# The consensus problem for opinion dynamics with local average random
interactions

M. Gianfelice∘, G. Scola#
${}^{{{}^{\circ}}}$Dipartimento di Matematica e Informatica
Università della Calabria
Campus di Arcavacata
Ponte P. Bucci - cubo 30B
I-87036 Arcavacata di Rende
<EMAIL_ADDRESS>
#Dipartimento di Matematica e Fisica
Università degli Studi Roma Tre
Largo S. Leonardo Murialdo, 1
00146 Roma
<EMAIL_ADDRESS>
###### Abstract

We study the consensus formation for an agents based model, generalizing that
originally proposed by Krause [Kr], by allowing the communication channels
between any couple of agents to be switched on or off randomly, at each time
step, with a probability law depending on the proximity of the agents’
opinions. Namely, we consider a system of agents sharing their opinions
according to the following updating protocol. At time $t+1$ the opinion
$X_{i}\left(t+1\right)\in\left[0,1\right]$ of agent $i$ is updated at the
weighted average of the opinions of the agents communicating with it at time
$t.$ The weights model the confidence level an agent assign to the opinions of
the other agents and are kept fixed by the system dynamics, but the set of
agents communicating with any agent $i$ at time $t+1$ is randomly updated in
such a way that the agent $j$ can be chosen to belong to this set
independently of the other agents with a probability that is a non increasing
function of $\left|X_{i}\left(t\right)-X_{j}\left(t\right)\right|.$ This
condition models the fact that a communication among the agents is more likely
to happen if their opinions are close. We prove that the system reaches
consensus, i.e. as the time tends to infinity the agents’ opinions will reach
the same value exponentially fast.

###### Contents

  1. 1 Introduction, notations and results
    1. 1.1 Notations
      1. 1.1.1 Graphs
    2. 1.2 Description of the model and results
      1. 1.2.1 Beliefs dynamics
      2. 1.2.2 Communication channels dynamics
      3. 1.2.3 Results for the finite system
      4. 1.2.4 Results for the large system
  2. 2 Finite system evolution
    1. 2.1 Consensus
    2. 2.2 Invariant measures for $\mathfrak{T}$ and $\mathbf{T}$
    3. 2.3 Emergence of consensus
      1. 2.3.1 Asymptotic estimate of $\mathbb{E}_{0}\left[W\left(t\right)\right]$
    4. 2.4 Convergence to the stationary measure of $\left\\{\chi_{t}\right\\}_{t\in\mathbb{Z}_{+}}$ and $\left\\{\mathfrak{y}_{t}\right\\}_{t\in\mathbb{Z}_{+}}$
  3. 3 Large system evolution

AMS subject classification:

91D30,60J20.

Keywords and phrases:

Opinion dynamics, agent-based models, multi-agents systems, consensus,
emergent behaviour, synchronization, complex networks.

Acknowledgement:

M. Gianfelice is partially supported by G.N.A.M.P.A..
G. Scola acknowledges financial support from MIUR, PRIN 2017 project MaQuMA,
PRIN201719VMAST01.

##  1 Introduction, notations and results

Opinion dynamics is a topic in applied mathematics which has witnessed a
growing interest in the last decades. This is due to the possibility to
describe the emergence of collective phenomena such as the reaching of
consensus in a community of peers by means of simple models of interacting
agents [CFL]. The literature on the subject concentrates mainly on two
families of models: those cast in the framework of interacting particle
systems (IPS) e.g. the voter model and the majority-vote process [Li], the
Axelrod model and its generalizations [La], the Deffuant-Weisbuch model
[DNAW], [Ha], [HH], and those belonging to the family of coupled dynamical
systems (CDS) e.g. [Kr] (see [FF] for linear models).

The common feature of these models is that the interactions among the agents
are designed in such a way that an agent adjust its opinion to that of its
neighbours and that agents are more likely to interact with those sharing
similar opinions. On the other hand, the main difference between IPS type
models and CDS type models, aside form the fact that IPS type models typically
evolve in continuous time while CDS type models evolve in discrete time, is
that in CDS type models all the agents can change the value of their opinion
at each time step while, in IPS type models, at a given time (usually at the
tick of a Poisson clock), only the elements of a randomly chosen subset of
agents are allowed to change the value of their opinion, and the rule under
which the opinion of the selected agents are updated, being stochastic in
general (see [CF] for a stochastic version of the Deffuant model), can be
deterministic too [DNAW], [HH].

