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\begin{document}

\title{Diffusive decay of the environment viewed by the particle}





\date{\today}
%\thanks{Supported by Modal'X}


\author[P. de Buyer]{Paul de Buyer}
\address[P. de Buyer]{Université Paris Ouest Nanterre La Défense - Modal'X, 200 avenue
de la République 92000 Nanterre, France}
\email{debuyer@math.cnrs.fr}

\author[J.-C. Mourrat]{Jean-Christophe Mourrat}
\address[J.-C. Mourrat]{ENS Lyon, CNRS, 46 all\'ee d'Italie, 69007 Lyon, France}
\email{jean-christophe.mourrat@ens-lyon.fr}


\keywords{Random Walks in Random Environments, Algebraic convergence to equilibrium,
Homogenization, Functional inequalities}

\subjclass{60K37, 60J27, 76M50}

\maketitle

\begin{abstract}
We prove an optimal diffusive decay of the environment viewed by the
particle in random walk among random independent conductances, with,
as a main assumption, finite second moment of the conductance. Our
proof, using the analytic approach of Gloria, Neukamm and Otto, is
very short and elementary. 
\end{abstract}

\section{Introduction}

In this paper we are interested in the speed of convergence of the
environment viewed by the particle in the context of random walk among
random independent conductances. We refer to \cite{biskup2011recent}
for an introduction. We will first introduce the model and then state
our main theorem.

Consider the $d$-dimensional lattice $(\mathbb{Z}^{d},\mathbb{B}^{d})$,
$d\geq1$, with $\mathbb{B}^{d}$ the set of unoriented edges connecting
any two points of $\mathbb{Z}^{d}$ at Euclidian distance 1 and set
$\Omega:=[0,+\infty)^{\mathbb{B}^{d}}$. Given an \emph{environment}
$\omega=\left(\omega_{x,y}\right)_{\{x,y\}\in\mathbb{B}^{d}}\in\Omega$
%If $e=\left(x,y\right)\in\mathbb{B}^{d}$, we may
%write $\omega_{e}=\omega_{x,y}$. By construction, the model is symetric,
%i.e., $\omega_{x,y}=\omega_{y,x}$. 
we shall consider the associated Markov process $\left(X_{t}^{\omega}\right)_{t\geqslant0}$
with jump rate between $x$ and $y$ given by the \emph{conductance}
$\omega_{x,y}$ and write $P_{x}^{\omega}$ for its law starting from
$x\in\mathbb{Z}^{d}$. In many places we may simply write $\left(X_{t}\right)_{t\geqslant0}$
for simplicity. %(and $E_{x}^{\omega}$ for the corresponding expectation). 
%%When we want to
%%fix an environment, we will note it $m$ and call it a walk scheme,
%%this will be used to evaluate the behaviour $\left(X_{t}^{m}\right)_{t\geqslant0}$.
%The infinitesimal generator of the process and its associated semi-group, acting on, say, local functions 
%$f \colon \mathbb{Z}^{d} \times \Omega \to \mathbb{R}$ are given, for any $x,\omega \in \mathbb{Z}^{d} \times \Omega$ by
%$$
%\mathcal{L}^{\omega}f\left(x,\omega\right)=\sum_{\left|z\right|=1}\omega_{x,x+z}\left(f\left(x+z,\omega\right)-f\left(x,\omega\right)\right) 
%$$
%and
%$$
%P_{t}^{\omega}f\left(x,\omega\right)=\exp\left(\mathcal{L}t\right)f\left(x,\omega\right)=\sum_{y\in\mathbb{Z}^{d}}P_{x}^{\omega}\left(X_{t}=y\right)f\left(y,\omega\right) .
%$$
The environment itself will also be a random variable. In fact, throughout
the paper, we will assume that the conductances $\omega_{x,y}$, $\{x,y\}\in\mathbb{B}^{d}$,
are i.i.d.\ with common law $\mu$, whose support is included in
$[1,\infty)$. The law of the environment is therefore the product
measure $\mathbb{P}:=\mu^{\mathbb{B}^{d}}$ and we denote by $\mathbb{E}$
the associated expectation. %The law of the conductances are i.i.d. following a law $\mu$ such
%that the law environnement that is the random variable, we can write
%its law $\mathbb{P}=\mu^{\mathbb{B}^{d}}$ and $\mathbb{E}$ the associated
%expectation. 
Since $\mathbb{P}$ is a product measure, standard percolation arguments
guarantee that that the Markov process $\left(X_{t}\right)_{t\geqslant0}$
is well defined for almost all $\omega\in\Omega$, for all times,
see \textit{e.g.}\ \cite{kipnis1986central, DFGW, Mo99}.
Moreover it is reversible with respect to the counting measure, \textit{i.e.}\ for
all $x,y\in\mathbb{Z}^{d}$, it holds $P_{x}^{\omega}\left(X_{t}=y\right)=P_{y}^{\omega}\left(X_{t}=x\right)$.

Now, define the translation operators $\left(\theta_{z}\right)_{z\in\mathbb{Z}^{d}}$
given by $\left(\theta_{z}\omega\right)_{x,y}:=\omega_{x+z,y+z}$.
Then the Markov process $\left(\omega\left(t\right)\right)_{t\geqslant0}:=\left(\theta_{X_{t}}\omega\right)_{t\geqslant0}$,
called \emph{the environment viewed by the particle}, is reversible
with respect to $\mathbb{P}$.

%Finally, for all $x,y\in\mathbb{Z}^{d}$ and $\omega\in\Omega$, one
%have the following equality : $P_{0}^{\theta_{x}\omega}\left(X_{t}=y\right)=P_{x}^{\omega}\left(X_{t}=y+x\right)$.


The aim of this short paper is to give an optimal quantitative bound
on the decay to equilibrium of $\left(\omega\left(t\right)\right)_{t\geqslant0}$.

In order to state our theorem, we need some more notations. A function
$f\colon\mathbb{Z}^{d}\times\Omega\rightarrow\mathbb{R}$ is said
to be \emph{translation invariant} if $f\left(x,\omega\right)=f\left(0,\theta_{x}\omega\right)$
for all $x\in\mathbb{Z}^{d}$ and \emph{local} if $f(0,\cdot)$ depends
only on a finite set of conductances. The smallest set satisfying
that property is called the support of $f$ and is denoted by $supp(f)$,
while $\#supp\left(f\right)$ stands for the size of the support (\textit{i.e.}\ the
number of sites of $\mathbb{Z}^{d}$ contained in $supp(f)$). For
any translation invariant function $f$, we set $\mathbb{E}[f]:=\mathbb{E}[f(0,\cdot)]$.

Our main theorem is the following.

\begin{thm} \label{thm:MainTheorem} Let $d\geq3$. Assume that the
law $\mu$ of the conductance has support in $[1,\infty)$ and finite
second moment $\mathbb{E}\left[\omega_{\cdot,\cdot}^{2}\right]<\infty$.
Then there exists a constant $C>0$ that depends only on $d$ such
that for all local translation invariant function $f\colon\mathbb{Z}^{d}\times\Omega\rightarrow\mathbb{R}$
with $\mathbb{E}\left[f\right]=0$, all $x\in\mathbb{Z}^{d}$, it
holds %, for any $t>0$, we have :
\begin{equation}
\mathbb{E}\left(E_{x}^{\omega}\left[f\left(X_{t},\omega\right)\right]^{2}\right)\leqslant C\mathbb{E}\left[\omega_{\cdot,\cdot}^{2}\right]\#supp\left(f\right)^{2}\frac{\mathbb{E}\left[f^{2}\right]}{t^{d/2}}\qquad\forall t>0.\label{eq:main}
\end{equation}


\end{thm}

Let us comment on Theorem \ref{thm:MainTheorem}.

We first observe that similar results already exist in the literature.
One of us \cite{Mo99} proved a polynomial decay in dimension $1$ with
exponent $1/2$, in dimension 2 with behavior $\log t/t$, in dimension
3 to 6 with $t^{-1}$ and in dimension 7 and higher with $t^{\frac{-d}{2}+2}$.
Moreover, in the right hand side of \eqref{eq:main} appears a much
stronger norm than only the $L^{2}$ norm (the sum of the $L^{\infty}$
norm of $f$ and the so-called triple norm $|||f|||$ that involves
the infinite sum of $L^{\infty}$-norm of local gradients of $f$,
a natural norm largely used in the statistical mechanics literature,
see for example \cite{martinelli} and also \cite{bertini,janvresse,roberto}).
On the other hand, the results of \cite{Mo99} hold without the assumption on
the finiteness of the second moment of the conductance. More recently,
Gloria, Neukamm and Otto \cite{gloria} obtained, among other results,
a polynomial decay $t^{-\frac{d}{2}-1}$ when the function $f$ is
the divergence of some other function. Because we follow closely their
approach, our result is not far from theirs, although they need to
assume that conductances are bounded, \textit{i.e.}\ that $\mu$
has support contained in a compact set and that the function $f$
considered is in $L^{p}$, where $p$ is very large and not explicit. Finally,
we mention that \cite{yurinskii,cudna,gloria-otto11,gloria-otto12}
are related papers that deal with estimates on the diffusion matrix
and we refer to \cite{new-gno,am} and references therein for recent
results on homogenization of random operators.

