(s,v)\right| }{
      \sqrt{2\varepsilon
      | \log \varepsilon |}}
      \le \sqrt{  1+(\ell+1)\eta  },
   \end{equation}
   where the supremum is taken over all
   $s\in [1+\ell\eta , 1+(\ell+1)\eta]$
   and $u,v\in[0,1]$ such that $|u-v|\le\varepsilon$.
   For the $s$ in question, $1+(\ell+1)\eta \le s(1+\eta)$.
   Since there are only finitely many integers $\ell$
   to consider (namely, $0\le\ell\le e/\eta$),
   \begin{equation}
      \limsup_{\varepsilon\to 0^+}\sup_{s\in[1,e]}
      \sup_{\scriptstyle u,v\in[0,1]:\atop\scriptstyle
      |u-v|\le\varepsilon}
      \frac{ \left| W(s,u)-W(s,v)\right| }{
      \sqrt{2s\varepsilon | \log \varepsilon |}}
      \le \sqrt{ 1+\eta}.
   \end{equation}
   Let $\eta\downarrow 0$ along a rational sequence
   to deduce that a.s.,
   \begin{equation}\label{unif:UB}
      \limsup_{\varepsilon\to 0^+}\sup_{s\in[1,e]}
      \sup_{\scriptstyle u,v\in[0,1]:\atop\scriptstyle
      |u-v|\le\varepsilon}
      \frac{ \left| W(s,u)-W(s,v)\right| }{
      \sqrt{2s\varepsilon | \log \varepsilon |}}
      \le 1.
   \end{equation}
   This proves half of (\ref{unif:goal'}).

   \subsection{The Lower Bound}
   Notice that for any $p\in(0,1)$ fixed, all
   integers $n\ge 1$, and all $s>0$,
   \begin{equation}\begin{split}
      &\IP\left\{ \sup_{\scriptstyle
         u,v\in[0,1]:\atop\scriptstyle |u-v|\le n^{-1}}
         \left| W(s,u) - W(s,v) \right| \le \sqrt{
         \frac {2ps}{n} \log n}
         \right\}\\
      & \quad \le \IP\left\{ \max_{
         0\le j\le n-1}
         \left| W \left( s, \frac{j+1}{n}
         \right) - W \left( s, \frac{j}{n}
         \right) \right| \le \sqrt{
         \frac {2ps}{n} \log n}
         \right\}\qquad\qquad\ \ \\
      & \quad = \left( 1 - \IP \left\{
         |\mathcal{N}|  > \sqrt{2p\log n}\right\}\right)^n,
   \end{split}\end{equation}
   where $\mathcal{N}$ denotes a standard normal variable.
   Consequently, as $n\to\infty$,
   \begin{equation}
   \IP\left\{ \sup_{\scriptstyle
      u,v\in[0,1]:\atop\scriptstyle |u-v|\le n^{-1}}
      \left| W(s,u) - W(s,v) \right| \le \sqrt{
      \frac {2ps}{n} \log n}
      \right\} \le \exp\left(
      -n^{1-p+o(1)}\right),
   \end{equation}
   where $o(1)$ goes to $0$ uniformly in $s>0$.
   Let
   \begin{equation}
      \sigma_n:= \left\{ 1+jn^{-4}:\
      0\le j\le (n+1)^4\right\}
   \end{equation}
   to see that as $n\to\infty$,
   \begin{equation}
      \sum_{n=1}^\infty
      \IP\left\{
      \min_{s\in\sigma_n}
      \sup_{\scriptstyle
      u,v\in[0,1]:\atop\scriptstyle |u-v|\le n^{-1}}
      \frac{ \left| W(s,u) - W(s,v) \right|
      }{\sqrt{s}} \le \sqrt{
      \frac {2p}{n} \log n}
      \right\} <+\infty.
   \end{equation}
   Since $p\in(0,1)$ is arbitrary,
   by the Borel--Cantelli, and
   after  applying another interpolation
   argument involving (\ref{eq:kolmogorov})
   and yet another monotonicity argument,
   we conclude that a.s.,
   \begin{equation}
      \liminf_{\varepsilon\to 0^+} \inf_{s\in[1,e]}
      \sup_{\scriptstyle u,v\in[0,1]:\atop
      \scriptstyle |u-v|\le \varepsilon}
      \frac{\left|W(s,u)-W(s,v)\right|}{
      \sqrt{2s \varepsilon| \log \varepsilon |}}
      \ge 1.
   \end{equation}
   Together with (\ref{unif:UB}), this yields
   (\ref{unif:goal'}) whence (\ref{unif:goal}).
\end{proof}

\section{Nowhere-Differentiability}

A classical result of R.\@ E.\@ A.\@ C.\@ Paley,
N.\@ Wiener and A.\@ Zygmund (\cite{paley-et-al})
states that almost all continuous functions
are nowhere-differentiable;
see also~\cite{dvoretzky-et-al}. This has been extended
in various directions in \cite{CR},
a consequence of which is the following;
see~\cite[Theorem 2]{fukushima} for
an analytical proof of most of this theorem.

\begin{theorem}[M.\@ Cs\"org\H{o} and
    P.\@ R\'ev\'esz~{\cite[Theorem 3]{CR}}]\label{th:CR}
    Quasi-every $f\in\Omega$ is
    nowhere-differentiable.
\end{theorem}

We mention---without
proof---the following uniform modulus
of nondifferentiability that is a two-parameter
extension of the result of \cite{CR:79}:
With probability one, as $\varepsilon\to 0^+$,
\begin{equation}
    \inf_{t\in[0,T]}
    \sup_{u\in[0,\varepsilon]}
    \frac{\left| U(s,t+u)-U(s,t)\right|}{
    \sqrt{\varepsilon / |\log\varepsilon|}}
    \longrightarrow \frac{\pi}{\sqrt{8}}  ,
\end{equation}
\emph{uniformly} for all $s\in[0,1]$.
In particular, the Cs\"org\H{o}--R\'ev\'esz modulus of
nondifferentiability holds for quasi-all
continuous functions.