In this paper we present a model of consensus formation which, although it
represents a modification of that originally proposed by Krause, it is defined
by an updating rule of the agents opinions that recalls those characterizing
IPS type models. In particular, at each time step, firstly the set of
neighboring peers of any agent $i$ is selected at random in such a way that
the events that any two distinct agents belong to the neighbourhood of $i$ are
independent. Then, each agent update the value of its opinion to the average
value of the opinions of its neighbours.

More precisely, we consider a collection of agents which form the set of
vertices of a directed graph $G:=\left(V,E\right),E\subseteq V\times V.$ We
assume that agent $u$ communicate with agent $v$ if the directed edge
$\left(u,v\right)$ is in $E.$ Each agent hold an opinion (_belief_)
represented by a variable taking values in $\left[0,1\right].$ Agent’s beliefs
evolve in time in such a way that the opinion $X_{u}\left(t+1\right)$ of the
agent labelled by the graph vertex $u$ at time $t+1$ is updated at the
weighted average of the beliefs $X_{v}\left(t\right)$ of the agents
communicating with $u$ at time $t.$ The weights appearing in the just
mentioned average represent the quality of the information exchange among the
agents and do not change in time. On the other hand, the set of the agents
communicating with agent $u$ at time $t$ is randomly chosen according to a
probability distribution which gives more chance to a communication exchange
between agent $u$ and agent $v$ to happen if the value of their opinion are
close. It is also natural to assume that the information exchange an agent has
with itself is always maximal.

In the rest of the section we introduce the notation used throughout the
paper, formally display the definition of the model and present the results
obtained in the following sections about the emergence of consensus for the
finite size system as well as the extension of these results when the size of
the system is very large.

We stress that, as in [Kr], in this work, the dynamics of the state of the
edges of $G,$ representing the communication channels amid the agents, is
synchronous, i.e. they are all updated at each time step. On the other hand,
one may wish to modify the system’s dynamics by letting the state of the edges
of $G$ to evolve under an asynchronous dynamics. In the case of a finite size
system, this can be realized, for example, by updating at each time step the
state of just one edge sampled uniformly at random among the elements of $E$
and leaving unchanged the states of the other edges. In fact, it seems that
the techniques used to analyse the emergence of consensus in the synchronous
case do not apply to this particular case. Therefore, the discussion about the
possibility to reach consensus for the opinion exchange model characterized by
the same updating rule for the values of the agents’ opinions presented in
this paper, but subject to the asynchronous evolution of the communication
exchange among the agents just described, are deferred to a forthcoming paper.

We also remark that the differences between the way, synchronous or
asynchronous, one chooses to update the state of the communication links
shared by the agents reflects in the large system limit analysis. As a matter
of fact, for the model where the state of just one communication channel is
updated at a time, as the one just described, with a suitable modification of
the interaction among the agents, the evolution of the system, in the large
system limit, can be analysed looking at the evolution of the empirical
distribution of the agents’ opinions, in the spirit of the kinetic limit for
models of flocking (see e.g. [GO], [CCH]), as it has already been discussed
for the stochastic version of Deffuant model in [CF] and more recently for
other IPS models [AM]. On the contrary, this is not possible in models where
the state of all the communication channels are update at once as for the one
studied in this paper. However, if the size of the neighborhood of the agents
is finite it is possible to define directly the evolution of the system on
very large, possibly infinite, graphs as in [La] and [HH].

###  1.1 Notations

If $A$ is a set and $B\subseteq A,\mathbf{1}_{B}$ denotes the indicator
function of $A$ and $B^{c}:=A\backslash B.$ Let $\mathcal{P}\left(A\right)$
the set of the subsets of $A.$ For any $k\geq 1,$ we set
$\mathcal{P}_{k}\left(A\right):=\left\\{B\in\mathcal{P}\left(A\right):\left|B\right|=k\right\\}$
and denote by $\mathcal{P}_{0}\left(A\right):=\bigvee\limits_{k\geq
1}\mathcal{P}_{k}\left(A\right)$ the set of finite subsets of $A.$

If $A$ is a metric space, $\mathcal{B}\left(A\right)$ denotes its Borel
$\sigma$algebra. We also denote by $BM\left(A\right)$ the Banach space of
bounded measurable functions on $A,$ by $C\left(A\right)$ the Banach space of
real-valued continuous functions on $A$ and by $\left\|\cdot\right\|$ the sup-
norm. Moreover, if $Lip\left(A\right)$ is the Banach space of real-valued
bounded Lipschitz functions on $A,$ for any $\varphi\in Lip\left(A\right),$ we
denote its norm by $\left\|\varphi\right\|_{L}.$

For $A,A^{\prime}$ metric spaces, $BL\left(A,A^{\prime}\right)$ denotes the
space of bounded linear operators on $A$ with values in $A^{\prime}.$ If
$A^{\prime}=A$ we set $BL\left(A,A\right):=BL\left(A\right).$ In particular,
if $\mathbb{V}$ is a finite set, we denote by $St\left(\mathbb{V}\right)$ the
convex subset of $BL\left(\mathbb{R}^{\mathbb{V}}\right)$ of stochastic
matrices.

