# Using Random Walks to Establish Wavelike Behavior in an FPUT System with
Random Coefficients

Joshua A. McGinnis, J. Douglas Wright

###### Abstract

We consider a linear Fermi-Pasta-Ulam-Tsingou lattice with random spatially
varying material coefficients. Using the methods of stochastic homogenization
we show that solutions with long wave initial data converge in an appropriate
sense to solutions of a wave equation. The convergence is strong and both
almost sure and in expectation, but the rate is quite slow. The technique
combines energy estimates with powerful classical results about random walks,
specifically the law of the iterated logarithm.

##  1 Introduction

We prove an almost sure convergence result for solutions of the following one-
dimensional random polymer linear Fermi-Pasta-Ulam-Tsingou (FPUT) lattice in
the long wave limit:

$\displaystyle
m(j)\ddot{u}(j)=k(j)\left[u(j+1)-u(j)\right]-k(j-1)\left[u(j)-u(j-1)\right].$
(1.1)

Here $j\in{\bf{Z}}$, $u=u(j,t)\in{\bf{R}}$ and $t\in{\bf{R}}$. We choose the
coefficients $m(j)$ (which we refer to as “the masses”) to be independent and
identically distributed (i.i.d.) random variables contained almost surely in
some intervals $[a_{m},b_{m}]\subset{\bf{R}}^{+}$ with standard deviation
$\sigma_{m}$. We similarly take the coefficients $1/k(j)$ (“the springs”) to
be i.i.d. with support in $[a_{k},b_{k}]\subset{\bf{R}}^{+}$ and deviation
$\sigma_{k}$. This system is well-understood when these coefficients are
either constant or periodic with respect to $j$ [8], but for the random
problem most of what is known is formal or numerical [6, 9].

For initial conditions whose wavelength is $O(1/\epsilon)$, with
$\epsilon\in(0,1)$, we prove that the $\ell^{2}$ norm of the difference
between true solutions and appropriately scaled solutions to the wave equation
is at most $O\left(\sqrt{\log\log(1/\epsilon)}\right)$ for times of
$O(1/\epsilon)$ for almost every realization. While such an absolute error
diverges as $\epsilon\to 0^{+}$, it happens that this is enough to establish
an almost sure convergence result within the “coarse-graining” setting used in
[8] to study the (multi-dimensional) periodic problem. In addition to the
almost sure convergence, we are able to provide estimates on the mean of the
error in terms of $\sigma_{m}$ and $\sigma_{k}$ and prove convergence in mean.

The articles [2, 5] study the nonlinear FPUT lattice with periodic
coefficients. These show that soliton-like solutions exist for very large time
scales using Korteweg-de Vries (KdV) approximations. The authors of [5] used
the so called multiscale method of homogenization, a by-now classical tool
with a long history in PDE for deriving effective equations, see [1]. In this
paper, we carry out a very similar approach in deriving and proving the
results; however, our expansions only result in an effective wave equation,
not the KdV equation. In our setting, since the coefficients are random, it is
necessary to average over the entire lattice. The law of large numbers implies
this average is equal to the expectation, so the speed of the approximate
solution depends on the expectation of the random variables. The probability
theory hinges upon classical but extremely powerful asymptotic analysis of
random walks, namely the law of iterated logarithms, as well as basic
martingale theory.

We denote a doubly infinite sequence $\\{x(j)\\}_{j\in{\bf{Z}}}$ by $x$. Let
$S^{\pm}$ be the shift operators which act on sequences $f=\\{f(j)\\}$ as

$(S^{\pm}f)(j)\coloneqq f(j\pm 1),$

and the operators $\delta^{+}$ and $\delta^{-}$, the left and right difference
operators, are

$\displaystyle(\delta^{+}f)(j)$ $\displaystyle:=f(j+1)-f(j)$
$\displaystyle(\delta^{-}f)(j)$ $\displaystyle:=f(j)-f(j-1).$

Defining

$\displaystyle r$ $\displaystyle\coloneqq\delta^{+}u$ $\displaystyle p$
$\displaystyle\coloneqq\dot{u},$

we convert our second order equation (1.1) to the system

$\displaystyle\dot{r}$ $\displaystyle=\delta^{+}p$ (1.2)
$\displaystyle\dot{p}$ $\displaystyle=\dfrac{1}{m}\delta^{-}(kr).$

For the remainder of the paper, we work with (1.2).

Here is the idea of our ultimate result. Suppose that the initial conditions
for (1.2) have the following long wave form:

$r(j,0)=\Phi(\epsilon j)/k(j)\quad\text{and}\quad p(j,0)=\Psi(\epsilon j)$

where $\Phi,\Psi:{\bf{R}}\to{\bf{R}}$ are suitably smooth, of somewhat rapid
decay, and $\epsilon\in(0,1)$. Then the solution $p$ of (1.2) has

${\mathcal{L}}[p](X/\epsilon,\tau/\epsilon)\longrightarrow P_{0}(X,\tau)$

as $\epsilon\to 0^{+}$ where $P_{0}$ solves the wave equation
$\partial_{\tau}^{2}P_{0}=c^{2}\partial_{X}^{2}P_{0}$. The operator
${\mathcal{L}}$ interpolates the sequence $p$ into a function on ${\bf{R}}$.
It is defined below, as is the wave speed $c$. The convergence is strong in
$L^{2}({\bf{R}})$ and is both almost sure and in expectation. A similar
convergence holds for $r$.

The paper is organized as follows. We carry out the multiscale expansion in
Section 2 and derive effective equations and approximate solutions. In Section
3 we dive into the analysis of various smooth, rapidly decaying functions
which are sampled at integers and multiplied componentwise by random walks.
These estimates are necessary to control the error and here is where most the
probability theory is needed. In Section 4 we provide the rigorous estimates
of the error. We introduce coarse-graining and prove the convergence results
in Section 5. In Section 6 we provide numerical simulations as evidence that
our estimates are good ones i.e. they are not vast overestimates.

##  2 Homogenization and derivation of the effect wave equation

In this section we homogenize the equation following closely what is done in
[5]. First, we define “residuals”, which quantify how close some function is
to a true solution. For any functions $\widetilde{r}(j,t)$ and
$\widetilde{p}(j,t)$ put

$\displaystyle\text{Res}_{1}(\widetilde{r},\widetilde{p})$
$\displaystyle:=\delta^{+}\widetilde{p}-\partial_{t}\widetilde{r}$ (2.1)
$\displaystyle\ \text{Res}_{2}(\widetilde{r},\widetilde{p})$
$\displaystyle:=\dfrac{1}{m}\delta^{-}(k\widetilde{r})-\partial_{t}\widetilde{p}.$

We look for approximate long wave solutions of the form

$\displaystyle\widetilde{r}(j,t)$
$\displaystyle=\widetilde{r}_{\epsilon}(j,t):=R(j,\epsilon j,\epsilon t)$
(2.2) $\displaystyle\ \widetilde{p}(j,t)$
$\displaystyle=\widetilde{p}_{\epsilon}(j,t):=P(j,\epsilon j,\epsilon t),$

where $R=R(j,X,\tau)$ and $P=P(j,X,\tau)$ are maps

${\bf{Z}}\times{\bf{R}}\times{\bf{R}}\to{\bf{R}}.$

In the periodic-coefficient problem studied in [5], it was necessary to assume
that these functions are periodic in the ${\bf{Z}}$ slot, but this needs to be
exchanged in the random case. Here, we make a “sublinear growth” assumption
that makes averaging possible:

$\lim\limits_{|j|\to\infty}R(j,X,\tau)/j=\lim\limits_{|j|\to\infty}P(j,X,\tau)/j=0.$
(2.3)

The following lemma is crucial to the derivation of the effective equations.

######  Lemma 2.1.

There exists an $f=\left\\{f(j)\right\\}_{j\in{\bf{Z}}}$ satisfying both

$\lim\limits_{|j|\to\infty}f(j)/j\to 0$ (2.4)

and

$(\delta^{\pm}f)(j)=g(j)$

if and only if

$\lim\limits_{N\to\infty}\dfrac{1}{N}\sum\limits_{i=0}^{N-1}g(i)=\lim\limits_{N\to\infty}\dfrac{1}{N}\sum\limits_{i=1}^{N}g(-i)=0.$

###### Proof.

We only give a proof for “$\delta^{+}$”. $\Rightarrow$ Since
$g(j)=f(j+1)-f(j)$ we get

$\lim_{N\to\infty}\dfrac{1}{N}\sum\limits_{i=0}^{N-1}g(i)=\lim_{N\to\infty}\dfrac{1}{N}\sum\limits_{i=0}^{N-1}[f(i+1)-f(i)]=\lim_{N\to\infty}\dfrac{f(N)-f(0)}{N}=0$

by assumption (2.4). Proof of the second such equality is analogous.

$\Leftarrow$ If we choose $f(0)=0$, it is readily checked that

$f(j)=\sum_{k=0}^{j-1}g(i)\quad\text{and}\quad f(-j)=-\sum_{k=1}^{j}g(-i)$
(2.5)

for $j>0$ solves $\delta^{+}f=g$. Then

$\lim\limits_{j\to\infty}\dfrac{f(j)}{j}=\lim\limits_{j\to\infty}\dfrac{\sum\limits_{i=0}^{j-1}g(i)}{j}=0.$

It is likewise seen that

$\lim_{j\to\infty}\dfrac{f(-j)}{-j}=0$

by using the formula for $f(-j)$. ∎

Now we continue with the homogenization procedure. We must understand how
$\delta^{\pm}$ act on functions of the type (2.2). The following expansions
are found in [5]. If $u(j)=U(j,\epsilon j),$ then

$\delta^{\pm}u(j)=\sum\limits_{n\geq 0}\epsilon^{n}\delta_{n}^{\pm}U$

where

$\delta_{0}^{\pm}:={\mathchoice{{\ooalign{\hbox{\raise
7.12962pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.12962pt\hbox{$\displaystyle\widehat{\vrule
width=0.0pt,height=6.94444pt\vrule
height=0.0pt,width=4.44444pt}$}}}}\cr\hbox{$\displaystyle\delta$}}}}{{\ooalign{\hbox{\raise
7.12962pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.12962pt\hbox{$\textstyle\widehat{\vrule width=0.0pt,height=6.94444pt\vrule
height=0.0pt,width=4.44444pt}$}}}}\cr\hbox{$\textstyle\delta$}}}}{{\ooalign{\hbox{\raise
6.43518pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.43518pt\hbox{$\scriptstyle\widehat{\vrule width=0.0pt,height=4.8611pt\vrule
height=0.0pt,width=3.1111pt}$}}}}\cr\hbox{$\scriptstyle\delta$}}}}{{\ooalign{\hbox{\raise
5.97221pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.97221pt\hbox{$\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=3.47221pt\vrule
height=0.0pt,width=2.22221pt}$}}}}\cr\hbox{$\scriptscriptstyle\delta$}}}}}^{\pm}\
\text{and}\ \delta_{n}^{\pm}:=\dfrac{(\pm
1)^{n+1}}{n!}{\mathchoice{{\ooalign{\hbox{\raise
7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\displaystyle\widehat{\vrule
width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=6.70831pt}$}}}}\cr\hbox{$\displaystyle
S$}}}}{{\ooalign{\hbox{\raise 7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\textstyle\widehat{\vrule width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=6.70831pt}$}}}}\cr\hbox{$\textstyle
S$}}}}{{\ooalign{\hbox{\raise 6.40926pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.40926pt\hbox{$\scriptstyle\widehat{\vrule width=0.0pt,height=4.78333pt\vrule
height=0.0pt,width=4.69582pt}$}}}}\cr\hbox{$\scriptstyle
S$}}}}{{\ooalign{\hbox{\raise 5.9537pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.9537pt\hbox{$\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=3.41666pt\vrule
height=0.0pt,width=3.35414pt}$}}}}\cr\hbox{$\scriptscriptstyle
S$}}}}}^{\pm}\partial_{X}^{n}.$

Here ${\mathchoice{{\ooalign{\hbox{\raise
7.12962pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.12962pt\hbox{$\displaystyle\widehat{\vrule
width=0.0pt,height=6.94444pt\vrule
height=0.0pt,width=4.44444pt}$}}}}\cr\hbox{$\displaystyle\delta$}}}}{{\ooalign{\hbox{\raise
7.12962pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.12962pt\hbox{$\textstyle\widehat{\vrule width=0.0pt,height=6.94444pt\vrule
height=0.0pt,width=4.44444pt}$}}}}\cr\hbox{$\textstyle\delta$}}}}{{\ooalign{\hbox{\raise
6.43518pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.43518pt\hbox{$\scriptstyle\widehat{\vrule width=0.0pt,height=4.8611pt\vrule
height=0.0pt,width=3.1111pt}$}}}}\cr\hbox{$\scriptstyle\delta$}}}}{{\ooalign{\hbox{\raise
5.97221pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.97221pt\hbox{$\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=3.47221pt\vrule
height=0.0pt,width=2.22221pt}$}}}}\cr\hbox{$\scriptscriptstyle\delta$}}}}}^{\pm}$
and ${\mathchoice{{\ooalign{\hbox{\raise
7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\displaystyle\widehat{\vrule
width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=6.70831pt}$}}}}\cr\hbox{$\displaystyle
S$}}}}{{\ooalign{\hbox{\raise 7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\textstyle\widehat{\vrule width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=6.70831pt}$}}}}\cr\hbox{$\textstyle
S$}}}}{{\ooalign{\hbox{\raise 6.40926pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.40926pt\hbox{$\scriptstyle\widehat{\vrule width=0.0pt,height=4.78333pt\vrule
height=0.0pt,width=4.69582pt}$}}}}\cr\hbox{$\scriptstyle
S$}}}}{{\ooalign{\hbox{\raise 5.9537pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.9537pt\hbox{$\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=3.41666pt\vrule
height=0.0pt,width=3.35414pt}$}}}}\cr\hbox{$\scriptscriptstyle S$}}}}}^{\pm}$
act only on the first slot; they are analogous to partial derivatives with
respect to $j$. Precisely,

