Some properties of the rate function of quenched large deviations for random walk in random environment
In this paper, we are interested in some questions of Greven and den Hollander \cite{GD2} about the rate function $\Iq$ of quenched large deviations for random walk in random environment. By studying the hitting times of RWRE, we prove that in the recurrent case, $\lim_{\theta\to 0^+}(\Iq)''(\theta )=+\infty$, which gives an affirmative answer to a conjecture of Greven and den Hollander \cite{GD2}. We also establish a comparison result between the rate function of quenched large deviations for a diffusion in a drifted Brownian potential, and the rate function for a drifted Brownian motion with the same speed.
devulder_largedeviations_article.tex
\documentclass[a4paper,12pt]{article}
\usepackage{theorem,amsmath,amssymb}
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%\usepackage{graphics}
\usepackage{multicol}

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\newtheorem{theo}{Theorem}[section]
\newtheorem{prop}[theo]{Proposition}
\newtheorem{lemma}[theo]{Lemma}
\newtheorem{hyp}[theo]{Hypothesis}
\newtheorem{cor}[theo]{Corollary}
\newtheorem{rappel}[theo]{Rappel}
\newtheorem{rem}[theo]{Remark}
\newtheorem{fact}[theo]{Fact}
\newtheorem{deff}[theo]{Definition}

\newcommand{\eqv}{\Longleftrightarrow}

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\def\cad{c'est � dire }
\def\dmin{d_{min}}
\def\a{\alpha}
\def\b{\beta}
\def\D{\Delta}
\def\d{\delta}
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\def\ep{\varepsilon}
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\def\l{\lambda}
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\def\P{\mathbb{P}}
\def\Q{\mathbb{Q}}
\def\R{\mathbb{R}}
\def\T{\mathcal{T}}
\def\t{\theta}
\def\Z{\mathbb{Z}}

\def\ta{\tau_1}
\def\tb{\tau_{-1}}
\def\c{c_{1,n}}
\def\cc{c_{3,n}}
\def\ccc{ELIMINE}
\def\cccc{c_{2,n}}
\def\dd{c_{4,n}}
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\def\dddd{\cccc}
\def\ee{c_{6,n}}
\def\eee{\ddd}

\def\W{\widetilde{W}}
\def\WW{\widehat{W}}
\def\V{\widetilde{V}}
\def\VV{\widehat{V}}

\def\EA{E_{n}}
\def\EE{E_{1,n}}
\def\EEE{E_{2,n}}
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\def\Gg{E_{7,n}}
\def\GG{E_{2,n}'}
\def\GGG{E_{3,n}'}
\def\Hh{E_{6,n}'}
\def\HH{E_{6,n}}

\def\po{P_{\omega}}
\def\elo{E_{\lambda,\omega}}
\def\eo{E_{\omega}}
\def\Iq{I^q_{\eta}}
\def\fo{\F_{\omega}}
\def\op{\o^+}
\def\om{\o^-}

\def\omin{\omega_{min}}
\def\omax{\omega_{max}}
\def\ra{\rightarrow}
\def\lra{\longrightarrow}
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%\def\i0{\mathcal{I}_{0}}
\def\var{\mathbb{V}ar}
\def\bc{Borel--Cantelli }
\def\bt{Bienaym�--Tch�bychev }
\def\omin{\omega_{min}}
\def\j1{J_{\Lambda}^<}

\def\Tb{\overline{T}}
\def\Ttm{\tilde{T}_{-1,m_n}}
\def\Tbm{\overline{T}_{-1,m_n}}
\def\Td{\overline{T}_{-1,b_n}}

\def\pas{$\mathbb{P}$--a.s. }
\def\poas{$\po$--a.s. }

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\def\k{\kappa}
\def\jk{J_{\k}}
\def\fk{F_{\k}}
\def\gk{\Gamma_{\k}}
\def\ik{I_{\k}}
\def\fv{\phi_{v_{\k}}}

\newcommand{\goto}[3]{p_{\omega}(#1\leadsto #2,#3)}

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\title{Some properties of the rate function of quenched large
deviations for random walk in random environment}

\author{Alexis Devulder
\footnote{Laboratoire de Probabilit\'es et Mod\`eles Al\'eatoires,
Universit\'e Paris VI, $4$ Place Jussieu, Case 188, F-$75252$
Paris Cedex $05$, France. E-mail: devulder@ccr.jussieu.fr.}}

\begin{document}

%\date{}

\maketitle

\begin{abstract}
In this paper, we are interested in some questions of Greven and
den Hollander \cite{GD2} about the rate function $\Iq$ of quenched
large deviations for random walk in random environment. By
studying the hitting times of RWRE, we prove that in the recurrent
case, $\lim_{\theta\to 0^+}(\Iq)''(\theta )=+\infty$, which gives
an affirmative answer to a conjecture of Greven and den Hollander
\cite{GD2}. We also establish a comparison result between the rate
function of quenched large deviations for a diffusion in a drifted
Brownian potential, and the rate function for a drifted Brownian
motion with the same speed.
\end{abstract}

\noindent\textbf{\sc{Key Words:}}\emph{ Random walk in random
environment, Large deviations.}

\bigskip
\noindent\textbf{AMS $(2000)$ Classification:} 60K37, 60F10,
60J60.




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%                                                              %
%                           SECTION 1                          %
%                                                              %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



\mysection{Introduction}\label{intro}
\subsection{Presentation of the model}
We consider a collection of independent and identically
distributed random variables $(\o_i)_{i\in\Z}$. A realization of
these variables is called an environment. Given an environment
$\o:=(\o_i)_{i\in\Z}$, we consider the random walk
$(S_n)_{n\in\N}$ defined by $S_0=0$ and
\begin{equation*}
\po(S_{n+1}=k|S_n=i) =\left\{ {\begin{array}{ll}
 \o_i  & \text{ if } k=i+1,\\
 1-\o_i & \text{ if } k=i-1,\\
 0 & \text{ otherwise.}
\end{array}}
\right.
\end{equation*}
The process $(S_n)_{n\in\N}$ is called a random walk in random
environment, abbreviated RWRE. This model has many applications in
physics, see for example Hughes \cite{Hug}. Let $\eta$ denote the
law of $(\o_i)_{i\in\Z}$. We call $\po$ the quenched law, whereas
$\P(.):=\int \po(.)\eta(d\o)$ is the annealed law. For technical
reasons, we assume that there exists an $\e_0>0$ such that
\begin{equation}
\eta(\e_0\leq\o_0\leq 1-\e_0)=1.\label{lip}
\end{equation}
For $i\in\Z$, let $\rho_i=\frac{1-\o_i}{\o_i}$. Solomon \cite{S1}
proved that the RWRE $(S_n)_{n\in\N}$ is \pas recurrent if and
only if
\begin{equation}
\int(\log\rho_0)\eta(d\o)=0.\label{rec}
\end{equation}
In order to avoid the degenerate case of simple random walk, we
assume in the following that
\begin{equation}
\text{Var}(\log \rho_0):=\sigma^2>0.\label{var}
\end{equation}
Sinai \cite{S3} showed that in the recurrent case, the random
environment considerably slows down the walk. More precisely, he
proved that if \eqref{rec} and \eqref{var} are satisfied, there
exists a nondegenerate non--Gaussian random variable $b_{\infty}$
such that
\begin{equation}
\sigma^2 \frac{S_n}{(\log
n)^2}\overset{\mathcal{L}}{\underset{n\to+\infty}{\longrightarrow
}}b_{\infty},\label{sinai}
\end{equation}
where $\overset{\mathcal{L}}{\longrightarrow}$ denotes convergence
in law under $\P$.