In fact, M.\@ Cs\"org\H{o} and
P.\@ R\'ev\'esz~\cite[Theorem 3]{CR}
proved the much stronger theorem that the Brownian
sheet is nowhere-differentiable along \emph{any line
in the plane.} Here we have specialized this
result to the simpler case where the lines
are parallel to one of the axes. A consequence of this
more general theorem of~\cite{CR} is that the
level sets of the Brownian sheet a.s. do not contain
straight-line segments. More recently,
R.\@ C.\@ Dalang and T.\@ S.\@ Mountford
have discovered a striking generalization of this
fact:

\begin{theorem}[
   R.\@ C.\@ Dalang and T.\@ S.\@ Mountford~\cite{DM}]
   With probability one,
   the level curves of the Brownian sheet
   do not contain
   any curve that is differentiable somewhere.
\end{theorem}

\begin{proof}[Proof of Theorem \ref{th:CR}]
    Motivated by the analysis of
    \cite{CR}, our strategy
    will be to show that
    if $T>0$ is held fixed, then with probability one,
    \begin{equation}\label{eq:4.1}
        \inf_{s\in[0,1]}\ \limsup_{n\to\infty}
        \inf_{t\in[0,T]}\
        \sup_{u\in[0,n^{-1}]} \frac{
        \left| U(s,t+u) - U(s,t)
        \right| }{n^{-1}} =+\infty.
    \end{equation}
    This implies Theorem \ref{th:CR}. With this in mind,
    let us first recall the following estimate: If $B$ denotes
    standard Brownian motion, then
    there exists a constant
    $p>0$ such that for all $x>0$,
    $-\log \IP \left\{ \sup_{0\le t \le 1}
    |B(t)| \le x \right\}$ is bounded below by
    $(p x^2)^{-1}$; cf. K.\@ L.\@ Chung
    \cite[Theorem 2]{chung}.
    Equivalently, for any $n\ge 1$ and all $t,a>0$,
    \begin{equation}
        \IP \left\{ \sup_{u\in[0,n^{-1}]}
        \left| B  \left( t+u \right) - B(t) \right|
        \le \frac{a}{n} \right\} \le
        \exp\left( -\frac{n}{p a^2} \right),
    \end{equation}
    for the same constant $p$ whose
    valued does not depend on our choice of $(a,t,n)$.
    [In fact, the optimal choice is $p= 8\pi^{-2}$.]
    Consequently,
    \begin{equation}\begin{split}
        &\IP \left\{ \inf_{t\in[0,T]}\
            \sup_{u\in[0,n^{-1}]}
            \left| B  \left( t+u \right) - B(t) \right|
            \le \frac{a}{n} \right\} \\
        &\quad \le \IP \left\{
            \min_{0\le j\le n^4}\
            \sup_{u\in[0,n^{-1}]}
            \left| B  \left(  j n^{-4}T +u \right) -
            B \left( jn^{-4}T \right)
            \right| \le \frac{2a}{n} \right\}\qquad\qquad\\
        &\qquad +\IP \left\{
            \sup \left| B(u) - B(v) \right|\ge
            \frac{a}{2n} \right\} \\
        &\quad \le (1+n^4)
            \exp\left( -\frac{n}{4p a^2}
            \right) + \left( \frac{n}{2a}\right)^2
            \IE \left[ \sup
            \left| B(u) - B(v) \right|^2 \right],
            \qquad\qquad
    \end{split}\end{equation}
    where the last two suprema
    are over all $u,v\in[0,1]$ such that
    $|u-v|\le n^{-4}T$.
    By (\ref{eq:kolmogorov}), the last expectation is
    seen to be no more than $A_2Tn^{-4}$.
    Consequently,
    \begin{equation}
        \sum_{n=1}^\infty
        \IP \left\{ \inf_{t\in[0,T]}\
        \sup_{r\in[0,n^{-1}]}
        \left| B  \left( t+r \right) - B(t) \right|
        \le \frac{a}{n} \right\}  < +\infty.
    \end{equation}
    An interpolation argument improves this condition
    to the following one for the
    Brownian sheet $W$:
    \begin{equation}
        \sum_{n=1}^\infty
        \IP \left\{ \inf_{s\in[0,e]}\
        \inf_{t\in[0,T]}\
        \sup_{r\in[ 0,n^{-1}]}
        \left| W  \left( s,t+r \right) - W(s,t) \right|
        \le \frac{a}{n} \right\}<+\infty,
    \end{equation}
    from which we can easily deduce (\ref{eq:4.1}).
\end{proof}

\section{Quadratic Variation}

We now come to the theorem that started much
of the interest in quasi-sure analysis. Namely,
D.\@ Williams's quasi-sure refinement of the
classical theorem of P.\@ L\'evy that
states that for almost every continuous function $f$,
at time $t$ the function $f$ has finite quadratic
variation $t$; cf. the appendix of~\cite{meyer:82}.

\begin{theorem}[D.\@ Williams {\cite[Appendix]{meyer}}]
   For each $n=1,2,\ldots$,
   let $0=\pi_{0,n}<\pi_{1,n}<\cdots<\pi_{n,n}=1$ denote
   a partition of $[0,1]$ such that
   $\sum_n\max_{1\le j\le n}(\pi_{j,n}-\pi_{j-1,n})<+\infty$.
   Then quasi-every $f\in\Omega$ has
   the following property: For all $t\in[0,1]$,
   \begin{equation}
      \lim_{n\to\infty} \sum_{j=1}^n \Big|
      f(\pi_{j,n} t) - f(\pi_{j-1,n}t) \Big|^2
      =t.
   \end{equation}
\end{theorem}

\begin{proof}
   In the interest of saving space, I will prove the slightly weaker
   statement
   that for each fixed $t>0$, with probability one,
   \begin{equation}\label{QV}
      \lim_{n\to\infty} \sup_{1\le s\le e}
      \left| \sum_{j=1}^n \big| W(s,\pi_{j,n} t)-
      W(s,\pi_{j-1,n} t) \big|^2 - st \right|=0.
   \end{equation}
   The proof that is to follow can be enhanced,
   using similar ideas, to show that in fact
   the above holds outside a single null set,
   uniformly for all $t\in[0,1]$, which yields the
   full statement of the theorem.