\smallskip{}


Then, we observe that the power in \eqref{eq:main} is the best possible,
and we note that it can be obtained under stronger assumptions in \cite{gloria}.
On the other hand, we believe that the result should hold without
the assumption on the finiteness of the second moment of the conductance,
which appears as a technical fact in our proof. The second moment
$\mathbb{E}\left[f^{2}\right]$ is also best possible. Indeed, a smaller
norm would imply some regularization effect that would in turn imply
a spectral gap estimate which is known not to hold. For the same reason,
there must be some dependence, in the right hand side of \eqref{eq:main},
on the size of the support of $f$. Finally, we observe that the assumption
on the support of $\mu$ is also necessary since it is known, for
some specific choice of function $f$, with conductances taking small
values, that the decay of the heat-kernel coefficients can be very
slow, see \cite{berger2008anomalous}, which would interfer with the
decay in \eqref{eq:main}.

\smallskip{}


As for the proof, diffusive scaling is usually obtained, including
other settings on graphs such as interacting particle systems (Kawasaki
dynamics), using functional inequalities of Nash type,
see \textit{e.g.}\ \cite{liggett,bertini,Mo99}, Harnack \cite{delmotte}
or weak Poincaré inequalities \cite{bobkov,roberto}, see also \cite{bakry,saloff}.
Induction techniques can also be used \cite{janvresse,cesi}, or specific
aspects of the model such as attractivness \cite{deuschel}. However,
none of these techniques seem, to the best of our knowledge, to apply
to our setting. Here instead, we will mainly follow the PDEs ideas
from \cite{gloria}, but with many simplifications due to our non
specific choice of class of functions.

\smallskip{}


As a result we believe that our approach of the decay to equilibrium
of the environment viewed by the particle improves upon known results
into different directions (class of function, assumption
on the conductance, short and elementary proof), but, as a counterpart,
we are not able to extend the result of Gloria, Neukamm and Otto \cite{gloria}
to unbounded conductances for divergence functions. 

\medskip{}


The paper is organized as follows. In the next subsection we will
give some more notations. Then, we will give the proof of Theorem
\ref{thm:MainTheorem}. Such a proof will rely on a series of lemmas
that we will prove later on. Finally, in the last section, using the
theory of completely monotonic functions, we comment on a possible
short way of extending the results of Gloria, Neukamm and Otto \cite{gloria}
to our setting.


\subsection{Notations}

In this section we give some more notations.

Given an environment $\omega\in\Omega$, the infinitesimal generator
of the process $(X_{t})_{t\geq0}$ and its associated semi-group,
acting on functions $f\colon\mathbb{Z}^{d}\times\Omega\to\mathbb{R}$
are given, for any $x,\omega\in\mathbb{Z}^{d}\times\Omega$ by 
\[
\mathcal{L}^{\omega}f\left(x,\omega\right)=\sum_{\left|z\right|=1}\omega_{x,x+z}\left(f\left(x+z,\omega\right)-f\left(x,\omega\right)\right), \notag
\]
respectively by 
\[
P_{t}^{\omega}f\left(x,\omega\right)=\exp\left(\mathcal{L}^{\omega}t\right)f\left(x,\omega\right)=\sum_{y\in\mathbb{Z}^{d}}P_{x}^{\omega}\left(X_{t}=y\right)f\left(y,\omega\right). \notag
\]


In some situations, we may want to work with a given fixed environment.
In that case, and to emphasize this fact, we shall use the letter
$m$ instead of $\omega$ and call that given environment \emph{the
walk scheme}. This will be used in particular to evaluate the behavior
of $\left(X_{t}^{m}\right)_{t\geqslant0}$.

In many places we shall use the following equality $P_{0}^{\theta_{x}\omega}\left(X_{t}=y\right)=P_{x}^{\omega}\left(X_{t}=y+x\right)$
that holds for all $x,y\in\mathbb{Z}^{d}$ and all $\omega\in\Omega$.

%Define the translation operators $\left(\theta_{x}\right)_{x\in\mathbb{Z}^{d}}$
%actin
%Finally, for all $x,y\in\mathbb{Z}^{d}$ and $\omega\in\Omega$, one
%have the following equality : $P_{0}^{\theta_{x}\omega}\left(X_{t}=y\right)=P_{x}^{\omega}\left(X_{t}=y+x\right)$.
For simplicity of notation, we set $f_{t}\left(x,\omega\right):=P_{t}^{\omega}f\left(x,\omega\right)=E_{x}^{\omega}f\left(X_{t},\omega\right)$,
$t\geq0$, $x\in\mathbb{Z}^{d}$, $\omega\in\Omega$, where $E_{x}^{\omega}$
is the mean associated to $P_{x}^{\omega}$, and in many places we
shall omit the dependence in $\omega$ when there is no ambiguity.

Next, we define three different gradients. Denote by $e_{1},\dots,e_{d}$
the canonical orthonormal basis of $\mathbb{Z}^{d}$ (\textit{i.e.}\ for
all $i$, $e_{i}=(0,\dots,0,1,0,\dots,0)$ with $1$ at the $i^{th}$
coordinate), set $e_{-i}=-e_{i}$, $i=1,\dots,d$ and $e_{0}:=(0,\dots,0)$
for the origin. With that notation in hand, we define the following
local gradient of $g\colon\mathbb{Z}^{d}\rightarrow\mathbb{R}$: 
\[
D_{i}g(y):=g\left(y+e_{i}\right)-g\left(y\right)\qquad-d\leqslant i\leqslant d,\quad y\in\mathbb{Z}^{d}. \notag
\]
In particular, if $f\colon\mathbb{Z}^{d}\times\Omega\rightarrow\mathbb{R}$,
then $D_{i}f\left(y,\omega\right)=f\left(y+e_{i},\omega\right)-f\left(y,\omega\right)$,
$\omega\in\Omega$, and $D_{i}P_{x}^{\omega}\left(X_{t}=y\right)$
stands for the gradient applied to the mapping $y\mapsto P_{x}^{\omega}\left(X_{t}=y\right)$.
Similarly, 
\[
\nabla^{i}P_{x}^{m}\left(X_{t}=y\right):=P_{x+e_{i}}^{m}\left(X_{t}=y\right)-P_{x}^{m}\left(X_{t}=y\right)\qquad-d\leqslant i\leqslant d,\quad x,y\in\mathbb{Z}^{d}. \notag
\]
Therefore, the infinitesimal generator of a random walk with scheme
$m$ can be written as 
\[
\mathcal{L}^{m}f\left(x,\omega\right)=\sum_{-d\leqslant i\leqslant d}m_{x,x+e_{i}}D_{i}f\left(x,\omega\right)\qquad x\in\mathbb{Z}^{d},\quad\omega\in\Omega. \notag
\]


Finally, for $x\in\mathbb{Z}^{d}$, let $a\left(x\right)=\left\{ \omega_{x,x+e_{i}};\,1\leqslant i\leqslant d\right\} $,
$\overline{a}\left(x\right)=\{\omega_{e},e\in\mathbb{B}^{d}\}\setminus a(x)=\cup_{y\neq x}a\left(y\right)$
and $\mathbb{E}^{(x)}$ be the conditional expectation given $\overline{a}\left(x\right)$.
Then, for $y\in\mathbb{Z}^{d}$, we set 
\[
\partial_{x}f\left(y,\omega\right):=f\left(y,\omega\right)-\mathbb{E}^{(x)}\left[f\left(y,\omega\right)\right] \notag
\]
so that $\mathbb{E}^{(x)}[(\partial_{x}f)^{2}]$ is nothing else than
the variance of $f$ with respect to the conditional expectation $\mathbb{E}^{(x)}$.
Two sites $x,y$ are \emph{neighbors}, a property we denote by $x\sim y$,
if $\{x,y\}\in\mathbb{B}^{d}$. Also, given $f\colon\Omega\rightarrow\mathbb{R}$,
we say that $x$ is a neighbor of $y$ with respect to $f$, and write
$x\underset{f}{\sim}y$, if $a\left(x-y\right)\cap supp\left(f\right)\neq\emptyset$
(observe that this is not an equivalence relation). In particular,
observe that, if $x$ is not a neighbor of $y$ with respect to a
local translation invariant function $f$ (\textit{i.e.}\ if $a\left(x-y\right)\cap supp\left(f\right)=\emptyset$)
then $\partial_{x}f\left(y,\omega\right)=0$.