It is moreover known (Solomon~\cite{S1}) that the RWRE
$(S_n)_{n\in\N}$ satisfies a law of large numbers: there exists
$v\in ]-1,1[$ such that $\lim_{n\to\infty} S_n/n=v$ $\P$--a.s. In
addition, $v$ is strictly positive if and only if $\int
\rho_0\eta(d\o)<1$.
%Under our assumption
%$\int\log\rho_0(\o)\eta(d\o)\ge 0$, we have $v\in [-1, 0]$.

The RWRE $(S_n)_{n\in\N}$ satisfies furthermore a quenched large
deviation principle with deterministic convex rate function $\Iq$
(see Greven and den Hollander~\cite{GD2}). This means there exists
a nonnegative convex function $\Iq$ such that $\eta$--a.s. for any
measurable set $A$,
\begin{eqnarray*}
    \liminf_{n\to\infty}\frac{1}{n}\log\po\left(\frac{S_n}{n}\in A\right)
 &\ge& -\inf_{x\in A^{\circ}}\Iq(x),
    %\label{gdinf}
    \\
    \limsup_{n\to\infty}\frac{1}{n}\log\po\left(\frac{S_n}{n}\in A\right)
 &\le& -\inf_{x\in \overline{A}}\Iq(x),
%    \label{gdsup}
\end{eqnarray*}
where $A^{\circ}$ denotes the interior of $A$ and $\overline{A}$
is the closure of $A$.

For more details on RWRE, we refer to Zeitouni \cite{Z1}.

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


\subsection{Results}
In this paper, we are interested in some questions raised by
Greven and den Hollander about quenched large deviations for RWRE.
First, we answer their Open problem 2 (see \cite{GD2}, p. 1389;
see also den Hollander \cite{H3} p. 80), and prove that

\begin{theo}\label{th1}
Under \eqref{lip}, \eqref{rec} and \eqref{var}, the rate function
$\Iq$ for quenched large deviations of the RWRE satisfies
\begin{equation}
\lim_{\theta\ra 0^+} (\Iq)''(\theta)=+\infty.\label{pb1}
\end{equation}
\end{theo}

This is what Greven and den Hollander conjectured. Observe that
this result is coherent with the subdiffusive behaviour of Sinai's
walk \eqref{sinai}. We mention that the corresponding problem for
Brox--type diffusions (see Brox~\cite{B3}), for which the rate
functions can be explicitly computed, has already been solved by
Taleb (see \cite{Ta}).

%In the recurrent case, la d\'eriv\'ee seconde de la fonction de
%taux en $0$ n'est pas toujours infinie, contrairement au cas
%discret; cela d\'epend de la valeur du drift de l'environnement.

In order to prove Theorem \ref{th1}, it is useful to study the
hitting times of $(S_n)_{n\in\N}$. Let us define, for $a\in\Z$,
\begin{equation*}
\tau_a:=\inf\{n>0, S_n=a\}.
\end{equation*}
We show the following estimate:

\begin{prop}\label{moments}
For each $\alpha\in \R_+^*$,
\begin{equation*}
\E(\tau_1^\a
e^{-r\tau_1})=\left(\frac{1}{r}\right)^{\a+o(1)},\qquad r\to 0^+.
\end{equation*}
\end{prop}

\bigskip
We are also interested in Open problem 3 of Greven and den
Hollander (\cite{GD2}, p. 1389): they conjectured that in the case
$\int \rho_0 \eta(d\o)<1$ (i.e., $v>0$), the quenched rate
function $\Iq$ of the RWRE is smaller than
%(or equal to)
the rate function of the simple random walk on $\Z$ with the same
speed $v$. That is, they conjectured that $\forall x>v,\
\Iq(x)<\widehat{I}_{\langle \rho\rangle }(x)$, where
$\widehat{I}_{\langle\rho\rangle}$ is the rate function of a usual
nearest neighbour random walk with speed $v$.

Unfortunately, we have not been able to answer this question, but
we solve the corresponding problem for Brox--type diffusion (see
Brox, \cite{B3}). For $\k\geq 0$, we define the random potential
\begin{equation*}
W_{\k}(x):=W(x)-\frac{\k}{2}x,
\end{equation*}
where $(W(x),x\in\R)$ is a standard two-sided Brownian motion. We
consider a diffusion~$X$ in the random potential $W_{\k }$, which
is defined as the solution to the formal stochastic differential
equation
\begin{equation*}
\left\{
\begin{array}{l}
\textnormal{d}X(t)=\textnormal{d}\b(t)-\frac{1}{2}W'(X(t))\textnormal{d}t,\\
X(0)=0,
\end{array}\right.
\end{equation*}
where $(\b(t), t\geq 0)$ is a Brownian motion independent of $W$.
More precisely, $X$ is a diffusion process whose conditional
generator given $W_{\k}$ is
\begin{equation*}
%\mathcal{L}_{W_{\k}}(x):=
\frac{1}{2}e^{W_{\k}(x)}\frac{\partial}{\partial
x} \left(e^{-W_{\k}(x)}\frac{\partial}{\partial x}\right).
\end{equation*}
This diffusion can be considered as the continuous time analogue
of RWRE and share many properties with it. See for example Shi
\cite{S2} for the relations between these two processes. For
instance, Kawazu and Tanaka \cite{KT1} established a law of large
numbers for $X$. That is, $\lim_{t\to\infty} X_t/t=v_{\k}$, where
$v_{\k}=\frac{(\k-1)^+}{4}$ is $>0$ if and only if $\k>1$.
Moreover, Taleb \cite{Ta} proved that $X$ satisfies quenched and
annealed large deviation principles. Let $\jk$ denote the rate
function of quenched large deviations of~$X$ (see \eqref{quotient}
below for more details). We compare $\jk$ with the function
\begin{equation*}
J_{v_{\k}}^B(x):=\frac{1}{2}(x-v_{\k})^2,
\end{equation*}
which is the rate function of large deviations of the drifted
Brownian motion $(B_t+v_{\k} t,\ t\in\R_+)$. We prove

\begin{theo}\label{comptaux}
If $\k>1$, then
\begin{equation*}
\forall x>v_{\k},\qquad \jk(x)< J_{v_{\k}}^B(x).
\end{equation*}
\end{theo}

\bigskip
Interestingly, we obtain as a by-product an inequality for the
modified Bessel functions
%of the third kind $K_{\nu}$ (Macdonald's function),
which might be new:

\begin{prop}\label{prop14}
Let $K_{\nu}$ be the modified Bessel function of index $\nu$. We
have,

\begin{equation*}
\forall \nu>0,\ \forall y>0,\qquad
\frac{K_{\nu}(y)}{K_{\nu+1}(y)}<\frac{1}{y}\left(\sqrt{y^2
+\nu^2}-\nu\right).
\end{equation*}
\end{prop}

\bigskip
The rest of the paper is organized as follows: in Section
\ref{milieu} we build environments $\EA$ for which the hitting
time of~$-1$ by $(S_n)_{n\in\N}$, denoted by $\tau_{-1}$, will be
large. We give an estimation of $\tau_{-1}$ for $\o\in \EA$ in
Section~\ref{proba}. In Section \ref{simpl}, we prove Theorem
\ref{th1} and Proposition \ref{moments}. Finally,
Section~\ref{sectbrox} is devoted to the proofs of
Theorem~\ref{comptaux} and Proposition~\ref{prop14}.