   We define the following,
   all the time keeping $t\in[0,1]$
   fixed:
   \begin{equation}
      \theta_j(s)  := \Big| W(s,\pi_{j,n} t)-
         W(s,\pi_{j-1,n} t) \Big|^2.
   \end{equation}
   A few lines of calculations then show the existence of
   a universal constant $K_1$ such that
   $\IE \{ |\theta_j(s')-\theta_j(s) |^8 \}
   \le K_1 |s-s'|^4 (\pi_{j,n}-\pi_{j-1,n})^4.$
   Thus, by the Kolmogorov continuity theorem
   (\cite[Chapter 5, Exercise~2.5.1]{MPP}), we can find a
   universal constant $K_2$ such that for all $\eta\in(0,1)$,
   $j=1,\ldots,n$, and $n=1,2,\ldots$,
   \begin{equation}\label{eq:mod}
       \IE\left\{ \max_{\scriptstyle
       s,s'\in[1,e]:\atop\scriptstyle
       |s-s'|\le\eta}
       \left| \theta_j(s)-\theta_j(s') \right|^8 \right\}
       \le K_2 \eta^3 (\pi_{j,n}-\pi_{j-1,n})^4.
   \end{equation}
   Now choose an equipartition $\mathbb{S}_n$ of $[1,e]$ with
   $\text{mesh}(\mathbb{S}_n)\to 0$ at a rate to be
   described shortly, and note that for any
   fixed $\delta>0$,
   \begin{equation}
      \IP\left\{ \max_{s\in \mathbb{S}_n} \left| V_n(s,t)
      \right| \ge \delta \right\} \le \frac{\#(\mathbb{S}_n)
      }{ \delta^2} \max_{s\in \mathbb{S}_n} \text{Var}( V_n(s,t)),
   \end{equation}
   where
   \begin{equation}
      V_n(s,t) := \sum_{j=1}^n \left|
      W(s,\pi_{j,n}t)-W(s,\pi_{j-1,n}t)\right|^2-st.
   \end{equation}
   On the other hand, $V_n(s,t)$ is a sum of $n$ i.i.d.
   random variables, and a simple computation yields
   a universal constant $K_3$ such that uniformly
   for all $s\le e$,
   $\text{\rm Var}(V_n(s,t))\le K_3\sum_{j=1}^n
   (\pi_{j,n}-\pi_{j-1,n})^2\le K_3 \max_{1\le j\le n}
   (\pi_{j,n}-\pi_{j-1,n})):= K_3 \|\Pi\|_n.$
   This yields
   \begin{equation}\label{eq:max1}
      \IP\left\{ \max_{s\in \mathbb{S}_n} \left| V_n(s,t)
      \right| \ge \frac{\delta}{2} \right\} \le
      \frac{K_3}{\delta^2}
      \#(\mathbb{S}_n) \|\Pi\|_n.
   \end{equation}
   Thus, for all $n$ large,
   \begin{equation}\label{eq:max2}\begin{split}
       \IP&\left\{ \sup_{s\in[1,e]} \left| V_n(s,t)
          \right| \ge \frac{\delta}{2} \right\} \cr
       &\le \frac{K_3}{\delta^2}
          \#(\mathbb{S}_n) \|\Pi\|_n
       + \IP \left\{ \max_{\scriptstyle s,s'\in
         [1,e]\atop\scriptstyle
         |s-s'|\le\text{mesh}(\mathbb{S}_n)} \sum_{j=1}^n
         \left| \theta_j(s)-\theta_j(s')\right|\ge \frac{\delta}{4}
         \right\}\cr
       &\le \frac{K_3}{\delta^2}
          \#(\mathbb{S}_n) \|\Pi\|_n
          + K_2 \left(\frac{4}{\delta}\right)^8
          \left[ \text{mesh}(\mathbb{S}_n)\right]^3 \sum_{j=1}^n
          (\pi_{j,n}-\pi_{j-1,n})^4\\
       &\le \frac{K_3}{\delta^2}
          \#(\mathbb{S}_n) \|\Pi\|_n
          + K_2 \left(\frac{4}{\delta}\right)^8
          \left[ \text{mesh}(\mathbb{S}_n)\right]^3 \|
          \Pi\|_n^3, \hskip1.3in
   \end{split}\end{equation}
   thanks to (\ref{eq:max1}).
   Since $\sum_n \|\Pi\|_n<\infty$
   we can always choose $\mathbb{S}_n$ such that
   the left-hand side of the preceding display is summable
   (in $n$), and this proves (\ref{QV}).
\end{proof}

\section{W.\@ S.\@ Kendall's Theorem}

Upto this point, we have adopted the viewpoint
that the Brownian sheet (equivalently,
the Ornstein--Uhlenbeck process) is a natural
diffusion on the space of continuous functions.
While this viewpoint provides us with a great
deal of insight about the sheet, it completely ignores
the effect of the geometry of the parameter space
on the process. The following is a delightful example of
the subtle effect of the geometry of the parameter
space, and was discovered by W.\@ S.\@ Kendall.

\begin{theorem}[W.\@ S.\@ Kendall
   ~{\cite[Theorem 1.1]{kendall}}]\label{th:kendall}
   If $s,t>0$ are arbitrary but fixed,
   then with probability one, the
   level curve of $W$ that goes through $(s,t)$ is
   totally disconnected.
\end{theorem}

\begin{remark}
   One can restate Theorem~\ref{th:kendall}
   in the following manner:
   Consider the level curve of $W$ that goes
   through $(s,t)$, and let
   \begin{equation}\label{CC}\begin{split}
      \Gamma(s,t) := &\text{ The connected
         component of }\Big\{ (u,v):\
         W(u,v)=W(s,t) \Big\}\\
      &\text{ that contains }(s,t).
   \end{split}\end{equation}
   By the continuity of $W$, this definition is
   perfectly well-defined, and
   Theorem~\ref{th:kendall} asserts
   that with probability one,
   $\Gamma(s,t)=\{(s,t)\}$. In other words,
   the excursion at the level $W(s,t)$ corresponding
   to the time-point $(s,t)$ is trivial. See
   the works of R.\@ C.\@ Dalang and J.\@ B.\@ Walsh~\cite{DW1,DW2}
   who present very precise descriptions of the local structure
   of the excursions of the Brownian sheet.
\end{remark}

\begin{remark}
   The null set in question depends on the
   choice of $(s,t)$. Moreover, a little thought shows
   that Theorem~\ref{th:kendall} cannot hold
   a.s. simultaneously
   for all $(s,t)$ in any given open set. In this
   sense, this result is optimal.
\end{remark}

The proof of Kendall's theorem rests on the
following zero-one law; it is a two-parameter
analogue
of the infinite-dimensional zero-law described
earlier in Corollary~\ref{co:0-1}.

\begin{lemma}[S.\@ Orey and W.\@ E.\@ Pruitt~{
   \cite[p.~140]{OP}}]\label{le:0-1}
   For any given $s,t>0$, the following
   $\sigma$-algebra is trivial:
   \begin{equation}
      \mathfrak{G}(s,t) :=
      \bigcap_{\varepsilon \in\mathbb{Q}_+}
        \sigma \Big\{ W(u,v)-W(s,t):\
        \left| (s,t)-(u,v) \right| <\varepsilon \Big\},
   \end{equation}
   where $|\cdots|$ denotes the $\ell^\infty$-norm on
   $\mathbb{R}^2$ for the sake of concreteness.
\end{lemma}

\begin{proof} \textbf{(Sketch)}
   This requires ideas that are very close to those
   introduced in the proof of
   Corollary~\ref{co:0-1}. Thus, we will
   indicate only the essential differences between
   the two proofs.

