More blueprint but only very sketchy.

blueprint/src/Green_Fourier.tex

@@ -20,26 +20,74 @@ \section{Fourier transform of the regularized Green's function}
 
 \section{Explicit formula for the Fourier transform}
 
+\begin{lemma}
+  \label{lem:Markovian_Green_Fourier}
+  \uses{def:Green_Fourier_transf, def:iid_random_walk}
+  For a time-homogeneous random walk $\RW = \big(\RW(t)\big)_{t \in \bN}$ on $\bZ^d$
+  with step distribution $p \colon \bZ^d \to [0,1]$,
+  the Fourier transform of the Green's function is
+  \begin{align*}
+  \Freg{r} (\theta) = \Big(1 - r \sum_{u\in \bZ^d} p(u) e^{\ii u \cdot \theta}\Big)^{-1} .
+  \end{align*}
+\end{lemma}
+\begin{proof}
+\ldots
+\end{proof}
+
+\begin{lemma}
+  \label{lem:SRW_Green_Fourier}
+  \uses{def:Green_Fourier_transf, def:simple_random_walk}
+  For the simple random walk $\RW = \big(\RW(t)\big)_{t \in \bN}$ on $\bZ^d$,
+  the Fourier transform of the Green's function is
+  \begin{align*}
+  \Freg{r} (\theta) = \frac{1}{1-\frac{r}{d} \sum_{j=1}^d \cos(\theta_j)} .
+  \end{align*}
+\end{lemma}
+\begin{proof}
+\uses{lem:Markovian_Green_Fourier}
+\ldots
+\end{proof}
+
 \section{Inversion of the discrete Fourier transform}
 
 \begin{lemma}
   \label{lem:Green_Fourier_inverse}
   \uses{def:Green_Fourier_transf}
   For any $x \in \bZ^d$ and $0 \le r < 1$, we have
   \begin{align*}
-  \Greg{r} (\theta)
-  = \frac{1}{(2\pi)^d} \iint_{\Fbox} e^{-\ii x \cdot \theta} \, \Freg{r} (\theta) \, \ud^d \theta .
+  \Greg{r} (x)
+  = \frac{1}{(2\pi)^d} \iint_{\Fbox} e^{-\ii x \cdot \theta} \, \Freg{r} (\theta) \; \ud^d \theta .
   \end{align*}
 \end{lemma}
 \begin{proof}
+\uses{lem:sum_Green_function}
 \ldots
 \end{proof}
 
 Recall that we are interested in $\EX[L]$, where $L$ is the number of visits to the
 origin by the random walk. Lemma~\ref{lem:Green_function_nonregularized_limit}
 states that $\EX[L]$ is the increasing limit of $\Greg{r}(\vec{0})$ as $r \nearrow 1$,
 and Lemma~\ref{lem:Green_Fourier_inverse} gives a formula for $\Greg{r}(\vec{0})$
-in terms of the Fourier transform.
+as the integral of the Fourier transform: $\Greg{r}(\vec{0}) = \frac{1}{(2\pi)^d} I_r$,
+where
+\begin{align}
+I_r = \iint_{\Fbox} \Freg{r} (\theta) \; \ud^d \theta .
+\end{align}
+
+\begin{corollary}
+  \label{cor:recurrence_iff_finite_limit_integral}
+  \uses{def:expectation_recurrence, def:Green_Fourier_transf}
+  A random walk $\RW = \big(\RW(t)\big)_{t \in \bN}$ on $\bZ^d$
+  is expectation recurrent if and only if $\lim_{r \nearrow 1} \, I_r = +\infty$.
+  In other words, $\RW$ is expectation transient if and only if
+  $\lim_{r \nearrow 1} \, I_r \, < \, +\infty$.
+\end{corollary}
+\begin{proof}
+\uses{lem:Green_Fourier_inverse, lem:Green_function_nonregularized_limit}
+\ldots
+\end{proof}
+
+
 
