# Bounds on the Quenched Pressure and Main Eigenvalue of the Ruelle Operator
for Brownian Type Potentials

$(\mathrm{L.C.})^{2}$

###### Abstract

In this paper we consider a random potential derived from the Brownian motion.
We obtain upper and lower bounds for the expected value of the main eigenvalue
of the associated Ruelle operator and for its quenched topological pressure.
We also exhibit an isomorphism between the space $C(\Omega)$ endowed with its
standard norm and a proper closed subspace of the Skorokhod space which is
used to obtain a stochastic functional equation for the main eigenvalue and
for its associated eigenfunction.

Key-words: Brownian motion, One-dimensional lattice, Quenched Pressure, Random
potentials, Ruelle operator, Thermodynamic formalism.
MSC2010: 37A50, 37A60, 60J65, 82B05, 97K50.

##  1 Introduction

The basic idea of the Ruelle operator is based on the transfer matrix method
introduced by Kramers and Wannier and independently by Montroll to explicitly
compute partition functions in Statistical Mechanics. This operator was
introduced in 1968 by David Ruelle [Rue68] and was used to describe local to
global properties of systems with infinite range interactions. Among other
things, Ruelle used this operator to prove the existence of a unique DLR-Gibbs
measure for a lattice gas system depending on infinitely many coordinates. The
acronym DLR stands for Dobrushin, Lanford and Ruelle.

With the advent of the Markov partitions due to Y. Sinai, remarkable
applications of this operator to Hyperbolic dynamical systems on compact
manifolds were further obtained by Ruelle, Sinai and Bowen, see [Rue68, Sin72,
Bow08]. Since its creation, this operator remains to have a great influence in
many fields of pure and applied Mathematics. It is a powerful tool to study
topological dynamics, invariant measures for Anosov flows, statistical
mechanics in one-dimension, meromorphy of the Zelberg and Ruelle dynamical
zeta functions, multifractal analysis, conformal dynamics in one dimension and
fractal dimensions of horseshoes, just to name a few. For those topics, we
recommend, respectively, [Bar11, Bow08, Bow79, Mañ90, MM83, PP90, Rue02,
Sin72] and references therein.

In order to describe our contributions, we shall start by briefly recalling
the classical setting of the Ruelle operator. A comprehensive exposition of
this subject can be found in [Bal00, Bow08, PP90].

We now introduce the state space and the dynamics that will guide the
following discussions. In this paper, the state space will be the Bernoulli
space $\Omega=\\{0,1\\}^{\mathbb{N}}$ and the dynamics is given by
$\sigma:\Omega\to\Omega$, the left shift map. When topological concept are
concerned, the state space $\Omega$ is regarded as a metric space where the
distance between points $x$ and $y\in\Omega$ is given by

$d(x,y)=\frac{1}{2^{N}},\text{ where }N=\inf\\{i\in\mathbb{N}:x_{i}\neq
y_{i}\\}.$

We shall remark that $\Omega$ is a compact metric space when endowed with this
metric. All of the results presented in this paper can be naturally extended
to $\\{0,1,\dots,m-1\\}^{\mathbb{N}}$, $m\geq 2$, but for the sake of
simplicity, we restrain ourselves to the binary case. A summary of the changes
required to study this more general case are presented in Section 7.

Back to the case $\Omega=\\{0,1\\}^{\mathbb{N}}$, the set of all real
continuous functions defined over $\Omega$ will be denoted by $C(\Omega)$ and
the Ruelle transfer operator, or simply the Ruelle operator, associated to a
continuous function $f:\Omega\to\mathbb{R}$ will be denoted by
$\mathscr{L}_{f}$. This operator acts on the space of all continuous functions
sending $\varphi$ to $\mathscr{L}_{f}(\varphi)$ which is defined for any
$x\in\Omega$ by the following expression

$\mathscr{L}_{f}(\varphi)(x)=\sum_{a=0,1}\exp(f(ax))\varphi(ax),$

where $ax\equiv(a,x_{1},x_{2},\ldots)$. The function $f$ is usually called
potential.

In this work a major role will be played by $C^{\gamma}(\Omega)$, the space of
$\gamma$-Hölder continuous functions, with the Hölder exponent $\gamma$
satisfying $0<\gamma<1/2$. This space is the set of all functions
$f:\Omega\to\mathbb{R}$ satisfying

$\mathrm{Hol}(f)=\sup_{x,y\in\Omega:\,x\neq
y}\dfrac{|f(x)-f(y)|}{d(x,y)^{\gamma}}<+\infty.$

Equipped with its standard norm $\|\cdot\|_{\gamma}$ given by
$C^{\gamma}(\Omega)\ni
f\mapsto\|f\|_{\gamma}\equiv\|f\|_{\infty}+\mathrm{Hol}(f)$, the space
$C^{\gamma}(\Omega)$ is a Banach space, see [PP90] for more details.

Furthermore, we present the suitable version of the Ruelle-Perron-Fröbenius
Theorem that will be required for our discussions. See [Bal00, Bow08, PP90]
for a proof.

######  Theorem A (Ruelle-Perron-Fröbenius, RPF).

If $f$ is a potential in $C^{\gamma}(\Omega)$ for some $0<\gamma<1$, then
$\mathscr{L}_{f}:C^{\gamma}(\Omega)\to C^{\gamma}(\Omega)$ have a simple
positive eigenvalue of maximal modulus $\lambda_{f}$ with a corresponding
strictly positive eigenfunction $h_{f}$ and a unique Borel probability measure
$\nu_{f}$ on $\Omega$ such that,

  * i)

the remainder of the spectrum of $\mathscr{L}_{f}:C^{\gamma}(\Omega)\to
C^{\gamma}(\Omega)$ is contained in a disc of radius strictly smaller than
$\lambda_{f}$;

  * ii)

the probability measure $\nu_{f}$ satisfies
$\mathscr{L}^{*}_{f}(\nu_{f})=\lambda_{f}\nu_{f}$, where $\mathscr{L}^{*}_{f}$
denotes the dual of the Ruelle operator;

  * iii)

for all continuous functions $\varphi\in C(\Omega)$ we have

$\lim_{n\to\infty}\left\|\frac{\mathscr{L}^{n}_{f}(\varphi)}{\lambda_{f}^{n}}-h_{f}\int_{\Omega}\varphi\,d\nu_{f}\right\|_{\infty}=0.$

Whereas in the classical theory of the Ruelle operator, $f$ is some fixed
Hölder, Walters [Wal07, Bou01] or Bowen [Bow08, Wal01] potential, here we are
interested in obtaining almost certain results for random potentials.