   We can think of Brownian sheet as
   the distribution function of white noise.
   Namely, let $\dot{\mathrm{W}}$
   denote one-dimensional white noise
   spread over $\mathbb{R}^2$ and \emph{define}
   the Brownian sheet $W$ as
   \begin{equation}\label{white-noise}
      W(u,v) :=\dot{\mathrm{W}}
      \Big( [0,u]\times[0,v] \Big),
      \qquad\forall u,v\ge 0.
   \end{equation}
   [The process $\dot{\mathrm{W}}$ is a
   well-defined vector-valued random measure with
   values in $L^2(\IP)$;
   cf.~\cite[p. 283--285]{walsh:84} and
   ~\cite[Chapter 5, \S1.3]{MPP} for more details.]
   In this way, we can write
   \begin{equation}
      \mathfrak{G}(s,t) = \bigcap_{\varepsilon\in
      \mathbb{Q}_+} \sigma \left\{
      \dot{\mathrm{W}}\Big(
      [0,s+\varepsilon] \times
      [0,t+\varepsilon] \setminus [0,s-\varepsilon] \times
      [0,t-\varepsilon]\Big) \right\}.
   \end{equation}
   Now suppose that $R$ is
   a rectangle with sides parallel to the axes,
   and that $R$ does not intersect the
   annulus,
   $[0,s+\varepsilon] \times
   [0,t+\varepsilon]\setminus[0,s-\varepsilon] \times
   [0,t-\varepsilon]$. Then the elementary properties
   of white noise show us that
   $\dot{\mathrm{W}}(R)$
   is independent of $\mathfrak{G}(s,t)$.
   To finish, consider a finite union of such $R$'s, and
   ``take limits'' in  a manner similar to what we
   did in the proof of Corollary~\ref{co:0-1}.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{th:kendall}]
   We may assume without loss of too much
   generality that $s=t=1$. Consider
   \begin{equation}
      \mathrm{B}(r):= \left\{ (x,y)\in\mathbb{R}^2:\
      \left| (x,y)-(1,1) \right| \le r \right\},
   \end{equation}
   which is the $\ell^\infty$-ball of radius
   $r>0$ about $(1,1)\in\mathbb{R}^2$.
   Also let $\partial \mathrm{B}(r)$ denote its Euclidean
   boundary; this is the perimeter
   boundary of the square of side $2r$ centered at
   $(1,1)$. Let
   \begin{equation}\label{J}\begin{split}
      \mathsf{J}(r)  := &\left\{\omega:\
         W(1,1) > \sup_{(u,v) \in \partial
         \mathrm{B}(r)} W(u,v) \right\}\\
      =& \left\{\omega:\
         W(1,1) > \sup_{(u,v) \in \partial
         \mathrm{B}(r)} W\left(ru+1-r,rv+1-r\right) \right\}.
   \end{split}\end{equation}
   Then the theorem follows at once from
   Lemma~\ref{le:0-1} and the following:
   \begin{equation}\label{kendall:goal}
      \liminf_{r\to 0^+} \IP
      \{ \mathsf{J} (r) \} >0.
   \end{equation}
   For then it follows that with probability one,
   infinitely many of the $\mathsf{J}(n^{-1})$'s
   occur, and clearly this
   does the job. Therefore, it suffices to prove
   (\ref{kendall:goal}). By
   (\ref{white-noise}), we can write the following
   path decomposition: For any $r\in(0,1)$ fixed,
   \begin{equation}\label{XYZ}\begin{split}
      W(1-r+ur,1-r+vr) = &\sqrt{(1-r)r}\Big[
         X(u)+Y(v)\Big]\\
      &+ rZ(u,v)-W(1-r,1-r),
      \end{split}\quad\forall u,v\ge 0,\qquad
   \end{equation}
   where $X$ and $Y$ are standard Brownian motions,
   $Z$ is a standard Brownian sheet, and the three
   are independent from one another as well as
   from $W(1-r,1-r)$. Indeed, here are the formulas
   for $(X,Y,Z)$ in terms of the white noise $\dot{\mathrm{W}}$
   of (\ref{white-noise}):
   \begin{equation}\label{eq:decompose}\begin{split}
      X(u) := & \frac{1}{\sqrt{(1-r)r}}
         \dot{\mathrm{W}} \Big( [1-r,1-r+ur]\times[0,1-r]
         \Big)\\
      Y(v) := & \frac{1}{\sqrt{(1-r)r}}
         \dot{\mathrm{W}}\Big( [0,1-r]\times[1-r,1-r+vr]
         \Big)\\
     Z(u,v) := &\frac{1}{r}\dot{\mathrm{W}} \Big(
         [1-r,1-r+ur]\times [1-r,1-r+vr]\Big).
   \end{split}\end{equation}
   We only need these formulas to check the
   assertions about $(X,Y,Z)$, and this
   is only a matter of checking a few covariances. In light of
   the second equality in (\ref{J}), we have
   $\omega\in\mathsf{J}(r)$ if and only if
   \begin{equation}\begin{split}
      &\sqrt{(1-r)r}\Big[ X(1)+Y(1)\Big] + rZ(1,1)\\
      &\hskip1in > \sqrt{(1-r)r} \Big[ X(u)+Y(v) \Big]
         +rZ(u,v),\hskip1in\
      \end{split}
   \end{equation}
   simultaneously for all $(u,v)\in\partial \mathrm{B}(1)$.
   But thanks to the sample function continuity
   of $Z$,
   the supremum of $|Z(u,v)|$ over
   all $(u,v)\in\partial \mathrm{B}(1)$ is bounded almost surely.
   Thus, we can divide the preceding display
   by $\sqrt{r}$ and let $r\to 0^+$ to see that
   \begin{equation}
      \lim_{r\to 0^+}\IP\{\mathsf{J}(r)\}
      =\IP\Big\{  \forall (u,v)\in\partial \mathrm{B}(1):\
      X(1)+Y(1) > X(u)+Y(v) \Big\},
   \end{equation}
   and it is easy to see that the latter probability
   is strictly positive. To see this, let
   $X_i:= X(i)$, $Y_i:= Y(i)$ ($i=1,2$). Also let
   $X_2^+:= \sup_{0\le u\le 2}X(u)$, and
   $Y_2^+:= \sup_{0\le v\le 2}Y(v)$, and note that
   the latter probability is equal to
   \begin{equation}\begin{split}
      &\IP \Big\{ X_1 +Y_1 >X_2 + Y_2^+,
         X_1 + Y_1 >X_2^+ + Y_2 \Big\} \\
      &\ge \IP \Big\{ X_1-X_2 \ge 1, X_1 - X_2^+ \ge -1 \Big\}
         \times \IP \Big\{
         Y_2^+-Y_1 \le 1 , Y_2-Y_1\le -1 \Big\},
   \end{split}\end{equation}
   and this is easily seen to be positive.
   This verifies (\ref{kendall:goal}) and the result
   follows.
\end{proof}