$\displaystyle{\mathchoice{{\ooalign{\hbox{\raise
7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\displaystyle\widehat{\vrule
width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=6.70831pt}$}}}}\cr\hbox{$\displaystyle
S$}}}}{{\ooalign{\hbox{\raise 7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\textstyle\widehat{\vrule width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=6.70831pt}$}}}}\cr\hbox{$\textstyle
S$}}}}{{\ooalign{\hbox{\raise 6.40926pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.40926pt\hbox{$\scriptstyle\widehat{\vrule width=0.0pt,height=4.78333pt\vrule
height=0.0pt,width=4.69582pt}$}}}}\cr\hbox{$\scriptstyle
S$}}}}{{\ooalign{\hbox{\raise 5.9537pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.9537pt\hbox{$\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=3.41666pt\vrule
height=0.0pt,width=3.35414pt}$}}}}\cr\hbox{$\scriptscriptstyle
S$}}}}}^{+}(U)(j,X)$ $\displaystyle:=U(j+1,X),$
$\displaystyle{\mathchoice{{\ooalign{\hbox{\raise
7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\displaystyle\widehat{\vrule
width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=6.70831pt}$}}}}\cr\hbox{$\displaystyle
S$}}}}{{\ooalign{\hbox{\raise 7.09259pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.09259pt\hbox{$\textstyle\widehat{\vrule width=0.0pt,height=6.83331pt\vrule
height=0.0pt,width=6.70831pt}$}}}}\cr\hbox{$\textstyle
S$}}}}{{\ooalign{\hbox{\raise 6.40926pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.40926pt\hbox{$\scriptstyle\widehat{\vrule width=0.0pt,height=4.78333pt\vrule
height=0.0pt,width=4.69582pt}$}}}}\cr\hbox{$\scriptstyle
S$}}}}{{\ooalign{\hbox{\raise 5.9537pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.9537pt\hbox{$\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=3.41666pt\vrule
height=0.0pt,width=3.35414pt}$}}}}\cr\hbox{$\scriptscriptstyle
S$}}}}}^{-}(U)(j,X)$ $\displaystyle:=U(j-1,X),$
$\displaystyle{\mathchoice{{\ooalign{\hbox{\raise
7.12962pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.12962pt\hbox{$\displaystyle\widehat{\vrule
width=0.0pt,height=6.94444pt\vrule
height=0.0pt,width=4.44444pt}$}}}}\cr\hbox{$\displaystyle\delta$}}}}{{\ooalign{\hbox{\raise
7.12962pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.12962pt\hbox{$\textstyle\widehat{\vrule width=0.0pt,height=6.94444pt\vrule
height=0.0pt,width=4.44444pt}$}}}}\cr\hbox{$\textstyle\delta$}}}}{{\ooalign{\hbox{\raise
6.43518pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.43518pt\hbox{$\scriptstyle\widehat{\vrule width=0.0pt,height=4.8611pt\vrule
height=0.0pt,width=3.1111pt}$}}}}\cr\hbox{$\scriptstyle\delta$}}}}{{\ooalign{\hbox{\raise
5.97221pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.97221pt\hbox{$\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=3.47221pt\vrule
height=0.0pt,width=2.22221pt}$}}}}\cr\hbox{$\scriptscriptstyle\delta$}}}}}^{+}(U)(j,X)$
$\displaystyle:=U(j+1,X)-U(j,X),\ $
$\displaystyle{\mathchoice{{\ooalign{\hbox{\raise
7.12962pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.12962pt\hbox{$\displaystyle\widehat{\vrule
width=0.0pt,height=6.94444pt\vrule
height=0.0pt,width=4.44444pt}$}}}}\cr\hbox{$\displaystyle\delta$}}}}{{\ooalign{\hbox{\raise
7.12962pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.12962pt\hbox{$\textstyle\widehat{\vrule width=0.0pt,height=6.94444pt\vrule
height=0.0pt,width=4.44444pt}$}}}}\cr\hbox{$\textstyle\delta$}}}}{{\ooalign{\hbox{\raise
6.43518pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.43518pt\hbox{$\scriptstyle\widehat{\vrule width=0.0pt,height=4.8611pt\vrule
height=0.0pt,width=3.1111pt}$}}}}\cr\hbox{$\scriptstyle\delta$}}}}{{\ooalign{\hbox{\raise
5.97221pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.97221pt\hbox{$\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=3.47221pt\vrule
height=0.0pt,width=2.22221pt}$}}}}\cr\hbox{$\scriptscriptstyle\delta$}}}}}^{-}(U)(j,X)$
$\displaystyle:=U(j,X)-U(j-1,X).$

Let

$(E_{M}^{\pm}u)(j):=(\delta^{\pm}u)(j)-\sum\limits_{n=0}^{M}\epsilon^{n}(\delta^{\pm}_{n}U)(j,\epsilon
j)$

be the error made by truncating the series expansion of $\delta^{\pm}u$ after
$M$ terms. Thus the lowest power of $\epsilon$ we see in the error term is
$\epsilon^{M+1}$.

We further assume that our approximate solutions $R$ and $P$ themselves have
expansions in $\epsilon$:

$R(j,X,\tau)=R_{0}(j,X,\tau)+\epsilon R_{1}(j,X,\tau)\quad\text{and}\quad
P(j,X,\tau)=P_{0}(j,X,\tau)+\epsilon P_{1}(j,X,\tau).$ (2.6)

Of course $R_{i}(j,X,\tau)$ and $P_{i}(j,X,\tau)$ meet (2.4). Using the above
expansion, we directly compute
$\text{Res}_{1}(\widetilde{r}_{\epsilon},\widetilde{p}_{\epsilon})$:

$\displaystyle\text{Res}_{1}(\widetilde{r}_{\epsilon},\widetilde{p}_{\epsilon})$
$\displaystyle=\delta_{0}^{+}P_{0}+\epsilon\delta^{+}_{1}P_{0}+E_{1}^{+}(P_{0})$
(2.7)
$\displaystyle+\epsilon\delta_{0}^{+}P_{1}+\epsilon^{2}\delta^{+}_{1}P_{1}+\epsilon
E_{1}^{+}(P_{1})$
$\displaystyle-\epsilon\partial_{\tau}R_{0}-\epsilon^{2}\partial_{\tau}R_{1}.$

Here we have used the expansion for $\delta^{+}$. Similarly

$\displaystyle\text{Res}_{2}(\widetilde{r}_{\epsilon},\widetilde{p}_{\epsilon})$
$\displaystyle=\frac{1}{m}(\delta_{0}^{-}kR_{0}+\epsilon\delta^{-}_{1}kR_{0}+E_{1}^{-}(kR_{0})$
(2.8)
$\displaystyle+\epsilon\delta_{0}^{-}kR_{1}+\epsilon^{2}\delta^{-}_{1}kR_{1}+\epsilon
E_{1}^{-}(kR_{1})$
$\displaystyle-\epsilon\partial_{\tau}P_{0}-\epsilon^{2}\partial_{\tau}P_{1}).$

Next set

$Q_{i}\coloneqq kR_{i}.$ (2.9)

We choose $P_{0},P_{1},Q_{0}$ and $Q_{1}$ so that the $O(1)$ and $O(\epsilon)$
terms in (2.7) and (2.8) vanish. We get

$\displaystyle{\mathchoice{{\ooalign{\hbox{\raise
7.12962pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.12962pt\hbox{$\displaystyle\widehat{\vrule
width=0.0pt,height=6.94444pt\vrule
height=0.0pt,width=4.44444pt}$}}}}\cr\hbox{$\displaystyle\delta$}}}}{{\ooalign{\hbox{\raise
7.12962pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.12962pt\hbox{$\textstyle\widehat{\vrule width=0.0pt,height=6.94444pt\vrule
height=0.0pt,width=4.44444pt}$}}}}\cr\hbox{$\textstyle\delta$}}}}{{\ooalign{\hbox{\raise
6.43518pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.43518pt\hbox{$\scriptstyle\widehat{\vrule width=0.0pt,height=4.8611pt\vrule
height=0.0pt,width=3.1111pt}$}}}}\cr\hbox{$\scriptstyle\delta$}}}}{{\ooalign{\hbox{\raise
5.97221pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.97221pt\hbox{$\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=3.47221pt\vrule
height=0.0pt,width=2.22221pt}$}}}}\cr\hbox{$\scriptscriptstyle\delta$}}}}}^{+}P_{0}$
$\displaystyle=0$ ($O(1)$)
$\displaystyle\dfrac{1}{m}{\mathchoice{{\ooalign{\hbox{\raise
7.12962pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.12962pt\hbox{$\displaystyle\widehat{\vrule
width=0.0pt,height=6.94444pt\vrule
height=0.0pt,width=4.44444pt}$}}}}\cr\hbox{$\displaystyle\delta$}}}}{{\ooalign{\hbox{\raise
7.12962pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.12962pt\hbox{$\textstyle\widehat{\vrule width=0.0pt,height=6.94444pt\vrule
height=0.0pt,width=4.44444pt}$}}}}\cr\hbox{$\textstyle\delta$}}}}{{\ooalign{\hbox{\raise
6.43518pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.43518pt\hbox{$\scriptstyle\widehat{\vrule width=0.0pt,height=4.8611pt\vrule
height=0.0pt,width=3.1111pt}$}}}}\cr\hbox{$\scriptstyle\delta$}}}}{{\ooalign{\hbox{\raise
5.97221pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.97221pt\hbox{$\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=3.47221pt\vrule
height=0.0pt,width=2.22221pt}$}}}}\cr\hbox{$\scriptscriptstyle\delta$}}}}}^{-}Q_{0}$
$\displaystyle=0$

and

$\displaystyle{\mathchoice{{\ooalign{\hbox{\raise
7.12962pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.12962pt\hbox{$\displaystyle\widehat{\vrule
width=0.0pt,height=6.94444pt\vrule
height=0.0pt,width=4.44444pt}$}}}}\cr\hbox{$\displaystyle\delta$}}}}{{\ooalign{\hbox{\raise
7.12962pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.12962pt\hbox{$\textstyle\widehat{\vrule width=0.0pt,height=6.94444pt\vrule
height=0.0pt,width=4.44444pt}$}}}}\cr\hbox{$\textstyle\delta$}}}}{{\ooalign{\hbox{\raise
6.43518pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.43518pt\hbox{$\scriptstyle\widehat{\vrule width=0.0pt,height=4.8611pt\vrule
height=0.0pt,width=3.1111pt}$}}}}\cr\hbox{$\scriptstyle\delta$}}}}{{\ooalign{\hbox{\raise
5.97221pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.97221pt\hbox{$\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=3.47221pt\vrule
height=0.0pt,width=2.22221pt}$}}}}\cr\hbox{$\scriptscriptstyle\delta$}}}}}^{+}P_{1}$
$\displaystyle=\dfrac{1}{k}\partial_{\tau}Q_{0}-S^{+}\partial_{X}P_{0}$
($O(\epsilon)$) $\displaystyle{\mathchoice{{\ooalign{\hbox{\raise
7.12962pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.12962pt\hbox{$\displaystyle\widehat{\vrule
width=0.0pt,height=6.94444pt\vrule
height=0.0pt,width=4.44444pt}$}}}}\cr\hbox{$\displaystyle\delta$}}}}{{\ooalign{\hbox{\raise
7.12962pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.12962pt\hbox{$\textstyle\widehat{\vrule width=0.0pt,height=6.94444pt\vrule
height=0.0pt,width=4.44444pt}$}}}}\cr\hbox{$\textstyle\delta$}}}}{{\ooalign{\hbox{\raise
6.43518pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.43518pt\hbox{$\scriptstyle\widehat{\vrule width=0.0pt,height=4.8611pt\vrule
height=0.0pt,width=3.1111pt}$}}}}\cr\hbox{$\scriptstyle\delta$}}}}{{\ooalign{\hbox{\raise
5.97221pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.97221pt\hbox{$\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=3.47221pt\vrule
height=0.0pt,width=2.22221pt}$}}}}\cr\hbox{$\scriptscriptstyle\delta$}}}}}^{-}Q_{1}$
$\displaystyle=m\partial_{\tau}P_{0}-S^{-1}\partial_{X}Q_{0}.$

From ($O(1)$) we learn that $P_{0}$ and $Q_{0}$ do not depend on $j$, i.e.