######  Definition 1 (Random Potentials).

Let $(\Psi,\mathcal{F},\mathbb{P})$ be a complete probability space. A random
potential is a map $X:\Psi\times\Omega\to\mathbb{R}$ satisfying for any fixed
$x\in\Omega$ that $\psi\mapsto X(\psi,x)$ is $\mathcal{F}$-measurable. A
random potential $X$ is said to be a continuous random potential if $x\mapsto
X(\psi,x)\in C(\Omega)$ for $\mathbb{P}$-almost all $\psi\in\Psi$.

Although some of our results could be stated for a large class of random
potentials, we focus on the random potentials of Brownian type (see the next
section for the definition of such potentials).

When moving into the realm of random potentials the basic theory of the
Thermodynamic Formalism needs to be reconstructed from scratch. In [FS88]
random potentials were considered and this is the setting of our work. Latter,
more general settings were introduced. The theory was extended to cover
countable random Markov subshifts of finite type, allowing randomness to be
considered also in the adjacency matrix, we refer the reader to [BG95, Bog96,
DKS08, FS88, GK00, Kif08, KL06, MSU11], for this more general setting.

As mentioned above the randomness here is considered exclusively in the
potential. Since this is simpler setting, for the reader’s convenience we
proved here, in full details for the Brownian potential, that important
quantities such as the spectral radius, the topological pressure and the main
eigenvalue of the Ruelle operator can be associated to random variables with
finite first moment in Wiener spaces. At the end of Section 2 we explain how
to obtain similar results for Brownian potential using the general setting
considered in [Kif08, KL06].

Working with random potentials, we are led to consider questions similar in
spirit to the ones we ask when studying disordered systems in Statistical
Mechanics, for instance the existence of annealed and quenched free energy and
pressure, respectively, see [MPV87]. Our approach provides a new way to obtain
the existence and finiteness of quenched topological pressure defined by

$P^{\mathrm{quenched}}(X)\equiv\mathbb{E}\left[\sup_{\mu\in\mathscr{P}_{\sigma}(\Omega)}\left\\{h(\mu)+\int_{\Omega}X\,d\mu\right\\}\right],$

(note that the existence of the rhs above is itself a deep question) as well
as the identity
$P^{\mathrm{quenched}}(X)=\mathbb{E}\left[\lim_{n\to\infty}n^{-1}\log\mathscr{L}^{n}_{X}(1)(\sigma^{n}(x))\right],\
\forall x\in\Omega$ for a large class of continuous random potential $X$ not
invoking the classical replica trick. On the other hand, despite the progress
achieved with this method, it has the disadvantage of not being able to
exhibit explicitly these quantities.

Our main results are Theorems 1, 2, the expression (3), and the bounds for the
quenched pressure in Section 6.

The paper is organized as follows. In Section 2 we introduce the so-called
Brownian potential and we prove that the main eigenvalue for the random Ruelle
operator associated to the Brownian potential almost surely exists. We also
prove in this section that phase transition for this random potential, in the
sense of the Definition 2, is almost surely absent. In Section 3 and 4 we
prove a Banach isomorphism between $C(\Omega)$ and a certain closed subspace
of the Skorokhod space, denoted by $\mathscr{D}[0,1]$. With the help of this
isomorphism we obtain a stochastic functional equation satisfied by both the
main eigenvalue and its associated eigenfunction. Then a representation of the
main eigenvalue in terms of the Brownian motion and a limit of a quotient
between linear combinations of log-normal random variables are also obtained.
Section 5 is devoted to obtain upper and lower bounds for the expected value
of the main eigenvalue and the main idea is the use of the reflection
principle of the Brownian motion to overcome the complex combinatorial problem
arising from the representation obtained in Section 4. In Section 6 we use the
results of Sections 2-5 to give a proof of existence and finiteness of the
quenched topological pressure. In Section 7 we briefly discuss some natural
extensions of our results, as to more general alphabets and to other random
potentials. We also explain the difficulties in improving our estimates and
obtaining the existence of the annealed pressure.

##  2 Brownian Type Potentials

Let $\\{B_{t}:t\in[0,1]\\}$ be the standard one-dimensional Brownian motion,
i.e., a stochastic process with continuous paths almost surely, stationary
independent increments and $B_{t}\sim\mathcal{N}(0,t)$ for each $t\geq 0$, see
[KS91, Lig10].

The idea is to use the almost certain continuity of its paths $t\mapsto B_{t}$
to define an almost certain continuous random potential on the symbolic space
$\Omega=\\{0,1\\}^{\mathbb{N}}$.

The construction of this random potential is as follows. We consider the
function (abusing notation) $t:\Omega\to[0,1]$ from the symbolic space to the
closed unit interval $[0,1]$, defined for each $x\in\Omega$ by the expression
$t(x)=\sum_{n\geq 1}2^{-n}x_{n}$. The Brownian potential is the random mapping
from $\Omega$ to $\mathbb{R}$ given by

$\Omega\ni x\mapsto B_{t(x)}.$

Figure 1: Samples of the Brownian potential.
$a_{1}\dots a_{n}^{\infty}:=(a_{1},\dots,a_{n},a_{n},a_{n},\dots)$

From the very definition of the Brownian motion follows that the Brownian
potential is almost certain a continuous potential over the symbolic space
$\Omega$. Therefore we have almost certain a well defined Ruelle operator
$\mathscr{L}_{B_{t(\cdot)}}$ sending $C(\Omega)$ to itself. This “random”
operator takes a continuous function $\varphi$ to another continuous function
$\mathscr{L}_{B_{t(\cdot)}}(\varphi)$ that is given for each $x\in\Omega$ by

$\displaystyle\mathscr{L}_{B_{t(\cdot)}}(\varphi)(x)=\sum_{y\in\Omega:\sigma(y)=x}\exp(B_{t(y)})\varphi(y).$
(1)

We observe that rhs of (1) let it clear that
$\mathscr{L}_{B_{t(\cdot)}}(\varphi)(x)$ is a random variable, for any choice
of $x\in\Omega$ and $\varphi\in C(\Omega)$.