\section{Criterion for Hitting Points}
Thus far, we have only touched upon results that
hold (or do not hold) for quasi-every
one-dimensional function. This in turn has led us
to the one-dimensional Brownian sheet. Now we turn to
results in higher dimensions.
With this in mind, let $W$
denote the Brownian sheet in $d$ dimensions
and $\mathfrak{m}_d$ the standard Lebesgue
measure on the Lebesgue-measurable subsets
of $\mathbb{R}^d$. Stated in terms of Wiener measure,
this yields the following.

We begin with the classical fact that
$d$-dimensional Brownian can hit points if
and only if $d=1$.

\begin{theorem}[P.\@ L\'evy]
   The following are equivalent: Given any $x\in\mathbb{R}^d$,
   \begin{enumerate}
      \item[(i)] Almost every continuous
                 $f:[0,1]\to\mathbb{R}^d$
                 avoids $\{x\}$; i.e., $x\not\in f([0,1])$.
      \item[(ii)] Almost every continuous
                 $f:[0,1]\to\mathbb{R}^d$
                 has a Lebesgue-null range; i.e.,
                 $\mathfrak{m}_d(f([0,1]))=0$.
      \item[(iii)] $d\ge 2$.
   \end{enumerate}
\end{theorem}

For the above conditions (i) and (ii) to hold for
quasi-every function $f$, one needs the stronger
condition that $d\ge 4$. Indeed, S.\@ Orey and W.\@ E.\@ Pruitt
have proven the following result.

\begin{theorem}[S.\@ Orey and W.\@ E.\@ Pruitt~
    {\cite[Theorems 3.3 and 3.4]{OP}}]
    \label{thm:hitpoints}
    The following are equivalent: Given any
    fixed $x\in\mathbb{R}^d$:
    \begin{enumerate}
        \item[(i)] With probability one, $d$-dimensional
             Brownian sheet does not hit $\{x\}$.
        \item[(ii)] The random set $W(\mathbb{R}^2_+)$
             has zero $d$-dimensional Lebesgue measure.
        \item[(iii)] $d \ge 4$.
        \item[(iv)] Quasi-every $d$-dimensional
             continuous function avoids $x$.
    \end{enumerate}
\end{theorem}

Let $U(s,t):= e^{-s/2} W(e^s,t)$ as before, and
note that (iv) is equivalent to the
following:
\begin{equation}
   \IP\left\{ \exists s\in[0,1]:\ \text{ for some }
   t>0,\ U(s,t)=x  \right\}=0,\qquad
   \forall x\in\mathbb{R}^d.
\end{equation}
On the other hand, by the Cameron--Martin
formula, the law of $\{ U(s,t);\,
s,t\in[0,1]\}$ is mutually absolutely continuous
with respect to the law of
$\{ W(s,t);\, s\in[1,e], t\in[0,1]\}$; cf.~\cite{cameron-martin}.
Therefore, (iv)$\Leftrightarrow$(i), and we only
need to prove the equivalence of (i)--(iii).
I will describe most of this proof in three steps.

\begin{proof}[Proof of (i){\boldmath
    $\Rightarrow$}(ii)]
    By Fubini's theorem,
    \begin{equation}
       \IE\left\{ \mathfrak{m}_d \Big( W([0,1]^2) \Big)
       \right\} = \int_{\mathbb{R}^d}
       \IP\left\{ x\in W([0,1]^2) \right\}\, dx=0,
    \end{equation}
    thanks to (i). Scaling then shows
    that with probability one,
    $\mathfrak{m}_d(W(\mathbb{R}^2_+))=0$.
\end{proof}