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\mysection{Construction of the event {$\hbox{\slshape\bfseries
E}_{\hbox{\slshape\bfseries \small n}}$} } \label{milieu}

In this section we build a set of environments $\EA$, such that
$\P(\EA)$ is not ``too small'' and that for $\o\in \EA$,
$\tau_{-1}$ is almost $n$ (we prove this last assertion in Section
\ref{proba}).

Throughout Sections \ref{milieu} and \ref{proba}, we fix an
$\e>0$. The constants $C_i$, $0\leq i\leq 10,$ depend only on
$\eta$ and $\e$, whereas $\theta$ and $\delta$ depend only on
$\eta$. The events $\EA$, $E_{i,n}$ and $E_{i,n}'$ depend on
$\eta$ and $\e$, but we omit to write $\e$.

We give some notation in Subsection \ref{sub20}. Subsection
\ref{sub21} is devoted to the construction of $\EA$. We give an
estimation of $\P(\EA)$ in Subsection \ref{sub22}, and study some
of the properties of $\o\in \EA$ in Subsection~\ref{sub23}.


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


\subsection{Some notation}\label{sub20}

We define the potential $V$ as follows:

\begin{deff}Let
\begin{equation*} V(n):=\sum_{i=1}^n\log\rho_i=\sum_{i=1}^n \log\frac{1-\o_i}{\o_i},
\qquad n\in\Z,
\end{equation*}
where by convention, $\sum_{i=1}^0 x_i=0$ and $\sum_{i=1}^n x_i
=-x_0-x_{-1}-\dots -x_{n+1}$ if $n$ is (strictly) negative.
\end{deff}

We define a valley for the potential (see Sinai, \cite{S3}):
\begin{deff}\label{defval}
Let $a<m<b$. $(a,m,b)$ is a valley if
\begin{equation*}
\begin{array}{ll}
\forall a\leq i\leq m, & \quad V(m)\leq V(i)\leq V(a), \nonumber\\
\forall m\leq i \leq b, & \quad V(m)\leq V(i)\leq V(b).\nonumber
\end{array}
\end{equation*}
Its depth is defined as $\min\{V(a)-V(m),V(b)-V(m)\}$.
\end{deff}



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


\subsection{Building {$\hbox{\slshape\bfseries
E}_{\hbox{\slshape\bfseries \small n}}$}}\label{sub21}

In this subsection, we build a valley $(0, m_n, b_n)$ for the
potential $V$, so that the RWRE will stay for a ``good'' amount of
time in this valley with ``large probability''.

As $\int(\log\rho_0)\eta(d\o)=0$ and $\sigma>0$, there exists a
real number $\d>0$ such that
\begin{equation*}
\P(-2\delta\leq \log \rho_0\leq -\delta):=\exp(-\theta)>0.
\end{equation*}
%Rigorously speaking, $\c$, $\cccc$ should be integers but we do
%not take their integer parts to simplify the notation.
Now we set
\begin{equation*}
\begin{array}{rclrcl} \e'& := &\e\d,
& \c & := & \lfloor \e \log n\rfloor,\\

 \cccc & := &  \lfloor \log n\rfloor^2,
& \cc & := & \d \c, \\

\dd & := &(1-10\e') \log n,&
\beta & := & \frac{1-9\e'}{1-10\e'},\\

\ddd & := & \frac{\e'}{2} \log n, &
\ee & := & 2 \log n.\\
\end{array}
\end{equation*}

\noindent For $i\in\Z$, define $\V(i)=V(i+\c)-V(\c)$ and
$\VV(i)=\V(i+\cccc)-\V(\cccc)$. We consider

\begin{eqnarray*}
\EE & := & \left\{\forall 0\leq i\leq \c, \qquad -2\d i\leq
V(i)\leq -\d i \right\},\\
\EEE & := & \{\V(\cccc)\in [-\b\dd,-\dd]\},\\
\Ff & := & \left\{\forall 0< i \leq \cccc,\qquad
\left|\V(i)-\frac{i}{\cccc}
\V(\cccc)\right|\leq \ddd \right\},\\
\FF & := & \{\VV(\dddd)\in[\ee,2\ee]\},\\
\FFF & := & \left\{\forall 0< i\leq \dddd,\qquad
\left|\VV(i)-\frac{i}{\dddd}\VV(\dddd)\right|\leq \eee \right\}.
\end{eqnarray*}
Finally, let
\begin{equation*}
\EA:=\EE\cap \EEE\cap \Ff\cap \FF\cap \FFF.
\end{equation*}
When $\o\in \EA$, we say the environment is ``good''. On $\EE$,
the potential $V(i)$ decreases almost linearly for $i\in[0,\c]$
(this will enable the walk $(S_n)_{n\in\N}$ to go quickly to $\c$
before hitting $-1$ with large probability). On $\EEE\cap \Ff$,
$V$ stays within a tunnel of height $2\ddd$, and sinks to
$V(\c)-\dd$. On $\FF\cap \FFF$, $V$ stays within another tunnel of
height $2\ddd$, and moves up to positive values. These comments on
$E_n$ are represented in the Figure \ref{figure1} ($b_n$ and $m_n$
are defined in Subsection~\ref{sub23}).

\bigskip

\begin{figure}[htbp]
\begin{center}
\setlength{\unitlength}{1cm}
\begin{picture}(12.5,7)
\footnotesize \psfrag{y}{\footnotesize $j$}
\psfrag{x}{\footnotesize $i$} \psfrag{c1+c2}{\footnotesize
$\c+\cccc$} \psfrag{c1+2c2}{\footnotesize $\c+2\cccc$}
\psfrag{c1}{\footnotesize $\c$} \psfrag{c5}{\footnotesize $\ddd$}
\psfrag{-1}{\footnotesize $-1$}\psfrag{-c3}{\footnotesize
$-\cc$}\psfrag{-2c3}{\footnotesize $-2\cc$}
\psfrag{-c4}{\footnotesize $V(\c)-\dd$}
\psfrag{-bc4}{\footnotesize $V(\c)-\beta\dd$}
\psfrag{c6}{\footnotesize $V(\c+\cccc)+\ee$}
\psfrag{2c6}{\footnotesize $V(\c+\cccc)+2\ee$}
\psfrag{m}{\footnotesize $m_n$} \psfrag{b}{\footnotesize $b_n$}

\put(0,0){\includegraphics[width=12.5cm,height=7cm]{devulder_largedeviations_figure}}
\end{picture}
\end{center}\caption{Example of the potential $V$ for a ``good'' environment $\o\in E_n$}
\label{figure1}\end{figure}

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


\subsection{Probability of {$\hbox{\slshape\bfseries
E}_{\hbox{\slshape\bfseries \small n}}$}}\label{sub22}
\begin{lemma}\label{en}
There exists a constant $C_0>0$ which depends only on $\eta$ and
$\e$, such that for $n$ large enough,
\begin{equation*}
P(\EA)\geq C_0 n^{-\theta\e}.
\end{equation*}
\end{lemma}