######  Proposition 1.

The spectral radius $\lambda_{B}$ of the random Ruelle operator
$\mathscr{L}_{B_{t(\cdot)}}$ is a non-negative random variable.

###### Proof.

Of course, the only issue here is about measurability and we show how to
overcome this by using the Gelfand Formula and expression (1). Indeed, by
applying iteratively the expression (1) one can see that
$\mathscr{L}_{B_{t(\cdot)}}^{n}(\varphi)(x)$ is a random variable for any
choice of $\varphi\in C(\Omega)$ and $x\in\Omega$. For any fixed
$n\in\mathbb{N}$ and $\varphi\in C(\Omega)$ there exists, by the definition of
the supremum, a sequence $(x_{k})$ in $\Omega$ such that
$\|\mathscr{L}_{B_{t(\cdot)}}^{n}(\varphi)\|_{\infty}=\lim_{k\to\infty}|\mathscr{L}_{B_{t(\cdot)}}^{n}(\varphi)(x_{k})|$
so proving that $\|\mathscr{L}_{B_{t(\cdot)}}^{n}(\varphi)\|_{\infty}$ is a
random variable. Since the Gelfand’s formula ensures that the spectral radius
of $\mathscr{L}_{B_{t(\cdot)}}$ is given by

$\lambda_{B}=\lim_{n\to\infty}(\sup\\{\|\mathscr{L}_{B_{t(\cdot)}}^{n}(\varphi)\|_{\infty}:\varphi\in
C(\Omega)\ \text{and}\ \|\varphi\|_{\infty}=1\\})^{\frac{1}{n}}$

we can argue similarly as above to conclude that $\lambda_{B}$ is a random
variable. ∎

######  Proposition 2.

The random variable $\lambda_{B}$ is the main eigenvalue of the random Ruelle
operator $\mathscr{L}_{B_{t(\cdot)}}$ almost surely. Moreover this operator
has a positive continuous eigenfunction $h_{B}$ and, additionally, $h_{B}(x)$
is a random variable, for each $x\in\Omega$.

###### Proof.

A typical trajectory of the Brownian motion in the unit interval $[0,1]$ is
$\gamma$-Hölder continuous function for any $\gamma<1/2$, see [KS91, Lig10]. A
simple computation shows that $|t(x)-t(y)|\leq d(x,y)$, where $d(x,y)=2^{-N}$
and $N=\inf\\{n:x_{j}=y_{j}\ \text{for}\ 1\leq j<n\ \text{and}\ x_{n}\neq
y_{n}\\}$. Therefore almost certain we have

$|B_{t(x)}-B_{t(y)}|\leq\mathrm{const.}|t(x)-t(y)|^{\gamma}\leq\mathrm{const.}(d(x,y))^{\gamma}.$

So our random potentials $x\mapsto B_{t(x)}$ are almost certain
$\gamma$-Hölder continuous for any $\gamma<1/2$. From the classical Ruelle-
Perron-Fröbenius Theorem follows that $\lambda_{B}$, the random spectral
radius of $\mathscr{L}_{B_{t(\cdot)}}$, is actually the main eigenvalue and it
has geometric multiplicity one almost surely. From item iii) of Theorem A we
have for each $x\in\Omega$ that

$\lambda_{B}^{-n}\cdot\mathscr{L}^{n}_{B_{t(\cdot)}}(1)(x)\to h(x).$

Since the lhs above is a random variable it follows that $h(x)$ is also a
random variable. ∎

Before proceed we recall some elementary facts about dual of the Ruelle
operator associated to a deterministic general continuous potential. As we
mentioned before $(\Omega,d)$ is a compact metric space so one can apply the
Riesz-Markov Theorem to prove that $C^{*}(\Omega)$ is isomorphic to
$\mathscr{M}_{s}(\Omega)$, the space of all signed Radon measures. Therefore
we can define $\mathscr{L}^{*}_{f}$, the dual of the Ruelle operator, as the
unique continuous map from $\mathscr{M}_{s}(\Omega)$ to itself satisfying for
each $\gamma\in\mathscr{M}_{s}(\Omega)$ the following identity

$\displaystyle\int_{\Omega}\mathscr{L}_{f}(\varphi)\,d\gamma=\int_{\Omega}\varphi\,d(\mathscr{L}^{*}_{f}\gamma)\qquad\forall\varphi\in
C(\Omega).$ (2)

From the positivity of $\mathscr{L}_{f}$ follows that the map
$\gamma\mapsto\mathscr{L}^{*}_{f}(\gamma)/\mathscr{L}^{*}_{f}(\gamma)(1)$
sends the space of all Borel probability measures $\mathscr{P}(\Omega)$ to
itself. Since $\mathscr{P}(\Omega)$ is convex set and compact in the weak
topology (which is Hausdorff in this case) and the mapping
$\gamma\mapsto\mathscr{L}^{*}_{f}(\gamma)/\mathscr{L}^{*}_{f}(\gamma)(1)$ is
continuous, the Tychonoff-Schauder Theorem ensures the existence of at least
one Borel probability measure $\nu_{f}$ such that
$\mathscr{L}^{*}_{f}(\nu_{f})=\mathscr{L}^{*}_{f}(\nu_{f})(1)\nu_{f}.$ The
eigenvalue $\lambda_{f}\equiv\mathscr{L}^{*}_{f}(\nu_{f})(1)$ is positive and
equals to the spectral radius of $\mathscr{L}_{f}$ and therefore independent
of the choice of the fixed point $\nu_{f}$, see [Bal00, Bow08, PP90].

If $X$ is a continuous random potential, then the Ruelle operator
$\mathscr{L}_{X}$ sends $C(\Omega)$ to itself almost surely. We claim that for
any continuous random potential $X$ the spectral radius of the operator
$\mathscr{L}_{X}$, denoted by $\lambda_{X}$ is a random variable. In fact, the
claim follows from the arguments given in the proof of Proposition 1 replacing
$B_{t(\cdot)}$ by $X$.