We denote by $\mathbb{E}$ the expected value of a random element when there is
no need to specify the probability space on which it is defined and
consequently write $\mathbb{P}\left\\{B\right\\}$ for the expected value of
the indicator function $\mathbf{1}_{B}$ of an event $B\subseteq A.$ The same
notation will be also kept when considering conditional expectations and
conditional probabilities. Besides, given a $\sigma$algebra $\mathcal{A}$ of
subsets of $A,$ we denote by $\mathfrak{P}\left(A,\mathcal{A}\right)$ the set
of probability measures on $\left(A,\mathcal{A}\right).$ If
$\mu\in\mathfrak{P}\left(A,\mathcal{A}\right),spt\mu$ denotes the support of
$\mu,$ and, for
$\mu,\nu\in\mathfrak{P}\left(A,\mathcal{A}\right),\left\|\mu-\nu\right\|$
denotes the total variation distance between the two measures.

Let $\mathbb{A}:=A^{\mathbb{V}}$ and denote by
$\mathbf{a}:=\left\\{a_{v}\right\\}_{v\in\mathbb{V}}.$ If $\mathbb{A}$ is a
poset w.r.t. the partial order: $\mathbf{a}\leq\mathbf{a}^{\prime}$ if
$a_{v}\leq a_{v}^{\prime},$ for any $v\in\mathbb{V},$ we say that a real-
valued function $\varphi$ on $\mathbb{A}$ is _non-decreasing_ if
$\varphi\left(\mathbf{a}\right)\leq\varphi\left(\mathbf{a}^{\prime}\right)$
whenever $\mathbf{a}\leq\mathbf{a}^{\prime}.$ Given two probability measures
$\mathbb{P},\mathbb{P}^{\prime}$ on
$\left(\mathbb{A},\mathcal{A}^{\otimes\mathbb{V}}\right)$ we say that
$\mathbb{P}$ is stochastically dominated by $\mathbb{P}^{\prime},$ and denote
this property by $\mathbb{P}\overset{st}{\leq}\mathbb{P}^{\prime},$ if for any
bounded non-decreasing function $\varphi,\mathbb{E}\left[\varphi\right]=\int
d\mathbb{P}\left(a\right)\varphi\left(a\right)\leq\int
d\mathbb{P}^{\prime}\left(a\right)\varphi\left(a\right)=\mathbb{E}^{\prime}\left[\varphi\right].$
Moreover, if $\mathbb{A}$ is finite, a probability measure $\mathbb{P}$ on
$\left(\mathbb{A},\mathcal{P}\left(\mathbb{A}\right)\right)$ is called
_irreducible_ if starting from any element of $\mathbb{A}$ with positive
$\mathbb{P}$-probability one can reach any other element with positive
$\mathbb{P}$-probability via successive coordinate changes without passing
through elements with zero $\mathbb{P}$-probability [GHM], [Gr].

If $\mathbb{A}:=A^{\mathbb{N}},$ for any $N\in\mathbb{N},$ we set
$\mathbf{a}_{N}:=\left(a_{i},..,a_{N}\right)$ and denote by
$\mathcal{C}\left(\mathbb{A}\right)$ the _cylinder_ $\sigma$_algebra_ that is
the $\sigma$algebra generated by the cylinder subsets

$\mathbf{C}_{N}\left(B\right):=\left\\{\mathbf{a}\in\mathbb{A}:\mathbf{a}_{N}\in
B\right\\}\ ,$ (1)

with $B\subseteq A^{N}$ if $A$ is a discrete set, while
$B\in\mathcal{B}\left(A^{N}\right)=\mathcal{B}\left(A\right)^{\otimes N}$ if
$A$ is a metric space.

If $\mathfrak{L}\left(\mathbb{A}\right)$ denotes the algebra of real-valued
bounded local (cylinder) functions on $\mathbb{A}$ we denote by
$\mathfrak{\bar{L}}\left(\mathbb{A}\right)$ the space of real-valued bounded
quasilocal functions on $\mathbb{A}$ that is is the closure in topology of
uniform convergence of the algebra of cylinder functions [Ge].