\bigskip
\noindent {\bf Proof:} First, observe that
\begin{eqnarray}
\P(\EE) & \geq & \P\left(\forall 1\leq i\leq \c,\quad
-2\d\leq\log\rho_i\leq-\d \right)\nonumber\\
& \geq & \exp(-\theta {\e\log n})\geq n^{-\theta\e}.\label{E2}
\end{eqnarray}

According to the Koml\'os--Major--Tusn\'ady strong approximation
theorem (see \cite{KMT}), possibly in an enlarged probability
space, there exists a coupling for  $\o$ and a standard Brownian
motion $W$, and (strictly) positive constants $C_1$, $C_2$ and
$C_3$ such that for all $N\geq 1$,
\begin{equation*}
\P\left(\sup_{1\leq i\leq N} |\V(i)-\sigma W(i)|\geq C_1\log
N\right) \leq \frac{C_2}{N^{C_3}}.
\end{equation*}
Define
\begin{equation*}
\Hh:=\left\{\sup_{1\leq i\leq 3(\log n)^2} |\V(i)-\sigma W(i)|\geq
C_1\log [3(\log n)^2]\right\}.
\end{equation*}
We have,
\begin{equation*}
\P(\Hh)\leq \frac{C_2}{(3(\log n)^2)^{C_3}}.
\end{equation*}
We then consider the following events:
\begin{eqnarray*}
\GG & := & \left\{\sigma W(\cccc)\in\left[-\b\dd+\frac{\e'}{4}\log
n,
-\dd-\frac{\e'}{4} \log n \right]\right\},\\
\GGG & := & \left\{\forall 0\leq t\leq \cccc,\qquad \left|\sigma
W(t)-\frac{t}{\cccc}\sigma W(\cccc)\right|\leq
\ddd-\frac{\e'}{4}\log n\right\}.
\end{eqnarray*}
We notice by scaling that there exists $C_4>0$ such that
$\P[\GG\cap \GGG]\geq 2 C_4$ for $n$ large enough. Since
$\log[3(\log n)^2]=o(\log n)$, we have for large $n$,
\begin{eqnarray}
\P(\EEE\cap \Ff) & \geq & \P[\GG\cap \GGG\cap
(\Hh)^c] \nonumber \\
& \geq  & \P(\GG\cap \GGG)-\frac{C_2}{[3(\log n)^2]^{C_3}}\nonumber \\
& \geq & C_4.\label{c10}
\end{eqnarray}
Similarly, there exists a constant $C_5>0$ such that
\begin{equation}\label{c11}
\P(\FF\cap \FFF)\geq C_5
\end{equation}
for $n$ large enough. Since  $\EE$, $\EEE\cap \Ff$ and $\FF\cap
\FFF$ are independent, we obtain Lemma \ref{en} by combining
\eqref{E2}, \eqref{c10} and \eqref{c11}. \hfill$\Box$


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


\subsection{Properties of a ``good'' environment}\label{sub23}
Let $\o\in E_n$. We define the integers $b_n$ and $m_n$ such that
\begin{eqnarray*}
b_n & := &\inf\{k\in\N, \ k>0, \quad V(k) \geq 0\}, \\
m_n & := & \inf\{k>0 ,\quad V(k)=\inf_{0\leq\ell\leq b_n}
V(\ell)\}.
\end{eqnarray*}
Note  that $(0,m_n,b_n)$ is a valley  (in the sense of Definition
\ref{defval}) with depth $-V(m_n)$, and that
\begin{eqnarray*}
V(m_n) & \in & [-2\cc-\b\dd-\ddd, -\cc-\dd],\\
%W_{b_n} & \in & [-2\cc-\b\dd+\ee, -\cc-\dd+2\ee+\eee],\\
%H_{a_n,m_n} & = & 2\ddd,\\
%H_{m_n,b_n} & = & W_{b_n}-W_{a_n},\\
%H_{m_n,a_n} & = & W_{a_n}-W_{m_n},\\
%H_{b_n,m_n} & = & 2\eee.
m_n & \in & \left[\c+\cccc-\frac{\ddd\cccc}{\dd},
\c+\cccc+\frac{\eee\dddd}{\ee}\right].
\end{eqnarray*}
In particular, we have for $\e'$ small enough and $n$ large
enough, \begin{equation}\label{mn}
\begin{array}{rcccl}
-\d+(1-9\e') \log n   & \leq & -V(m_n) & \leq &  (1-6\e') \log n, \\
(1-\e') (\log n)^2 & \leq & m_n      & \leq & (1+\e') (\log n)^2 .
\end{array}
\end{equation}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                              %
%                       SECTION 3                              %
%                                                              %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%




\mysection{Probability that
%{$\hbox{\slshape\bfseries $\greek t
%$}_{\hbox{\slshape\bfseries \small n}}$}
$\tau_{-1}$ has a
``good'' length }\label{proba}

This section is devoted to the proof of the following result:

\begin{lemma}\label{encadrement}
There exists a constant $C_6>0$, depending only on $\eta$ and
$\e$, such that for all large $n$,
\begin{equation*}
\forall\o\in \EA,\qquad
\po\left(n^{1-10\e'}\leq\tau_{-1}<n\right)\geq C_6.
\end{equation*}
\end{lemma}

In Subsection \ref{sub31}, we show that when $\o\in \EA$, with a
large quenched probability, the RWRE goes quickly to the bottom
$m_n$ of the valley $(0,m_n,b_n)$ without hitting $-1$. In
Subsection \ref{sub32}, we prove that with a large quenched
probability, after hitting $m_n$, the RWRE stays in $\N$ during
almost $n$ units of time and then hits $-1$ for the first time.


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


\subsection{Going to the bottom
{$\hbox{\slshape\bfseries m}_{\hbox{\slshape\bfseries \small n}}$}
of the valley}\label{sub31}

\begin{lemma} \label{lemma32}There exists a constant $C_7>0$, depending
only on $\eta$ and $\e$, such that
$$\forall \o\in \EA,\qquad \po(
\tau_{m_n}<\tau_{-1} )\geq C_7.$$
\end{lemma}

\bigskip
\noindent {\bf Proof:} Let $\o\in \EA$. Since $\EE\subset \EA$,
\begin{equation}\label{minorationA1}
\sum_{i=0}^{\c-1}\exp(V(i))\leq \sum_{i=0}^{+\infty}\exp(-\d i)
\leq \frac{1}{1-e^{-\d}}.
\end{equation}
Furthermore,
\begin{equation*}
\forall \c\leq i\leq m_n,\qquad V(i) \leq  -\d\c+\ddd\leq
-\frac{\e'}{2}\log n+\d.
\end{equation*}
Then, for all large $n$,
\begin{equation}\label{minorationA2}
0  \leq \sum\limits_{i=\c+1}^{m_n-1}\exp(V(i)) \leq 2\cccc
n^{-\e'/2}e^{\d} \leq 1.
\end{equation}
Accordingly (see Zeitouni \cite{Z1} p. 196),
\begin{equation*} \po(\tau_{m_n}<\tau_{-1})
=  \frac{\exp(V(-1))}{\sum_{k=-1}^{m_n-1}\exp(V(k))}
 \geq
\frac{\frac{\e_0}{1-\e_0}}{\frac{1-\e_0}{\e_0}+\frac{1}{1-e^{-\d}}+1}:=C_7>0.
\end{equation*}
\hfill $\Box$