\section{Variance decay for unbounded conductances: proof of Theorem \ref{thm:MainTheorem}}

In this section we prove Theorem \ref{thm:MainTheorem}. The idea,
following \cite{gloria}, is to decompose the variance $\mathbb{E}\left(E_{x}^{\omega}\left[f\left(X_{t},\omega\right)\right]^{2}\right)$,
using Efron-Stein's inequality, into an infinite sum of terms of the
type $\mathbb{E}\left[\left(\partial_{y}P_{t}f\right)^{2}\right]$,
which, by Duhamel's formula, are split into two different terms that
need to be analyzed separately (the core of the proof). The point
in using Duhamel's Formula is to commute the operators $P_{t}$ and
$\partial_{y}$. The proof ends by applying some sort of Gronwall Lemma.
%We observe that, in \cite{gloria}, the authors use, as a first step, a Poincaré inequality for Glauber dynamics, on the lattice, which gives the same result as the Efron-Stein Inequality. (this is really the same inequality.)

\begin{proof}{[}Proof of Theorem \ref{thm:MainTheorem}{]} Let $f\colon\Omega\to\mathbb{R}$
be a local translation invariant function with $\mathbb{E}[f]=0$,
and assume, by homogeneity and for simplicity, that $\mathbb{E}[f^{2}]=1$.
Following \cite{gloria}, we apply Efron-Stein's Inequality and the
Duhamel formula (that we recall below, in Lemma \ref{froid}, for
completeness) to bound $\mathbb{E}\left[f_{t}^{2}\right]$: 
\begin{align}
\mathbb{E}\left[f_{t}^{2}\right] & \leqslant\mathbb{E}\left[\sum_{y\in\mathbb{Z}^{d}}\left(\partial_{y}P_{t}f\right)^{2}\right]=\mathbb{E}\left[\sum_{y\in\mathbb{Z}^{d}}\left(P_{t}\partial_{y}f+\int_{0}^{t}P_{t-s}h_{s}\left(y,0,\omega\right)ds\right)^{2}\right]\notag\label{MainProof1}\\
 & \leqslant2\mathbb{E}\left[\sum_{y\in\mathbb{Z}^{d}}\left(P_{t}\partial_{y}f\right)^{2}\right]+2\mathbb{E}\left[\sum_{y\in\mathbb{Z}^{d}}\left(\int_{0}^{t}P_{t-s}h_{s}\left(y,0,\omega\right)ds\right)^{2}\right].
\end{align}
where $h_{s}\left(x,y,\omega\right):=\mathbb{E}^{(y)}\left[\mathcal{L}f_{s}\left(x,\omega\right)\right]-\mathcal{L}\mathbb{E}^{(y)}\left[f_{s}\left(x,\omega\right)\right]$,
$x,y\in\mathbb{Z}^{d}$, $\omega\in\Omega$, $s\geq0$. Next we analyze
each term of the right hand side of the latter separately and start
with the first one.

First recall that if $a\left(y-x\right)\cap supp\left(f\right)=\emptyset$
then $\partial_{y}f\left(x,\omega\right)=0$. Hence 
\begin{align}
\mathbb{E}\left[\sum_{y\in\mathbb{Z}^{d}}\left(P_{t}\partial_{y}f\right)^{2}\right] & =\mathbb{E}\left[\sum_{y}\left(\sum_{x\in\mathbb{Z}^{d}}P_{0}^{\omega}\left(X_{t}=x\right)\partial_{y}f\left(x,\omega\right)\right)^{2}\right]\notag\label{MainProof2}\\
 & \leqslant\#supp(f)\sum_{y}\sum_{x:y\underset{f}{\sim}x}\mathbb{E}\left[P_{0}^{\omega}\left(X_{t}=x\right)^{2}\partial_{y}f\left(x,\omega\right)^{2}\right].
\end{align}
By invariance by translation we have 
\begin{align*}
\mathbb{E}\left[P_{0}^{\omega}\left(X_{t}=x\right)^{2}\partial_{y}f\left(x,\omega\right)^{2}\right] & =\mathbb{E}\left[P_{0}^{\theta_{-x}\omega}\left(X_{t}=x\right)^{2}\partial_{y}f\left(x,\theta_{-x}\omega\right)^{2}\right]\\
 & =\mathbb{E}\left[P_{0}^{\omega}\left(X_{t}=-x\right)^{2}\partial_{y-x}f\left(0,\omega\right)^{2}\right].
\end{align*}
Therefore, changing variables (set $x'=y-x$ and $y'=x'-y$), it holds
\begin{align*}
\mathbb{E}\left[\sum_{y\in\mathbb{Z}^{d}}\left(P_{t}\partial_{y}f\right)^{2}\right] & \leqslant\#supp(f)\sum_{y}\sum_{x:y\underset{f}{\sim}x}\mathbb{E}\left[P_{0}^{\omega}\left(X_{t}=-x\right)^{2}\partial_{y-x}f\left(0,\omega\right)^{2}\right]\notag\\
 & \leqslant\#supp(f)\sum_{y}\sum_{x':x'\underset{f}{\sim}0}\mathbb{E}\left[P_{0}^{\omega}\left(X_{t}=x'-y\right)^{2}\partial_{x'}f\left(0,\omega\right)^{2}\right]\notag\\
 & \leqslant\#supp(f)\sum_{x':x'\underset{f}{\sim}0}\sum_{y'}\mathbb{E}\left[P_{0}^{\omega}\left(X_{t}=y'\right)^{2}\partial_{x'}f\left(0,\omega\right)^{2}\right].
\end{align*}
Finally, using Lemma $\ref{pro:DecroissanceProbaMarche}$ below and
the fact that $\mathbb{E}\left[\partial_{x'}f\left(0,\cdot\right)^{2}\right]\leqslant2\mathbb{E}\left[f\left(0,\cdot\right)^{2}\right]=2\mathbb{E}\left[f^{2}\right]=2$,
we conclude that, for some constant $C$ that depends only on $d$,
\begin{equation}
\mathbb{E}\left[\sum_{y\in\mathbb{Z}^{d}}\left(P_{t}\partial_{y}f\right)^{2}\right]\leqslant C\frac{\#supp\left(f\right)^{2}}{\left(t+1\right)^{d/2}}\label{term1}
\end{equation}