\begin{proof}[Proof of (ii){\boldmath
    $\Rightarrow$}(iii)]
    We will use the Fourier-analytical ideas
    of~\cite{kahanebook}, and prove that if (iii)
    fails, then so will (ii).
    Thus, let us assume that $d\le 3$, and consider
    the \emph{occupation (or sojourn) measure},
    \begin{equation}\label{eq:OccMeas}
       \sigma(A) :=
       \int_0^\infty\int_0^\infty
       e^{-s-t} \mathbf{1}_A\left(
       W(s,t) \right) \, ds\, dt.
    \end{equation}
    Its Fourier transform is given by
    \begin{equation}
       \widehat{\sigma}(\xi) =
       \int_0^\infty\int_0^\infty
       e^{-s-t} e^{i\xi\cdot W(s,t)}
       \, ds\, dt,\qquad\forall\xi\in\mathbb{R}^d.
    \end{equation}
    Our strategy is to show that with probability one,
    $\widehat{\sigma}\in L^2(\mathbb{R}^d)$.
    If so, then by the Plancherel theorem,
    $\sigma$ is a.s. absolutely continuous
    with respect to $\mathfrak{m}_d$, and
    $\left(\frac{d\sigma}{d\mathfrak{m}_d}\right)\in
    L^2(\mathbb{R}^d)$ almost surely. But the fact that
    $\sigma\left(W(\mathbb{R}^2_+) \right)
    =1$ implies that
    \begin{equation}
       \mathfrak{m}_d(W(\mathbb{R}^2_+)) =
       \int_{W(\mathbb{R}^2_+)} \frac{d\sigma (\xi)}{
       d\mathfrak{m}_d}\, \mathfrak{m}_d(d\xi)
       =1.
    \end{equation}
    Thus,
    (ii)$\Rightarrow$(iii) follows once we
    show that
    $\IE\{\|\widehat{\sigma}
    \|^2_{L^2(\mathbb{R}^d)}\}<+\infty$. The latter
    expectation is equal to the following:
    \begin{equation}\begin{split}
       &\IE \left\{ \| \widehat{\sigma}
          \|^2_{L^2(\mathbb{R}^d)} \right\}\\
       &\quad= \int_{\mathbb{R}^2_+}
          \int_{\mathbb{R}^2_+}
          \int_{\mathbb{R}^d}
          e^{-s_1-s_2-t_1-t_2}
          \IE\left\{ e^{
          i\xi\cdot\left[ W(s_1,s_2) - W(t_1,t_2)
          \right] } \right\}\, d\xi\, ds\, dt\\
       &\quad= \int_{\mathbb{R}^2_+}
          \int_{\mathbb{R}^2_+}
          \int_{\mathbb{R}^d}
          e^{-s_1-s_2-t_1-t_2}
          \exp \left( -\frac{|\xi|^2 \tau^2(s,t)}{2}
          \right) \, d\xi\, ds\, dt,\hskip.8in
    \end{split}\end{equation}
    where $\tau^2(s,t):=
    \text{Var}[ W(s_1,s_2) -
    W(t_1,t_2) ]$ is the $\mathfrak{m}_d$-measure
    of the set difference between the rectangles
    $[0,s_1]\times [0,s_2]$ and
    $[0,t_1]\times [0,t_2]$. A picture will convince
    you that no matter how the two said rectangles
    are situated, we always have the bound,
    $\tau^2(s,t) \ge (s_2\wedge t_2) |s_1-t_1|
    +(s_1\wedge t_1) |s_2-t_2|.$
    Thus,
    \begin{equation}\begin{split}
       &\IE \left\{ \| \widehat{\sigma}
          \|^2_{L^2(\mathbb{R}^d)} \right\}\\
       &\displaystyle
          \quad\le 4\iint_{0\le s_1\le t_1}
          \iint_{0\le s_2\le t_2}
          \int_{\mathbb{R}^d}
          e^{-s_1-s_2-t_1-t_2}\\
       &\displaystyle\qquad\times
          \exp\left(-\frac{|\xi|^2 \left[
          s_2(t_1-s_1) + s_1(t_2-s_2)\right]
          }{2} \right)\, d\xi\, ds\, dt\\
       &\displaystyle
          \quad= 4\int_{\mathbb{R}^2_+}
          \int_{\mathbb{R}^2_+}
          \int_{\mathbb{R}^d}
          e^{-2s_1-2s_2-t_1-t_2}
          \exp\left(-\frac{|\xi|^2 \left[
          s_2t_1 + s_1t_2\right]
          }{2} \right)\, d\xi\, ds\, dt.\qquad
    \end{split}\end{equation}
    We can integrate $[dt]$ to deduce that
    $\IE \{ \| \widehat{\sigma}
    \|^2_{L^2(\mathbb{R}^d)} \}\le 4
    \int_{\mathbb{R}^d}
    \mathcal{Q}(\xi) \, d\xi,$
    where
    \begin{equation}
        \mathcal{Q}(\xi) := \int_{\mathbb{R}^2_+}
        e^{-2(s_1+s_2)}\left(1+
        \frac{|\xi|^2s_1}{2}\right)^{-1}\cdot
        \left(1+\frac{|\xi|^2s_2}{2}\right)^{-1}\, ds.
    \end{equation}
    Evidently, $\mathcal{Q}(\xi)$ is bounded, and is
    $O(|\xi|^{-4})$ as $|\xi|\to\infty$. Therefore,
    whenever $d< 4$, then
    $\mathcal{Q}\in L^1(\mathbb{R}^d)$, and so $\IE\{ \|
    \widehat{\sigma}
    \|^2_{L^2(\mathbb{R}^d)} \}<+\infty$, as asserted.
\end{proof}

\begin{proof}[Partial Proof of (iii){\boldmath$\Rightarrow$}(i)]
    This is the most interesting, as well as difficult, part
    of Theorem~\ref{thm:hitpoints}, and
    I will present a proof that is valid in the ``supercritical''
    regime $d\ge 5$. When $d=4$, the known proofs are much longer
    and not included here.
    Clearly, it suffices to show that whenever $0<a<b$,
    $\mathrm{P}\{x\in W([a,b]^2)\}=0$. Without loss of too much
    generality, we will do this for $a=1$ and $b=2$.