We denote by $\po^x$ and $\eo^x$ the probability and expectation
of $(S_n)_{n\in\N}$, starting at site $x$ and conditioned on the
environment $\o$. We have (see Zeitouni, \cite{Z1}, p. 250)
\begin{fact}
\label{fait} If $a<x<b$,
\begin{equation}
\eo^x(\tau_a    \wedge\tau_b)\leq
\sum_{k=x}^{b-1}\sum_{\ell=a}^{k}\frac{\exp[V(k)-V(\ell)]}{\o_{\ell}}.
\end{equation}
\end{fact}

\bigskip
We can now give an upper bound for the hitting time of $m_n$ if
the RWRE hits $m_n$ before $-1$:
\begin{lemma}\label{lemma34}
There exists a constant $C_8>0$ such that
\begin{equation*}
\forall\o\in \EA,\qquad \po(\tau_{m_n}<\tau_{-1}\textnormal{ and }
\tau_{m_n}\leq n^{4\e'})\geq C_8.
\end{equation*}
\end{lemma}

\bigskip
\noindent {\bf Proof:} According to Fact \ref{fait} and
\eqref{mn}, we obtain for $\o\in \EA$,
\begin{eqnarray*}
\eo(\tb\wedge\tau_{m_n}) & \leq &
\sum_{k=0}^{m_n-1}\sum_{\ell=-1}^{k}
\frac{\exp[V(k)-V(\ell)]}{\o_{\ell}}\\
& \leq & \frac{1}{\e_0} (m_n+1)^2\exp(\cc+2\ddd)\\
& \leq & \frac{1}{\e_0}[(1+\e')(\log n)^2+1]^2 n^{2\e'}\leq
n^{3\e'}.
\end{eqnarray*}
Now, by Chebyshev's inequality,
\begin{eqnarray*}
\po(\tb\wedge\tau_{m_n}\geq n^{4\e'}) \leq
n^{-4\e'}\eo(\tb\wedge\tau_{m_n})\leq n^{-\e'}.
\end{eqnarray*}
As a consequence, recalling Lemma \ref{lemma32},
\begin{eqnarray*}
\po(\tau_{m_n}<\tau_{-1}\text{ and } \tau_{m_n}\leq n^{4\e'})
& = & \po(\tau_{m_n}<\tau_{-1})-\po(n^{4\e'}<\tau_{m_n}<\tb)\\
& \geq & C_7-n^{-\e'}\geq C_7/2:=C_8
\end{eqnarray*}
for $n$ large enough. \hfill$\Box$


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


\subsection{Leaving the valley}\label{sub32}
First, we give a majoration of the exit time from the valley
$(0,m_n,b_n)$.

\begin{lemma}\label{major} There exists a constant $C_9>0$,
depending only on $\eta$ and $\e$, such that
\begin{equation*}
\forall\o\in \EA,\qquad \po^{m_n}(\tau_{-1}\leq n^{1-5\e'})\geq
C_9.
\end{equation*}
\end{lemma}

\bigskip
\bigskip
\noindent {\bf Proof:} Let $\o\in E_n$. The probability to leave
the valley $(0,m_n,b_n)$ on the left is
\begin{eqnarray}
\po^{m_n}(\tau_{-1}<\tau_{b_n}) & = &
\frac{1}{\frac{\sum_{k=-1}^{m_n-1}\exp(V(k))}
{\sum_{k=m_n}^{b_n-1}\exp(V(k))}+1} \nonumber\\
& \geq &
\frac{1}{\left(1+\frac{1}{1-e^{-\d}}+\frac{1-\e_0}{\e_0}\right)\frac{1-\e_0}{\e_0}+1}
:=2C_9,\label{e1p10}
\end{eqnarray}
due to \eqref{minorationA1} and \eqref{minorationA2}, and since
$\exp(V(b_n))\geq 1$.

Moreover, Fact \ref{fait} gives (by symmetry), recalling
\eqref{mn},
\begin{eqnarray*}
\eo^{m_n}(\tau_{b_n}\wedge\tb) & \leq & \sum_{k=0}^{m_n}\
\sum_{\ell=k}^{b_n}
\frac{\exp[V(k-1)-V(\ell-1)]}{\e_0}\\
& \leq & \frac{1}{\e_0} (b_n+1)^2\frac{1-\e_0}{\e_0}\exp[V(0)-V(m_n)]\\[1mm]
& \leq & (3\log^2 n)^2 \exp[(1-6\e') \log n]\e_0^{-2}\\
& \leq & n^{1-11\e'/2}
\end{eqnarray*}
for $n$ large enough. Then Chebyshev's inequality yields
\begin{equation*}
\po^{m_n}(n^{1-5\e'}<\tau_{b_n}\wedge\tb)\leq n^{-\e'/2}.
\end{equation*}
Consequently, for all environment $\o\in \EA$, recalling
\eqref{e1p10},
\begin{eqnarray*}
\po^{m_n}(\tau_{-1}  \leq  n^{1-5\e'})& \geq &
\po^{m_n}(\tau_{-1}\leq \tau_{b_n} \text{ and } \tau_{-1}\leq
n^{1-5\e'})\\
& \geq & 2C_9-n^{-\e'/2}\geq C_9
\end{eqnarray*}
for $n$ large enough. $\hfill\Box$
\bigskip

Now we give a lower bound for $\tau_{-1}$.

\begin{lemma}\label{minor}
We have,
\begin{equation*}
\inf_{\o\in
E_n}\po^{m_n}(\tau_{-1}>n^{1-10\e'})\underset{n\to\infty}
{\longrightarrow}1.
\end{equation*}
\end{lemma}

\bigskip
\noindent {\bf Proof:} Let $\o\in E_n$. To establish Lemma
\ref{minor}, we use another argument of Sinai's proof.
%(see Zeitouni \cite{Z1}, p. 63)
When the RWRE is located at $m_n-1$, the probability that it hits
$-1$ before going to $m_n$ is
\begin{eqnarray*}
\po^{m_n-1}(\tau_{-1}<\tau_{m_n})
& = & \frac{\exp (V(m_n-1))}{\sum_{k=-1}^{m_n-1}\exp (V(k))}\\
&  \leq & \exp(V(m_n-1)) \\
&  \leq &
\left(\frac{1-\e_0}{\e_0}\right)\frac{e^{\d}}{n^{1-9\e'}},
\end{eqnarray*}
due to \eqref{mn}. Similarly, we have
\begin{equation*}
\po^{m_n+1}(\tau_{b_n}<\tau_{m_n}) \leq
\left(\frac{1-\e_0}{\e_0}\right)\frac{e^{\d}}{n^{1-9\e'}}:=
\frac{C_{10}}{n^{1-9\e'}} .
\end{equation*}
As the RWRE is recurrent, we can consider the $\lfloor
n^{1-10\e'}\rfloor $ first excursions away from $m_n$, which are
independent under $\po$. More precisely, let us define recursively
\begin{equation*}
\left\{
\begin{array}{lcl}
\tau_{m_n}^{(1)} & := & \tau_{m_n},\\[2mm]
\tau_{m_n}^{(k+1)} & := & \inf\{\ell>\tau_{m_n}^{(k)},\quad
S_{\ell}=m_n\}, \quad k\geq 1,
\end{array}
\right.
\end{equation*}
and consider the set
\begin{equation*}
\Gg:=\left\{ \forall 1\leq k\leq \lfloor n^{1-10\e'}\rfloor,\quad
\tau_{m_n}^{(k)}<\tau_{-1}\wedge\tau_{b_n}\right\}.
\end{equation*}
We obtain
\begin{eqnarray*}
\po^{m_n}(\Gg^c)\leq \lfloor
n^{1-10\e'}\rfloor\po^{m_n\pm1}(\tb\wedge\tau_{b_n}<\tau_{m_n})
\leq C_{10}n^{-\e'}.
\end{eqnarray*}
Now, on $\Gg$, the RWRE $(S_i)_{i\geq \tau_{m_n}}$ stays in
$[0,b_n]$ during the first $\lfloor n^{1-10\e'}\rfloor$ excursions
away from $m_n$, hence $\tau_{-1}>n^{1-10\e'}$. Therefore,
\begin{equation*}
\forall\o\in E_n,\qquad \po^{m_n}(\tau_{-1}>n^{1-10\e'})\geq
\po^{m_n}(\Gg)\geq  1-C_{10}n^{-\e'}.
\end{equation*}
 $\hfill\Box$