$P_{0}(j,X,\tau)=\bar{P}_{0}(X,\tau)\ \text{and}\
Q_{0}(j,X,\tau)=\bar{Q}_{0}(X,\tau).$ (2.10)

If there are to be solutions $P_{1}$ and $Q_{1}$ or ($O(\epsilon)$) which
satsify (2.4), Lemma 2.1 tells us we must have

$\displaystyle\lim_{N\to\infty}\dfrac{1}{N}\sum_{j=0}^{N-1}\left[\frac{1}{k(j)}\partial_{\tau}\bar{Q}_{0}-\partial_{X}\bar{P}_{0}\right]=\lim_{N\to\infty}\dfrac{1}{N}\sum_{j=1}^{N}\left[\frac{1}{k(-j)}\partial_{\tau}\bar{Q}_{0}-\partial_{X}\bar{P}_{0}\right]=0$
(2.11)
$\displaystyle\lim_{N\to\infty}\dfrac{1}{N}\sum_{j=0}^{N-1}\left[m(j)\partial_{\tau}\bar{P}_{0}-\partial_{X}\bar{Q}_{0}\right]=\lim_{N\to\infty}\dfrac{1}{N}\sum_{j=1}^{N}\left[m(-j)\partial_{\tau}\bar{P}_{0}-\partial_{X}\bar{Q}_{0}\right]=0.$

Since $\bar{P}_{0}$ and $\bar{Q}_{0}$ do not depend upon $j$ these can be
rewritten as

$\left[\lim_{N\to\infty}\dfrac{1}{N}\sum_{j=0}^{N-1}\frac{1}{k(j)}\right]\partial_{\tau}\bar{Q}_{0}=\lim_{N\to\infty}\left[\dfrac{1}{N}\sum_{j=1}^{N}\frac{1}{k(-j)}\right]\partial_{\tau}\bar{Q}_{0}=\partial_{X}\bar{P}_{0}$

and

$\left[\lim_{N\to\infty}\dfrac{1}{N}\sum_{j=0}^{N-1}m(j)\right]\partial_{\tau}\bar{P}_{0}=\left[\lim_{N\to\infty}\dfrac{1}{N}\sum_{j=1}^{N}m(-j)\right]\partial_{\tau}\bar{P}_{0}=\partial_{X}\bar{Q}_{0}.$

The law of large numbers tells us that

$\lim_{N\to\infty}\dfrac{1}{N}\sum_{j=0}^{N-1}m(j)=\lim_{N\to\infty}\dfrac{1}{N}\sum_{j=1}^{N}m(-j)={\mathbb{E}}[m]=:\bar{m}$
(2.12)

and

$\lim_{N\to\infty}\frac{1}{N}\sum_{j=0}^{N-1}\frac{1}{k(j)}=\lim_{N\to\infty}\frac{1}{N}\sum_{j=0}^{N-1}\frac{1}{k(-j)}={\mathbb{E}}\left[\frac{1}{k}\right]=:\frac{1}{\widetilde{k}}$
(2.13)

almost surely, since $m$ and $k$ are sequences of i.i.d. random variables. To
be clear ${\mathbb{E}}[\cdot]$ is the expectation of a random variable. And so
we find that

$\displaystyle\partial_{\tau}\bar{Q}_{0}$
$\displaystyle=\widetilde{k}\partial_{X}\bar{P}_{0}$ (2.14)
$\displaystyle\partial_{\tau}\bar{P}_{0}$
$\displaystyle=\dfrac{1}{\bar{m}}\partial_{X}\bar{Q}_{0}.$

From this, out pops the effective wave equation

$\partial^{2}_{\tau}\bar{Q}_{0}=c^{2}\partial^{2}_{X}\bar{Q}_{0}$

with wave-speed

$c:=\sqrt{{\widetilde{k}/\bar{m}}}.$

We can use d’Alemberts formula to get $\bar{Q}_{0}$ and subsequently find
$\bar{P}_{0}$ from its relation to $\bar{Q}_{0}$:

$\displaystyle\bar{Q}_{0}(X,\tau)$ $\displaystyle=A(X-c\tau)+B(X+c\tau)$
(2.15) $\displaystyle\bar{P}_{0}(X,\tau)$
$\displaystyle=\frac{1}{\sqrt{\widetilde{k}\bar{m}}}(-A(X-c\tau)+B(X+c\tau)).$

The functions $A$ and $B$ will ultimately be determined by the initial
conditions for (1.2) in a fashion that is consistent with (2.2).

At this point we have computed the effective wave equation but we must also
determine the full form of ${P}_{1}$ and ${Q}_{1}$. Using (2.10) and (2.14) in
($O(\epsilon)$) we get

$\displaystyle{\mathchoice{{\ooalign{\hbox{\raise
7.12962pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.12962pt\hbox{$\displaystyle\widehat{\vrule
width=0.0pt,height=6.94444pt\vrule
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7.12962pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.12962pt\hbox{$\textstyle\widehat{\vrule width=0.0pt,height=6.94444pt\vrule
height=0.0pt,width=4.44444pt}$}}}}\cr\hbox{$\textstyle\delta$}}}}{{\ooalign{\hbox{\raise
6.43518pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.43518pt\hbox{$\scriptstyle\widehat{\vrule width=0.0pt,height=4.8611pt\vrule
height=0.0pt,width=3.1111pt}$}}}}\cr\hbox{$\scriptstyle\delta$}}}}{{\ooalign{\hbox{\raise
5.97221pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.97221pt\hbox{$\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=3.47221pt\vrule
height=0.0pt,width=2.22221pt}$}}}}\cr\hbox{$\scriptscriptstyle\delta$}}}}}^{+}P_{1}$
$\displaystyle=\left(\dfrac{\widetilde{k}}{k}-1\right)\partial_{X}\bar{P}_{0}$
$\displaystyle{\mathchoice{{\ooalign{\hbox{\raise
7.12962pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.12962pt\hbox{$\displaystyle\widehat{\vrule
width=0.0pt,height=6.94444pt\vrule
height=0.0pt,width=4.44444pt}$}}}}\cr\hbox{$\displaystyle\delta$}}}}{{\ooalign{\hbox{\raise
7.12962pt\hbox{\scalebox{1.0}[-1.0]{\lower
7.12962pt\hbox{$\textstyle\widehat{\vrule width=0.0pt,height=6.94444pt\vrule
height=0.0pt,width=4.44444pt}$}}}}\cr\hbox{$\textstyle\delta$}}}}{{\ooalign{\hbox{\raise
6.43518pt\hbox{\scalebox{1.0}[-1.0]{\lower
6.43518pt\hbox{$\scriptstyle\widehat{\vrule width=0.0pt,height=4.8611pt\vrule
height=0.0pt,width=3.1111pt}$}}}}\cr\hbox{$\scriptstyle\delta$}}}}{{\ooalign{\hbox{\raise
5.97221pt\hbox{\scalebox{1.0}[-1.0]{\lower
5.97221pt\hbox{$\scriptscriptstyle\widehat{\vrule
width=0.0pt,height=3.47221pt\vrule
height=0.0pt,width=2.22221pt}$}}}}\cr\hbox{$\scriptscriptstyle\delta$}}}}}^{-}Q_{1}$
$\displaystyle=\left(\dfrac{m}{\bar{m}}-1\right)\partial_{X}\bar{Q}_{0}.$

Define $\chi_{m}$ and $\chi_{k}$ as the solutions to

$\displaystyle\delta^{+}\chi_{k}=\dfrac{\widetilde{k}}{k}-1\ \text{and}\
\delta^{-}\chi_{m}=\dfrac{m}{\bar{m}}-1.$

Using formula (2.5) in Lemma 2.1, we can solve explicitly for $\chi_{k}$ and
$\chi_{m}.$ They are

$\displaystyle\chi_{k}(j)$
$\displaystyle=\sum_{i=0}^{j-1}\left[\dfrac{\widetilde{k}}{k(i)}-1\right]\
\text{and}\
\chi_{k}(-j)=\sum_{i=1}^{j}\left[1-\dfrac{\widetilde{k}}{k(i)}\right]$ (2.16)
$\displaystyle\chi_{m}(j)$
$\displaystyle=\sum_{i=0}^{j-1}\left[\dfrac{m(i)}{\bar{m}}-1\right]\
\text{and}\ \chi_{k}(-j)=\sum_{i=1}^{j}\left[1-\dfrac{m(i)}{\bar{m}}\right].$

Observe that $\displaystyle\dfrac{\widetilde{k}}{k}-1$ and
$\dfrac{m}{\bar{m}}-1$ are mean zero random variables and as such $\chi_{k}$
and $\chi_{m}$ are classical random walks. The expression for $Q_{1}$ and
$P_{1}$ can be given in terms of $\chi_{k}$ and $\chi_{m}$:

$\displaystyle Q_{1}(j,X,\tau)$
$\displaystyle=\chi_{m}(j)\partial_{X}\bar{Q}_{0}(X,\tau)$ (2.17)
$\displaystyle P_{1}(j,X,\tau)$
$\displaystyle=\chi_{k}(j)\partial_{X}\bar{P}_{0}(X,\tau).$

We need to know estimates for the norm of $P_{1}$ and $Q_{1}$ so that we can
estimate the residuals. Results are given in the next section. Here is an
important preview of what we find: the growth rates for random walks
ultimately imply that the terms $\epsilon P_{1}$ and $\epsilon R_{1}$ in (2.6)
are, despite appearances, not actually $O(\epsilon)$. This in turn implies
that the residuals are not as small as their formal derivation (namely
$O(\epsilon^{2})$) would lead one to believe. This is the main technical
complication in this article and the key difference between the random problem
we study here and the periodic or constant coefficient problems studied in
[8].

Before moving on, we now spell out our long wave approximation in detail.
Putting together (2.2), (2.6), (2.9), (2.10), (2.15) and (2.17) we see that

$\begin{split}\widetilde{r}_{\epsilon}(j,t)&={1\over
k(j)}\left(A(\epsilon(j-ct))+B(\epsilon(j+ct))\right)+\epsilon{\chi_{m}(j)\over
k(j)}\left(A^{\prime}(\epsilon(j-ct))+B^{\prime}(\epsilon(j+ct))\right)\\\
\widetilde{p}_{\epsilon}(j,t)&={1\over\sqrt{\widetilde{k}\bar{m}}}\left(-A(\epsilon(j-ct))+B(\epsilon(j+ct))\right)+\epsilon{\chi_{k}(j)\over\sqrt{\widetilde{k}\bar{m}}}\left(-A^{\prime}(\epsilon(j-ct))+B^{\prime}(\epsilon(j+ct))\right).\end{split}$
(2.18)

##  3 Probabilistic estimates

In this section we provide tools which will allow us to compute the $\ell^{2}$
norms of the residuals for all $|t|\leq T_{0}/\epsilon$. The first subsection
deals with almost sure and realization dependent estimates by making use of
the the law of iterated logarithms (LIL). The second subsection provides
estimates on the expectation of the norms using martingale inequalities.

###  3.1 Almost Sure Estimates

One can find the statement of the LIL in [3] and more details can be found in
[4]. Here we present the theorem in a form convenient to us.

######  Theorem 3.1.

(The Law of Iterated Logarithms) Suppose $y(j)$ ($j\in{\bf{Z}}$) are i.i.d
random variables with mean zero and ${\mathbb{E}}[y^{2}]=\sigma^{2}$. Define
the (two-sided) random walk $\chi$ via

$\chi(j)\coloneqq\sum_{i=0}^{j-1}y(i)\quad\text{and}\quad\chi(-j)\coloneqq\sum_{i=1}^{j}y(-i)$
(3.1)

for $j>0$ and $\chi(0)=0$.

Then

$\limsup_{|j|\to\pm\infty}\frac{\pm\chi(j)}{\sqrt{2|j|\log\log(|j|)}}\overset{a.s.}{=}\sigma.$

The LIL is an extremely sharp description of a random walk. It says that, with
a probability of one, the magnitude of $\chi(j)$ exceeds the curve
$\sigma\sqrt{2j\log\log(j)}$ (by any fixed amount) only a finite number of
times but comes arbitrarily near it an infinite number of times. Here is how
we use the LIL:

######  Corollary 3.2.

For almost every realization of $\\{k(j)\\}$ and $\\{m(j)\\}$ there is a
finite positive constant $C_{\omega}=C_{\omega}(k,m)$ for which

$|\chi_{k}(j)|+|\chi_{m}(j)|\leq C_{\omega}\sqrt{|j|\log\log(|j|+e)}$

for all $j\in{\bf{Z}}$.

######  Remark 1.

The constant $C_{\omega}$ is almost surely finite by the LIL, but it may be
extremely large. There is no way to determine its magnitude except in very
special circumstances. Note, however, it does not depend on $\epsilon$.

######  Remark 2.

In this paper, we use a small modification of the usual “big $C$” notation. If
a constant in an estimate depends on the particular realization of the
coefficients we mark it as “$C_{\omega}$.” If it does not, we omit the
subscript $\omega$. All such constants $C_{\omega}$ are always almost surely
finite. No such constants will ever depend on $\epsilon$.

###### Proof.

The LIL implies that for almost every realization of $\\{k(j)\\}$ there is a
natural number $N_{k}$ such that

$|\chi_{k}(j)|\leq 2\widetilde{k}\sigma_{k}\sqrt{2|j|\log\log(|j|)}$

when $|j|\geq N_{k}$. Then put

$C_{k}:=\max\left\\{{2\widetilde{k}\sigma_{k}\sqrt{2}},\max_{0<|j|\leq
N_{k}}{\chi_{k}(j)\over\sqrt{|j|\log\log(|j|)}}\right\\}.$

It follows that $|\chi_{k}(j)|\leq C_{k}\sqrt{|j|\log\log(|j|)}\leq
C_{k}\sqrt{|j|\log\log(|j|+e)}$ for all $j$. The same argument can be used to
estimate $\chi_{m}$. ∎

Given the growth rate in the LIL, we introduce a new norm fashioned to absorb
it:

$\|F\|_{H^{s}_{LIL}}:=\sum_{i=0}^{s}\|(1+|\cdot|\log\log(|\cdot|+e))^{1/2}F^{(i)}\|_{L^{2}}.$

The space $H^{s}_{LIL}$ will be the completion of $L^{2}$ with respect to this
norm. Similarly, we also introduce

$\|F\|_{H^{s}_{sr}}:=\sum_{i=0}^{s}\|(1+|\cdot|)^{1/2}F^{(i)}\|_{L^{2}}$

and the space $H^{s}_{sr}$. Note that
$\|F\|_{H^{s}}\leq\|F\|_{H^{s}_{sr}}\leq\|F\|_{H^{s}_{LIL}}$ where $H^{s}$ is
the usual $L^{2}$-based Sobolev space of functions ${\bf{R}}\to{\bf{R}}$ which
are weakly $s$-times differentiable.