######  Definition 2 (Phase Transition).

Let $X$ be a continuous random potential and $\lambda_{X}$ the random spectral
radius of $\mathscr{L}_{X}$. We say that the random potential $X$ does not
present phase transition if the eigenvalue problem
$\mathscr{L}_{X}^{*}\nu=\lambda_{X}\nu$ has a unique solution in
$\mathscr{P}(\Omega)$ almost surely. Otherwise, we say that $X$ presents phase
transition.

######  Corollary 1 (Absence of Phase Transition for Brownian Potentials).

Almost surely the dual of the Ruelle operator $\mathscr{L}^{*}_{B_{t(\cdot)}}$
has one eigenprobability associated to $\lambda_{B}$.

###### Proof.

From the Theorem 1 follows that $\lambda_{B}$ is both the spectral radius and
the main eigenvalue of $\mathscr{L}_{B_{t(\cdot)}}$. So the corollary follows
from the almost certain $\gamma$-Hölder continuity of the mapping $\Omega\ni
x\mapsto B_{t(x)}$, obtained in the proof of Theorem 1, together with the item
ii) of Theorem A. ∎

##  3 $\boldsymbol{C(\Omega)\simeq\mathscr{D}[0,1]}$

Let us denote by $\mathscr{D}$ the subset of the unit interval $[0,1]$ such
that all of its points have exactly two distinct binary expansion
representations, i.e.,

$\mathscr{D}\equiv\\{s\in[0,1]:\\#t^{-1}(s)=2\\}.$

If $t(x)\in\mathscr{D}$ then there is a unique $y\in\Omega\setminus\\{x\\}$
such that $t(x)=t(y)$. Moreover such $x$ and $y$ are always comparable in the
lexicographic order. When $x$ is smaller than $y$ in this order we indicate
this by using the notation $x\prec_{\mathrm{Lex}}y$.

We now introduce a proper subset of the Skorokhod space that will play a
fundamental role in this section. More details on the Skorokhod space can be
found in [Bil99]. The functions belonging to the subspace mentioned above are
càdlàg on the unit interval but only allowed to jump on the set $\mathscr{D}$.
We use the convenient notation $\mathscr{D}[0,1]$ to denote this subspace and
it is characterized as follows

$\mathscr{D}[0,1]\equiv\left\\{F:[0,1]\to\mathbb{R}:\ \begin{array}[]{c}F\
\text{is continuous in}\ [0,1]\setminus\mathscr{D}\ \text{and}\ \forall
t\in\mathscr{D}\\\ \text{the left limit}\ \lim_{s\uparrow t}F(s)\
\text{exists};\\\ \text{the right limit}\ \lim_{s\downarrow
t}F(s)=F(t).\end{array}\right\\}.$

We want to relate this space to $C(\Omega)$ and to this end we introduce the
following linear operator $\Theta:\mathscr{D}[0,1]\to C(\Omega)$ which sends a
function $F\in\mathscr{D}[0,1]$ to a function $f\in C(\Omega)$ whose its
definition is

  * •

$f(x)=F(t(x))$ if $x\notin t^{-1}(\mathscr{D})$ and

  * •

if $x\in t^{-1}(\mathscr{D})$ then there exists a unique
$y\in\Omega\setminus\\{x\\}$ such that $t(x)=t(y)$. If
$x\prec_{\mathrm{Lex}}y$ we put $f(x)=\lim_{s\uparrow t(x)}F(s)$ and
$f(y)=\lim_{s\downarrow t(y)}F(s)=F(t(y))$.

The linearity of $\Theta$ is obvious, but the statement $\Theta(F)\in
C(\Omega)$ whenever $F\in\mathscr{D}[0,1]$ requires a proof and this is the
content of the next proposition.

######  Proposition 3.

For every $F\in\mathscr{D}[0,1]$ we have that $\Theta(F)\in C(\Omega)$.

###### Proof.

Let $x\in\Omega$ be an arbitrary point and $F\in\mathscr{D}[0,1]$. In order to
prove the continuity of $\Theta(F)$ at $x$ we split the analysis in two cases:
whether $F$ is continuous or not in $t(x)$. Suppose that $F$ is continuous in
$t(x)$. In this case independently whether $t(x)$ belongs or not to
$\mathscr{D}$ we have $\Theta(F)(x)=F(t(x))$. Given $\varepsilon>0$ follows
from the continuity of $F$ that there is $\delta>0$ such that if
$|s-t(x)|<\delta$ then $|F(s)-F(t(x))|<\varepsilon$. From the continuity of
$t$ it is possible to find $\eta>0$ such that if $d(y,x)<\eta$ then
$|t(x)-t(y)|<\delta$. Suppose by contradiction that $\Theta(F)$ is not
continuous at $x$. So there is a sequence $y_{n}\in\Omega$ such that $y_{n}\to
x$ but
$|\Theta(F)(y_{n})-\Theta(F)(x)|=|\Theta(F)(y_{n})-F(t(x))|\geq\varepsilon$
for all $n\in\mathbb{N}$. Because of the previous observations for $n$ large
enough $\Theta(F)(y_{n})\neq F(t(y_{n}))$ otherwise we reach a contradiction.
So the only possibility is $\Theta(F)(y_{n})=\lim_{s\uparrow t(y_{n})}F(s)$
for all $n\geq n_{0}$. Again, we reach a contradiction since for large enough
$n$ we have $t(y_{n})$ close to $t(x)$ and therefore by taking $s$ close
enough to $t(x)$ we have $F(s)$ close to $F(t(x))$ which is a contradiction
with $|\Theta(F)(y_{n})-F(t(x))|\geq\varepsilon$.