    Let us fix $\varepsilon\in(0,1)$ and an integer $n\ge 1$,
    and consider the covering $[1,2]^2=\cup_{i,j=0}^n I_{i,j}$,
    where $I_{i,j}:= [1+(i/n),1+(i+1)/n]
    \times [1+(j/n),1+(j+1)/n]$. Now if there exists $(s,t)\in I_{i,j}$
    such that $|W(s,t)-x|\le\varepsilon$, then
    \begin{equation}\begin{split}
       &\left| W\left( 1+\frac{i}{n},1+\frac{j}{n}\right)
          - x\right|\\
       &\quad \le  \varepsilon + \sup_{0\le u,v\le \frac1n}
          \left| W\left( 1+\frac{i}{n},1+\frac{j}{n}\right)
          - W\left( 1+\frac{i}{n}+u,1+\frac{j}{n}+v\right)\right|\\
       &\quad := \varepsilon + \delta_{i,j;n}.
    \end{split}\end{equation}
    Because of the white noise representation~(\ref{white-noise}) of $W$,
    $\delta_{i,j;n}$ is independent of $W(1+in^{-1},1+jn^{-1})$.
    Moreover, the probability density of
    $W(1+in^{-1},1+jn^{-1})$ is uniformly
    bounded above by one. Therefore,
    \begin{equation}\label{eq:supercrit}\begin{split}
       \mathrm{P}\left\{ \exists (s,t)\in I_{i,j}:\
          \left| W(s,t)-x\right| \le \varepsilon \right\}
          & \le C_d\mathrm{E} \left[ \left( \varepsilon +
          \delta_{i,j;n}\right)^d \right]\\
       &\le 2^d C_d
          \mathrm{E}\left[\varepsilon^d +\delta_{i,j;n}^d \right],
    \end{split}\end{equation}
    where $C_d$ denotes the volume of the unit ball in $\mathbb{R}^d$.
    On the other hand, by the white noise representation of $W$,
    \begin{equation}\begin{split}
       &W\left( 1+\frac{i}{n}+u,1+\frac{j}{n}+v\right)-
          W\left( 1+\frac{i}{n},1+\frac{j}{n}\right)\\
      &\quad =\sqrt{1+\frac{j}{n}}\ B(u) +
          \sqrt{1+\frac{i}{n}}\ B'(v) +Z(u,v),
    \end{split}\end{equation}
    where $B$, $B'$, and $Z$ are independent, $B$ and $B'$
    are Brownian motions, and $Z$ is a Brownian sheet.
    Consequently, we can take absolute values and maximize over
    $u,v\le n^{-1}$ to see that given $0\le i,j\le n^{-1}$,
    \begin{equation}
       \delta_{i,j;n}\le \sqrt2 \sup_{u\in[0,1/n]}
       |B(u)| + \sqrt2 \sup_{v\in[0,1/n]} |B'(v)|
       +\sup_{u,v\in[0,1/n]} |Z(u,v)|.
    \end{equation}
    This and scaling show the existence of a constant
    $K_d$ such that $\mathrm{E}[\delta_{i,j;n}^d]\le K_d n^{-d/2}.$
    Thus, according to~(\ref{eq:supercrit}),
    \begin{equation}
       \mathrm{P}\left\{ \exists (s,t)\in I_{i,j}:\
       \left| W(s,t)-x\right| \le \varepsilon \right\}
       \le  2^d C_d K_d \left[\varepsilon^d + n^{-d/2} \right].
    \end{equation}
    We can sum this over all $0\le i,j\le n$ to see that
    \begin{equation}
       \mathrm{P}\left\{ \exists (s,t)\in [1,2]^2:\
       \left| W(s,t)-x\right| \le \varepsilon \right\}
       \le  2^d C_d K_d (n+1)^2
       \left[\varepsilon^d + n^{-d/2} \right].
    \end{equation}
    Because this is valid for all $n\ge 1$, we can choose $n := \lfloor
    \varepsilon^{-2}\rfloor$, and deduce that
    \begin{equation}
       \sup_{x\in\mathbb{R}^d}\mathrm{P}\left\{ \exists (s,t)\in [1,2]^2:\
       \left| W(s,t)-x\right| \le \varepsilon \right\}
       = O\left( \varepsilon^{d-4} \right),\qquad(\varepsilon\to 0).
    \end{equation}
    In particular, if $d\ge 5$, then with
    probability one, $x\not\in W([1,2]^2)$, as claimed.
\end{proof}

\section{The O-U Process on Wiener Space and Two Open Problems}

In this section, I very briefly sketch the connection between
the process $Y$ and symmetric forms. This is the starting point
for the introduction of the methods of potential theory; an
area which is not the focus of the present article. I then conclude
this discussion by presenting two of my favorite
open problems in this general area.

We have seen already that the Ornstein--Uhlenbeck process
$Y$ of (\ref{Y}) is a stationary diffusion on $\Omega$
(the space of all real continuous functions on $[0,1]$)
whose stationary measure is Wiener's measure. Moreover, in
the usual notation of Markov processs, we have
the following for all continuous $\phi:\Omega\to\mathbb{R}_+$,
$s>0$, and $x\in\Omega$:
\begin{equation}\begin{split}
   T_s \phi (x) & :=  \IE
      \big[ \phi(Y_s)\, \big|\, Y_0=x\big] =
      \IE \left[ \phi \Big( e^{-s/2} W(e^s,\bullet) \Big)\,\Big|\,
      W(1,\bullet)=x\right]\\
   &=\IE \left[ \phi \Big( e^{-s/2} \Big[
      W(e^s,\bullet) - W(1,\bullet)\Big] +e^{-s/2}x\Big)\right]\\
   &=\IE \left[ \phi \Big( e^{-s/2} \Big[
      W(e^s-1,\bullet) \Big] +e^{-s/2}x\Big)\right]\\
   &=\IE \left[ \phi \Big( \sqrt{1-e^{-s}}
      W(1,\bullet) +e^{-s/2}x\Big)\right].
\end{split}\end{equation}
On the other hand, $W(1,\bullet)$ is just a Brownian motion,
and this leads to the following.

\begin{lemma}[Mehler's Formula]
   If $\mu$ denote the Wiener measure on
   the classical Wiener space $(\Omega,\mathcal{B}(\Omega))$,
   then the transition semigroup of the diffusion $Y$
   are given by the operator-formula: For all $s>0$ and all
   continuous functions $x\in\Omega$,
   \begin{equation}
      T_s f(x) = \int_\Omega f\left(
      \sqrt{1-e^{-s}} y + e^{-s/2}x \right)\, \mu(dy).
   \end{equation}
\end{lemma}It is also a simple matter to check
that $T_s$ is a symmetric semigroup on
the classical Wiener space; i.e., that
$\left\langle g,T_s f\right\rangle_\Omega
=\left\langle T_s g,f\right\rangle_\Omega$,
where $\langle u,v\rangle_\Omega$ is the
covariance form, $\int_\Omega uv\, d\mu$.

Thus, the standard theory of symmetric Markov processes
constructs a Dirichlet form $\mathcal{E}$ for $Y$
killed at an exponential rate that is formally
defined as follows:
\begin{equation}
   \mathcal{E}(f,g) := \lim_{s\to 0^+}
   \left\langle  s^{-1} ( f -T_s f )
   ,g\right\rangle_\Omega +\langle
   f ,g\rangle_\Omega;
\end{equation}
cf. \cite[(1.3.15) and Theorem 1.4]{fukushima-et-al}.
Now given any open set $G\subseteq\Omega$
one has the following identity; it relates the capacity
of \S1 to the Dirichlet forms of \cite{fukushima-et-al}:
\begin{equation}\label{cap}
   \text{\rm Cap}(G) = \inf \Big\{
   \mathcal{E}(f,f);\, f\in
   \text{\rm dom}(\mathcal{E}),\
   f\ge 1\ \mu\text{-a.e. on }G
   \Big\},
\end{equation}
where $\text{\rm dom}(\mathcal{E})$ denotes the domain
of $\mathcal{E}$; i.e., all $f\in L^2(\Omega)$
such that $\mathcal{E}(f,f)<+\infty.$
Furthermore, for a general set $A\subseteq\Omega$,
$\text{\rm Cap}(A)=\inf\{\text{\rm Cap}(G):\
A\subseteq G\text{ open},\}$.
The latter remarks are proved in
\cite[p. 164]{fukushima},
and are another starting point for the analytic
treatment of many of the
quasi-sure results within the references.