Combining Lemmas \ref{major} and \ref{minor}, we get
\begin{equation*}
\forall \o\in E_n,\qquad \po^{m_n}(n^{1-10\e'}<\tau_{-1}\leq
n^{1-5\e'})\geq \frac{C_9}{2}
\end{equation*}
for $n$ large enough. Recalling Lemma \ref{lemma34}, this ends the
proof of Lemma \ref{encadrement}.\hfill$\Box$



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                                %
%                          SECTION 4                             %
%                                                                %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%




\mysection{Proofs of Theorem \ref{th1} and Proposition
\ref{moments}} \label{simpl}

In this section, we use the results of the previous sections to
prove Theorem~\ref{th1} and Proposition~\ref{moments}.

\subsection{Proof of Proposition \ref{moments}}
Let $\a\in\R_+^*$ and
\begin{equation*}
M_{\a}:=\sup_{x\in\R_+}(x^{\a}e^{-x})\in(0,+\infty).
\end{equation*}
Then,
\begin{equation}\label{minoreq}
\forall r>0, \qquad \E[\tb^{\alpha} \exp(-r\tb)]\leq M_\alpha
r^{-\alpha}.
\end{equation}

Now we give a lower bound for $\eo[\tb^\alpha \exp(-r \tb)]$. For
any $0<a<1$ and any $\o$,
\begin{equation*}
\eo[\tb^{\a}\exp(-\tb/n)]\geq e^{-1} n^{\a a} \po(n^a\leq\tb\leq
n).\end{equation*}

\noindent Thus, by Lemma \ref{encadrement}, for any $\e>0$, taking
$a=1-10\e'=1-10\d\e$,
\begin{equation*}
\forall \o\in \EA,\qquad
 \eo[\tb^{\alpha} \exp(-\tb/n)]\geq
C_6 e^{-1}n^{\a(1-10\d\e)}.
\end{equation*}
Integrating this inequality on $\EA$, and in view of Lemma
\ref{en}, we get, for all large $n$,
\begin{equation*}
\E[\tb^{\alpha} \exp(-\tb/n)]\geq C_6 C_0
e^{-1}n^{\a(1-10\d\e)-\theta\e}.
\end{equation*}
Since $\e>0$ can be arbitrary small, this, together with
\eqref{minoreq}, yields
\begin{equation*}
\E(\tau_{-1}^\a
e^{-r\tau_{-1}})=\left(\frac{1}{r}\right)^{\a+o(1)},\qquad r\to
0^+.
\end{equation*}
By symmetry, we can replace $\tau_1$ by $\tau_{-1}$, which gives
Proposition \ref{moments}.\hfill$\Box$


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


\subsection{Proof of Theorem \ref{th1}}
\indent It is known (see den Hollander \cite{H3}, p. 80), that
(\ref{pb1}) is equivalent to
\begin{equation}
\lim_{r\ra 0^-} \frac{[\log \l]''(r)}{\{[\log \l]'(r)\}^3}=0,
\label{pb2}
\end{equation}
where
\begin{equation}
\log\lambda(r)=\E[\log\eo(e^{r\tau_{1}})].
\end{equation}
Note that
\begin{eqnarray*}
f(r) & := &  \frac{[\log \l]''(r)}{\{[\log \l]'(r)\}^3}\\
& \leq & \frac{\E\left(\frac{\eo(\tau_1^2
e^{r\tau_1})}{\eo(e^{r\tau_1})}\right)}
{\left[\E\left(\frac{\eo(\tau_1 e^{r
\tau_1})}{\eo(e^{r\tau_1})}\right)\right]^3}:=g(r).
\end{eqnarray*}
Moreover, due to assumption (\ref{lip}), we have, for all $-1<r<0$
and for all environment $\o$,
\begin{equation*}
\e_0 e^{-1}\leq\o_0e^{r}\leq\eo(\exp(r\tau_1))\leq 1.
\end{equation*}
As a consequence, for $-1<r<0$,
\begin{equation*}
g(r)\leq \frac{e}{\e_0}
\frac{\E[\tau_1^2\exp(r\tau_1)]}{\left\{\E[\tau_1
\exp(r\tau_1)]\right\}^3}:=h(r).
\end{equation*}
Furthermore, $f(r)\geq 0$ (by the Cauchy--Schwarz inequality).
%, or by convexity of~$\Iq$
 Now, according to Proposition \ref{moments},

$$h(r) = |r|^{1+o(1)}\underset{r\to0^-}{\longrightarrow 0},$$
which proves \eqref{pb2} and thus Theorem \ref{th1}.\hfill$\Box$









%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                               %
%                      SECTION 5                                %
%                                                               %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{Comparison between rate functions}\label{sectbrox}

In this section we consider the diffusion $X$ in the random
potential $W_{\k}$ and assume $\k>1$. In this case,
$v_{\k}=(\k-1)/4$. We know (see Taleb \cite{Ta}) that the rate
function $\jk$ of quenched large deviations for $X$ can be written
as $\jk(x)=x \ik(1/x)$ for $x> 0$, where
\begin{equation}\label{quotient}
\ik(u)=\sup_{\l\geq 0}(\gk(\l)-\l u),
\end{equation}
and $\gk$ can be expressed in terms of modified Bessel functions
(see \eqref{bessel} below).

Let
\begin{equation*}
\fv(\l):=\sqrt{2\l+{v_{\k}}^2}-{v_{\k}}.
\end{equation*}
We first show that $\gk(\l)<\fv(\l)$ for large $\l$. Then we use a
differential equation satisfied by $\gk$ to prove that this
inequality is true on $\R^*_+$. Finally, we prove Theorem
\ref{comptaux} and Proposition \ref{prop14}.