Now we assume that $F$ has a jump at $t(x)$. In this case by the definition of
$\mathscr{D}[0,1]$ there exist exactly one point $y\in\Omega\setminus\\{x\\}$
such that $t(x)=t(y)$. We give the argument for $x\prec_{\mathrm{Lex}}y$ on
the other case the analysis is similar. Note that

$\begin{array}[]{rl}y=&(x_{1},\ldots,x_{n},1,0,0,\ldots)\\\
x=&(x_{1},\ldots,x_{n},0,1,1,\ldots).\end{array}$

From the definition we have

$\Theta(F)(x)=\lim_{s\uparrow t(x)}F(s).$

Given $\varepsilon>0$ there is $\delta>0$ such that for all $s$ satisfying
$t(x)-\delta<s\leq t(x)$ we have $|F(s)-\Theta(F)(x)|<\varepsilon$. Note that
for every $z\in\Omega$ such that $d(z,x)\leq 2^{n+1}$ we have $t(z)\leq t(x)$.
By taking $z$ close enough to $x$ and proceed similarly to the previous case
one can show that $\Theta(F)(z)$ is close to $\Theta(F)(x)$, thus proving the
continuity of $\Theta(F)$ at $x$. Since $x$ is arbitrary the proof is
complete. ∎

######  Theorem 1.

The operator $\Theta$ is a Banach isomorphism (linear isometry) between the
Banach spaces $(\mathscr{D}[0,1],\|\cdot\|_{\infty})$ and
$(C(\Omega),\|\cdot\|_{\infty})$.

###### Proof.

We first construct the inverse operator $\Theta^{-1}$ and then we proceed to
show that $\Theta$ is an isometry.

Let $f\in C(\Omega)$ be an arbitrary continuous function. If
$s\in[0,1]\setminus\mathscr{D}$ we define $F(s)=f(t^{-1}(s))$. If
$s\in\mathscr{D}$ then $t^{-1}(s)=\\{x,y\\}$ with $x\neq y$. If
$x\prec_{\mathrm{Lex}}y$ we put $F(s)=f(y)$. We claim that
$F\in\mathscr{D}[0,1]$. Indeed, if $s\in[0,1]\setminus\mathscr{D}$, then $s$
has a unique binary expansion $t^{-1}(s)=(x_{1},\ldots,x_{n},\ldots)$. For
each $n\in\mathbb{N}$ and $j=1,\ldots,2^{n}$ we consider the following
collection of open intervals

$I_{n}^{j}=\left(\frac{j-1}{2^{n}},\frac{j}{2^{n}}\right)$

Note that $\mathscr{D}\cup\\{0,1\\}=\cup_{n\geq 1}\cup_{j=1}^{2^{n}}\partial
I_{n}^{j}$. Note that if $s\in[0,1]\backslash\mathscr{D}$, then $s$ belongs to
the interior of infinitely many $I_{n}^{j}$’s. Given $\varepsilon>0$ there is
$\delta>0$ such that for all $y\in\Omega$ satisfying $d(t^{-1}(s),y)<\delta$
we have $|f(t^{-1}(s))-f(y)|<\varepsilon$. As mentioned before we can choose
$n$ as large as we please so that $s\in I_{n}^{j}$, for some
$j=1,\ldots,2^{n}$. From the definition of $I_{n}^{j}$ the first $n$ terms of
the binary expansion of any $r\in I_{n}^{j}$ coincides with
$x_{1},\ldots,x_{n}$. For $n$ such that $2^{-n}<\delta$ we have
$d(t^{-1}(s),t^{-1}(r))<\delta$ and therefore for all $r\in I_{n}^{j}$ we have
$|F(r)-F(s)|\leq\max_{z\in t^{-1}(r)}|f(z)-f(x)|<\varepsilon$.

Suppose that $t(x)\in\mathscr{D}$ and $\\{x,y\\}=t^{-1}(x)$. Without loss of
generality we can assume that $x\prec_{\mathrm{Lex}}y$. For $n\geq n_{0}$ we
have $t(x)\in\partial I_{n}^{j}\cap\partial I_{n}^{j+1}$ for a unique value of
$j\in\\{1,\ldots,2^{n}\\}$. Every element in $t^{-1}(I_{n}^{j})$ has its first
$n$ digits in the binary expansion equals to $x_{1},x_{2},\ldots,x_{n}$.
Therefore some small neighborhood of $x$ is sent by $t$ in
$I_{n}^{j}\cup\\{t(x)\\}$. From the continuity of $f$ at $x$ it follows that
the small neighborhood defined above is sent in a $\varepsilon$ neighborhood
of $f(x)$ which proves the existence of the left limit of $F$ at $t(x)$. The
right continuity of $F$ at $t(x)\in\mathscr{D}$ is proved in similar way.

Notice that the argument in the two previous paragraphs also proves that the
mapping

$\begin{array}[]{rl}C(\Omega)\ni f\mapsto&F\\\
&s\mapsto\begin{cases}f(t^{-1}(s)),&\text{if}\
s\in[0,1]\setminus\mathscr{D};\\\ f(y),&\text{if}\ t^{-1}(s)=\\{x,y\\}\
\text{and}\ x\prec_{\mathrm{Lex}}y.\end{cases}\end{array}$

is a left inverse of $\Theta$. Of course, $\Theta$ is right inverse of this
operator so $\Theta$ is a bijection.

Remains to prove that $\Theta$ is an isometry. From its very definition we
have

$\|\Theta(F)\|_{\infty}=\sup_{s\in[0,1]}|\Theta(F)(s)|\leq\|F\|_{\infty}.$

If the supremum above is attained at certain point
$s\in[0,1]\setminus\mathscr{D}$, take $x\in\Omega$ such that $t(x)$, then we
have $\|F\|_{\infty}=|F(s)|=|\Theta(F)(x)|\leq\|\Theta(F)\|_{\infty}$. If the
supremum is not attained for some $s\in\mathscr{D}$ the conclusion is
analogous, otherwise $\|F\|_{\infty}=\lim_{n\to\infty}\lim_{r\uparrow
s_{n}}|F(r)|=\lim_{n\to\infty}|\Theta(F)(x_{n})|$, where $x_{n}$ is the
smallest point, with respect to the lexicographic order, in $t^{-1}(s_{n})$
and the proposition follows. ∎

######  Remark 1 (Wiener Spaces).

Recall that the classical Wiener space on the closed unit interval $[0,1]$,
notation $W[0,1]$, is the set of all real continuous functions
$F:[0,1]\to\mathbb{R}$ such that $F(0)=0$. Consider the closed subspace of
$C(\Omega)$ given by

$W(\Omega)\equiv\\{f\in C(\Omega):f(0^{\infty})=0,\ f(x)=f(y)\
\text{whenever}\ t(x)=t(y)\\},$

where the mapping $t:\Omega\to\mathbb{R}$ was previously defined. It is easy
to see that for every $F\in W[0,1]$ its composition with $t$ give us a mapping
$F\circ t\in W(\Omega)$. Moreover the linear mapping $F\mapsto F\circ t$ is an
isometry from $W[0,1]$ to $W(\Omega)$.