If $\mathbb{V}$ is denumerable we denote by $\mathcal{E}_{\mathbb{A}}$ the
product $\sigma$algebra $\mathcal{A}^{\otimes\mathbb{V}}$ on
$\mathbb{A}:=A^{\mathbb{V}}.$

####  1.1.1 Graphs

We recall some basic definition of graph theory useful to give a mathematical
definition of consensus for the system. The connection of graph theory with
Markov chains will be exploited in the next section. We refer the reader to
basic textbooks such as [Bo] and [St] for an account on this subject.

A directed graph $G$ is a ordered pair of sets $\left(V,E\right)$ where $V$ is
a finite set called _set of vertices_ and $E\subseteq V\times V$ is called
_set of edges_ or _bonds._ $G^{\prime}=\left(V^{\prime},E^{\prime}\right)$
such that $V^{\prime}\subseteq V$ and
$E^{\prime}\subseteq\left(V^{\prime}\times V^{\prime}\right)\cap E$ is said to
be a _subgraph_ of $G$ and this property is denoted by $G^{\prime}\subseteq
G.$ If $G^{\prime}\subseteq G,$ we denote by $V\left(G^{\prime}\right)$ and
$E\left(G^{\prime}\right)$ respectively the set of vertices and the collection
of the edges of $G^{\prime}.\ \left|V\left(G^{\prime}\right)\right|$ is called
the _order_ of $G^{\prime}$ while $\left|E\left(G^{\prime}\right)\right|$ is
called its _size_. Given $G_{1},G_{2}\subseteq G,$ we denote by $G_{1}\cup
G_{2}:=\left(V\left(G_{1}\right)\cup
V\left(G_{2}\right),E\left(G_{1}\right)\cup E\left(G_{2}\right)\right)\subset
G$ the _graph union_ of $G_{1}$ and $G_{2}.$ Moreover, we say that
$G_{1},G_{2}\subseteq G$ are _disjoint_ if $V\left(G_{1}\right)\cap
V\left(G_{2}\right)=\varnothing.$ For any $E^{\prime}\subseteq E,$ we denote
by $G\left(E^{\prime}\right):=\left(V,E^{\prime}\right)$ the _spanning_ graph
of $E^{\prime}.$ We also define

$V\left(E^{\prime}\right):=\left(\bigcup_{e\in E^{\prime}}e\right)\subset V\
.$ (2)

Given $V^{\prime}\subseteq V,$ we set

$E\left(V^{\prime}\right):=\left\\{e\in E:e\subset V^{\prime}\right\\}$ (3)

and denote by
$G\left[V^{\prime}\right]:=\left(V^{\prime},E\left(V^{\prime}\right)\right)$
that is called the subgraph of $G$ _induced_ or _spanned_ by $V^{\prime}.$

Two vertices $u,v$ are said to be _adjacents_ if belong to the same bond i.e.
if $V\left(e\right)=\left\\{u,v\right\\}.$ If $e=\left(u,v\right),$ $e$ is
said to be _outgoing_ from $u$ and _ingoing_ in $v.$ Let

$E_{v}^{-}:=\left\\{e\in E:e=\left(u,v\right),\ u\in V\right\\}\ ,\
E_{v}^{+}:=\left\\{e\in E:e=\left(v,u\right),\ u\in V\right\\}$ (4)