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


\subsection{Study in the neighbourhood of  $+\infty$}
According to Taleb (we mention that in Taleb~\cite{Ta}, p. 1178,
the expression $F_{\k}(\l)$ should be
$2(2\l)^{\k/2}K_{\k}[4\sqrt{2\l}]$, see for instance Magnus et
al., \cite{MOS} p. 85; this misprint has no consequence on the
results of \cite{Ta}), we have
\begin{equation}\label{bessel}
\forall \l\geq 0,\qquad
\gk(\l)=\sqrt{2\l}\frac{K_{\k-1}(4\sqrt{2\l})}{K_{\k}(4\sqrt{2\l})}.
\end{equation}
Using the ``series of the Hankel type'' (see Magnus et al.
\cite{MOS}, p. 139), we obtain
\begin{equation}\label{dl}
\gk(\l)=\sqrt{2\l}-\frac{1}{4}\left(\k-\frac{1}{2}\right)+O\left(\frac{1}{\sqrt{\l}}\right)
\qquad \l\to+\infty.
\end{equation}
This yields
\begin{equation*}
\gk(\l)-\fv(\l)\underset{\l\to+\infty}{\longrightarrow}-\frac{1}{8}.
\end{equation*}
Consequently, there exists $B>0$, such that
\begin{equation}
\forall\l\geq B,\qquad  \gk(\l)<\fv(\l). \label{vois}
\end{equation}


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


\subsection{Using a differential equation}
According to Taleb \cite{Ta}, $\gk$ is a solution of the
differential equation $x y'-2y^2-\k y=-4x$ on $(0,+\infty)$. It is
natural to introduce
\begin{equation}\label{eqdefA}
A(x):=x\fv'(x)-2\fv^2(x)-\k\fv(x)+4x=\frac{-x-v_{\k}^2+v_{\k}
\sqrt{2x+v_{\k}^2}}{\sqrt{2x+v_{\k}^2}}.
\end{equation}
In particular, $A(x)<0$ for all $x>0$.

Let us consider the set
\begin{equation*} E:=\{x>0,\quad  \gk(x)\geq
\fv(x)\}.
\end{equation*}
We prove that $E=\emptyset$. Indeed, let us assume that $E\neq
\emptyset$. According to~\eqref{vois}, $E\cap
[B,+\infty)=\emptyset$. Consequently, $E$ would have a supremum
$x_0\in (0,B]$. By continuity, $\gk(x_0)=\fv(x_0)$. Now,
\eqref{eqdefA} would yield
\begin{eqnarray*}
\fv'(x_0) & = & \frac{1}{x_0}[A(x_0)+2\fv^2(x_0)+\k\fv(x_0)-4x_0]\\
          & = & \frac{1}{x_0}[A(x_0)+2\gk^2(x_0)+\k\gk(x_0)-4x_0] \\
          & = & \frac{A(x_0)}{x_0}+\gk'(x_0)<\gk'(x_0).
\end{eqnarray*}
Consequently, there would exist an $\e>0$ such that
\begin{equation*}
\forall x\in [x_0,x_0+\e],\qquad \fv(x)<\gk(x).
\end{equation*}
Therefore, $[x_0,x_0+\e]\subset E$, which contradicts $x_0=\sup
E$. Hence $E=\emptyset$, which means that

\begin{equation} \forall \l> 0,\qquad  \gk(\l)<\fv(\l). \label{Q3}
\end{equation}


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


\subsection{Proofs of Theorem \ref{comptaux} and Proposition
\ref{prop14}}


It is easily seen that
\begin{equation*} \forall \l\geq 0,\quad
\inf_{0< u< \frac{1}{v_{\k}}} \left\{ \l u+ \frac{u}{2}
\left(\frac{1}{u}-{v_{\k}}\right)^2\right\}
=\sqrt{2\l+{v_{_k}}^2}-{v_{\k}}= \fv(\l). \label{Q4}
\end{equation*}
Thus \eqref{Q3} yields
\begin{equation}
\forall 0<u<\frac{1}{v_{\k}},\ \forall \l> 0,\qquad  \gk(\l)-\l u<
\frac{u}{2}\left(\frac{1}{u}-v_{\k}\right)^2.\label{Q2}
\end{equation}
Notice that \eqref{Q2} remains true for $\l=0$ since $\gk(0)=0$.
Now, fix $u\in(0,1/v_{\k})$. Recalling \eqref{dl}, it follows that
\begin{equation*}
\gk(\l)-\l u\underset{\l\to+\infty}{\longrightarrow}-\infty.
\end{equation*}
As the function $\l\mapsto [\gk(\l)-\l u]$ is continuous on
$\R_+$, it has a maximum on, say,  $\l_u\in \R_+$.
%We have on the one hand $\gk(0)=0$, and on the other hand
%$\gk'(\l)$ tends to $0$ on $+\infty$ (see Taleb, p1189). Hence,
%$\frac{\partial}{\partial \l} (\gk(\l)-\l u)$ tends to $-u<0$ on
%$+\infty$. Consequently, the function $\l\mapsto \gk(\l)-\l u$ is
%strictly negative for $\l$ large enough, and then has a  maximum
%on a real number $\l_u$.
Hence, by \eqref{Q2},
\begin{equation*}
\sup_{\l\geq 0} (\gk(\l)-\l u )=\gk(\l_u)-\l_u u
<\frac{u}{2}\left(\frac{1}{u}-v_{\k}\right)^2,
\end{equation*}
which can be written as, recalling \eqref{quotient}:
\begin{equation*}
\forall 0<u<\frac{1}{v_{\k}},\qquad
\ik(u)<\frac{u}{2}\left(\frac{1}{u}-v_{\k}\right)^2.
\end{equation*}
This is equivalent to
\begin{equation*}
\forall x>v_{\k},\qquad
\jk(x)=x\ik\left(\frac{1}{x}\right)<J_{v_{\k}}^B(x) =
\frac{1}{2}(x-v_{\k})^2,
\end{equation*}
proving Theorem \ref{comptaux}.

\bigskip
We notice that \eqref{Q3} can be written in terms of modified
Bessel functions, using \eqref{bessel}, which gives Proposition
\ref{prop14}.


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


\subsection{Remarks}

Recall that the rate function of large deviations of the standard
Brownian motion is $x\mapsto x^2/2$. By the same arguments as in
the case $\k>1$, we obtain for the transient case with zero speed
($0<\k\leq1$),

\begin{prop} (zero speed case),\\
\begin{tabular}{ll}
If $\k\in (0,1/2)$, &then $\forall x>0,\ \jk(x)>x^2/2$;\\
If $\k=1/2$, &then $\forall x>0,\ \jk(x)=x^2/2$;\\
If $\k\in (1/2,1]$, &then $\forall x>0,\ \jk(x)<x^2/2$.
\end{tabular}
\end{prop}

\bigskip
(The case $\k=1/2$ was obtained by Taleb, \cite{Ta}).

 We also notice that Proposition \ref{prop14} together with the formula
$K_{\nu-1}(z)-K_{\nu+1}(z)=-\frac{2\nu}{z}K_{\nu}(z)$ also give a
lower bound for $K_\nu/K_{\nu+1}$:
\begin{equation*}
\forall\nu>0, \forall y>0, \qquad \frac{K_{\nu}(y)}{K_{\nu+1}(y)}>
\frac{1}{y}\left[\frac{y^2}{\sqrt{y^2+(\nu+1)^2}-(\nu+1)}-2(\nu+1)\right].
\end{equation*}


\bigskip
\bigskip
\noindent {\bf \Large Acknowledgements}

 I am grateful to Zhan Shi
for several helpful discussions.