##  4 Stochastic Functional Equation for $\boldsymbol{\lambda_{B}}$

In this section we apply the results above obtained. The idea is to obtain a
stochastic functional equation using the operator $\Theta$ and Proposition 2
to semi-explicit representation of the main eigenvalue $\lambda_{B}$ of the
random Ruelle operator $\mathscr{L}_{B_{t(\cdot)}}$.

Since $h_{B}$ is a continuous function almost surely, we can associate to it a
càdlàg process $\\{X_{s}:0\leq s\leq 1\\}$ so that
$\Theta(X_{(\cdot)})=h_{B}$. By simple algebraic manipulation we can see that
$t(0x)=t(x)/2$ and $t(1x)=1/2+t(x)/2$. By applying the inverse of the operator
$\Theta$ on both sides of the above equation we get the following stochastic
functional equation

$\displaystyle\exp(B_{\frac{t}{2}})X_{\frac{t}{2}}+\exp(B_{\frac{1}{2}+\frac{t}{2}})X_{\frac{1}{2}+\frac{t}{2}}=\lambda_{B}X_{t}.$

Now we have to play with the properties of the Brownian motion and the Ruelle
operator to obtain the law of the main eigenvalue. Recalling that $B_{0}=0$
almost surely, so we get from the stochastic functional equation that

$X_{0}+\exp(B_{\frac{1}{2}})X_{\frac{1}{2}}=\lambda_{B}X_{0}.$

Therefore we get by isolating the eigenvalue (recall that $X_{s}>0$ for each
$s\in[0,1]$) the following equality a.s.

$\displaystyle\lambda_{B}=1+\exp(B_{\frac{1}{2}})\frac{X_{\frac{1}{2}}}{X_{0}}$
(3)

By using again the operator $\Theta$ one can see that
$X_{0}=h_{B}(0,0,\ldots)$ and $X_{1/2}=h_{B}(1,0,0,\ldots)$, recall that
$(0,1,1,\ldots)\prec_{\mathrm{Lex}}(1,0,0,\ldots)$. From the Theorem A we have
almost certain

$\frac{X_{\frac{1}{2}}}{X_{0}}=\frac{h_{B}(1,0,0,\ldots)}{h_{B}(0,0,\ldots)}=\lim_{n\to\infty}\frac{\mathscr{L}^{n}_{B_{t(\cdot)}}(1)(1,0,0,\ldots)}{\mathscr{L}^{n}_{B_{t(\cdot)}}(1)(0,0,0,\ldots)}.$

From where we conclude that

$\lambda_{B}=1+\exp(B_{\frac{1}{2}})\lim_{n\to\infty}\frac{\mathscr{L}^{n}_{B_{t(\cdot)}}(1)(1,0,0,\ldots)}{\mathscr{L}^{n}_{B_{t(\cdot)}}(1)(0,0,0,\ldots)}.$

##  5 Bounds on the Expected Value of $\boldsymbol{\lambda_{B}}$

In the deterministic case, when the potential is assumed to have Hölder,
Walters or Bowen regularity, it is easy to obtain lower and upper bounds of
the main eigenvalue of the Ruelle operator by using the supremum norm of the
potential. We recall that when the potential have such regularity the main
eigenvalue do exists and is actually equals to the spectral radius.

Since here we are considering random potentials lower and upper bounds for the
expected values, with respect to the Wiener measure, of the spectral radius is
a natural question to ask.

Opposed to the deterministic case the supremum norm of $B_{t(\cdot)}$ by
itself does not help in finding bounds to the spectral radius. The reason is
$\sup\\{|B_{t(x)}|:x\in\Omega\\}$ is an unbounded random variable. The
representation of $\lambda_{B}$ obtained in the last section is not suitable
to take expectations and very hard to bound due its complex combinatorial
nature. Therefore a different approach is needed to bound the expected value
of main eigenvalue and the idea is based on the reflection principle of the
Brownian motion.

######  Theorem 2.

Let $\lambda_{B}$ be the random variable defining the spectral radius of the
random Ruelle operator $\mathscr{L}_{B_{t(\cdot)}}$. Then $\lambda_{B}$ has
finite first moment, with respect to the Wiener measure, and moreover

$\exp(1/2)\leq\mathbb{E}[\lambda_{B}]\leq 4\exp{(1/2)}.$

###### Proof.

We begin with the lower bound. To lighten the notation we use the symbol
$1^{\infty}$ to denote the constant sequence $(1,1,1,\ldots)$. By the
definition of the supremum we have

$\displaystyle\sup_{\varphi:\|\varphi\|_{\infty}=1}\|\mathscr{L}_{B_{t(\cdot)}}^{n}(\varphi)\|_{\infty}$
$\displaystyle=\sup_{\varphi:\|\varphi\|_{\infty}=1}\
\sup_{x\in\Omega}|\mathscr{L}_{B_{t(\cdot)}}^{n}(\varphi)(x)|$
$\displaystyle=\sup_{\varphi:\|\varphi\|_{\infty}=1}\
\sup_{x\in\Omega}|\sum_{a_{1},\ldots,a_{n}}\exp(\sum_{j=0}^{n-1}B_{t(\sigma^{j}(a_{1}\ldots
a_{n}x))})\varphi(\sigma^{j}(a_{1}\ldots a_{n}x))|$
$\displaystyle\geq\exp\sum_{j=0}^{n-1}B_{t(1^{\infty})}$
$\displaystyle\geq\exp(nB_{1}).$

By taking the $n$-th root, the limit when $n\to\infty$ and the expectation in
the above inequality we get that

$\mathbb{E}[\lambda_{B}]\geq\mathbb{E}[\exp(B_{1})]=\exp(1/2).$

To get the upper bound we will take advantage of the reflection principle of
the Wiener process. We first observe that