As promised earlier, we conclude this paper by
presenting two open problems:
\begin{enumerate}
   \item[{\bf OP-1}]
      P.\@ Malliavin has introduced a parametric
      family of Gaussian capacities, one of which is the
      $\text{\rm Cap}$ of this and the first section.
      Is there a ``truly probabilistic'' description
      (i.e., one involving concrete random processes)
      of all of these capacities? If so, do the quasi-sure
      results of this paper continue to holds if $\text{\rm
      Cap}$ is everywhere replaced by any and all
      of the said capacities? For related results, see
      the results of M.\@ Takeda (\cite{takeda}).
   \item[{\bf OP-2}]
      One of the outstanding open problems of the geometry
      of Brownian sheet is the following: Let $\mathcal{L}$
      denote the complement of the
      zero-set of an $N$-parameter Brownian sheet. Does
      the complement of $\mathcal{L}$ have an infinite
      connected component? When $N=2$, this was answered
      in the negative
      by W.\@ S.\@ Kendall (\cite{kendall}), but the general question
      remains open.
\end{enumerate}

config: arxiv


TITLE: Linear algebra (abstract) prerequisites
QUESTION [2 upvotes]: I want to understand dynamic programing, so im figuring out the irreducible and minimal path i should take. The secuence is:
1. General topology (Munkres chapters 1-4).
2. Abstract algebra.
3. Linear algebra.
4. Functional analysis.
5. Calculus of variations.
6. Dynamic prigramming.
So, im asking for some books, in specific a book that allow me to jump from 2 to 3 in an eficient maner.
Thanks, and sorry for my english!

REPLY [3 votes]: I don’t think any of those things are necessary for dynamic programming. AFAIK just understanding proofs by induction and basic algebra are sufficient to prove correctness and complexity of such things.
But not don’t know a lot of deep dynamic programming. There are probably some things you do need those subjects for.
The most relevant thing you might want to look up ahead of time is what recursion is in computer science. Other than that, just look for introductory material on algorithms. Dynamic programming can be considered an algorithmic paradigm. I took a Coursera course on algorithms for fun, Stanford I think, and it was really well explained. You might try finding that course and see if a trial is available.
Needless to say it would be silly to insist on learning all those five things before studying dynamic programming. Most of them do not seem relevant to me, with  the partial exception being linear algebra. You should just pick up all the basics of linear programming first and then backfill those subjects as necessary.
Also, leaning abstract algebra before linear algebra is silly.  Do that the other way around.


TITLE: How are these integrals equal?
QUESTION [0 upvotes]: I am struggling to understand how the highlighted two integrals are equal to each other. I changed $\sin(2bx)$ into exponential form to help, but it doesn't seem to be helping at all. Also $b$ is just a number greater than $>0$.

REPLY [1 votes]: It comes from equating/comparing the real and imaginary parts (the imaginary part being 0) of the previous line.  Not sure where that line comes from though...
The imaginary parts must cancel, giving
$$
e^{b^2} \int_0^\infty e^{-x^2} \sin 2bx \ dx = \int_0^b e^{y^2} \ dy.
$$
Now shuffle the $e^{b^2}$ over to the other side by dividing it, giving the negated exponent.


TITLE: Convergence in mean of increasing random variables
QUESTION [0 upvotes]: Let $(X_n)_{ n \in \mathbb{N}}$ be a sequence of random variables such that $X_n$ takes the value $0$ or $n$ with probability $1$. Let $a>0$ be such that $E(X_n) < a$ for all $n \in \mathbb{N}$ and $E(X_n) \uparrow a$ as $n \rightarrow \infty$.

Can one conclude that $E|X_n - a| \rightarrow0$ as $n \rightarrow \infty$?

We can write $X_n = n 1_{ A_n }$, where $A_n$ is the set on which $X_n = n$. Then $E(X_n)= n P (A_n) \uparrow a < \infty$. This means that $P(A_n)$ has to somehow balance the $n$ by going to $0$.

REPLY [1 votes]: Certainly not, since when $n>2a$ we have $|X_n-a|\ge a$ with probability $1$, so that $E|X_n-a|\ge a$ at the very least. (I suspect that $E|X_n-a|\to 1+a$.)


TITLE: How does $\sigma’(x)$ simplify to $\sigma(x)(1-\sigma(x))$?
QUESTION [0 upvotes]: \begin{aligned}
\sigma^{\prime}(x) &=\frac{\partial}{\partial x} \frac{1}{1+e^{-x}} \\
&=\frac{e^{-x}}{\left(1+e^{-x}\right)^{2}} \\
&=\frac{1}{1+e^{-x}} \cdot \frac{e^{-x}}{1+e^{-x}} \\
&=\sigma(x)(1-\sigma(x))
\end{aligned}
I do not understand what is the last step in this derivative. How can you factor out the last term?
\begin{aligned}
\sigma(x)(1-\sigma(x))
\end{aligned}

REPLY [4 votes]: $$\frac{e^{-x}}{1+e^{-x}} = \frac{(1+e^{-x})-1}{1+e^{-x}} = 1 - \frac{1}{1+e^{-x}} = 1- \sigma(x)$$


TITLE: Find constant a and b for which $F(a,b) = \int_0^\pi \{ \sin x - (ax^2 + bx)\}^2 dx$ is a minimum.
QUESTION [0 upvotes]: I am not sure how to do this problem. I tried to look up similar problems but was unsuccessful at finding any. I would appreciate if someone could show me how to solve this problem. Here's the question again :
Find constants a and b for which $F(a,b) =\int_0^\pi \{ \sin x - (ax^2 + bx)\}^2 dx$ is a minimum. (You may use Wolfram or other tools to integrate)

REPLY [0 votes]: The function is
$$F(a,b) = {{10\,\pi^3\,b^2+\left(15\,\pi^4\,a-60\,\pi\right)\,b+6\,\pi^5\,a^2
 +\left(240-60\,\pi^2\right)\,a+15\,\pi}\over{30}}.$$
And the critical points are given by:
$$\frac{\partial F(a,b)}{\partial a} = {{15\,\pi^4\,b+12\,\pi^5\,a-60\,\pi^2+240}\over{30}} = 0,$$
$$\frac{\partial F(a,b)}{\partial b} = {{20\,\pi^3\,b+15\,\pi^4\,a-60\,\pi}\over{30}} = 0,$$
that is a $2\times 2$ linear system.