 % This allows to express the
 % \begin{corollary}

blueprint/src/content.tex

@@ -9,6 +9,7 @@
 %\newtheorem{theorem}{Theorem}
 
 \input{overview.tex}
+\input{random_walk.tex}
 \input{recurrence_and_transience.tex}
 \input{occupation.tex}
 \input{Green_Fourier.tex}

blueprint/src/integral_manipulation.tex

@@ -1,9 +1,81 @@
 \chapter{Treatment of the integral in the Fourier inversion}
 
+In this part, we analyze the integral
+\begin{align}
+I_r = \iint_{\Fbox} \Freg{r} (\theta) \; \ud^d \theta ,
+\end{align}
+where $\Freg{r} \colon \bR^d \to \bC$ is the Fourier transform of the
+regularized Green's function of a random walk $\RW$ on $\bZ^d$. By
+Corollary~\ref{cor:recurrence_iff_finite_limit_integral}, the finiteness
+of this integral in the limit $r \nearrow 1$ characterizes expectation
+transience of $\RW$.
+
 \section{Decomposition of the integral}
 
+\begin{lemma}
+  \label{lem:integral_decomposition}
+  \uses{def:Green_Fourier_transf}
+  For any $0 < \delta \le \pi$ we can write
+  \begin{align*}
+  I_r = J_r^{(\delta)} + K_r^{(\delta)},
+  \end{align*}
+  where the two parts are
+  \begin{align}
+  J_r^{(\delta)} = \; & \iint_{\Fbox \setminus B_\delta} \Freg{r} (\theta) \; \ud^d \theta \\
+  K_r^{(\delta)} = \; & \iint_{B_\delta} \Freg{r} (\theta) \; \ud^d \theta ,
+  \end{align}
+  where $B_\delta := \set{ \theta \in \bR^d \, \big| \, \|\theta\| < \delta}$ is the
+  ball of radius $\delta$ centered at $\vec{0} \in \bR^d$.
+\end{lemma}
+\begin{proof}
+Obvious, since $\Fbox = (\Fbox \setminus B_\delta) \cup B_\delta$
+is a disjoint union.
+\end{proof}
+
+
 \section{Dominated convergence away from the origin}
 
+TODO: Define non-degenerate step distribution (essentially $\sum_{u\in \bZ^d} p(u) e^{\ii u \cdot \theta} \ne 1$ for $\theta \ne 0$ modulo periodicity).
+
+\begin{lemma}
+  \label{lem:integral_away}
+  \uses{lem:integral_decomposition}
+  If $\RW$ is a time-homogeneous Markovian random walk with suitable
+  non-degeneracy conditions on its step distribution (to be written down more precisely),
+  then for any $0 < \delta \le \pi$ the limit
+  \begin{align*}
+  \lim_{r \nearrow 1} J_r^{(\delta)}
+  \end{align*}
+  exists and is finite (limit in $\bR$).
+\end{lemma}
+\begin{proof}
+Under the nondegeneracy conditions, on the compact set $\Fbox \setminus B_\delta$,
+the continuous integrand $\theta \mapsto \Freg{r}(\theta)$
+is bounded (and therefore dominated by a constant function) and
+it has the pointwise limit $\lim_{r \nearrow 1} \Freg{r}(\theta) = \Freg{1}(\theta)$.
+It therefore follows from the dominated convergence theorem
+that $\lim_{r \nearrow 1} J_r^{(\delta)} = J_1^{(\delta)} \in \bR$.
+\end{proof}
+
 \section{Monotone convergence near the origin}
 
+TODO: Think about the best conditions for step distribution under which monotone convergence can be applied (real-valuedness requires symmetricity of the step-distribution?!?).
+
+\begin{lemma}
+  \label{lem:integral_near}
+  \uses{lem:integral_decomposition}
+  If $\RW$ is a time-homogeneous Markovian random walk with suitable
+  symmetricity and integrability conditions on its step distribution (to be written down more precisely),
+  then there exists a $\delta_0 > 0$ such that for any $0 < \delta \le \delta_0$, the limit
+  \begin{align*}
+  \lim_{r \nearrow 1} K_r^{(\delta)}
+  \end{align*}
+  is increasing and exists in $[0,+\infty]$.
+\end{lemma}
+\begin{proof}
+\ldots
+\end{proof}
+
+
 \section{Characterizing finiteness of the integral}
+

blueprint/src/macros/common.tex

@@ -28,7 +28,7 @@
 \newcommand{\Greg}[1]{\G_{#1}}
 \newcommand{\F}{\widehat{\G}}
 \newcommand{\Freg}[1]{\F_{#1}}
-\newcommand{\Fbox}{(-\pi,\pi]^d}
+\newcommand{\Fbox}{[-\pi,\pi]^d}
 