$\displaystyle\sup_{\varphi:\|\varphi\|_{\infty}=1}\|\mathscr{L}_{B_{t(\cdot)}}^{n}(\varphi)\|_{\infty}$
$\displaystyle\leq\ \sup_{x\in\Omega}|\mathscr{L}_{B_{t(\cdot)}}^{n}(1)(x)|$
$\displaystyle=\sup_{x\in\Omega}\sum_{a_{1},\ldots,a_{n}}\exp\sum_{j=0}^{n-1}B_{t(\sigma^{j}(a_{1}\ldots
a_{n}x))}$ $\displaystyle\leq\sum_{a_{1},\ldots,a_{n}}\sup_{x\in\Omega}\
\exp\sum_{j=0}^{n-1}B_{t(\sigma^{j}(a_{1}\ldots a_{n}x))}$ $\displaystyle\leq
2^{n}\exp(n\max\\{B_{t}:0\leq t\leq 1\\}).$

By taking the $n$-th root and the limit when $n\to\infty$ in the last
inequality we get $\lambda_{B}\leq 2\exp(M_{1})$, where
$M_{1}\equiv\max\\{B_{t}:0\leq t\leq 1\\}$. From the reflection principle of
the Wierner process it follows that

$\mathbb{E}[\lambda_{B}]\leq 2\mathbb{E}[\exp(M_{1})]\leq
4\cdot\mathbb{E}[\exp{B_{1}}]=4\exp(1/2).\qed$

##  6 Existence and Finiteness of the Quenched Pressure

By using the $\gamma$-Hölder regularity of the Brownian potential $x\mapsto
B_{t(x)}$ and a classical result of Thermodynamic formalism, we have almost
certain

$\log\lambda_{B}=P(B_{t(\cdot)})=\sup_{\mu\in\mathscr{P}_{\sigma}(\Omega)}\left\\{h(\mu)+\int_{\Omega}B_{t(\cdot)}\,d\mu\right\\},$

where $P(\cdot)$ is the topological pressure, $\mathscr{P}_{\sigma}(\Omega)$
is the set of all Borel probability measures that are shift invariant and $h$
is the Kolmogorov-Sinai entropy. From (3) we know that
$\lambda_{B}=1+\exp(B_{1/2})(X_{1/2}/X_{0}),$ and $X_{1/2}/X_{0}\geq 0$.
Therefore we can conclude that $\log\lambda_{B}$ is non-negative random
variable and its expected value with respect to the Wiener measure is well
defined, at least as an extended real number.

Finally, by using the elementary inequality $\log\lambda_{B}\leq\lambda_{B}$
and the Theorem 2 we have that $\mathbb{E}[\log\lambda_{B}]<+\infty$, thus
proving the finiteness of the quenched pressure. Using once more the
$\gamma$-Hölder regularity of the Brownian potential we have almost certain
that

$\log\lambda_{B}=\lim_{n\to\infty}\frac{1}{n}\log\mathscr{L}^{n}_{B_{t(\cdot)}}(1)(\sigma^{n}(x)),\qquad\forall
x\in\Omega$

and therefore

$P^{\mathrm{quenched}}(B_{t(\cdot)})\equiv\mathbb{E}[\log\lambda_{B}]=\mathbb{E}\left[\lim_{n\to\infty}\frac{1}{n}\log\mathscr{L}^{n}_{B_{t(\cdot)}}(1)(\sigma^{n}(x))\right].$

Finally, by the Theorem 2 and the Jensen Inequality we get

$0\leq
P^{\mathrm{quenched}}(B_{t(\cdot)})=\mathbb{E}[\log\lambda_{B}]\leq\log\mathbb{E}[\lambda_{B}]\leq\log
4+\frac{1}{2}.$

##  7 Concluding Remarks

As mentioned, all the results presented in this paper can extended to the
space $\Omega=\\{0,\dots,m-1\\}^{\mathbb{N}}$. For such state spaces the map
$t:\Omega\to[0,1]$ is given by $(x_{1},x_{2}\dots)\mapsto\sum_{n\geq
1}m^{-n}x_{n}$. Analogous arguments can be used to prove the Proposition 1 and
Proposition 2. In the Section 3, the arguments would be completely analogous,
only changing appropriately the set $\mathscr{D}$.

In the Section 4, we would have a subtle change in the stochastic functional
equation. Now we have for every $a\in\\{0,\dots,m-1\\}$ and $x\in\Omega$,
$t(ax)=a/m+t(x)/m$, so a slightly different stochastic functional equation
shows up $\exp(B_{t/m})X_{t/m}+\exp(B_{1/m+t/m})X_{1/m+t/m}=\lambda_{B}X_{t}.$
Thus, by taking $t=0$, we recover the representation of the main eigenvalue

$\displaystyle\lambda_{B}$
$\displaystyle=1+\exp(B_{\frac{1}{m}})\lim_{n\to\infty}\frac{\mathscr{L}^{n}_{B_{t(\cdot)}}(1)(1,0,0,\ldots)}{\mathscr{L}^{n}_{B_{t(\cdot)}}(1)(0,0,0,\ldots)}.$

The technique employed in Theorem 2 provided the following bounds

$\exp(1/2)\leq\mathbb{E}[\lambda_{B}]\leq 2m\exp(1/2).$

Therefore, by using the same arguments, one can also prove the existence and
finiteness of the Quenched Pressure on this case.

Another natural question is whether we could study another random potential
defined by a continuous time stochastic process other than a Brownian Motion.
Note that the only property of the Brownian Motion used to guarantee the
existence of the main eigenvalue is the almost certain Hölder continuity of
the paths (in fact this is required only in $[0,1]\backslash\mathscr{D}$ ),
which is not an exclusive property of Brownian Motion. In the functional
stochastic equation, we only used that $B_{0}=0$ almost surely, but for the
general case all that is required is to add a constant to the potential. Here
we used the reflection principle and the stationarity of the increments to
bound $\mathbb{E}[\lambda_{B}]$ which also follow for some other processes.
Nevertheless if one has another technique to guarantee its finiteness, the
general sequence described in the article still holds and the existence and
finiteness of the Quenched Pressure is still guaranteed. Therefore, the
natural candidate to extend the results is Fractional Brownian motion, which
indeed leads to very analogous results.