Now we unveil the two main estimates we need to provide almost sure control of
the residuals.

######  Lemma 3.3.

For any $T_{0}>0$ and almost every realization of $\\{k(j)\\}$ and
$\\{m(j)\\}$ there is a finite positive constant
$C_{\omega}=C_{\omega}\left(k,m,T_{0}\right)$ for which $\epsilon\in(0,1/2)$
implies

$\sup_{|t|\leq T_{0}/\epsilon}\|\chi(\cdot)F(\epsilon(\cdot-
ct))\|_{\ell^{2}}\leq
C_{\omega}\epsilon^{-1}\sqrt{\log\log(\epsilon^{-1})}\|F\|_{H^{1}_{LIL}}$
(3.2)

and

$\sup_{|t|\leq T_{0}/\epsilon}\|\chi(\cdot)\delta^{\pm}F(\epsilon(\cdot-
ct))\|_{\ell^{2}}\leq
C_{\omega}\sqrt{\log\log(\epsilon^{-1})}\|F\|_{H^{2}_{LIL}}.$ (3.3)

In the above $\chi$ is either $\chi_{k}$ or $\chi_{m}$.

To prove these we need some calculus estimates.

######  Lemma 3.4.

For all $\epsilon\in(0,1)$, and $a,b\in{\bf{R}}$

$|a+b|\log\log(|a+b|+e)\leq|a|\log\log(2|a|+e)+|b|\log\log(2|b|+e)$

and

$\log\log(|x|+e)\leq\log(2\log(\epsilon|x|+e))+\log\log(\epsilon^{-1}+e).$

###### Proof.

The first inequality follows from the fact that $|x|\log\log(|x|+e)$ is a
convex function.

We will show the second inequality in two steps. First we show that

$\log\log(|x|+e)\leq\log(2\log(\epsilon|x|+\epsilon^{-1}+e)).$

Since $\log$ is monotonic, this inequality follows from

$|x|+e\leq(\epsilon|x|+\epsilon^{-1}+e)^{2},$

which is trivial.

Now we show that

$\log(2\log(\epsilon|x|+\epsilon^{-1}+e))\leq\log(2\log(\epsilon|x|+e))+\log\log(\epsilon^{-1}+e).$

Note that at $x=0$, equality holds. For $x\geq 0$ we have that

$\frac{d}{dx}\log(\epsilon x+\epsilon^{-1}+e)=\frac{\epsilon}{\epsilon
x+\epsilon^{-1}+e}$

and

$\frac{d}{dx}\log(\epsilon|x|+e)\log(\epsilon^{-1}+e)=\frac{\epsilon\log(\epsilon^{-1}+e)}{\epsilon
x+e}.$

Since

$\frac{\epsilon}{\epsilon x+\epsilon^{-1}+e}\leq{\epsilon\over\epsilon
x+e}\leq\frac{\epsilon\log(\epsilon^{-1}+e)}{\epsilon x+e},$

we see that $\log(\epsilon x+\epsilon^{-1}+e)$ grows more slowly than
$\log(\epsilon x+e)\log(\epsilon^{-1}+e)$. Since both functions are even, we
get by symmetry that

$2\log(\epsilon|x|+\epsilon^{-1}+e)\leq
2\log(\epsilon|x|+e)\log(\epsilon^{-1}+e).$

Taking $\log$ of both sides, we get the desired result. ∎

Now we can prove our key estimates.

###### Proof.

(Lemma 3.3) Take $\chi$ to be $\chi_{k}$ or $\chi_{m}$ and fix $T_{0}>0$.
Using Corollary 3.2

$\displaystyle\|\chi(\cdot)F(\epsilon(\cdot-ct))\|_{\ell^{2}}$
$\displaystyle=\left(\sum_{j\in{\bf{Z}}}\chi(j)^{2}F(\epsilon(j-ct))^{2}\right)^{1/2}$
$\displaystyle\leq
C_{\omega}\left(\sum_{j\in{\bf{Z}}}|j|\log\log(|j|+e)F(\epsilon(j-ct))^{2}\right)^{1/2}.$

The constant $C_{\omega}$ here depends upon the realization and any estimate
below will depend on the realization because of this step only.

Using the first inequality in Lemma 3.4 with $a=j-ct$ and $b=ct$ and the
triangle inequality we get

$\displaystyle\|\chi(\cdot)F(\epsilon(\cdot-ct))\|_{\ell^{2}}$
$\displaystyle\leq
C_{\omega}\left(\sum_{j\in{\bf{Z}}}|j-ct|\log\log(2|j-ct|+e)F(\epsilon(j-ct))^{2}\right)^{1/2}$
$\displaystyle+C_{\omega}\sqrt{|t|\log\log(2c|t|+e)}\|F(\epsilon(\cdot-
ct))\|_{\ell^{2}}.$

Call the two terms on the right $I$ and $II$. We estimate $II$ first.

Lemma 4.3 from [5] shows that

$\|F(\epsilon(\cdot-ct))\|_{\ell^{2}}\leq C\epsilon^{-1/2}\|F(\cdot-
ct)\|_{H^{1}}=C\epsilon^{-1/2}\|F\|_{H^{1}}$

and so

$II\leq C_{\omega}\epsilon^{-1/2}\sqrt{|t|\log\log(2c|t|+e)}\|F\|_{H^{1}}.$

Then

$\sup_{|t|\leq T_{0}/\epsilon}II\leq
C_{\omega}\epsilon^{-1}\sqrt{\log\log(2cT_{0}\epsilon^{-1}+e)}\|F\|_{H^{1}}.$

Routine features of the logarithm show that
$\log\log(2cT_{0}\epsilon^{-1}+e)\leq C\log\log(1/\epsilon)$ when
$\epsilon\in(0,1/2)$ and so we have

$\sup_{|t|\leq T_{0}/\epsilon}II\leq
C_{\omega}\epsilon^{-1}\sqrt{\log\log(1/\epsilon)}\|F\|_{H^{1}}.$

As for $I$, using the second inequality in Lemma 3.4 with $|x|=2|j-ct|$
followed by the triangle inequality gets us:

$\begin{split}I\leq&C_{\omega}\left(\sum_{j\in{\bf{Z}}}|j-ct|\log(2\log(2\epsilon|j-ct|+e))F(\epsilon(j-ct))^{2}\right)^{1/2}\\\
+&C_{\omega}\left(\sum_{j\in{\bf{Z}}}|j-ct|\log\log(\epsilon^{-1}+e)F(\epsilon(j-ct))^{2}\right)^{1/2}.\end{split}$

Then we multiply by $\sqrt{\epsilon/\epsilon}$ and do some algebra to get:

$\begin{split}I\leq&C_{\omega}\epsilon^{-1/2}\|\sqrt{\epsilon|\cdot-
ct|\log(2\log(2|\epsilon(\cdot-ct)|+e))}F(\epsilon(\cdot-ct))\|_{\ell^{2}}\\\
+&C_{\omega}\epsilon^{-1/2}\sqrt{\log\log(\epsilon^{-1}+e)}\|\sqrt{\epsilon|\cdot-
ct|}F(\epsilon(\cdot-ct))\|_{\ell^{2}}.\end{split}$

Applying Lemma 4.3 from [5] to tells us that

$\begin{split}\|\sqrt{\epsilon|\cdot-ct|\log(2\log(2|\epsilon(\cdot-
ct)|+e))}F(\epsilon(\cdot-
ct))\|_{\ell^{2}}\leq&C\epsilon^{-1/2}\|F\|_{H^{1}_{LIL}}\end{split}$

and

$\|\sqrt{\epsilon|\cdot-ct|}F(\epsilon(\cdot-ct))\|_{\ell^{2}}\leq
C\epsilon^{-1/2}\|F\|_{H^{1}_{sr}}$

and so we have,

$I\leq
C_{\omega}\epsilon^{-1}\sqrt{\log\log(\epsilon^{-1}+e)}\|F\|_{H^{1}_{LIL}}\leq
C_{\omega}\epsilon^{-1}\sqrt{\log\log(\epsilon^{-1})}\|F\|_{H^{1}_{LIL}}.$

Note that the right hand side does not depend on $t$ and so $\sup_{|t|\leq
T_{0}/\epsilon}I\leq C_{\omega}\epsilon^{-1}\sqrt{\log\log(\epsilon^{-1})}$
and all together we have shown (3.2).

It happens that (3.3) follows almost immediately from (3.2) with some operator
trickery. For functions $G:{\bf{R}}\to{\bf{R}}$ and $\epsilon\neq 0$ define
the operator ${\mathcal{A}}_{\epsilon}$ via

$({\mathcal{A}}_{\epsilon}G)(X):={1\over\epsilon}\int_{X}^{X+\epsilon}G(s)ds.$

We have

$\|{\mathcal{A}}_{\epsilon}G\|_{H}\leq C\|G\|_{H}$ (3.4)

where $H$ may be $H^{s}$, $H^{s}_{LIL}$ or $H^{s}_{sr}$. Here comes the
argument. First we use Jensen’s inequality to get:

$\|w(\cdot){\mathcal{A}}_{\epsilon}G\|_{L^{2}}^{2}=\int_{-\infty}^{\infty}w(X)^{2}\left(\dfrac{1}{\epsilon}\int_{X}^{X+\epsilon}G(s)ds\right)^{2}dX\leq\int_{-\infty}^{\infty}w(X)^{2}\dfrac{1}{\epsilon}\int_{X}^{X+\epsilon}G(s)^{2}dsdX.$

In the above $w(X)$ is a weight function. If we change the order of
integration we get

$\|w(\cdot){\mathcal{A}}_{\epsilon}G\|_{L^{2}}^{2}=\int_{-\infty}^{\infty}G(s)^{2}{1\over\epsilon}\int_{s-\epsilon}^{s}w(X)^{2}dXds.$

Let $\displaystyle b_{\epsilon}(s):={1\over\epsilon
w(s)^{2}}\int_{s-\epsilon}^{s}w(X)^{2}dX$ so we have

$\|w(\cdot){\mathcal{A}}_{\epsilon}G\|_{L^{2}}^{2}=\int_{-\infty}^{\infty}w(s)^{2}G(s)^{2}b_{\epsilon}(s)dXds\leq\|b_{\epsilon}\|_{L^{\infty}}\|w(\cdot)G\|^{2}_{L^{2}}.$

If $w(X)=1$, $w(X)=\sqrt{1+|X|}$ or $w(X)=\sqrt{1+|X|\log\log(|X|+e)}$ it is
easy to use the mean value theorem to show $\|b_{\epsilon}\|_{L^{\infty}}\leq
C$ when $\epsilon\in(0,1)$. With this, the last displayed inequality implies
(3.4) (A little calculus shows that when $w(X)=\sqrt{1+|X|}$, then
$\|b_{\epsilon}\|_{L^{\infty}}\leq 3/2$ and thus in (3.4) $C\leq 3/2$.)

Continuing on in the proof of (3.3), the fundamental theorem of calculus tells
us that $F(X+\epsilon)-F(X)=\epsilon({\mathcal{A}}_{\epsilon}F^{\prime})(X)$.
Thus:

$(\delta^{+}F)(\epsilon(j-ct))=F(\epsilon(j-ct)+\epsilon)-F(\epsilon(j-ct))=\epsilon({\mathcal{A}}_{\epsilon}F^{\prime})(\epsilon(j-ct)).$

In which case we see that

$\|\chi(\cdot)\delta^{+}F(\epsilon(\cdot-
ct))\|_{\ell^{2}}=\epsilon\|\chi(\cdot)({\mathcal{A}}_{\epsilon}F^{\prime})(\epsilon(\cdot-
ct))\|_{\ell^{2}}.$

We have produced an extra factor of $\epsilon$! Using (3.2) and (3.4)

$\|\chi(\cdot)\delta^{+}F(\epsilon(\cdot-
ct))\|_{\ell^{2}}\leq{C}_{\omega}\sqrt{\log\log(\epsilon^{-1})}\|{\mathcal{A}}_{\epsilon}F^{\prime}\|_{H^{1}_{LIL}}\leq{C}_{\omega}\sqrt{\log\log(\epsilon^{-1})}\|F\|_{H^{2}_{LIL}}.$

That is (3.3) and does it for this proof.