Next we focus on the second term in the right hand side of \eqref{MainProof1}.
Using Lemma \ref{pro:PropositionIntermediaire} we have 
\[
\mathbb{E}\left[\sum_{y\in\mathbb{Z}^{d}}\left(\int_{0}^{t}P_{t-s}h_{s}\left(y,0,\omega\right)ds\right)^{2}\right]=\mathbb{E}\left[\sum_{y\in\mathbb{Z}^{d}}\left(\int_{0}^{t}\sum_{i=1}^{d}D_{i}P_{0}^{\omega}\left(X_{t-s}=y\right)g_{s}\left(y,y,\omega,i\right)ds\right)^{2}\right]\notag
\]
so that, by Minkowski's integral inequality and the invariance by
translation, it holds 
\begin{align}
 & \mathbb{E}\left[\sum_{y\in\mathbb{Z}^{d}}\left(\int_{0}^{t}P_{t-s}h_{s}\left(y,0,\omega\right)ds\right)^{2}\right]^{\nicefrac{1}{2}}\notag\label{MainProof4}\\
 & \qquad\qquad\leqslant\sqrt{d}\int_{0}^{t}\left(\sum_{y}\sum_{i}\mathbb{E}\left[\left(D_{i}P_{0}^{\omega}\left(X_{t-s}=y\right)\right)^{2}g_{s}\left(y,y,\omega,i\right)^{2}\right]\right)^{\nicefrac{1}{2}}ds\notag\\
 & \qquad\qquad=\sqrt{d}\int_{0}^{t}\left(\sum_{y}\sum_{i}\mathbb{E}\left[\left(\nabla^{i}P_{0}^{\omega}\left(X_{t-s}=y\right)\right)^{2}g_{s}\left(0,0,\omega,i\right)^{2}\right]\right)^{\nicefrac{1}{2}}ds\notag\\
 & \qquad\qquad\leqslant\sqrt{2d}\int_{0}^{t}\left(\sum_{y}\sum_{i}\mathbb{E}\left[\left(\nabla^{i}P_{0}^{\omega}\left(X_{t-s}=y\right)\right)^{2}\mathbb{E}^{(0)}\left[\omega_{0,e_{i}}D_{i}f_{s}\left(0,\omega\right)\right]^{2}\right]\right.\notag\\
 & \qquad\qquad\quad+\left.\mathbb{E}\left[\omega_{0,e_{i}}^{2}\left(\nabla^{i}P_{0}^{\omega}\left(X_{t-s}=y\right)\right)^{2}\mathbb{E}^{(0)}\left[D_{i}f_{s}\left(0,\omega\right)\right]^{2}\right]\right)^{\nicefrac{1}{2}}ds
\end{align}
where $g_{s}\left(x,y,\omega,i\right):=\mathbb{E}^{(x)}\left[\omega_{y,y+e_{i}}D_{i}f_{s}\left(y,\omega\right)\right]-\omega_{y,y+e_{i}}\mathbb{E}^{(x)}\left[D_{i}f_{s}\left(y,\omega\right)\right]$,
$s\geq0$, $x,y\in\mathbb{Z}^{d}$, $\omega\in\Omega$ and $i=1,\dots,d$.
Therefore, using twice that $\left(a+b\right)^{2}\leqslant2a^{2}+2b^{2}$,
Proposition \ref{pro:DecroissanceProbaMarche} and Lemma \ref{lem:Lemmeclef}
guarantee that, for some constant $C$ that depends only on $d$,
it holds 
\begin{align}
 & \mathbb{E}\left[\sum_{y\in\mathbb{Z}^{d}}\left(\int_{0}^{t}P_{t-s}h_{s}\left(y,0,\omega\right)ds\right)^{2}\right]^{\nicefrac{1}{2}}\!\!\!\leqslant C\int_{0}^{t}\left(\left(t-s+1\right)^{-\frac{d}{2}}\sum_{i}\mathbb{E}\left[\mathbb{E}^{(0)}\left[\omega_{0,e_{i}}D_{i}f_{s}\left(0,\omega\right)\right]^{2}\right]\right.\notag\\
 & \qquad\qquad\quad+\left.\left(t-s+1\right)^{-d/2}\sum_{i}\mathbb{E}\left[\omega_{0,e_{i}}^{2}\mathbb{E}^{(0)}\left[D_{i}f_{s}\left(0,\omega\right)\right]^{2}\right]\right)^{1/2}ds\notag\\
 & \qquad\qquad\quad\leqslant\sqrt{2}C\int_{0}^{t}\left(t-s+1\right)^{-\nicefrac{d}{4}}\mathbb{E}\left[\omega_{0,e_{1}}^{2}\right]^{1/2}\left(-\partial_{s}\mathbb{E}\left[|f_{s}\left(0,\omega\right)|^{2}\right]\right)^{1/2}ds. \label{term2} \end{align}
Plugging \eqref{term1} and \eqref{term2} into \eqref{MainProof1},
we end up with 
\[
\mathbb{E}\left[f_{t}^{2}\right]^{\frac{1}{2}}\leqslant C'\#supp\left(f\right)\mathbb{E}\left[\omega_{0,e_{1}}^{2}\right]^{\frac{1}{2}}\left(\left(t+1\right)^{-\frac{d}{4}}+\int_{0}^{t}\left(t-s+1\right)^{-\frac{d}{4}}\left(-\partial_{s}\mathbb{E}\left[f_{s}^{2}\right]\right)^{\frac{1}{2}}ds\right)\notag
\]
for some constant $C'$ that depends only on $d$. The expected result
will finally follow from Lemma \ref{lem:Lemme15bis} with $a\left(t\right):=\mathbb{E}\left[f_{t}^{2}\right]^{\frac{1}{2}}$
and $\alpha=\frac{d}{4}$. Indeed, $a(t)$ is non-increasing since,
using classical computations for reversible Markov processes, $\partial_{t}\mathbb{E}\left[f_{t}^{2}\right]=2\mathbb{E}\left[f_{t}\mathcal{L}f_{t}\right]=-\sum_{i}\mathbb{E}\left[\omega_{0,e_{i}}(D_{i}f_{t})^{2}\right]\leq0$.
This ends the proof. \end{proof}

In the proof of Theorem \ref{thm:MainTheorem} we used the following
series of lemma. The first two lemmas are well known results from
Probability Theory and Analysis. The others are technical.

\begin{lem}{[}Efron-Stein's Inequality and the Duhamel formula{]}
\label{froid} The following holds. 
\begin{description}
\item [{(Efron-Stein's Inequality)}] Let $n>1$ and $f$ be a function
of $X_{1},...,X_{n}$, $n$ independent variables, then 
\[
\mathbb{V}ar\left(f\right)\leqslant\sum_{i=1}^{n}\mathbb{E}\left[\mathbb{V}ar^{(i)}\left(f\right)\right]\notag
\]
where $\mathbb{V}ar^{(i)}$ is the conditional variance given $\left\{ X_{1},...,X_{n}\right\} \backslash\left\{ X_{i}\right\} $. 
\item [{(Duhamel's Forumla)}] For all $t\geqslant0$ and almost all $\omega\in\Omega$
it holds 
\[
\partial_{y}P_{t}f\left(x,\omega\right)=P_{t}\partial_{y}f\left(x,\omega\right)+\int_{0}^{t}P_{t-s}h_{s}\left(x,y,\omega\right)ds \notag
\]
where $h_{s}\left(x,y,\omega\right):=\mathbb{E}^{(y)}\left[\mathcal{L}f_{s}\left(x,\omega\right)\right]-\mathcal{L}\mathbb{E}^{(y)}\left[f_{s}\left(x,\omega\right)\right]$,
$x,y\in\mathbb{Z}^{d}$, $\omega\in\Omega$, $s\geq0$. 
\end{description}
\end{lem}

\begin{proof} Efron-Stein's Inequality, also called tensorisation
of the variance (see \textit{e.g.}\ \cite[Proposition 1.4.1]{ane}),
is a well known result following from Cauchy-Schwarz' inequality.
See \cite{latala} for an extension to general $\phi$-entropy, see
also \cite{Ma07}.

As for the Duhamel formula, we observe that 
\[
\partial_{y}P_{t}f\left(x,\omega\right)=P_{t}\partial_{y}f\left(x,\omega\right)+\int_{0}^{t}\partial_{s}P_{t-s}\partial_{y}P_{s}f\left(x,\omega\right)ds \notag
\]
which leads to the expected result, since $h_{s}\left(x,y,\omega\right)=\left(\partial_{y}\mathcal{L}-\mathcal{L}\partial_{y}\right)P_{s}f$
and $\partial_{t}P_{t}=\mathcal{L}P_{t}=P_{t}\mathcal{L}$. \end{proof}

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\begin{lem} \label{pro:DecroissanceProbaMarche} There exists a constant
$C$ (that depends only on $d$) such that for 
all well-defined walk scheme $m$ and for all $t>0$ it holds 
\[
\sum_{y\in\mathbb{Z}^{d}}P_{0}^{m}\left(X_{t}=y\right)^{2}\leqslant C\left(t+1\right)^{-d/2}. \notag
\]
\end{lem}

\begin{proof} The proof of the this inequality is a consequence of
the reversibility and the fact that the invariant measure is the uniform
measure. Indeed 
\[
\sum_{y\in\mathbb{Z}^{d}}P_{0}^{m}\left(X_{t}=y\right)^{2}=\sum_{y\in\mathbb{Z}^{d}}P_{0}^{m}\left(X_{t}=y\right)P_{y}^{m}\left(X_{t}=0\right)=P_{0}^{m}\left(X_{2t}=0\right) \notag
\]
which gives the desired result combining \cite[Theorem 2.1]{CKS}
and \cite[Proposition 10.2]{Mo99} in its first arXiv version. \end{proof}

