From 7245c498986c59f7293d6ba6d5e57b5f1fd657a0 Mon Sep 17 00:00:00 2001 From: Kalle Date: Sun, 14 Jul 2024 12:49:27 +0300 Subject: [PATCH] More blueprint but only very sketchy. --- blueprint/src/Green_Fourier.tex | 54 +++++++++++++- blueprint/src/content.tex | 1 + blueprint/src/integral_manipulation.tex | 72 ++++++++++++++++++ blueprint/src/macros/common.tex | 2 +- blueprint/src/occupation.tex | 60 +-------------- blueprint/src/overview.tex | 2 +- blueprint/src/random_walk.tex | 82 +++++++++++++++++++++ blueprint/src/recurrence_and_transience.tex | 4 +- 8 files changed, 211 insertions(+), 66 deletions(-) create mode 100644 blueprint/src/random_walk.tex diff --git a/blueprint/src/Green_Fourier.tex b/blueprint/src/Green_Fourier.tex index 7a8d832..32ed296 100644 --- a/blueprint/src/Green_Fourier.tex +++ b/blueprint/src/Green_Fourier.tex @@ -20,6 +20,34 @@ \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} @@ -27,11 +55,12 @@ \section{Inversion of the discrete Fourier transform} \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} @@ -39,7 +68,26 @@ \section{Inversion of the discrete Fourier transform} 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} diff --git a/blueprint/src/content.tex b/blueprint/src/content.tex index 9a4c868..6abd4d1 100644 --- a/blueprint/src/content.tex +++ b/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} diff --git a/blueprint/src/integral_manipulation.tex b/blueprint/src/integral_manipulation.tex index 036f115..1a810a7 100644 --- a/blueprint/src/integral_manipulation.tex +++ b/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} + diff --git a/blueprint/src/macros/common.tex b/blueprint/src/macros/common.tex index 232f3d2..dbb0feb 100644 --- a/blueprint/src/macros/common.tex +++ b/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} diff --git a/blueprint/src/occupation.tex b/blueprint/src/occupation.tex index c1fd570..53bdff0 100644 --- a/blueprint/src/occupation.tex +++ b/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} diff --git a/blueprint/src/overview.tex b/blueprint/src/overview.tex index 1e2ad3d..242702a 100644 --- a/blueprint/src/overview.tex +++ b/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} diff --git a/blueprint/src/random_walk.tex b/blueprint/src/random_walk.tex new file mode 100644 index 0000000..186a777 --- /dev/null +++ b/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} diff --git a/blueprint/src/recurrence_and_transience.tex b/blueprint/src/recurrence_and_transience.tex index 802e4b2..03c1016 100644 --- a/blueprint/src/recurrence_and_transience.tex +++ b/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