be the set of edges respectively ingoing, outgoing from $v.$ We denote by
$\mathcal{N}^{-}\left(v\right):=\left(\cup_{e\in
E_{v}^{-}}V\left(e\right)\right)\subseteq V$ the _closed ingoing neighborhood_
of $v$ and by $\mathcal{N}^{+}\left(v\right):=\left(\cup_{e\in
E_{v}^{+}}V\left(e\right)\right)\subseteq V$ the _closed outgoing
neighborhood_ of $v.$ Moreover, for any $W\subset V,$ we set
$\mathcal{N}^{+}\left(W\right):=\cup_{v\in W}\mathcal{N}^{+}\left(v\right)$ to
be the closed outgoing neighborhood of $W.$ Given $v\in V,$ we set
$\mathcal{N}_{1}^{+}\left(v\right):=\mathcal{N}^{+}\left(v\right)$ and, for
$k\geq 2,\
\mathcal{N}_{k}^{+}\left(v\right):=\mathcal{N}^{+}\left(\mathcal{N}_{k-1}^{+}\left(v\right)\right)$
to be the _outgoing_ $k$_-neighborhood_ of $v.$ Given two vertices $u$ and
$v,$ $v$ is said to _communicate_ with $u$ if there exists $k\geq 1$ such that
$u\in\mathcal{N}_{k}^{+}\left(v\right).$ Therefore, $u,v\in V$ are said to be
_connected_ if one communicates with the other. Indeed, since if
$u\in\mathcal{N}_{k}^{+}\left(v\right)$ for some $k\geq 1,$ then
$u\in\mathcal{N}_{l}^{+}\left(v\right),\ \forall l>k,$ for $u$ and $v$ to be
connected there must be $k_{1},k_{2}\geq 1$ such that
$u\in\mathcal{N}_{k_{1}}^{+}\left(v\right)$ and
$v\in\mathcal{N}_{k_{2}}^{+}\left(u\right),$ that is
$u\in\mathcal{N}_{k_{1}\vee
k_{2}}^{+}\left(v\right),v\in\mathcal{N}_{k_{1}\vee k_{2}}^{+}\left(u\right).\
G$ is then said to be _strongly connected_ if any two distinct vertices are
connected. The maximal connected subgraphs of $G$ are called _components_ of
$G$ and to denote that $G^{\prime}\subset G$ is a component of $G$ we write
$G^{\prime}\sqsubset G.$

An example of directed graph is the one which can be associated to a Markov
chain. In this case, $V$ coincides with the set of states of the chain and,
denoting by $P$ the transition matrix associated to the chain,
$E=E\left(P\right):=\left\\{\left(u,v\right)\in V\times V:P_{u,v}>0\right\\}.$
Then, the directed graph associated to the Markov chain with transition matrix
$P$ is denoted by $G\left(P\right).$ Hence, the Markov chain and therefore $P$
are said to be _irreducible_ if and only if $G\left(P\right)$ is strongly
connected.

In the following, if $e=\left(u,v\right)$ is an edge of a directed graph
$\left(V,E\right),$ we will occasionally note $\bar{e}$ for the edge
$\left(v,u\right)\in E.$

###  1.2 Description of the model and results

In the following, unless differently specified, we will be concerned only with
graphs $G$ being subgraphs of the complete directed graph
$\mathbf{G}=\left(\mathbf{V},\mathbf{E}\right)$ of finite order where
$\mathbf{E}:=\left(\mathbf{V}\times\mathbf{V}\right).$

A bond (or edge) configuration is a map $\mathbf{E}\ni
e\longmapsto\omega_{e}\in\left\\{0,1\right\\}$ so that a bond $e$ is said to
be _open_ if $\omega_{e}=1.$ Setting $\forall
u,v\in\mathbf{V},\omega_{u,v}:=\omega_{e}\delta_{e,\left(u,v\right)}$ and
defining

$\Omega:=\left\\{\omega\in\left\\{0,1\right\\}^{\mathbf{E}}:\forall
u\in\mathbf{V}\ ,\ \omega_{u,u}=1\right\\}\ ,$ (5)

we define

$\Omega\ni\omega\longmapsto
E\left(\omega\right):=\left\\{e\in\mathbf{E}:\omega_{e}=1\right\\}\in\mathcal{P}\left(\mathbf{E}\right)$
(6)

and consequently

$G\left(\omega\right):=G\left(E\left(\omega\right)\right)\subseteq\mathbf{G}\
.$ (7)

We also set, $\forall v\in\mathbf{V},$

$\displaystyle E_{v}^{-}\left(\omega\right)$ $\displaystyle:=\left\\{e\in
E\left(\omega\right):e=\left(u,v\right),\ u\in\mathbf{V}\right\\}\ ,$ (8)
$\displaystyle E_{v}^{+}\left(\omega\right)$ $\displaystyle:=\left\\{e\in
E\left(t\right):e=\left(v,u\right),\ u\in\mathbf{V}\right\\}\ ,$ (9)
$\displaystyle\mathcal{N}^{\pm}\left(v,\omega\right)$
$\displaystyle:=\left(\cup_{e\in
E_{v}^{\pm}\left(\omega\right)}V\left(e\right)\right)\ ,$ (10) $\displaystyle\
\mathcal{N}_{1}^{+}\left(v,\omega\right)$
$\displaystyle:=\mathcal{N}^{+}\left(v,\omega\right)\ ;\
\mathcal{N}_{k}^{+}\left(v,\omega\right):=\mathcal{N}^{+}\left(\mathcal{N}_{k-1}^{+}\left(v,\omega\right)\right)\
,\;k\geq 2\ .$ (11)