 \newcommand{\walk}{\mathfrak{w}}
 \newcommand{\RW}{X}

blueprint/src/occupation.tex

@@ -1,62 +1,4 @@
-\chapter{Random walks and their occupations}
-
-
-
-\section{Random walks on the $d$-dimensional integer grid}
-
-\begin{definition}
-  \label{def:grid}
-  \lean{Grid}
-  \leanok
-  The $d$-dimensional integer \textbf{grid} is $\bZ^d$.
-\end{definition}
-
-A walk on the grid $\bZ^d$ is a function
-$\walk \colon \bN \to \bZ^d$, denoted $t \mapsto \walk(t)$).
-We construct walks from their sequences of steps as follows:
-
-\begin{definition}
-  \label{def:walk}
-  \lean{deftest}
-  \uses{def:grid}
-  \leanok
-  A sequence $(u_s)_{s \in \bN}$ of steps in $\bZ^d$
-  determines a \textbf{walk} $\walk \colon \bN \to \bZ^d$ by
-  \begin{align}\label{eq: RW def}
-  \walk(t) = \sum_{0 \le s < t} u_s .
-  \end{align}
-\end{definition}
-
-A random walk is constructed from a sequence of random steps.
-
-\begin{definition}
-  \label{def:random_walk}
-  \uses{def:walk}
-  \lean{RW}
-  \leanok
-  A sequence $(\xi_s)_{s \in \bN}$ of $\bZ^d$-valued
-  random variables (on some a probability space)
-  determines a \textbf{random walk} $\RW = \big(\RW(t)\big)_{t \in \bN}$ by
-  \[ \RW(t) = \sum_{0 \le s < t} \xi_s . \]
-\end{definition}
-
-\begin{lemma}
-  \label{lem:RW_mble}
-  \uses{def:random_walk}
-  \lean{RW.measurable}
-  The position $\RW(t)$ of a
-  random walk $\RW = \big(\RW(t)\big)_{t \in \bN}$
-  at any time $t \in \bN$ is a $\bZ^d$-valued random variable.
-\end{lemma}
-\begin{proof}
-We must prove that $\RW(t) \colon \Omega \to \bZ^d$ is measurable.
-Since each of the steps $\xi_s \colon \Omega \to \bZ^d$ is measurable by
-assumption and the position of the
-random walk is defined in~\eqref{eq: RW def} by summing the
-first $t$ steps, the measurability follows by induction
-on $t$ using the fact that sums of measurable $\bZ^d$-valued
-functions are measurable.
-\end{proof}
+\chapter{Occupations and Green's functions of random walks}
 
 
 

blueprint/src/overview.tex

@@ -45,6 +45,6 @@ \chapter{P\'olya's theorem}
 %   on the $d$-dimensional grid $\bZ^d$.
 \end{theorem}
 \begin{proof}
-\uses{lem:Green_function_nonregularized_limit}
+\uses{lem:recurrence_iff_finite_limit_integral, lem:SRW_Green_Fourier, lem:integral_near, lem:integral_away, cor:recurrence_iff_finite_limit_integral}
 \ldots
 \end{proof}