The obstacle one faces to obtain tighter lower bounds for
$\mathbb{E}[\lambda_{B}]$ is of combinatorial nature and the complicated sums
of dependents log-normal random variables that appear in the calculations and
have to be controlled. Some informal computations suggest that the annealed
pressure is also well defined for Brownian type potential and possibly given
by $\log(\mathrm{ess\,sup}\lambda_{B})$, but unfortunately the techniques
developed here seem not so helpful in attacking this problem.

## Acknowledgments

The authors gratefully acknowledge Eduardo A. Silva, Hannah Goozee and the
professors Mike Todd and Yuri Dumaresq Sobral for many helpful remarks and
suggestions on the early versions of this manuscript. We also thank to
professor Yuri Kifer for several useful references. Leandro Chiarini and
Leandro Cioletti acknowledge CNPq for financial support.

## References

  * [Bal00] V. Baladi.  Positive transfer operators and decay of correlations, volume 16 of Advanced Series in Nonlinear Dynamics.  World Scientific Publishing Co., Inc., River Edge, NJ, 2000.
  * [Bar11] L. Barreira.  Thermodynamic formalism and applications to dimension theory, volume 294 of Progress in Mathematics.  Birkhäuser/Springer Basel AG, Basel, 2011.
  * [BG95] T. Bogenschütz and V. M. Gundlach.  Ruelle’s transfer operator for random subshifts of finite type.  Ergodic Theory Dynam. Systems, 15(3):413–447, 1995.
  * [Bil99] P. Billingsley.  Convergence of probability measures.  Wiley Series in Probability and Statistics: Probability and Statistics. John Wiley & Sons, Inc., New York, second edition, 1999.  A Wiley-Interscience Publication.
  * [Bog96] T. Bogenschütz.  Stochastic stability of equilibrium states.  Random Comput. Dynam., 4(2-3):85–98, 1996.
  * [Bou01] T. Bousch.  La condition de Walters.  Ann. Sci. École Norm. Sup. (4), 34(2):287–311, 2001.
  * [Bow79] R. Bowen.  Hausdorff dimension of quasicircles.  Inst. Hautes Études Sci. Publ. Math., (50):11–25, 1979.
  * [Bow08] R. Bowen.  Equilibrium states and the ergodic theory of Anosov diffeomorphisms, volume 470 of Lecture Notes in Mathematics.  Springer-Verlag, Berlin, revised edition, 2008.  With a preface by David Ruelle, Edited by Jean-René Chazottes.
  * [DKS08] M. Denker, Y. Kifer, and M. Stadlbauer.  Thermodynamic formalism for random countable Markov shifts.  Discrete Contin. Dyn. Syst., 22(1-2):131–164, 2008.
  * [FS88] P. Ferrero and B. Schmitt.  Produits aléatoires d’opérateurs matrices de transfert.  Probab. Theory Related Fields, 79(2):227–248, 1988.
  * [GK00] V. M. Gundlach and Yu. Kifer.  Expansiveness, specification, and equilibrium states for random bundle transformations.  Discrete Contin. Dynam. Systems, 6(1):89–120, 2000.
  * [Kif08] Y. Kifer.  Thermodynamic formalism for random transformations revisited.  Stoch. Dyn., 8(1):77–102, 2008.
  * [KL06] Y. Kifer and P-D. Liu.  Random dynamics.  In Handbook of dynamical systems. Vol. 1B, pages 379–499. Elsevier B. V., Amsterdam, 2006.
  * [KS91] I. Karatzas and S. E. Shreve.  Brownian motion and stochastic calculus, volume 113 of Graduate Texts in Mathematics.  Springer-Verlag, New York, second edition, 1991.
  * [Lig10] T. M. Liggett.  Continuous time Markov processes, volume 113 of Graduate Studies in Mathematics.  American Mathematical Society, Providence, RI, 2010.  An introduction.
  * [Mañ90] R. Mañé.  The Hausdorff dimension of horseshoes of diffeomorphisms of surfaces.  Bol. Soc. Brasil. Mat. (N.S.), 20(2):1–24, 1990.
  * [MM83] H. McCluskey and A. Manning.  Hausdorff dimension for horseshoes.  Ergodic Theory Dynam. Systems, 3(2):251–260, 1983.
  * [MPV87] M. Mézard, G. Parisi, and M. A. Virasoro.  Spin glass theory and beyond, volume 9 of World Scientific Lecture Notes in Physics.  World Scientific Publishing Co., Inc., Teaneck, NJ, 1987.
  * [MSU11] V. Mayer, B. Skorulski, and M. Urbanski.  Distance expanding random mappings, thermodynamical formalism, Gibbs measures and fractal geometry, volume 2036 of Lecture Notes in Mathematics.  Springer, Heidelberg, 2011.
  * [PP90] W. Parry and M. Pollicott.  Zeta functions and the periodic orbit structure of hyperbolic dynamics.  Astérisque, (187-188):268, 1990.
  * [Rue68] D. Ruelle.  Statistical mechanics of a one-dimensional lattice gas.  Comm. Math. Phys., 9:267–278, 1968.
  * [Rue02] D. Ruelle.  Dynamical zeta functions and transfer operators.  Notices Amer. Math. Soc., 49(8):887–895, 2002.
  * [Sin72] Ja. G. Sinaĭ.  Gibbs measures in ergodic theory.  Uspehi Mat. Nauk, 27(4(166)):21–64, 1972.
  * [SP14] R. L. Schilling and L. Partzsch.  Brownian motion.  De Gruyter Graduate. De Gruyter, Berlin, second edition, 2014.  An introduction to stochastic processes, With a chapter on simulation by B. Böttcher.
  * [Wal01] P. Walters.  Convergence of the Ruelle operator for a function satisfying Bowen’s condition.  Trans. Amer. Math. Soc., 353(1):327–347 (electronic), 2001.
  * [Wal07] P. Walters.  A natural space of functions for the Ruelle operator theorem.  Ergodic Theory Dynam. Systems, 27(4):1323–1348, 2007.

Leandro Chiarini
Leandro Cioletti
Departamento de Matemática - UnB
Campus Universitário Darcy Ribeiro - Asa Norte
70910-900 Brasília - DF - Brazil.

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