Moreover, given a $\Omega$-valued random sequence
$\left\\{\omega\left(t\right)\right\\}_{t\geq 0}$ we set
$E\left(t\right):=E\left(\omega\left(t\right)\right),G\left(t\right):=G\left(\omega\left(t\right)\right)$
as well as, $\forall
v\in\mathbf{V},E_{v}^{\pm}\left(t\right):=E_{v}^{\pm}\left(\omega\left(t\right)\right)$
and $\forall k\geq
1,\mathcal{N}_{k}^{\pm}\left(v,t\right):=\mathcal{N}_{k}^{\pm}\left(v,\omega\left(t\right)\right).$

A _belief configuration_ is a map $\mathbf{V}\ni v\longmapsto
X_{v}\in\left[0,1\right].$ We set $\Xi:=\left[0,1\right]^{\mathbf{V}}$ and
consider the sequence $\left\\{X\left(t\right)\right\\}_{t\geq 0}$
representing the beliefs evolution in time.

####  1.2.1 Beliefs dynamics

The beliefs evolution is given by the system of equations

$\left\\{\begin{array}[c]{l}\begin{array}[c]{ll}X_{v}\left(t+1\right):=&\frac{\sum_{u\in\mathcal{N}^{-}\left(v,t\right)}r_{u,v}X_{u}\left(t\right)}{\sum_{u\in\mathcal{N}^{-}\left(v,t\right)}r_{u,v}}=\frac{\sum_{u\in\mathbf{V}}r_{u,v}\omega_{u,v}\left(t\right)X_{u}\left(t\right)}{\sum_{u\in\mathbf{V}}r_{u,v}\omega_{u,v}\left(t\right)}\\\
&=X_{v}\left(t\right)+\frac{\sum_{u\in\mathbf{V}}r_{u,v}\omega_{u,v}\left(t\right)\left[X_{u}\left(t\right)-X_{v}\left(t\right)\right]}{\sum_{u\in\mathbf{V}}r_{u,v}\omega_{u,v}\left(t\right)}\end{array}\\\
X_{v}\left(0\right)=X_{v}^{0}\end{array}\right.\;,\;v\in\mathbf{V}\ ,\ t\geq
0\ ,$ (12)

which, by (8), can be rewritten as

$\left\\{\begin{array}[c]{l}\begin{array}[c]{ll}X_{v}\left(t+1\right)=&\frac{\sum_{e\in
E_{v}^{-}\left(t\right)}r_{e}\sum_{u\in\mathbf{V}}\delta_{e,\left(u,v\right)}X_{u}\left(t\right)}{\sum_{e\in
E_{v}^{-}\left(t\right)}r_{e}}=X_{v}\left(t\right)+\frac{\sum_{e\in
E_{v}^{-}\left(t\right)}r_{e}\Delta_{e}X\left(t\right)}{\sum_{e\in
E_{v}^{-}\left(t\right)}r_{e}}\\\ &=X_{v}\left(t\right)+\frac{\sum_{e\in
E_{v}^{-}}r_{e}\omega_{e}\left(t\right)\Delta_{e}X\left(t\right)}{\sum_{e\in
E_{v}^{-}}r_{e}\omega_{e}\left(t\right)}=\frac{\sum_{e\in
E_{v}^{-}}r_{e}\omega_{e}\left(t\right)\sum_{u\in\mathbf{V}}\delta_{e,\left(u,v\right)}X_{u}\left(t\right)}{\sum_{e\in
E_{v}^{-}}r_{e}\omega_{e}\left(t\right)}\end{array}\\\
X_{v}\left(0\right)=X_{v}^{0}\end{array}\right.\;,\;v\in\mathbf{V}\ ,\ t\geq
0\ ,$ (13)

where, $\forall e\in\mathbf{E},$

$\Delta_{e}X\left(t\right):=\left(X_{u}\left(t\right)-X_{v}\left(t\right)\right)\mathbf{1}_{\left(v,u\right)}\left(e\right)$
(14)

and $r_{e}\in\left[0,1\right]$ is the communication rate between the agents
labelled by the the vertices incident in $e,$ namely
$r_{u,v}:=r_{e}\delta_{e,\left(u,v\right)}$ represents the confidence level
assigned by the agent $v$ to the belief of the agent $u.$