∎

Now we can prove:

######  Proposition 3.5.

Fix $A,B\in H^{3}_{LIL}$ and take $\widetilde{r}_{\epsilon}$ and
$\widetilde{p}_{\epsilon}$ as in (2.18). Fix $T_{0}>0$. Then for almost every
realization of $\\{k(j)\\}$ and $\\{m(j)\\}$ there is a finite positive
constant
$C_{\omega}=C_{\omega}\left(k,m,T_{0},\|A\|_{H^{3}_{LIL}},\|B\|_{H^{3}_{LIL}}\right)$
for which $\epsilon\in(0,1/2)$ implies

$\sup_{|t|\leq
T_{0}/\epsilon}\left(\|\operatorname{Res}_{1}(\widetilde{r}_{\epsilon},\widetilde{p}_{\epsilon})\|_{\ell^{2}}+\|\operatorname{Res}_{1}(\widetilde{r}_{\epsilon},\widetilde{p}_{\epsilon})\|_{\ell^{2}}\right)\leq
C_{\omega}\epsilon\sqrt{\log\log(1/\epsilon)}.$ (3.5)

###### Proof.

We prove the estimate for the piece involving $\operatorname{Res}_{1}$ as the
other part is all but identical. A tedious calculation shows that

$\displaystyle\text{Res}_{1}(\widetilde{r}_{\epsilon},\widetilde{p}_{\epsilon})$
$\displaystyle=\frac{1}{\sqrt{\widetilde{k}\bar{m}}}\left(-\delta^{+}[A(\epsilon(j-ct))]+\epsilon
A^{\prime}(\epsilon(j-ct))\right)$ (3.6)
$\displaystyle+\frac{1}{\sqrt{\widetilde{k}\bar{m}}}\left(-\delta^{+}[B(\epsilon(j+ct))]+\epsilon
B^{\prime}(\epsilon(j+ct))\right)$
$\displaystyle+\frac{c\epsilon^{2}\chi_{m}(j)}{k(j)}A^{\prime}(\epsilon(j-ct))+\frac{c\epsilon^{2}\chi_{m}(j)}{k(j)}B^{\prime}(\epsilon(j+ct))$
$\displaystyle-\frac{\epsilon\chi_{k}(j+1)}{\sqrt{\widetilde{k}\bar{m}}}\delta^{+}[A^{\prime}(\epsilon(j-ct))]-\frac{\epsilon\chi_{k}(j+1)}{\sqrt{\widetilde{k}\bar{m}}}\delta^{+}[B^{\prime}(\epsilon(j+ct))].$

The terms in the first two lines are fully deterministic and estimable using
Lemma 4.3 of [5]. Specifically the $\ell^{2}$ norm of each is controlled by

$C\epsilon^{3/2}\left(\|A\|_{H^{2}}+\|B\|_{H^{2}}\right)$

for $|t|\leq T_{0}/\epsilon$. This is dominated by the right hand side of
(3.5). Using (3.2) we see that the $\ell^{2}$ norm in the third line is
controlled by

$\epsilon^{2}\left(C_{\omega}\epsilon^{-1}\sqrt{\log\log(1/\epsilon)}\left(\|A^{\prime}\|_{H^{1}_{LIL}}+\|B^{\prime}\|_{H^{1}_{LIL}}\right)\right)$

for $|t|\leq T_{0}/\epsilon$. Again this is dominated by the right hand side
of (3.5). Similarly we use (3.3) to handle the terms in the last line, which
are controlled by

$\epsilon\left(C_{\omega}\sqrt{\log\log(1/\epsilon)}\left(\|A^{\prime}\|_{H^{2}_{LIL}}+\|B^{\prime}\|_{H^{2}_{LIL}}\right)\right).$

It is here we see why $H^{3}_{LIL}$ is needed in (3.5). ∎

######  Remark 3.

We quickly note that if the $k(j)$ and $m(j)$ are constant with respect to
$j$, one can easily chase through this proof and see that the estimate size of
the residuals decreases to $C\epsilon^{3/2}$. Likewise if the springs and
masses vary periodically, one finds that $\chi_{m}(j)$ and $\chi_{k}(j)$ are
in $\ell^{\infty}$ and then this proof would demonstrate the size of the
residuals is bounded by $C\epsilon^{1/2}$.

###  3.2 Boundedness in Mean

The almost sure boundedness does not provide us with any kind of description
for the $\omega$ dependent constant $C_{\omega}$. In this section we estimate
the error in mean, finding estimates in terms of $\sigma_{m}$ and
$\sigma_{k}$.

######  Lemma 3.6.

Let $y(j)$ and $\chi(j)$ be as in Theorem 3.1 and
$n\in{\bf{Z}}^{+}\cup\\{0\\}$. Consider the process

$W_{j}(n):=\chi(j+n)-\chi(j).$

Then, for every $j$, $W_{j}(n)$ is a martingale in the variable $n$ and, for
any $N>0$,

${\mathbb{E}}[\max_{0\leq n\leq N}(W_{j}(n))^{2}]\leq 4N\sigma^{2}.$ (3.7)

###### Proof.

From the definition of $\chi$, we have

$W_{j}(n)=y(j)+\cdots+y(j+n-1).$ (3.8)

Then

${\mathbb{E}}[|W_{j}(n)|]\leq C|n|.$

Conditioning upon $W_{j}(n)$ gives

$\displaystyle{\mathbb{E}}[W_{j}(n+1)|W_{j}(n)]={\mathbb{E}}[y(j+n)+W_{j}(n)|W_{j}(n)]=W_{j}(n).$

This proves that $W_{j}$ is a martingale. From the basic theory of martingales
this tells us $|W_{j}|^{2}$ is a submartingale. It follows from the $L^{p}$
maximum inequality, see [3], and a direct computation using (3.8)

${\mathbb{E}}\left[\max_{0\leq n\leq N}(W_{j}(n))^{2}\right]\leq
4{\mathbb{E}}[|W_{j}(N)|^{2}]=4N\sigma^{2}.$

∎

######  Remark 4.

We can define a similar process $W_{j}(n):=\chi(j-n)-\chi(j)$ which would have
exactly the same properties but with a different version of (3.8) i.e.

$W_{j}(n)=y(j-n)+\cdots+y(j-1).$

This symmetry allows us to handle positive and negative times with the same
argument.

We use the following corollary in the results that follow.

######  Corollary 3.7.

$\chi_{k}(j+n)-\chi_{k}(j)\quad\text{and}\quad\chi_{m}(j+n)-\chi_{m}(j)$

are martingales in $n$ with

${\mathbb{E}}\left[\max_{0\leq n\leq
N}\left(\chi_{k}(j+n)-\chi_{k}(j)\right)^{2}\right]\leq
4N\widetilde{k}\sigma_{k}^{2}\quad\text{and}\quad\
{\mathbb{E}}\left[\max_{0\leq n\leq
N}\left(\chi_{k}(j+n)-\chi_{k}(j)\right)^{2}\right]\leq
4N\dfrac{\sigma^{2}_{m}}{\bar{m}}.$

We have now gotten the necessary probability out of the way to prove the
following lemma, analogous to Lemma 3.3, but in expectation.

######  Lemma 3.8.

For any $T_{0}>0$ and $\epsilon\in(0,1/2)$ the following inequalities hold

${\mathbb{E}}\left[\sup_{|t|\leq T_{0}/\epsilon}\|\chi(\cdot)F(\epsilon(\cdot-
ct))\|_{\ell^{2}}\right]\leq
2\sqrt{2}\epsilon^{-1}\sigma\max\\{2\sqrt{|c|T_{0}},1\\}\|F\|_{H^{2}_{sr}}$
(3.9)

and

${\mathbb{E}}\left[\sup_{|t|\leq
T_{0}/\epsilon}\|\chi(\cdot)\delta^{\pm}F(\epsilon(\cdot-
ct))\|_{\ell^{2}}\right]\leq
3\sqrt{2}\sigma\max\\{2\sqrt{|c|T_{0}},1\\}\|F\|_{H^{3}_{sr}}.$ (3.10)

In the above $\chi$ is either $\chi_{k}$ or $\chi_{m}$ with $\sigma$ either
$\sigma_{k}\sqrt{\widetilde{k}}$ or $\dfrac{\sigma_{m}}{\sqrt{\bar{m}}}$
respectively.

###### Proof.

Without loss of generality (see Remark 4) let $t\in{\bf{R}}^{+}\cup\\{0\\}$.
Write $ct=\lfloor ct\rfloor+\alpha$ where $\alpha\in[0,1).$ Let $n\in{\bf{Z}}$
in the following. We start with the inequality

$\sup_{0\leq t\leq T_{0}/\epsilon}\|\chi(\cdot)F(\epsilon\cdot-\epsilon
ct)\|^{2}_{\ell^{2}}\leq\sup_{0\leq n\leq
cT_{0}/\epsilon,\alpha\in[0,1)}\sum_{j\in{\bf{Z}}}\chi(j)^{2}F(\epsilon
j-\epsilon n-\epsilon\alpha)^{2}.$

The inequality is due to the fact that for any $t\in[0,\lfloor
T_{0}/\epsilon\rfloor+1)$ there exists an $n\in[0,cT_{0}/\epsilon]$ and
$\alpha\in[0,1)$ s.t. $n+\alpha=ct$, which is a slightly greater range for $t$
than we initially cared about. Using the Mean Value Theorem, we have that

$F(\epsilon j-\epsilon n-\epsilon\alpha)=F(\epsilon j-\epsilon
n)-\epsilon\alpha F^{\prime}(x_{j})$

where $x_{j}\in(\epsilon j-\epsilon n-\epsilon\alpha,\epsilon j-\epsilon n).$
Substituting this in and using the basic inequality $(a+b)^{2}\leq
2(a^{2}+b^{2})$, we get

$\displaystyle\sup_{0\leq t\leq
T_{0}/\epsilon}\|\chi(\cdot)F(\epsilon\cdot-\epsilon ct)\|^{2}_{\ell^{2}}$
$\displaystyle\leq\sup_{0\leq n\leq
cT_{0}/\epsilon,\alpha\in[0,1)}\sum_{j\in{\bf{Z}}}\chi(j)^{2}(F(\epsilon
j-\epsilon n)-\epsilon\alpha F^{\prime}(x_{j}))^{2}.$
$\displaystyle\leq\sup_{0\leq n\leq
cT_{0}/\epsilon,\alpha\in[0,1)}2\sum_{j\in{\bf{Z}}}\chi(j)^{2}F(\epsilon
j-\epsilon n)^{2}+\chi(j)^{2}(\epsilon\alpha F^{\prime}(x_{j}))^{2}.$

Call the expectation of the first term $I$ and the second’s expectation $II$.
Note that $I$ does not depend upon $\alpha$. We find from a change of indices,
that

$\displaystyle I$ $\displaystyle={\mathbb{E}}\left[\sup_{0\leq n\leq
cT_{0}/\epsilon}2\sum_{j\in{\bf{Z}}}\chi(j+n)^{2}F(\epsilon j)^{2}\right]$
(3.11) $\displaystyle={\mathbb{E}}\left[\sup_{0\leq n\leq
cT_{0}/\epsilon}2\sum_{j\in{\bf{Z}}}(\chi(j+n)-\chi(j)+\chi(j))^{2}F(\epsilon
j)^{2}\right]$

Using the same basic inequality as above we get

$I\leq{\mathbb{E}}\left[\sup_{0\leq n\leq
cT_{0}/\epsilon}4\sum_{j\in{\bf{Z}}}\left(\chi(j)^{2}+(\chi(j+n)-\chi(j))^{2}\right)F(\epsilon
j)^{2}\right].$ (3.12)

The supremum sees only the term with $n$, and Fubini’s theorem allows the
expected value to pass through the sum. And so

$I\leq
4\sum_{j\in{\bf{Z}}}\left({\mathbb{E}}\left[\chi(j)^{2}\right]+{\mathbb{E}}\left[\sup_{0\leq
n\leq cT_{0}/\epsilon}(\chi(j+n)-\chi(j))^{2}\right]\right)F(\epsilon j)^{2}.$

A direct computation on the first term using the definition of $\chi$ and
using Corollary 3.7 on the second term we find

$I\leq
4\sum_{j\in{\bf{Z}}}\left(\sigma^{2}|j|+4\sigma^{2}cT_{0}\epsilon^{-1}\right)F(\epsilon
j)^{2}.$ (3.13)

According to Lemma 4.3 and 4.4 from [5], $I$ is dominated by

$8\epsilon^{-2}\sigma^{2}\max\\{4cT_{0},1\\}\|F\|^{2}_{H^{1}_{sr}}.$ (3.14)

Now we turn our attention to $II$. We can eliminate the $\alpha$ dependence by
taking $\alpha=1$ i.e. choose $\widetilde{x}_{j}$ s.t.

$F^{\prime}(\widetilde{x}_{j})=\max_{x\in[\epsilon j-\epsilon
n-\epsilon,\epsilon j-\epsilon n]}F^{\prime}(x).$

Then

$(\epsilon\alpha F^{\prime}(x_{j}))^{2}\leq(\epsilon
F^{\prime}(\widetilde{x}_{j}))^{2}.$

Shifting the index by $n$ we get

$II={\mathbb{E}}\left[\sup_{0\leq n\leq
cT_{0}/\epsilon}\sum_{j\in{\bf{Z}}}\chi(j+n)^{2}F^{\prime}(\widetilde{x}_{j+n})^{2}\right]$

where $\widetilde{x}_{j+n}\in[\epsilon j-\epsilon,\epsilon j]$ does not depend
on $n$. We therefore may relabel $\widetilde{x}_{j}=\widetilde{x}_{j+n}.$ We
use the same steps here as we used from to (3.12) to (3.13).

$II\leq
4\sum_{j\in{\bf{Z}}}(\sigma^{2}|j|+4\sigma^{2}cT_{0}\epsilon^{-1})\epsilon^{2}F^{\prime}(\epsilon\widetilde{x}_{j})^{2}.$

Again, by Lemma 4.3 from [5], $II$ is dominated by

$8\sigma^{2}\max\\{4cT_{0},1\\}\|F\|^{2}_{H^{2}_{sr}}.$ (3.15)

By (3.14) and (3.15) we have

${\mathbb{E}}\left[\sup_{0\leq t\leq
T_{0}/\epsilon}\|\chi(\cdot)F(\epsilon\cdot-\epsilon ct\|^{2}\right]\leq
8\epsilon^{-2}\sigma^{2}\max\\{4cT_{0},1\\}\|F\|^{2}_{H^{2}_{sr}}.$

An standard application of Jensen’s inequality yields

${\mathbb{E}}\left[\sup_{0\leq t\leq
T_{0}/\epsilon}\|\chi(\cdot)F(\epsilon\cdot-\epsilon ct\|\right]\leq
2\sqrt{2}\epsilon^{-1}\sigma\max\\{2\sqrt{cT_{0}},1\\}\|F\|_{H^{2}_{sr}}.$

This proves (3.9).