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\begin{lem} \label{pro:PropositionIntermediaire} Given $f\colon\mathbb{Z}^{d}\times\Omega\to\mathbb{R}$,
define $h_{s}\left(x,y,\omega\right):=\mathbb{E}^{(y)}\left[\mathcal{L}f_{s}\left(x,\omega\right)\right]-\mathcal{L}\mathbb{E}^{(y)}\left[f_{s}\left(x,\omega\right)\right]$
and $g_{s}\left(x,y,\omega,i\right)=\mathbb{E}^{(x)}\left[\omega_{y,y+e_{i}}D_{i}f_{s}\left(y,\omega\right)\right]-\omega_{y,y+e_{i}}\mathbb{E}^{(x)}\left[D_{i}f_{s}\left(y,\omega\right)\right]$,
$x,y\in\mathbb{Z}^{d}$, $\omega\in\Omega$, $s\geq0$ and $i=0,\dots,d$.
Then, for all $s,t>0$ and all $x$ it holds 
\[
P_{t}h_{s}\left(x,0,\omega\right)=-\sum_{i=1}^{d}D_{i}P_{0}\left(X_{t}=x\right)g_{s}\left(x,x,\omega,i\right). \notag 
\]
\end{lem}

\begin{proof} Recall the definition of $D_{i}$. On the one hand,
by definition, we have 
\[
\mathbb{E}^{(x)}\left[\mathcal{L}f_{s}\left(y,\omega\right)\right]=\sum_{i=1}^{d}\mathbb{E}^{(x)}\left[\omega_{y,y+e_{i}}\left(D_{i}f_{s}\left(y,\omega\right)\right)\right]-\mathbb{E}^{(x)}\left[\omega_{y,y-e_{i}}\left(D_{i}f_{s}\left(y,\omega\right)\right)\right] \notag
\]
and 
\[
\mathcal{L}^{\theta_{y}\omega}\mathbb{E}^{(x)}\left[f_{s}\left(y,\omega\right)\right]=\sum_{i=1}^{d}\omega_{y,y+e_{i}}\mathbb{E}^{(x)}\left[\left(D_{i}f_{s}\left(y,\omega\right)\right)\right]-\omega_{y,y-e_{i}}\mathbb{E}^{(x)}\left[\left(D_{i}f_{s}\left(\theta_{y-e_{i}}\omega\right)\right)\right]. \notag
\]
On the other hand, $\omega_{y,y+e_{i}}$ is $\bar{a}\left(x\right)$-measurable
iff $y\neq x$ and $\omega_{y,y-e_{i}}$ is $\bar{a}\left(x\right)$-measurable
iff $y-e_{i}\neq x$. Therefore, 
\[
h_{s}\left(x,y,\omega\right)=\begin{cases}
\sum_{i=1}^{d}g_{s}\left(x,x,\omega,i\right) & \mbox{if }y=x\\
-g_{s}\left(x,x,\omega,i\right) & \mbox{if }y-e_{i}=x,\; i=1,\dots,d\\
0 & \mbox{otherwise}. \notag
\end{cases}
\]
It finally follows that 
\begin{eqnarray*}
P_{t}h_{s}\left(x,\omega\right) & = & \sum_{y\in\mathbb{Z}^{d}}P\left(X_{t}=y\right)h_{s}\left(x,y,\omega\right)\\
 & = & P\left(X_{t}=x\right)h_{s}\left(x,x,\omega\right)-\sum_{i=1}^{d}P\left(X_{t}=x+e_{i}\right)h_{s}\left(x,x+e_{i},\omega\right)\\
 & = & \sum_{i=1}^{d}\left(P\left(X_{t}=x\right)-P\left(X_{t}=x+e_{i}\right)\right)g_{s}\left(x,x,\omega,i\right)
\end{eqnarray*}
which is the desired result. \end{proof}

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\begin{lem} \label{lem:Lemmeclef} Let $f\colon\mathbb{Z}^{d}\times\Omega\to\mathbb{R}$.
Then, for all $s>0$ it holds 
\begin{align*}
\sum_{i=1}^{d}\mathbb{E}\left[\left(\mathbb{E}^{(0)}\left[\omega_{0,e_{i}}D_{i}f_{s}\left(0,\omega\right)\right]\right)^{2}\right] & \leqslant-\mathbb{E}\left[\omega_{0,e_{1}}^{2}\right]\partial_{s}\mathbb{E}\left[f_{s}\left(0,\omega\right)^{2}\right],\\
\sum_{i=1}^{d}\mathbb{E}\left[\left(\omega_{0,e_{i}}\mathbb{E}^{(0)}\left[D_{i}f_{s}\left(0,\omega\right)\right]\right)^{2}\right] & \leqslant-\mathbb{E}\left[\omega_{0,e_{1}}^{2}\right]\partial_{s}\mathbb{E}\left[f_{s}\left(0,\omega\right)^{2}\right].
\end{align*}
\end{lem}

\begin{rem} The proof actually leads to a better bound in the first
inequality above. Indeed, one can replace the second moment of the
conductance by the first moment. %following more precise result
%$$
%\sum_{i=1}^d\mathbb{E}\left[\left(\mathbb{E}^{(0)}\left[\omega_{0,e_{i}}D_{i}f_{s}\left(0,%\omega\right)\right]\right)^{2}\right]
%\leqslant
%-\mathbb{E}\left[\omega_{0,e_{1}}\right]\partial_{s}\mathbb{E}\left[f_{s}\left(0,\omega\right)^{2}\right]
%$$
This refinement will anyway not be useful for our purpose. \end{rem}

\begin{proof}{[}Proof of Lemma \ref{lem:Lemmeclef}{]} Using Cauchy-Schwarz'
inequality and the fact that $\omega_{0,e_{i}}$ is $a\left(0\right)$-measurable,
we have 
\begin{align*}
\mathbb{E}\left[\left(\mathbb{E}^{(0)}\left[\omega_{0,e_{i}}D_{i}f_{s}\left(0,\omega\right)\right]\right)^{2}\right] & \leqslant\mathbb{E}\left[\mathbb{E}^{(0)}\left[\omega_{0,e_{i}}\right]\mathbb{E}^{(0)}\left[\omega_{0,e_{i}}\left(D_{i}f_{s}\left(0,\omega\right)\right)^{2}\right]\right]\\
 & =\mathbb{E}\left[\omega_{0,e_{i}}\right]\mathbb{E}\left[\omega_{0,e_{i}}\left(D_{i}f_{s}\left(0,\omega\right)\right)^{2}\right].
\end{align*}
Using that $\partial_{t}\mathbb{E}\left[f_{t}^{2}\right]=\partial_{t}\mathbb{E}\left[f_{t}(0,\omega)^{2}\right]=-\sum_{i=1}^{d}\mathbb{E}\left[\omega_{0,e_{i}}\left(D_{i}f_{s}\left(0,\omega\right)\right)^{2}\right]$
(a classical consequence of the reversibility) and summing over $i=1,\dots,d$,
we get 
\[
\sum_{i=1}^{d}\mathbb{E}\left[\left(\mathbb{E}^{(0)}\left[\omega_{0,e_{i}}D_{i}f_{s}\left(0,\omega\right)\right]\right)^{2}\right]\leqslant-\mathbb{E}\left[\omega_{0,e_{1}}\right]\partial_{s}\mathbb{E}\left[f_{s}\left(0,\omega\right)^{2}\right]\notag 
\]
which leads to the first inequality since $\omega_{0,e_{1}}\geq1$.

For the second inequality by conditioning we have 
\begin{align*}
\mathbb{E}\left[\left(\omega_{0,e_{i}}\mathbb{E}^{(0)}\left[D_{i}f_{s}\left(0,\omega\right)\right]\right)^{2}\right] & =\mathbb{E}\left[\mathbb{E}^{(0)}\left[\omega_{0,e_{i}}^{2}\right]\left(\mathbb{E}^{(0)}\left[D_{i}f_{s}\left(0,\omega\right)\right]\right)^{2}\right]\\
 & =\mathbb{E}\left[\omega_{0,e_{i}}^{2}\right]\mathbb{E}\left[D_{i}f_{s}\left(0,\omega\right)^{2}\right].
\end{align*}
Now, since $\omega_{0,e_{1}}\geq1$, one has $\sum_{i=1}^{d}\mathbb{E}\left[D_{i}f_{s}\left(0,\omega\right)^{2}\right]\leq\sum_{i=1}^{d}\mathbb{E}\left[\omega_{0,e_{1}}D_{i}f_{s}\left(0,\omega\right)^{2}\right]=-\partial_{s}\mathbb{E}\left[f_{s}(0,\omega)^{2}\right]$
which leads to the desired result and ends the proof of the lemma.
\end{proof}

The last lemma is related to Lemma 15 of \cite{gloria}. However,
due to our specific setting, considering $p=1/2$, its proof is a
bit simpler. We give it for completeness.