blueprint/src/random_walk.tex

@@ -0,0 +1,82 @@
+\chapter{Random walks}
+
+
+
+\section{Random walks on the $d$-dimensional integer grid}
+
+\begin{definition}
+  \label{def:grid}
+  \lean{Grid}
+  \leanok
+  The $d$-dimensional integer \textbf{grid} is $\bZ^d$.
+\end{definition}
+
+A walk on the grid $\bZ^d$ is a function
+$\walk \colon \bN \to \bZ^d$, denoted $t \mapsto \walk(t)$).
+We construct walks from their sequences of steps as follows:
+
+\begin{definition}
+  \label{def:walk}
+  \lean{deftest}
+  \uses{def:grid}
+  \leanok
+  A sequence $(u_s)_{s \in \bN}$ of steps in $\bZ^d$
+  determines a \textbf{walk} $\walk \colon \bN \to \bZ^d$ by
+  \begin{align}\label{eq: RW def}
+  \walk(t) = \sum_{0 \le s < t} u_s .
+  \end{align}
+\end{definition}
+
+A random walk is constructed from a sequence of random steps.
+
+\begin{definition}
+  \label{def:random_walk}
+  \uses{def:walk}
+  \lean{RW}
+  \leanok
+  A sequence $(\xi_s)_{s \in \bN}$ of $\bZ^d$-valued
+  random variables (on some a probability space)
+  determines a \textbf{random walk} $\RW = \big(\RW(t)\big)_{t \in \bN}$ by
+  \[ \RW(t) = \sum_{0 \le s < t} \xi_s . \]
+\end{definition}
+
+\begin{lemma}
+  \label{lem:RW_mble}
+  \uses{def:random_walk}
+  \lean{RW.measurable}
+  The position $\RW(t)$ of a
+  random walk $\RW = \big(\RW(t)\big)_{t \in \bN}$
+  at any time $t \in \bN$ is a $\bZ^d$-valued random variable.
+\end{lemma}
+\begin{proof}
+We must prove that $\RW(t) \colon \Omega \to \bZ^d$ is measurable.
+Since each of the steps $\xi_s \colon \Omega \to \bZ^d$ is measurable by
+assumption and the position of the
+random walk is defined in~\eqref{eq: RW def} by summing the
+first $t$ steps, the measurability follows by induction
+on $t$ using the fact that sums of measurable $\bZ^d$-valued
+functions are measurable.
+\end{proof}
+
+\begin{definition}
+  \label{def:iid_random_walk}
+  \uses{def:random_walk}
+  A random walk $\RW = \big(\RW(t)\big)_{t \in \bN}$ on $\bZ^d$
+  said to be \textbf{time-homogeneous Markovian} if its
+  steps $\big(\RW(t+1) - \RW(t)\big)_{t \in \bN}$
+  are independent and identically distributed.
+\end{definition}
+
+\begin{definition}
+  \label{def:simple_random_walk}
+  \uses{def:iid_random_walk}
+  A random walk $\RW = \big(\RW(t)\big)_{t \in \bN}$ on $\bZ^d$
+  is \textbf{simple} if it is time-homogeneous Markovian and its
+  steps are uniformly distributed on nearest neighbors on the grid:
+  \begin{align*}
+  \PR \big[ \RW(t+1) - \RW(t) = u \big] = \begin{cases}
+        \frac{1}{2d}  & \text{ if } \|u\| = 1 \\
+        0             & \text{ otherwise.}
+    \end{cases}
+  \end{align*}
+\end{definition}

blueprint/src/recurrence_and_transience.tex

@@ -63,7 +63,7 @@ \section{Equivalent conditions}
 
 \begin{lemma}
   \label{lem:recurrent_iff_expectation_recurrent}
-  \uses{def:recurrence, def:expectation_recurrence}
+  \uses{def:recurrence, def:expectation_recurrence, def:iid_random_walk}
   A Markovian random walk $\RW = \big(\RW(t)\big)_{t \in \bN}$ is recurrent
   %(in the sense of Definition~\ref{def:recurrence})
   if and only if it is
@@ -76,7 +76,7 @@ \section{Equivalent conditions}
 
 \begin{lemma}
   \label{lem:recurrent_iff_return_recurrent}
-  \uses{def:recurrence, def:Markovian_recurrence}
+  \uses{def:recurrence, def:Markovian_recurrence, def:iid_random_walk}
   A Markovian random walk $\RW = \big(\RW(t)\big)_{t \in \bN}$ is recurrent
   %(in the sense of Definition~\ref{def:recurrence})
   if and only if it is