The exact same trickery that was used in Lemma 3.3 works to prove (3.10).
Using (3.9) and then (3.4)

${\mathbb{E}}\left[\sup_{0\leq t\leq
T_{0}/\epsilon}\|\chi(\cdot)\delta^{+}F(\epsilon(\cdot-
ct))\|_{\ell^{2}}\right]\leq
3\sqrt{2}\sigma\max\\{2\sqrt{cT_{0}},1\\}\|F\|_{H^{3}_{sr}}.$

This shows (3.10). ∎

######  Remark 5.

The functions in this subsection are required to be once more differentiable
than the functions in the previous subsection, due to the use of the Mean
Value Theorem in the beginning of the proof of the previous lemma.

Now we can prove:

######  Proposition 3.9.

Fix $A,B\in H^{4}_{sr}$ and take $\widetilde{r}_{\epsilon}$ and
$\widetilde{p}_{\epsilon}$ as in (2.18). Fix $T_{0}>0$. For there exists a
positive constant
$C(\widetilde{k},\bar{m},a_{k},b_{k},a_{m},b_{m},T_{0},\|A\|_{H_{sr}^{4}},\|B\|_{H_{sr}^{4}})$
for which $\epsilon\in(0,1/2)$ implies

${\mathbb{E}}\left[\sup_{|t|\leq
T_{0}/\epsilon}\left(\|\operatorname{Res}_{1}(\widetilde{r}_{\epsilon},\widetilde{p}_{\epsilon})\|_{\ell^{2}}+\|\operatorname{Res}_{2}(\widetilde{r}_{\epsilon},\widetilde{p}_{\epsilon})\|_{\ell^{2}}\right)\right]\leq
C\epsilon\left(\epsilon^{1/2}+\max\\{\sigma_{m},\sigma_{k}\\}\right).$ (3.16)

###### Proof.

The proof begins the same way as the proof for Proposition 3.5 except now we
take expectation of (3.6). Since the first two lines of (3.6) are
deterministic, using Lemma 4.3 in [5], they are controlled by

$\epsilon^{3/2}\dfrac{\sqrt{2}}{\sqrt{\widetilde{k}\bar{m}}}\left(\|A\|_{H^{2}}+\|B\|_{H^{2}}\right).$

Next use (3.9) to control the third line with

$\dfrac{2\sqrt{2}\epsilon\sigma_{m}c\max\\{2\sqrt{|c|T_{0}},1\\}}{a_{k}\sqrt{\bar{m}}}\left(\|A^{\prime}\|_{H_{sr}^{2}}+\|B^{\prime}\|_{H_{sr}^{2}}\right),$

which is dominated by (3.16). We use (3.10) to estimate the fourth line:

$\dfrac{3\sqrt{2}\epsilon\sigma_{k}\max\\{2\sqrt{|c|T_{0},}1\\}}{\sqrt{\bar{m}}}\left(\|A^{\prime}\|_{H_{sr}^{3}}+\|B^{\prime}\|_{H_{sr}^{3}}\right).$

As before, the estimate for $\operatorname{Res}_{2}$ follows a parallel
argument and is omitted. ∎

##  4 Error estimates

In this section we prove rigorous estimates using “energy” arguments, similar
to [5, 2, 10].

###  4.1 The energy argument

Let $r$ and $p$ be a true solution to (1.2) and take
$\widetilde{r}_{\epsilon}$ and $\widetilde{p}_{\epsilon}$ as in (2.18). Define
error functions $\eta$ and $\xi$ implicitly by

$r=\widetilde{r}_{\epsilon}+\frac{\eta}{k}\quad\text{and}\quad
p=\widetilde{p}_{\epsilon}+\xi.$ (4.1)

It is our goal to determine the size in $\ell^{2}$ of $\eta$ and $\xi$ during
the period $|t|\leq T_{0}/\epsilon$. To that end, insert (4.1) into (1.2) to
find that

$\displaystyle\frac{\dot{\eta}}{k}$
$\displaystyle=\delta^{+}\xi+\operatorname{Res}_{1}$ (4.2) $\displaystyle
m\dot{\xi}$ $\displaystyle=\delta^{-}\eta+\operatorname{Res}_{2}$

where
$\operatorname{Res}_{1}=\operatorname{Res}_{1}(\widetilde{r}_{\epsilon},\widetilde{p}_{\epsilon})$
and
$\operatorname{Res}_{2}=\operatorname{Res}_{2}(\widetilde{r}_{\epsilon},\widetilde{p}_{\epsilon})$
as in (2.1).

Next define the energy to be

$H(t)\coloneqq\frac{1}{2}\sum_{j\in{\bf{Z}}}\left[k(j)^{-1}\eta^{2}(j,t)+m(j)\xi^{2}(j,t)\right].$

Since we have assumed that the $k(j)$ and $m(j)$ are drawn from distributions
with support in $[a_{k},b_{k}]\subset{\bf{R}}^{+}$ and
$[a_{m},b_{m}]\subset{\bf{R}}^{+}$, respectively, a short calculation shows
that $\sqrt{H}$ is equivalent to $\|\eta,\xi\|_{\ell^{2}\times\ell^{2}}$ and
the constants of equivalence depend only on $a_{k},a_{m},b_{k}$ and $b_{m}$.
That is to say, the equivalence is realization independent.

Time differentiation of $H$ gives

$\dot{H}=\sum_{j\in{\bf{Z}}}\left[k^{-1}\eta\dot{\eta}+m\xi\dot{\xi}\right].$

Using (4.2)

$\dot{H}=\sum_{j\in{\bf{Z}}}\left[\eta(\delta^{+}\xi+\text{Res}_{1})+\xi(\delta^{-}\eta+\text{Res}_{2})\right].$

Summing by parts:

$\dot{H}=\sum_{j\in{\bf{Z}}}\left[\eta\text{Res}_{1}+\xi\text{Res}_{2}\right].$

Cauchy-Schwarz implies that

$\dot{H}\leq\|\text{Res}_{1},\text{Res}_{2}\|_{\ell^{2}\times\ell^{2}}\|\eta,\xi\|_{\ell^{2}\times\ell^{2}}.$

Then we use the equivalence of $\sqrt{H}$ and
$\|\eta,\xi\|_{\ell^{2}\times\ell^{2}}$ to get:

$\dot{H}\leq
C\|\text{Res}_{1},\text{Res}_{2}\|_{\ell^{2}\times\ell^{2}}\sqrt{H}.$

Set

$\Gamma_{\epsilon}\coloneqq\sup_{|t|\leq
T_{0}/\epsilon}\|\text{Res}_{1},\text{Res}_{2}\|_{\ell^{2}\times\ell^{2}},$

so $\dot{H}/\sqrt{H}\leq C\Gamma_{\epsilon}$. We integrate from $0$ to $t$

$2\sqrt{H(t)}\leq 2\sqrt{H(0)}+C\Gamma_{\epsilon}t.$

And so, for $t\leq T_{0}/\epsilon$, we have

$\sqrt{H(t)}\leq\sqrt{H(0)}+C\Gamma_{\epsilon}T_{0}\epsilon^{-1}.$

If we use the equivalence of the $\sqrt{H}$ and
$\|\eta,\xi\|_{\ell^{2}\times\ell^{2}}$ once again, we find that we have
proven

$\sup_{|t|\leq T_{0}/\epsilon}\|\eta(t),\xi(t)\|_{\ell^{2}\times\ell^{2}}\leq
C\|\eta(0),\xi(0)\|_{\ell^{2}\times\ell^{2}}+C\Gamma_{\epsilon}\epsilon^{-1}.$
(4.3)

A key feature of the above inequality is that the only place where the
specific realization of the springs and masses enters is through
$\Gamma_{\epsilon}$.

###  4.2 Almost sure error estimates

We can now prove our first main theorem, which is about almost sure estimation
of the absolute error:

######  Theorem 4.1.

Fix $\Phi,\Psi\in H^{3}_{LIL}$ and $T_{0}>0$. Let $r$ and $p$ be the solution
of (1.2) with initial data

$r(j,0)=\Phi(\epsilon j)/k(j)\quad\text{and}\quad p(j,0)=\Psi(\epsilon j).$

For almost every realization of $\\{k(j)\\}$ and $\\{m(j)\\}$ there is a
finite positive constant

$C_{\omega}=C_{\omega}(k,m,a_{k},b_{k},a_{m},b_{m},\|\Phi\|_{H^{3}_{LIL}},\|\Psi\|_{H^{3}_{LIL}})$

for which $\epsilon\in(0,1/2)$ implies

$\sup_{|t|\leq T_{0}/\epsilon}\left\|r(\cdot,t)-{1\over
k(\cdot)}\left(A(\epsilon(\cdot-
ct)+B(\epsilon(\cdot+ct)\right)\right\|_{\ell^{2}}\leq
C_{\omega}\sqrt{\log\log(1/\epsilon)}$

and

$\sup_{|t|\leq
T_{0}/\epsilon}\left\|p(\cdot,t)-{1\over\sqrt{\widetilde{k}\bar{m}}}\left(-A(\epsilon(\cdot-
ct)+B(\epsilon(\cdot+ct)\right)\right\|_{\ell^{2}}\leq
C_{\omega}\sqrt{\log\log(1/\epsilon)}.$

In the above

$A(X):={1\over 2}\Phi(X)-{\sqrt{\widetilde{k}\bar{m}}\over
2}\Psi(X)\quad\text{and}\quad B(X):={1\over
2}\Phi(X)+{\sqrt{\widetilde{k}\bar{m}}\over 2}\Psi(X).$

######  Remark 6.

In the case where the masses and springs vary periodically instead of
randomly, the size of the error decreases to $C\epsilon^{1/2}$; in fact the
proof we supply in a moment together with Remark 3 suffices to demonstrate
this. Likewise, if the masses and springs are constant a slightly modified
version of the proof can be used to decrease the error to $C\epsilon^{3/2}$.
It is this extra wiggle room in the error in these cases which opens the door
to longer time scales and KdV-like approximations.

###### Proof.

Form $\widetilde{r}_{\epsilon}$ and $\widetilde{p}_{\epsilon}$ from the
functions $A$ and $B$ as specified in (2.18) and $\eta$ and $\xi$ as in (4.1).
A bit of algebra shows that

$\eta(j,0)=\epsilon{\chi_{m}(j)}\left(A^{\prime}(\epsilon
j)+B^{\prime}(\epsilon
j)\right)\quad\text{and}\quad\xi(j,0)=\epsilon{\chi_{k}(j)\over\sqrt{\widetilde{k}\bar{m}}}\left(-A^{\prime}(\epsilon
j)+B^{\prime}(\epsilon j)\right).$ (4.4)

Using (3.2) in a very crude way, we see that almost surely

$\|\eta(0),\xi(0)\|_{\ell^{2}\times\ell^{2}}\leq
C_{\omega}\sqrt{\log\log(1/\epsilon)}$

with the constant depending on $\|A\|_{H^{2}_{LIL}}$ and
$\|B\|_{H^{2}_{LIL}}$. We estimated $\Gamma_{\epsilon}$ in Proposition 3.5 and
found that $\Gamma_{\epsilon}\leq
C_{\omega}\epsilon\sqrt{\log\log(1/\epsilon)}$ when $\epsilon\in(0,1/2)$
almost surely. Therefore (4.3) gives

$\sup_{|t|\leq T_{0}/\epsilon}\|\eta(t),\xi(t)\|_{\ell^{2}\times\ell^{2}}\leq
C_{\omega}\sqrt{\log\log(1/\epsilon)}.$

To finish the proof we note that the triangle inequality tells us

$\begin{split}&\left\|r(\cdot,t)-{1\over k(\cdot)}\left(A(\epsilon(\cdot-
ct))+B(\epsilon(\cdot+ct))\right)\right\|_{\ell^{2}}\\\
\leq&\left\|r(t)-\widetilde{r}_{\epsilon}(t)\right\|_{\ell^{2}}+\left\|\widetilde{r}_{\epsilon}(\cdot,t)-{1\over
k(\cdot)}\left(A(\epsilon(\cdot-
ct))+B(\epsilon(\cdot+ct))\right)\right\|_{\ell^{2}}\\\
\leq&C\|\eta(t)\|_{\ell^{2}}+C\epsilon\|{\chi_{m}(\cdot)}A^{\prime}(\epsilon(\cdot-
ct))\|+C\epsilon\|{\chi_{m}(\cdot)}B^{\prime}(\epsilon(\cdot-
ct))\|_{\ell^{2}}.\end{split}$

The terms involve $A$ and $B$ can be estimated using (3.2) by
$C_{\omega}\sqrt{\log\log(1/\epsilon)}$ so we find

$\sup_{|t|\leq T_{0}/\epsilon}\left\|r(\cdot,t)-{1\over
k(\cdot)}\left(A(\epsilon(\cdot-
ct))+B(\epsilon(\cdot+ct))\right)\right\|_{\ell^{2}}\leq
C_{\omega}\sqrt{\log\log(1/\epsilon)}.$

The remaining estimate in the Theorem 4.1 is shown by a parallel argument and
is omitted. ∎

It may seem like the estimates in Theorem 4.1 are utterly useless since the
size of the error diverges as $\epsilon\to 0^{+}$. But the error in that
theorem is the absolute error; the relative error does in fact vanish in the
limit.