%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%


\begin{lem}{[}\cite{gloria}{]} \label{lem:Lemme15bis} Let $\alpha>1/2$
and $a\colon\mathbb{R}^{+}\to\mathbb{R}^{+}$ be a $\mathcal{C}^{1}$
non-increasing function. Let $b(t)=\sqrt{-2a'(t)a(t)}$, $t\geq0$.
Assume that 
\begin{align*}
a\left(t\right) & \leqslant C\left(\left(t+1\right)^{-\alpha}+\int_{0}^{t}\left(t-s+1\right)^{-\alpha}b\left(s\right)ds\right)\qquad\forall t\geq0
\end{align*}
for some constant $C$. Then there exists a constant $C_{\alpha}$
that depends only on $\alpha$ such that $a\left(t\right)\leqslant C_{\alpha}\max(C,a(0))\left(t+1\right)^{-\alpha}$.
\end{lem}

\begin{proof} Throughout the proof we use that $u\lesssim v$ if
there exists a constant $A$ that depends only on $\alpha$ such that
$u\leq Av$. The expected result will follow from the fact that, for
some $t_{o}>0$ that will be chosen later on, $[t_{o},\infty)\ni t\mapsto\Lambda\left(t\right):=\sup_{t_{o}\leq s\leqslant t}\left(s+1\right)^{\alpha}a\left(s\right)$
is a bounded function, bounded by $C'\max(C,a(0))$ for some $C'$
depending only on $\alpha$. Indeed, since $a$ is non-increasing
$a(s)\leq a(0)$ for any $s\in[0,t_{o}]$ which, together with the
bound on $\Lambda$ would lead to the desired conclusion.

Our starting point is the following inequality obtained using that
$a$ is non-increasing. 
\begin{align}
a\left(t\right) & \leqslant\frac{2}{t}\int_{\frac{t}{2}}^{t}a\left(u\right)du\leqslant\frac{2C}{t}\int_{\frac{t}{2}}^{t}\frac{du}{\left(u+1\right)^{\alpha}}+\frac{2}{t}\int_{\frac{t}{2}}^{t}\int_{0}^{u}\frac{b(s)dsdu}{\left(u+1-s\right)^{\alpha}}\leq\frac{C}{(\frac{t}{2}+1)^{\alpha}}\nonumber \\
 & +\frac{2}{t}\int_{\frac{t}{2}}^{t}\int_{0}^{T}\frac{b(s)dsdu}{(u+1-s)^{\alpha}}+\frac{2}{t}\int_{\frac{t}{2}}^{t}\int_{T}^{u/2}\frac{b(s)dsdu}{(u+1-s)^{\alpha}}+\frac{2}{t}\int_{\frac{t}{2}}^{t}\int_{u/2}^{u}\frac{b(s)dsdu}{(u+1-s)^{\alpha}}\nonumber \\
 & =\frac{C}{(\frac{t}{2}+1)^{\alpha}}+I+II+III \label{start26}
\end{align}
where $T\in[1,t/4]$ is a parameter that will be chosen later on.
In order to bound $I,II$ and $III$, we will repeatedly use the following
bounds, that holds for all $T_{1}<T_{2}$ and whose proof is given
below 
\begin{equation}
\int_{T_{1}}^{T_{2}}b\left(s\right)ds\lesssim\begin{cases}
\sqrt{T_{2}-T_{1}}a\left(T_{1}\right)\\
\Lambda\left(T_{2}\right)T_{1}^{\frac{1}{2}-\alpha} & \mbox{ if }T_{1}>0.
\end{cases}\label{estimebs}
\end{equation}
To prove the first inequality, we use Cauchy-Schwarz' inequality.
Namely 
\begin{align*}
\frac{1}{T_{2}-T_{1}}\left(\int_{T_{1}}^{T_{2}}b\left(s\right)ds\right)^{2} & \leqslant\int_{T_{1}}^{T_{2}}b^{2}\left(s\right)ds=-\int_{T_{1}}^{T_{2}}\frac{d}{ds}a^{2}\left(s\right)ds\leqslant a\left(T_{1}\right)^{2}.
\end{align*}
To prove the second inequality, we repeatedly use the latter. set
$N=\log_{2}\left(\nicefrac{T_{2}}{T_{1}}\right)$, then we have 
\begin{align*}
\int_{T_{1}}^{T_{2}}b\left(s\right)ds & =\sum_{n=0}^{N-1}\int_{2^{n}T_{1}}^{2^{n+1}T_{1}}b\left(s\right)ds\leqslant\sum_{n=0}^{N-1}\sqrt{2^{n}T_{1}}a\left(2^{n}T_{1}\right)\leqslant\sum_{n=0}^{N-1}\left(2^{n}T_{1}\right)^{\frac{1}{2}-\alpha}\Lambda\left(2^{n}T_{1}\right)\\
 & \leqslant\Lambda\left(T_{2}\right)T_{1}^{\frac{1}{2}-\alpha}\sum_{n=0}^{N-1}\left(2^{\frac{1}{2}-\alpha}\right)^{n}\lesssim\Lambda\left(T_{2}\right)T_{1}^{\frac{1}{2}-\alpha}.
\end{align*}


Now, since $T\leq t/4$ and thanks to \eqref{estimebs}, we have 
\[
I:=\frac{2}{t}\int_{\frac{t}{2}}^{t}\int_{0}^{T}\frac{b(s)dsdu}{(u+1-s)^{\alpha}}\leqslant\frac{2}{t}\int_{\frac{t}{2}}^{t}\int_{0}^{T}\frac{b(s)dsdu}{(\frac{t}{4}+1)^{\alpha}}=\frac{\int_{0}^{T}b(s)ds}{(\frac{t}{4}+1)^{\alpha}}\lesssim\frac{\sqrt{T}a(0)}{(t+1)^{\alpha}}.
\notag \]
Again using \eqref{estimebs} we have 
\[
II:=\frac{2}{t}\int_{\frac{t}{2}}^{t}\int_{T}^{u/2}\frac{b(s)dsdu}{(u+1-s)^{\alpha}}\leq\frac{2}{t}\int_{\frac{t}{2}}^{t}\frac{\int_{T}^{u/2}b(s)ds}{(\frac{t}{4}+1)^{\alpha}}du\leq\frac{\frac{2}{t}\int_{\frac{t}{2}}^{t}\Lambda(\frac{u}{2})T^{\frac{1}{2}-\alpha}du}{(\frac{t}{4}+1)^{\alpha}}\lesssim\frac{\Lambda(t)T^{\frac{1}{2}-\alpha}}{(t+1)^{\alpha}}.
\notag \]
In order to bound the third term,
which is more intricate, we first use the Fubini Theorem, noting that $\mathds{1}_{ t/2 \leqslant u \leqslant t } \mathds{1}_{ u/2 \leqslant s \leqslant u} \leqslant \mathds{1}_{t/4\leqslant s \leqslant t} \mathds{1}_{s\leqslant u \leqslant t}$, 
and that $\alpha > 1/2$ to get 
\begin{eqnarray}
III & \leqslant & \frac{2}{t}\int_{\frac{t}{4}}^{t}\int_{s}^{t}\left(u+1-s\right)^{-\alpha}dub\left(s\right)ds =  \frac{2}{t}\int_{\frac{t}{4}}^{t}\int_{0}^{t-s}\left(t'+1\right)^{-\alpha}dt'b\left(s\right)ds\notag\\
 & \leqslant & \frac{2}{t}\int_{0}^{t}\left(t'+1\right)^{-\alpha}dt'\times\int_{\frac{t}{4}}^{t}b\left(s\right)ds \lesssim (t+1)^{-\frac{1}{2}-\alpha}\Lambda\left(t\right)\notag
\end{eqnarray}
Therefore, plugging the previous bounds on $I,II$
and $III$ into \ref{start26}, it holds 
\[
(1+t)^{\alpha}a(t)\leq A\sqrt{T}\max(C,a(0))+A\Lambda(t)\left(T^{\frac{1}{2}-\alpha}+\frac{\phi(t)}{\sqrt{t}}\right)
\notag \]
for some constant $A$ that depends only on $\alpha$. Now, since
$\alpha>1/2$, there exist $t_{o}\geq4$ and $T\geq1$ such that for
all $t\geq t_{o}$, $\Phi(t)/\sqrt{t}\leq1/(4A)$ and $T^{\frac{1}{2}-\alpha}\leq1/(4A)$
so that, taking the supremum and using the monotonicity of $\Lambda$,
it holds $\Lambda(t)\leq A\sqrt{T}\max(C,a(0))+\frac{1}{2}\Lambda(t)$
for all $t\geq t_{o}$ which leads to the desired conclusion. The
proof of the lemma is complete. \end{proof}


\section{Additional remarks}


\subsection{Completely monotonic functions}

In this section, we prove some results on $P_{x}^{m}\left(X_{t}=y\right)$,
for a given walk scheme $m$, using the notion of completely monotonic
functions. Recall that a function $f\colon(0,\infty)\to\mathbb{R}$
is said to be completely monotonic if it possesses derivatives $f^{(n)}$
of all orders and if $(-1)^{n}f^{(n)}(x)\geq0$ for all $x>0$ and
all $n=0,1,2,\dots$ (see \textit{e.g.} \cite{feller,miller}).