####  1.2.2 Communication channels dynamics

For any $v\in\mathbf{V},$ the $\mathcal{P}\left(\mathbf{E}\right)$-valued
sequence $\left\\{E_{v}^{-}\left(t\right)\right\\}_{t\geq 0},$ as well as the
$\mathbf{G}$-valued sequence $\left\\{G\left(t\right)\right\\}_{t\geq 0},$ are
constructed by $\left\\{\omega\left(t\right)\right\\}_{t\geq 0}$ through the
random evolution described by the collection of regular conditional
probabilities

$\displaystyle\mathbb{P}\left\\{\omega_{e}\left(t+1\right)=\omega_{e}^{\prime}|X\left(t\right)\right\\}$
$\displaystyle=\delta_{\omega_{e}^{\prime},1}p\left(\left|\Delta_{e}X\left(t\right)\right|\right)+\delta_{\omega_{e}^{\prime},0}\left(1-p\left(\left|\Delta_{e}X\left(t\right)\right|\right)\right)$
(15)
$\displaystyle=\omega_{e}^{\prime}p\left(\left|\Delta_{e}X\left(t\right)\right|\right)+\left(1-\omega_{e}^{\prime}\right)\left(1-p\left(\left|\Delta_{e}X\left(t\right)\right|\right)\right)\
,\;e\in\mathbf{E}\ ,$

where $X\left(t\right)\in\Xi$ is the belief configuration at time $t\geq 0$
and $p:\left[0,1\right]\circlearrowleft$ is a nonincreasing function such that
$p\left(0\right)=1.$

Notice that, for any $t\geq 0,$ given $e,f\in\mathbf{E}$ such that
$\Delta_{e}X\left(t\right)=\Delta_{f}X\left(t\right),$ the r.v.’s
$\omega_{e}\left(t+1\right)$ and $\omega_{\bar{e}}\left(t+1\right)$ have the
same conditional probabilities w.r.t. $X\left(t\right).$ In particular this
holds for $e=\left(u,v\right)$ and $f=\bar{e}=\left(v,u\right),$ although the
edge configurations $\omega_{e}$ and $\omega_{\bar{e}}$ are different in
general.

In the following we will consider $\forall e\in\mathbf{E},r_{e}>0.$ As a
matter of fact, since the $r_{e}$’s are fixed, we can restrict ourselves to
consider instead of $\mathbf{G}$ each component of its spanning subgraph
$\mathbf{G}_{r}:=G\left(\mathbf{E}_{r}\right),$ where

$\mathbf{E}_{r}:=\left\\{e\in\mathbf{E}:r_{e}>0\right\\}\ ,$ (16)

because, by (12), if $G_{1},G_{2}\sqsubset\mathbf{G}_{r}$ the evolution of the
beliefs labeled by the vertices of $G_{1}$ is never affected by those labeled
by the vertices of $G_{2}.$

Moreover, since it is reasonable to assume that the agents put maximal
confidence on their own beliefs, we can set $r_{\left(u,u\right)}=1,$ for any
$u\in\mathbf{V.}$

####  1.2.3 Results for the finite system

Let $\mathbf{V}$ be of finite order. If $X^{0}\in\Xi$ is such that $\forall
v\in\mathbf{V},X_{v}^{0}=x\in\left[0,1\right],$ then by (12)
$X\left(t\right)=X^{0},\forall t\geq 0.$ Hence these configuration, called
_consensus_ configurations, are stationary for the system evolution.

In the next section we will prove the following result.

###### Theorem 1

The agents system reaches consensus for any realization of the initial value
of the noise $\omega_{0}\in\Omega$ and any initial configuration
$X^{0}\in\left\\{X\in\Xi:\Gamma\left(X\right)>0\right\\}.$

Where $\Gamma:\Xi\longrightarrow[0,1)$ is defined in (41).

Moreover, we will also prove that, the random sequence
$\left\\{\left(X\left(t\right),\omega\left(t\right)\right)\right\\}_{t\geq 0}$
started at
$\left(X^{0},\omega^{0}\right)\in\left\\{X\in\Xi:\Gamma\left(X\right)>0\right\\}\times\Omega$
in the limit as $t$ tends to infinity weakly converges at geometric rate to
$\left(X^{\infty},\bar{1}\right),$ where $X^{\infty}\in\Xi$ is such that, for
any $v\in\mathbf{V},X_{v}^{\infty}=x,$ for some $x\in\left[0,1\right],$ and
$\bar{1}$ is the element of $\Omega$ such that all its entries are equal to
$1.$ As a byproduct of this result we will obtain that the random sequence
$\left\\{\left(X\left(2t\right),X\left(2t+1\right)\right)\right\\}_{t\geq 0}$
started at
$\left(X^{0},X^{1}\right)\in\left\\{X\in\Xi:\Gamma\left(X\right)>0\right\\}\times\Xi$
will also weakly converge, in the limit as $t$ tends to infinity, to
$\left(X^{\infty},X^{\infty}\right)$ at geometric rate.