######  Corollary 4.2.

Under the same conditions as in Theorem 4.1 we almost surely have

$\lim_{\epsilon\to 0^{+}}\sup_{|t|\leq
T_{0}/\epsilon}{\left\|r(\cdot,t)-{1\over k(\cdot)}\left(A(\epsilon(\cdot-
ct))+B(\epsilon(\cdot+ct))\right)\right\|_{\ell^{2}}\over\|r(t)\|_{\ell^{2}}}=0$

and

$\lim_{\epsilon\to 0^{+}}\sup_{|t|\leq
T_{0}/\epsilon}{\left\|p(\cdot,t)-{1\over\sqrt{\widetilde{k}\bar{m}}}\left(-A(\epsilon(\cdot-
ct))+B(\epsilon(\cdot+ct))\right)\right\|_{\ell^{2}}\over\|p(t)\|_{\ell^{2}}}=0.$

###### Proof.

The reverse triangle inequality gives

$\|r(t)\|_{\ell^{2}}\geq\left\|{1\over k(\cdot)}\left(A(\epsilon(\cdot-
ct))+B(\epsilon(\cdot+ct))\right)\right\|_{\ell^{2}}-\left\|r(\cdot,t)-{1\over
k(\cdot)}\left(A(\epsilon(\cdot-
ct)+B(\epsilon(\cdot+ct)\right)\right\|_{\ell^{2}}.$

Using Lemma 4.3 from [5] for the first term and Theorem 4.1 for the second we
obtain

$\|r(t)\|_{\ell^{2}}\geq
C\epsilon^{-1/2}-C_{\omega}\sqrt{\log\log(1/\epsilon)}$

for all $|t|\leq T_{0}/\epsilon$. This is positive for $\epsilon$ small enough
and so we get the first limit in the corollary by dividing the absolute error
for $r$ in Theorem 4.1 by this estimate and taking the limit. The second limit
is analogous. ∎

###  4.3 Error estimate in mean

We can now prove our second main theorem, which is an estimate of the mean of
the error.

######  Theorem 4.3.

Fix $\Phi,\Psi\in H^{4}_{sr}$ and $T_{0}>0.$ Let $r$ and $p$ be the solution
of (1.2) with initial data

$r(j,0)=\Phi(\epsilon j)/k(j)\quad\text{and}\quad p(j,0)=\phi(\epsilon j).$

There exists a positive constant
$C(\widetilde{k},\bar{m},a_{k},b_{k},a_{m},b_{m},T_{0},\|A\|_{H^{4}_{sr}},\|B\|_{H^{4}_{sr}})$
for which $\epsilon\in(0,1/2)$ implies

${\mathbb{E}}\left[\sup_{|t|\leq T_{0}/\epsilon}\left\|r(\cdot,t)-{1\over
k(\cdot)}\left(A(\epsilon(\cdot-
ct)+B(\epsilon(\cdot+ct)\right)\right\|_{\ell^{2}}\right]\leq
C\left(\epsilon^{1/2}+\max\\{\sigma_{m},\sigma_{k}\\}\right)$

and

${\mathbb{E}}\left[\sup_{|t|\leq
T_{0}/\epsilon}\left\|p(\cdot,t)-{1\over\sqrt{\widetilde{k}\bar{m}}}\left(-A(\epsilon(\cdot-
ct)+B(\epsilon(\cdot+ct)\right)\right\|_{\ell^{2}}\right]\leq
C\left(\epsilon^{1/2}+\max\\{\sigma_{m},\sigma_{k}\\}\right).$

In the above

$A(X):={1\over 2}\Phi(X)-{\sqrt{\widetilde{k}\bar{m}}\over
2}\Psi(X)\quad\text{and}\quad B(X):={1\over
2}\Phi(X)+{\sqrt{\widetilde{k}\bar{m}}\over 2}\Psi(X).$

###### Proof.

Begin as in the proof of Theorem 4.1. Using (3.9) on (4.4)

$\|\eta(0),\xi(0)\|_{\ell^{2}\times\ell^{2}}\leq
C\max\\{\sigma_{m},\sigma_{k}\\}$

with constant $C$ depending on $\|A\|_{H^{3}_{sr}}$ and $\|B\|_{H^{3}_{sr}}.$
Proposition 3.9 gives us

${\mathbb{E}}\left[\Gamma_{\epsilon}\right]\leq
C\epsilon\left(\epsilon^{1/2}+\max\\{\sigma_{m},\sigma_{k}\\}\right)$

when $\epsilon\in(0,1/2)$. Therefore (4.3) gives

${\mathbb{E}}\left[\sup_{|t|\leq
T_{0}/\epsilon}\|\eta(t),\xi(t)\|_{\ell^{2}\times\ell^{2}}\right]\leq
C\left(\epsilon^{1/2}+\max\\{\sigma_{m},\sigma_{k}\\}\right).$

To finish the proof we note that the triangle inequality tells us

$\begin{split}&{\mathbb{E}}\left[\sup_{|t|\leq
T_{0}/\epsilon}\left\|r(\cdot,t)-{1\over k(\cdot)}\left(A(\epsilon(\cdot-
ct))+B(\epsilon(\cdot+ct))\right)\right\|_{\ell^{2}}\right]\\\
\leq&{\mathbb{E}}\left[\sup_{|t|\leq
T_{0}}\left\|r(t)-\widetilde{r}_{\epsilon}(t)\right\|_{\ell^{2}}\right]+{\mathbb{E}}\left[\sup_{|t|\leq
T_{0}}\left\|\widetilde{r}_{\epsilon}(\cdot,t)-{1\over
k(\cdot)}\left(A(\epsilon(\cdot-
ct))+B(\epsilon(\cdot+ct))\right)\right\|_{\ell^{2}}\right]\\\
\leq&{\mathbb{E}}\left[\sup_{|t|\leq
T_{0}}C\|\eta(t)\|_{\ell^{2}}\right]+{\mathbb{E}}\left[\sup_{|t|\leq
T_{0}}C\epsilon\|{\chi_{m}(\cdot)}A^{\prime}(\epsilon(\cdot-
ct))\|\right]+{\mathbb{E}}\left[\sup_{|t|\leq
T_{0}}C\epsilon\|{\chi_{m}(\cdot)}B^{\prime}(\epsilon(\cdot-
ct))\|_{\ell^{2}}\right].\end{split}$

The $C$ depends on $a_{k},b_{k},a_{m}$, and $b_{m}$, which are fixed, so we
may pull it out of the expected value. The terms that involve $A$ and $B$ can
be estimated using (3.9) by $C\max\\{\sigma_{k},\sigma_{m}\\}$ so we find

${\mathbb{E}}\left[\sup_{|t|\leq T_{0}/\epsilon}\left\|r(\cdot,t)-{1\over
k(\cdot)}\left(A(\epsilon(\cdot-
ct))+B(\epsilon(\cdot+ct))\right)\right\|_{\ell^{2}}\right]\leq
C\left(\epsilon^{1/2}+\max\\{\sigma_{k},\sigma_{m}\\}\right).$

The remaining estimate in the Theorem 4.3 is shown by a parallel argument and
is omitted.

∎

##  5 Coarse-graining

We now prove strong convergence results using the ideas of coarse-graining
from [8]. We need quite a few tools. Letting $f:{\bf{Z}}\to{\bf{R}}$ and
$g,u,v:{\bf{R}}\to{\bf{R}}$ define

$\displaystyle F[f](\kappa)$
$\displaystyle:=\frac{1}{2\pi}\sum_{j\in{\bf{Z}}}e^{-ij\kappa}f(j)$
$\displaystyle F^{-1}[g](j)$
$\displaystyle:=\int_{-\pi}^{\pi}g(\kappa)e^{i\kappa j}$
$\displaystyle\mathcal{F}[u](\xi)$
$\displaystyle:=\frac{1}{2\pi}\int_{{\bf{R}}}u(x)e^{-i\xi x}dx$
$\displaystyle\mathcal{F}^{-1}[v](x)$
$\displaystyle:=\int_{{\bf{R}}}v(\xi)e^{i\xi x}d\xi$
$\displaystyle\theta_{\phi}(\kappa)$
$\displaystyle:=\begin{cases}1&\kappa\in(-\phi,\phi)\\\
0&\text{else}\end{cases}$ $\displaystyle\mathcal{L}[f](x)$
$\displaystyle:=\mathcal{F}^{-1}[\theta_{\pi}(\cdot)F[f](\cdot)](x)$
$\displaystyle\mathcal{S}[u](j)$ $\displaystyle:=u(j).$

These are, in order, the Fourier Transform for sequences, its inverse, the
Fourier transform of functions ${\bf{R}}\to{\bf{R}}$, its inverse, the
indicator function of $(-\phi,\phi)$, a “low pass” interpolation operator, and
a sampling operator. To be clear, in the above $x,\xi,\kappa\in{\bf{R}}$ and
$j\in{\bf{Z}}$, always.

The operator $\mathcal{L}$ converts a sequence $f$ defined on ${\bf{Z}}$ to a
new function defined on ${\bf{R}}.$ The sampling function $\mathcal{S}$
returns a sequence from a function defined on ${\bf{R}}$. It is an easy
exercise to show that $\mathcal{SL}[f](j)=f(j)$ so it is clear $\mathcal{L}$
is an interpolation operator. Another essential property is the following.

######  Lemma 5.1.

Let $f$ be a sequence in $\ell^{2}$. Then

$\|f\|_{\ell^{2}}=2\pi\|\mathcal{L}[f]\|_{L^{2}({\bf{R}})}.$

###### Proof.

By Plancherel’s theorem:

$\|F[f]\|_{L^{2}(-\pi,\pi)}=\frac{1}{2\pi}\|f\|_{\ell^{2}}.$

Then by a slightly different Plancherel’s theorem:

$\|\mathcal{F}^{-1}[\theta_{\pi}F[f]]\|_{L^{2}({\bf{R}})}=\|\theta_{\pi}F[f]\|_{L^{2}({\bf{R}})}=\|F[f]\|_{L^{2}(-\pi,\pi)}$

completing the proof. ∎

We need one more lemma before we can state the strong convergence results. It
states that for a smooth enough function, the more frequently it is sampled,
the more is interpolation looks like the original function.

######  Lemma 5.2.

Let $f:{\bf{R}}\to{\bf{R}}$ be in $H^{s}$ with $s>1/2$ and put
$f_{\epsilon}(x)=f(\epsilon x)$. Then

$\lim_{\epsilon\to
0^{+}}\|\mathcal{L}\mathcal{S}[f_{\epsilon}](\cdot/\epsilon)-f\|_{L^{2}}=0.$

###### Proof.

From their definitions we have

$\mathcal{L}{{\mathcal{S}}}[f_{\epsilon}](x)=\frac{1}{2\pi}\int_{-\pi}^{\pi}(\sum_{j\in{\bf{Z}}}e^{-i\kappa
j}f(\epsilon j))e^{i\kappa
x}d\kappa=\frac{1}{2\pi}\int_{-\pi}^{\pi}(\sum_{j\in{\bf{Z}}}e^{-i\frac{\kappa}{\epsilon}\epsilon
j}f(\epsilon j))e^{i\kappa x}d\kappa.$

Changing variables with $u=\kappa/\epsilon$ we get

$\mathcal{L}{{\mathcal{S}}}[f_{\epsilon}](x)=\frac{1}{2\pi}\int_{-\pi/\epsilon}^{\pi/\epsilon}(\sum_{j\in{\bf{Z}}}\epsilon
e^{-iu\epsilon j}f(\epsilon j))e^{iu\epsilon x}du.$

Exchanging the sum and integral and then computing the integral.

$\mathcal{L}{{\mathcal{S}}}[f_{\epsilon}](x)=\sum_{j\in{\bf{Z}}}f(\epsilon
j)\operatorname{sinc}(x-j).$

This $\operatorname{sinc}$ is the normalized $\operatorname{sinc}$ function.

Now put
$\widetilde{f}_{\epsilon}(X):=\mathcal{F}^{-1}[\theta_{\pi/\epsilon}\mathcal{F}[f]](X).$
$\widetilde{f}_{\epsilon}$ is a band limited approximation of $f$. Using
Plancherel’s theorem

$\|f-\widetilde{f}_{\epsilon}\|^{2}_{L^{2}}=\|\mathcal{F}[f]-\theta_{\pi/\epsilon}\mathcal{F}[f]\|^{2}_{L^{2}}=\int_{\kappa>|\pi/\epsilon|}\left|\mathcal{F}[f](\kappa)\right|^{2}d\kappa.$

Since $f\in H^{s}$ we have by Cauchy-Schwarz, when $\epsilon\in(0,1)$:

$\begin{split}\|f-\widetilde{f}_{\epsilon}\|^{2}_{L^{2}}=&\int_{\kappa>|\pi/\epsilon|}{1\over|\kappa|^{2s}}|\kappa|^{2s}\left|\mathcal{F}[f](\kappa)\right|^{2}d\kappa\\\
\leq&\left(\int_{|\kappa|\geq\pi/\epsilon}|\kappa|^{-2s}d\kappa\right)^{1/2}\left(\int_{|\kappa|\geq\pi/\epsilon}|\kappa|^{2s}\left|\mathcal{F}[f](\kappa)\right|^{2}d\kappa\right)^{1/2}\\\
\leq&C\epsilon^{{s-1/2}}\|f\|_{H^{s}}.\end{split}$

Since $s>1/2$ we see that $\lim_{\epsilon\to
0^{+}}\|f-\widetilde{f}_{\epsilon}\|_{L^{2}}=0$.