\begin{prop} \label{cor:CorollaireDeriveeCM} Let $C,\alpha>0$.
Assume that $f\colon(0,\infty)\to\mathbb{R}$ is a completely monotonic
function satisfying for all $t>0$, $f(t)\leqslant\frac{C}{t^{\alpha}}$.
Then, for all $t>0$, it holds $-f'(t)\leq\frac{C'}{t^{\alpha+1}}$
for some constant $C'$ that depends only on $C$ and $\alpha$. \end{prop}

\begin{rem} At the price of some technicalities, the above result
can be extended to more general decay (\textit{i.e.} replacing $1/t^{\alpha}$
by some general completely monotonic function $g$ with $\lim_{\infty}g=0$).
\end{rem}

\begin{proof} Without loss of generality assume that $f\left(0\right)=1$.
It is well known (see \cite{feller,widder}) that $f$ is the Laplace
transform of a positive random variable $X$, namely that, for all
$t>0$, $f\left(t\right)=\mathbb{E}\left[e^{-tX}\right]$ where $\mathbb{E}$
denotes the mean of the law of $X$. Using the Markov Inequality we
get for all $\lambda>0$ and all $x>0$ 
\[
\mathbb{P}\left(X\leqslant x\right)\leqslant\mathbb{E}\left[e^{-\lambda X}\right]e^{\lambda x}\leqslant C\exp\left(\lambda x-\alpha\log\lambda\right).
\notag \]
Optimizing over the $\lambda>0$ we get (since $\inf_{\lambda>0}\{\lambda x-\alpha\log\lambda\}=\alpha-\alpha\log\alpha+\alpha\log x$
(the minimum being reached at $\lambda=\alpha/x$)) $\mathbb{P}\left(X\leqslant x\right)\leqslant Ce^{\alpha}\left(\frac{x}{\alpha}\right)^{\alpha}$.
Therefore, using Fubini's theorem, for all $t>0$ it holds 
\begin{align*}
-f'(t) & =\mathbb{E}\left[Xe^{-tX}\right]=\mathbb{E}\left[X\int_{X}^{\infty}te^{-tx}dx\right]=\int_{0}^{\infty}te^{-tx}\mathbb{E}\left[X\mathds{1}_{X\leq x}\right]dx\\
 & \leq\int_{0}^{\infty}txe^{-tx}\mathbb{P}(X\leq x)dx\leq\frac{C_{\alpha}}{t^{\alpha}}\int_{0}^{\infty}(tx)^{\alpha+1}e^{-tx}dx=\frac{C_{\alpha}\Gamma(\alpha+2)}{t^{\alpha+1}}
\end{align*}
which ends the proof. \end{proof}

\begin{cor} \label{pro:DecroissanceProbaMarche2} There exists a
constant $C$ such that for all well-defined walk scheme $m$
and all $t>0$ it holds $\sum_{y\in\mathbb{Z}^{d},\left|z\right|=1}\left(P_{0}^{m}\left(X_{t}=y\right)-P_{0}^{m}\left(X_{t}=y+z\right)\right)^{2}\leqslant C/\left(t+1\right)^{\frac{d}{2}+1}$.
\end{cor}

\begin{proof} For $p\geq1$, set $\Vert f\Vert_{p}^{p}:=\sum_{x}f\left(x\right)^{p}$
and, given an operator $P$ acting on functions, $\Vert P\Vert_{p\rightarrow q}=\sup\Vert Pf\Vert_{q}$,
where the supremum is taken over all $f$ with $\Vert f\Vert_{p}=1$.

As in lemma \ref{pro:DecroissanceProbaMarche}, it is known that there exists a constant $C$ that
does not depend on the walk scheme $m$ such that $\Vert P_{t}^{m}\Vert_{1\rightarrow2}\leqslant C/t^{d/4}$.
This implies that for all $y\in\mathbb{Z}^{d}$ and $t\geqslant0$,
$\sum_{x}{P_{y}^{m}}\left(X_{t}=x\right)^{2}\leqslant1$. Therefore
the quantity we want to bound is finite.

Then, we observe that $t\mapsto\Vert P_{t}^{m}f\Vert_{2}^{2}$ is
a completely monotonic function. Hence, using Proposition \ref{cor:CorollaireDeriveeCM},
there exists a constant $C'$ that depends only on $d$ and $C$ such
that for all $f\in\ell^{1}$, $\frac{d}{dt}\Vert P_{t}^{m}f\Vert_{2}^{2}\leqslant C'/t^{\frac{d}{2}+1}$.

On the other hand, by definition of $\mathcal{L}^{m}$ and $P_{t}^{m}$,
we have 
\begin{equation}
\frac{d}{dt}\Vert P_{t}^{m}f\Vert_{2}^{2}=-\sum_{x,|z|=1}m_{x,x+z}\left(P_{t}^{m}f\left(x+z\right)-P_{t}^{m}f\left(x\right)\right)^{2}\label{eq:fm2}
\end{equation}
Observe that for $f=\mathds{1}_{\{0\}}$ (the function that equal
1 at $x$ and 0 elsewhere), we have $f\in\ell^{1}$ and $P_{t}^{m}f(x)=\sum_{y\in\mathbb{Z}^{d}}P_{x}^{m}(X_{t}=y)f(y)=P_{0}^{m}(X_{t}=x)$
which plugged in $\eqref{eq:fm2}$ gives the desired result. \end{proof}


\subsection{Gloria, Neukamm and Otto with a fixed walk scheme}

Using the monotonic function approach above, we can prove a stronger
decay of the variance of the environment view by the particle, when
the function $f$ is the divergence of an other function, but only
when the walk scheme $m$ is fixed. This is a result (much weaker
but) in the spirit of \cite{gloria}.

\begin{prop} There exists a constant $C>0$ such that for almost all
walk scheme $m$, all $t\geqslant0$ and all function $f=D_{i}g$,
where $-d\leqslant i\leqslant d$, $g$ is local, translation-invariant
and $\mathbb{E}[g^{2}]<\infty$, it holds $\mathbb{E}\left[\left(P_{t}^{m}f\left(0,\omega\right)\right)^{2}\right]\leqslant C\#supp(g)^{2}\frac{\mathbb{E}[g^{2}]}{(t+1)^{\frac{d}{2}+1}}$.
\end{prop}

\begin{proof} Using the Cauchy-Schwarz Inequality and that $\mathbb{E}\left[g\left(x,\omega\right)g\left(y,\omega\right)\right]=0$
as soon as $supp\left(g(x,\omega)\right)\cap supp\left(g(y,\omega)\right)=\emptyset$
we have 
\begin{align*}
 & \mathbb{E}\left[\left(P_{t}^{m}f\left(0,\omega\right)\right)^{2}\right]=\mathbb{E}\left[\left(\sum_{x\in\mathbb{Z}^{d}}\left(P_{0}^{m}\left(X_{t}=x\right)-P_{0}^{m}\left(X_{t}=x+e_{i}\right)\right)g\left(y,\omega\right)\right)^{2}\right]\\
 & \leqslant\#supp\left(g\right)^{2}\sum_{x\in\mathbb{Z}^{d}}\left(P_{0}^{m}\left(X_{t}=x\right)-P_{0}^{m}\left(X_{t}=x+e_{i}\right)\right)^{2}\mathbb{E}\left[g^{2}\right]\notag
\end{align*}
which, combined with Corollary \ref{pro:DecroissanceProbaMarche2}
gives the expected result. \end{proof}


\subsection{Discussion about the polynomial decay}
In the introduction, we told that $t^{-d/2}$ is the optimal decay for 
local functions in $L^2$. In \cite{Mo99}, it is suggested that, using spectral theory, we can construct a non-local function $f$ such that $\mathbb{E}[f^{2}_{t}]$ decays as fast or as slow as we want. We note here that one can construct a local function with
a faster decay than $t^{-d/2}$. Indeed, because $\mathbb{E}[f^{2}_{t}]$ is a completely monotonic function, 
it is a consequence of corrolary \ref{cor:CorollaireDeriveeCM}.
For example, consider a function $g\in L^2$ such that $f=\mathcal{L}g$, then
$$\mathbb{E}[f^{2}_{t}]=\mathbb{E}[(\mathcal{L}g_{t})^{2}]=\partial_{t}\partial_{t}\mathbb{E}[g_{t}^{2}]\leqslant C_{g} t^{-d/2-2} $$
Iterating the process, considering $f=\mathcal{L}^{n}g$, we have that $\mathbb{E}[f^{2}_{t}]\leqslant C_{g}t^{-d/2-2n}$.
Note that in this case, $f$ might not be in $L^{2}$, even if $f_{t}$ would.

\subsection*{Acknowledgement} We thank Cyril Roberto and Julien Bureaux.