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                       %
%                        A MODIFIER                     %
%                                                       %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%





%\subsection{A Modifier}
%\begin{itemize}
%\item Changer les notations des $\ccc$, $E_{i,n}$,etc.
%\item verifier s'il ne faudrait pas conditionner par
%$\tau_{m_n}<\tb$, etc.
%\item refaire une bonne fois pour toutes les calculs a la
%Sinai, la def de V, les probas de sortie, etc.
%\item reverifier la dependance des constantes par rapport a $\eta$
%\item faire un dessin ?
%et $\e_0$ uniquement.
%%\item donner les reultats de grandes deviations dont on a besoin.
%%\item introduire le modele de Brox.
%\end{itemize}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                         %
%                     BIBLIO                              %
%                                                         %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%




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%
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%    {\it Probab. Theory Related Fields} {\bf 98}, (1994) 91--112.
%
%\bibitem{B1}
%    Biggins, J.D.:
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%    {\it J. Appl. Probab.} {\bf 14}, (1977) 630--636.
%

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%\bibitem{GD}
%    Greven, A. and den Hollander, F.:
%    Branching random walk in random environment: phase transitions
%              for local and global growth rates.
%    {\it Probab. Theory Related Fields} {\bf 91}, (1992) 195--249.%
%
\bibitem{GD2}
    Greven, A. and den Hollander, F.:
    Large deviations for a random walk in random environment.
    {\it Ann. Probab.} {\bf 22}, (1994) 1381--1428.

%\bibitem{H2}
%    Hammersley, J.M.:
%    Postulates for subadditive processes.
%    {\it Ann. Probab.} {\bf 2}, (1974) 652--680.%
%

\bibitem{H3} den Hollander, F.: {\it Large Deviations}. Fields Institute
Monographs, {\bf 14}, AMS, Providence, RI, 2000.

\bibitem{Hug} Hughes, B.D.: {\it Random Walks and Random Environment,
vol. II: Random Environments}. Oxford Science Publications,
Oxford, 1996.


\bibitem{KT1} Kawazu, K. and Tanaka, H.: A diffusion process in a
Brownian environment with drift. {\it J. Math. Soc. Japan} {\bf
49}, (1997) 189--211.

\bibitem{KMT} Koml{\'o}s, J., Major, P. and Tusn{\'a}dy, G.: An
approximation of partial sums of independent {${\rm RV}$}'s
              and the sample {${\rm DF}$. {I}}.
              {\it Z. Wahrsch. Verw. Gebiete}  {\bf 32}, (1975) 111--131.

\bibitem{MOS}
Magnus, W., Oberhettinger, F. and Soni, R.P.: {\it Formulas and
Theorems for the Special Functions of Mathematical Physics}, Third
enlarged edition. Springer, New York, 1966.



\bibitem{S2}
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     Synth\`eses, {\bf 12}, (2001) 53--74.

\bibitem{S3}
    Sinai, Ya. G.:
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\bibitem{S1}
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\bibitem{Ta} Taleb, M.: Large deviations for a {B}rownian motion in a drifted
              {B}rownian potential,  {\it Ann. Probab.} {\bf 29}, (2001) 1173--1204.

\bibitem{Z1}
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    Lectures notes on random walks in random environment.
    In: {\it Lect. Notes Math.}, Springer, Berlin,  1837, pp.
    193--312}, (2004).

     %%% end of "baselineskip=14pt"

\end{thebibliography}




\end{document}

math0502316.extracted_bib
\begin{thebibliography}{99}

{\baselineskip=14pt

%\bibitem{AN1}
%    Athreya, K.B. and Ney, P.E.:
%    {\it Branching Processes}.
%    Springer, New York, 1972.
%
%\bibitem{BP1}
%    Benjamini, I. and Peres, Y.:
%    Tree-indexed random walks on groups and first passage percolation.
%    {\it Probab. Theory Related Fields} {\bf 98}, (1994) 91--112.
%
%\bibitem{B1}
%    Biggins, J.D.:
%    Chernoff's theorem in the branching random walk.
%    {\it J. Appl. Probab.} {\bf 14}, (1977) 630--636.
%

\bibitem{B3} Brox, Th.: A one-dimensional diffusion process in a {W}iener
medium. {\it Ann. Probab.} {\bf 14}, (1986) 1206--1218.

\bibitem{CGZ}
    Comets, F., Gantert, N. and Zeitouni, O.:
    Quenched, annealed and functional large deviations for
    one-dimensional random walk in random environment.
    {\it Probab. Th. Related Fields} {\bf 118}, (2000) 65--114.

%\bibitem{GD}
%    Greven, A. and den Hollander, F.:
%    Branching random walk in random environment: phase transitions
%              for local and global growth rates.
%    {\it Probab. Theory Related Fields} {\bf 91}, (1992) 195--249.%
%
\bibitem{GD2}
    Greven, A. and den Hollander, F.:
    Large deviations for a random walk in random environment.
    {\it Ann. Probab.} {\bf 22}, (1994) 1381--1428.

%\bibitem{H2}
%    Hammersley, J.M.:
%    Postulates for subadditive processes.
%    {\it Ann. Probab.} {\bf 2}, (1974) 652--680.%
%

\bibitem{H3} den Hollander, F.: {\it Large Deviations}. Fields Institute
Monographs, {\bf 14}, AMS, Providence, RI, 2000.

\bibitem{Hug} Hughes, B.D.: {\it Random Walks and Random Environment,
vol. II: Random Environments}. Oxford Science Publications,
Oxford, 1996.


\bibitem{KT1} Kawazu, K. and Tanaka, H.: A diffusion process in a
Brownian environment with drift. {\it J. Math. Soc. Japan} {\bf
49}, (1997) 189--211.

\bibitem{KMT} Koml{\'o}s, J., Major, P. and Tusn{\'a}dy, G.: An
approximation of partial sums of independent {${\rm RV}$}'s
              and the sample {${\rm DF}$. {I}}.
              {\it Z. Wahrsch. Verw. Gebiete}  {\bf 32}, (1975) 111--131.

\bibitem{MOS}
Magnus, W., Oberhettinger, F. and Soni, R.P.: {\it Formulas and
Theorems for the Special Functions of Mathematical Physics}, Third
enlarged edition. Springer, New York, 1966.



\bibitem{S2}
     Shi, Z.:
     Sinai's walk via stochastic calculus. Panoramas et
     Synth\`eses, {\bf 12}, (2001) 53--74.

\bibitem{S3}
    Sinai, Ya. G.:
    The limiting behavior of a one-dimensional random walk in a random
    medium, (English translation), {\it Th. Probab. Appl.} {\bf 27},
    (1982) 256--268.

\bibitem{S1}
    Solomon, F.:
    Random walks in a random environment.
    {\it Ann. Probab.} {\bf 3}, (1975) 1--31.%

\bibitem{Ta} Taleb, M.: Large deviations for a {B}rownian motion in a drifted
              {B}rownian potential,  {\it Ann. Probab.} {\bf 29}, (2001) 1173--1204.

\bibitem{Z1}
    Zeitouni, O.:
    Lectures notes on random walks in random environment.
    In: {\it Lect. Notes Math.}, Springer, Berlin,  1837, pp.
    193--312}, (2004).

     %%% end of "baselineskip=14pt"

\end{thebibliography}