Since $\widetilde{f}_{\epsilon}$ is band limited, it is exactly equal to its
cardinal series, see [7],

$\widetilde{f}_{\epsilon}(X)=\sum_{j\in{\bf{Z}}}f(\epsilon
j)\operatorname{sinc}(X/\epsilon-j).$

But this is exactly equal to
${\mathcal{L}}{\mathcal{S}}[f_{\epsilon}](X/\epsilon)$. Therefore we have
shown that

$\lim_{\epsilon\to
0^{+}}\|\mathcal{L}\mathcal{S}(f_{\epsilon})(\cdot/\epsilon)-f(\cdot)\|_{L^{2}}=0.$

∎

Here is our first course-graining result:

######  Theorem 5.3.

Fix $\Phi,\Psi\in H^{3}_{LIL}$ and $T_{0}>0$. Let $r$ and $p$ be the solution
of (1.2) with initial data

$r(j,0)=\Phi(\epsilon j)/k(j)\quad\text{and}\quad p(j,0)=\Psi(\epsilon j).$

Put

$Q_{\epsilon}(X,\tau)={\mathcal{L}}[kr(\cdot,\tau/\epsilon)](X/\epsilon)\quad\text{and}\quad
P_{\epsilon}(X,\tau)={\mathcal{L}}[p(\cdot,\tau/\epsilon)](X/\epsilon).$

Suppose that $Q_{0}(X,\tau)$ and $P_{0}(X,\tau)$ solve (2.14) with initial
data $Q_{0}(X,0)=\Phi(X)$ and $P_{0}(X,0)=\Psi(X)$. Then, almost surely,

$\lim_{\epsilon\to 0^{+}}\sup_{|\tau|\leq
T_{0}}\left(\|Q_{\epsilon}(X,\tau)-Q_{0}(X,\tau)\|_{L^{2}}+\|P_{\epsilon}(X,\tau)-P_{0}(X,\tau)\|_{L^{2}}\right)=0.$

###### Proof.

We show the limit for $\|P_{\epsilon}(X,\tau)-P_{0}(X,\tau)\|_{L^{2}}$ as the
other is all but identical. By the triangle inequality we have

$\|P_{\epsilon}(\cdot,\tau)-P_{0}(\cdot,\tau)\|_{L^{2}}\leq\|P_{\epsilon}(\cdot,\tau)-{\mathcal{L}}{\mathcal{S}}[P_{0}(\epsilon\cdot,\tau)](\cdot/\epsilon)\|_{L^{2}}+\|{\mathcal{L}}{\mathcal{S}}[P_{0}(\epsilon\cdot,\tau)](\cdot/\epsilon)-P_{0}(\cdot,\tau)\|_{L^{2}}.$

The second term vanishes as $\epsilon\to 0^{+}$ by virtue of Lemma 5.2. (In
fact, given (2.15) one sees that this convergence happens uniformly for all
$\tau\in{\bf{R}}$.)

For the first term we do a change of variables $X=\epsilon x$ and
$\tau=\epsilon t$ to get

$\|P_{\epsilon}(\cdot,\tau)-{\mathcal{L}}{\mathcal{S}}[P_{0}(\epsilon\cdot,\tau)](\cdot/\epsilon)\|_{L^{2}}=\sqrt{\epsilon}\|P_{\epsilon}(\epsilon\cdot,\epsilon
t)-{\mathcal{L}}{\mathcal{S}}[P_{0}(\epsilon\cdot,\epsilon
t)](\cdot)\|_{L^{2}}.$

Then we use the definition of $P_{\epsilon}$ and Lemma 5.1 to get

$\|P_{\epsilon}(\cdot,\tau)-{\mathcal{L}}{\mathcal{S}}[P_{0}(\epsilon\cdot,\tau)](\cdot/\epsilon)\|_{L^{2}}={1\over
2\pi}\sqrt{\epsilon}\|p(\cdot,t)-{\mathcal{S}}[P_{0}(\epsilon\cdot,\epsilon
t)]\|_{\ell^{2}}.$

Using (2.15) and the formulas relating $\Phi$ and $\Psi$ to $A$ and $B$ in
Theorem 4.1 we see

${\mathcal{S}}[P_{0}(\epsilon\cdot,\epsilon
t)](j)={1\over\sqrt{\widetilde{k}\bar{m}}}\left(-A(\epsilon(j-ct)+B(\epsilon(j+ct)\right).$

Thus we can use the final estimate in Theorem 4.1 to get

$\sup_{|\tau|\leq
T_{0}}\|P_{\epsilon}(\cdot,\tau)-{\mathcal{L}}{\mathcal{S}}[P_{0}(\epsilon\cdot,\tau)](\cdot/\epsilon)\|_{L^{2}}\leq\sup_{|t|\leq
T_{0}/\epsilon}{1\over
2\pi}\sqrt{\epsilon}\|p(\cdot,t)-{\mathcal{S}}[P_{0}(\epsilon\cdot,\epsilon
t)]\|_{\ell^{2}}\leq C_{\omega}\sqrt{\epsilon\log\log(1/\epsilon)}.$ (5.1)

The right hand side goes to zero as $\epsilon\to 0^{+}$ and we are done.

∎

We have same result but the convergence is in mean:

######  Theorem 5.4.

Fix $\Phi,\Psi\in H^{4}_{sr}$ and $T_{0}>0$. Let $r$ and $p$ be the solution
of (1.2) with initial data

$r(j,0)=\Phi(\epsilon j)/k(j)\quad\text{and}\quad p(j,0)=\Psi(\epsilon j).$

Put

$Q_{\epsilon}(X,\tau)={\mathcal{L}}[kr(\cdot,\tau/\epsilon)](X/\epsilon)\quad\text{and}\quad
P_{\epsilon}(X,\tau)={\mathcal{L}}[p(\cdot,\tau/\epsilon)](X/\epsilon).$

Suppose that $Q_{0}(X,\tau)$ and $P_{0}(X,\tau)$ solve (2.14) with initial
data $Q_{0}(X,0)=\Phi(X)$ and $P_{0}(X,0)=\Psi(X)$. Then

$\lim_{\epsilon\to 0^{+}}{\mathbb{E}}\left[\sup_{|\tau|\leq
T_{0}}\left(\|Q_{\epsilon}(X,\tau)-Q_{0}(X,\tau)\|_{L^{2}}+\|P_{\epsilon}(X,\tau)-P_{0}(X,\tau)\|_{L^{2}}\right)\right]=0.$

###### Proof.

As before, we start with the triangle inequality

$\begin{split}&{\mathbb{E}}\left[\sup_{|\tau|\leq
T_{0}}\|P_{\epsilon}(\cdot,\tau)-P_{0}(\cdot,\tau)\|_{L^{2}}\right]\\\
\leq&{\mathbb{E}}\left[\sup_{|\tau|\leq
T_{0}}\|P_{\epsilon}(\cdot,\tau)-{\mathcal{L}}{\mathcal{S}}[P_{0}(\epsilon\cdot,\tau)](\cdot/\epsilon)\|_{L^{2}}\right]+{\mathbb{E}}\left[\sup_{|\tau|\leq
T_{0}}\|{\mathcal{L}}{\mathcal{S}}[P_{0}(\epsilon\cdot,\tau)](\cdot/\epsilon)-P_{0}(\cdot,\tau)\|_{L^{2}}\right].\end{split}$

The expected value does not see the second term, so it vanishes as
$\epsilon\to 0^{+}$ by virtue of Lemma 5.2. The same steps are valid up
through (5.1) only now we take expectation and use Theorem 4.3

${\mathbb{E}}\left[\sup_{|\tau|\leq
T_{0}}\|P_{\epsilon}(\cdot,\tau)-{\mathcal{L}}{\mathcal{S}}[P_{0}(\epsilon\cdot,\tau)](\cdot/\epsilon)\|_{L^{2}}\right]\leq
C\sqrt{\epsilon}\max\\{\sigma_{k},\sigma_{m}\\}.$

The right hand side vanishes as $\epsilon^{+}\to 0$. ∎

##  6 Simulations and Conclusion

We finish out the paper with supporting numerical simulations and a concluding
discussion.

###  6.1 Simulations

We present various numerical data supporting our results. In our experiments,
the springs $k$ are picked to be constant and the probability distribution of
the masses $m$ s.t. $\bar{m}=1.$. We choose initial conditions

$r(j)=e^{-(\epsilon j)^{2}}\quad\text{and}\quad p(j)=-e^{-(\epsilon j)^{2}}.$

From these

$A(X)=e^{-X^{2}}\quad\text{and}\quad B(X)=0.$

We numerically integrate (1.2) to get $r(j,t)$ and use this to calculate the
relative error which we call $\rho$

$\rho\coloneqq\sup_{0\leq t\leq
T_{0}/\epsilon}\dfrac{\|(r(\cdot,t)-A(\epsilon(\cdot-
ct))\|_{\ell^{2}}}{\|r(t)\|_{\ell^{2}}}.$

According to Corollary 4.2, for some $C_{\omega}$, $\rho$ will vanish to $0$
at least as fast as $C_{\omega}\sqrt{\epsilon\log\log(1/\epsilon)}.$ Seeing
the $\sqrt{\log\log(\/\epsilon)}$ is numerically challenging and we make no
claim that we do here. However, if it were to show up in the numerical
calculations, it would be best to factor it out, so we calculate

$\dfrac{\rho}{\sqrt{\log\log(1/\epsilon)}}.$

Now this should vanish at a rate no slower than $C_{\omega}\sqrt{\epsilon}$,
which on a log-log plot, should look like a straight line with a slope of
$1/2$. Anything with a slope greater than $1/2$ is vanishing at a faster rate.

We move onto the figures after one aside on the methods of integration used.
Since the total energy of the system is conserved, it is worth performing
experiments with a symplectic integrator. A six-step version of Yoshida’s
method, see [11], was initially used, as well as the standard four-step Runge-
Kutta method. As it turns out, these methods produce negligible differences
for the time scales studied, so most of the experiments below all use only the
four-step Runge-Kutta for the sake of computational efficiency.

Moving on, Figure 1 gives some numerical validations of our relative error
results, since the slope produced by the log-log plot is greater than $1/2$.
In this case, the realization of masses is the same for each $\epsilon$.
Figure 2 repeats the experiment in Figure 1 40 times, displaying the results
as a series of box plots. Figure 2, suggests the slope in Figure 1 is not a
statistical anomaly.

It is worth noting that the most important tool of our analysis is $\chi_{m}$.
For instance, it allows one to carry out similar analysis with many different
kinds of sequences of masses. If the average of the masses exists and one
knows the growth rate of $\chi_{m}$, then one can find an upper bound on the
error. For example, in Figure 4, we use a sequence of masses such that
$\chi(j)$ grows like $\sqrt{j}$. In particular, using two types of masses
$m_{1}$ and $m_{2}$, the following pattern works

$m_{1},m_{2},m_{1},m_{1},m_{2},m_{2},m_{1},m_{1},m_{1},m_{2},m_{2},m_{2}...$

We conjecture without providing arguments that if $\chi_{m}$ grows like
$|j|^{p}$, analysis would show that the relative error is bounded by an
$\epsilon^{1-p}$ order term, which in this case coincides with the numerical
results seen in Figure 4.

There are also hints in our work, see for example 3.9, that for fixed
$\epsilon$ and small $\sigma_{m}$, the mean of the error should be close to
that of the system where masses and springs are taken to be constant average.
Evidence for this is seen in Figure 5. When $\sigma_{m}$ is smallest, then
error from 40 trials, is concentrated around the error in the case of the
system being constant coefficient, which was numerically calculated to be
roughly $0.126$. In conclusion, the simulations are a strong affirmation of
our analytic results and that our bounds are at least close to optimal.

Relative Error for Fixed Random Masses

Figure 1: Figure 1 is a log-log plot of the relative error $\rho$ divided by
$\sqrt{\log\log(1/\epsilon)}.$

40 Random Experiments

Figure 2: Figure 2 is 10 box plots of 40 different realization of masses at 10
various epsilons. It is also log-log.

Periodic Masses

Figure 3: Figure 3 is a log-log plot of the relative error masses chosen
periodically

$\chi(j)$ Grows like $\sqrt{j}$

Figure 4: In Figure 4 masses are chosen so that $\chi(j)$ will grow like
$\sqrt{j}$.

Masses with Small $\sigma$

Figure 5: In Figure 5 $\epsilon$ is fixed and small while $\sigma$ is varied
and the absolute error is measured. When $\sigma$ is smallest, the data is
concentrated near the error for the constant coefficient case.

###  6.2 Conclusion

Our results are significant in several important ways regarding the
description of approximate waves in the random polymer linear FPUT system. We
have proven from first principles that that solutions to the wave equation are
good approximate solutions to the system studied here. We showed that the
absolute error only grows at most like $O(\log\log(1/\epsilon)$ almost surely
and is constant in mean, but also small in mean if the masses and springs have
small deviation. Using an interpolation operator with strong analytic
properties we were able to show that the interpolated approximate solutions
converged to interpolated true solutions in a relative sense a.s. and in
expectation. Such results provide a rigorous justification for claiming that
the relative error is made arbitrarily small by taking $\epsilon$ to be small.

The advantage of our method comes from the use of the random walk in capturing
the build up of error. Since random walks of independent variables are well
studied and sharp asymptotic estimates are known, we were able to use the
random walk to its full extent. Although it remains unproven if the error we
achieved is sharp, the numerical results suggest it is close, and it seems
nothing more about the asymptotics of the random walk, at least in the almost
sure sense, could be used to prove sharper bounds. It also remains unclear if
the random walk is an intrinsic part of the mechanics of the problem or if it
is only a useful fiction for modeling the error. To what extent could it be
further exploited here and in other models that have similar dynamics

With this work we have laid the foundation for a couple of questions. First,
can the error term be modeled by a random variable independent of $\epsilon$
with a nice probability distribution such as a Gaussian. There is also the
question as to whether the results can be extended to higher dimensions.
Probably most interesting, is what happens on larger time scales? In the
periodic setting, solitons are known to last up to times proportional to
$1/\epsilon^{3}$; however, it is not clear how to continue with the current
methodology as was done to derive the KdV equations in the periodic case. This
is mainly because one needs to make sense of $\lim_{n\to\infty}\sum_{|j|\leq
n}\chi(j)/n$, which, even if one optimistically replaces $\chi(j)$ with
$\sqrt{|j|}$, will diverge. This raises the question: is it is possible to
find an effective equation describing the the dynamics for longer times and
will these descriptions be statistical or is there room to achieve anything
more definite, like the high probability and almost sure results constructed
here?

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