\newcommand{\etalchar}[1]{$^{#1}$}
\begin{thebibliography}{DMFGW89}

\bibitem[ABC{\etalchar{+}}00]{ane}
C.~An\'e, S.~Blach\`ere, D.~Chafai, P.~Foug\`eres, I.~Gentil, F.~Malrieu,
  C.~Roberto, and G.~Scheffer.
\newblock {\em Sur les in\'egalit\'es de Sobolev logarithmiques}, volume~10 of
  {\em Panoramas et Syth\`ese}.
\newblock Soci\'et\'e Math\'ematique de France, Paris, 2000.

\bibitem[AM14]{am}
S.~Armstrong and J.-C. Mourrat.
\newblock A regularity theory for random elliptic operators.
\newblock Preprint, arXiv:1411.3668, 2014.

\bibitem[BBHK08]{berger2008anomalous}
Noam Berger, Marek Biskup, Christopher~E Hoffman, and Gady Kozma.
\newblock Anomalous heat-kernel decay for random walk among bounded random
  conductances.
\newblock In {\em Annales de l'Institut Henri Poincar\'e Probabilit\'es et
  Statistiques}, volume~44, page 374. Gauthier-Villars, 2008.

\bibitem[BCLSC95]{bakry}
D.~Bakry, T.~Coulhon, M.~Ledoux, and L.~Saloff-Coste.
\newblock Sobolev inequalities in disguise.
\newblock {\em Indiana Univ. Math. J.}, 44(4):1033--1074, 1995.

\bibitem[Bis11]{biskup2011recent}
Marek Biskup.
\newblock Recent progress on the random conductance model.
\newblock {\em Probability Surveys}, 8, 2011.

\bibitem[BZ99]{bertini}
L.~Bertini and B.~Zegarlinski.
\newblock Coercive inequalities for {K}awasaki dynamics. {T}he product case.
\newblock {\em Markov Process. Related Fields}, 5(2):125--162, 1999.

\bibitem[BZ10]{bobkov}
S.~Bobkov and B.~Zegarlinski.
\newblock Distributions with slow tails and ergodicity of {M}arkov semigroups
  in infinite dimensions.
\newblock In {\em Around the research of {V}ladimir {M}az'ya. {I}}, volume~11
  of {\em Int. Math. Ser. (N. Y.)}, pages 13--79. Springer, New York, 2010.

\bibitem[CCR05]{cesi}
N.~Cancrini, F.~Cesi, and C.~Roberto.
\newblock Diffusive long-time behavior of {K}awasaki dynamics.
\newblock {\em Electron. J. Probab.}, 10:no. 7, 216--249 (electronic), 2005.

\bibitem[CK08]{cudna}
M.~Cudna and T.~Komorowski.
\newblock A finite dimensional approximation of the effective diffusivity for a
  symmetric random walk in a random environment.
\newblock {\em J. Comput. Appl. Math.}, 213(1):186--204, 2008.

\bibitem[CKS87]{CKS}
E.~Carlen, S.~Kusuoka, and D.~Stroock.
\newblock {\em Upper bounds for symmetric Markov transition functions},
  volume~23.
\newblock 1987.

\bibitem[Del99]{delmotte}
T.~Delmotte.
\newblock Parabolic {H}arnack inequality and estimates of {M}arkov chains on
  graphs.
\newblock {\em Rev. Mat. Iberoamericana}, 15(1):181--232, 1999.

\bibitem[Deu94]{deuschel}
J.-D. Deuschel.
\newblock Algebraic {$L^2$} decay of attractive critical processes on the
  lattice.
\newblock {\em Ann. Probab.}, 22(1):264--283, 1994.

\bibitem[DMFGW89]{DFGW}
Anna De~Masi, Pablo~A Ferrari, Sheldon Goldstein, and William~David Wick.
\newblock An invariance principle for reversible markov processes. applications
  to random motions in random environments.
\newblock {\em Journal of Statistical Physics}, 55(3-4):787--855, 1989.

\bibitem[Fel71]{feller}
W.~Feller.
\newblock {\em An introduction to probability theory and its applications.
  {V}ol. {II}.}
\newblock Second edition. John Wiley \& Sons, Inc., New York-London-Sydney,
  1971.

\bibitem[GNO14a]{gloria}
A.~Gloria, S.~Neukamm, and F.~Otto.
\newblock Quantification of ergodicity in stochastic homogenization: optimal
  bounds via spectral gap on glauber dynamics.
\newblock To appear in Inventiones Mathematicae, 2014.

\bibitem[GNO14b]{new-gno}
A.~Gloria, S.~Neukamm, and F.~Otto.
\newblock A regularity theory for random elliptic operators.
\newblock Preprint, arXiv:1409.2678, 2014.

\bibitem[GO11]{gloria-otto11}
A.~Gloria and F.~Otto.
\newblock An optimal variance estimate in stochastic homogenization of discrete
  elliptic equations.
\newblock {\em Ann. Probab.}, 39(3):779--856, 2011.

\bibitem[GO12]{gloria-otto12}
A.~Gloria and F.~Otto.
\newblock An optimal error estimate in stochastic homogenization of discrete
  elliptic equations.
\newblock {\em Ann. Appl. Probab.}, 22(1):1--28, 2012.

\bibitem[JLQY99]{janvresse}
E.~Janvresse, C.~Landim, J.~Quastel, and H.~T. Yau.
\newblock Relaxation to equilibrium of conservative dynamics. {I}. {Z}ero-range
  processes.
\newblock {\em Ann. Probab.}, 27(1):325--360, 1999.

\bibitem[KV86]{kipnis1986central}
Claude Kipnis and SR~Srinivasa Varadhan.
\newblock Central limit theorem for additive functionals of reversible markov
  processes and applications to simple exclusions.
\newblock {\em Communications in Mathematical Physics}, 104(1):1--19, 1986.

\bibitem[Lig91]{liggett}
T.~Liggett.
\newblock {$L_2$} rates of convergence for attractive reversible nearest
  particle systems: the critical case.
\newblock {\em Ann. Probab.}, 19(3):935--959, 1991.

\bibitem[LO00]{latala}
R.~Lata{\l}a and K.~Oleszkiewicz.
\newblock Between {S}obolev and {P}oincar\'e.
\newblock In {\em Geometric aspects of functional analysis}, volume 1745 of
  {\em Lecture Notes in Math.}, pages 147--168. Springer, Berlin, 2000.

\bibitem[Mar99]{martinelli}
F.~Martinelli.
\newblock Lectures on {G}lauber dynamics for discrete spin models.
\newblock In {\em Lectures on probability theory and statistics
  ({S}aint-{F}lour, 1997)}, volume 1717 of {\em Lecture Notes in Math.}, pages
  93--191. Springer, Berlin, 1999.

\bibitem[Mas07]{Ma07}
P.~Massart.
\newblock {\em Concentration inequalities and model selection}, volume 1896 of
  {\em Lecture Notes in Mathematics}.
\newblock Springer, Berlin, 2007.
\newblock Lectures from the 33rd Summer School on Probability Theory held in
  Saint-Flour, July 6--23, 2003, With a foreword by Jean Picard.

\bibitem[Mou11]{Mo99}
J.-C. Mourrat.
\newblock Variance decay for functionals of the environment viewed by the
  particle.
\newblock {\em Ann. Inst. Henri Poincar\'e Probab. Stat.}, 47(1):294--327,
  2011.

\bibitem[MS01]{miller}
K.~S. Miller and S.~G. Samko.
\newblock Completely monotonic functions.
\newblock {\em Integral Transform. Spec. Funct.}, 12(4):389--402, 2001.

\bibitem[Rob10]{roberto}
C.~Roberto.
\newblock Slow decay of {G}ibbs measures with heavy tails.
\newblock {\em Probab. Theory Related Fields}, 148(1-2):247--268, 2010.

\bibitem[SC97]{saloff}
L.~Saloff-Coste.
\newblock Lectures on finite {M}arkov chains.
\newblock In {\em Lectures on probability theory and statistics
  ({S}aint-{F}lour, 1996)}, volume 1665 of {\em Lecture Notes in Math.}, pages
  301--413. Springer, Berlin, 1997.

\bibitem[Wid41]{widder}
D.~V. Widder.
\newblock {\em The {L}aplace {T}ransform}.
\newblock Princeton Mathematical Series, v. 6. Princeton University Press,
  Princeton, N. J., 1941.

\bibitem[Yur86]{yurinskii}
V.~V. Yurinski{\u\i}.
\newblock Averaging of symmetric diffusion in a random medium.
\newblock {\em Sibirsk. Mat. Zh.}, 27(4):167--180, 215, 1986.

\end{thebibliography}
\end{document}