####  1.2.4 Results for the large system

Let
$\mathbf{V}:=\mathbb{N},\mathbf{E}:=\left\\{\left(u,v\right)\in\mathbb{N}\times\mathbb{N}\right\\}$
and set $\Xi:=\left[0,1\right]^{\mathbb{N}}$ and $\Omega$ as in 5. Given
$N\in\mathbb{N},$ let
$\mathbf{V}_{N}:=\left\\{1,..,N\right\\}\subset\mathbb{N}$ and
$\mathbf{E}_{N}:=\left\\{\left(u,v\right)\in\mathbf{V}_{N}\times\mathbf{V}_{N}\right\\}$
we denote by $X_{N}:=\left(X_{1},..,X_{N}\right)$ the element of
$\Xi_{N}:=\left[0,1\right]^{N}$ representing the restriction of the beliefs
configuration $X\in\Xi$ to $\mathbf{V}_{N}$ and by $\omega_{N}$ the
restriction of the configuration $\omega\in\Omega$ to
$\Omega_{N}:=\left\\{0,1\right\\}^{\mathbf{E}_{N}}$ and, by (6), if
$E:=E\left(\omega\right),$ we set $E_{N}:=E\cap\mathbf{E}_{N}.$

Assuming that
$R:=\left\\{\left(u,v\right)\in\mathbf{V}\times\mathbf{V}:r_{u,v}>0\right\\}$
is finite, in the last section we prove that the random sequence
$\left\\{\left(X\left(2t\right),X\left(2t+1\right)\right)\right\\}_{t\geq 0}$
started at$\left(X^{0},X^{1}\right)\in\left\\{X^{\prime}\in\Xi\;:\;\right.$
$\left.\inf_{N\in\mathbb{N}}\Gamma\left(X_{N}^{\prime}\right)>0\right\\}\times\Xi,$
in the limit as $t$ tends to infinity, weakly converges at geometric rate to
$\left(X^{\infty},X^{\infty}\right).$

##  2 Finite system evolution

Therefore, the evolution of the system is given by the following algorithm:

###### Algorithm 2

  1. 1.

Label the elements of $\mathbf{V}$ from $1$ to $N$ in such a way that
$\mathbf{V}:=\left\\{1,..,N\right\\}$ and consequently label
$\left(i,j\right)$ the elements of $\mathbf{E}:=\mathbf{V}\times\mathbf{V},$
then go to the next step.

  2. 2.

Set
$t:=0,X\left(0\right)=\left(X_{1}\left(0\right),..,X_{N}\left(0\right)\right):=\left(X_{1}^{0},..,X_{N}^{0}\right)\in\left[0,1\right]^{N},\omega\left(0\right):=\left\\{\omega_{i,j}^{0}\right\\}_{\left(i,j\right)\in\mathbf{E}}\in\left\\{0,1\right\\}^{\mathbf{E}}$
such that $\forall i=1,..,N,\omega_{i,i}^{0}=1,$ and go to the next step.

  3. 3.

Set $i:=1$ and go to the next step.

    1. (a)

Set $j:=1$ and go to the next step.

    2. (b)

Compute
$p_{i,j}\left(t\right):=p\left(\left|X_{i}\left(t\right)-X_{j}\left(t\right)\right|\right)$
and form the vector

$\mathbf{p}\left(t\right):=\left(p_{1,1}\left(t\right),..,p_{1,N}\left(t\right),p_{2,1}\left(t\right),..,p_{2,N}\left(t\right)..,p_{i,1}\left(t\right),..,p_{i,j}\left(t\right)\right)$
(17)

and go to the step.

    3. (c)

Set $j:=j+1.$ If $j+1\leq N$ go back to step 3.b, otherwise go to the next
step.

    4. (d)

Set $i:=i+1.$ If $i+1\leq N$ go back to step 3.a, otherwise go to the next
step.

  4. 4.

Set $i:=1$ and go to the next step.

    1. (a)

Compute $X_{i}\left(t+1\right)$ according to (12) and form the vector
$X\left(t+1\right):=\left(X_{1}\left(t+1\right),\right.$
$\left...,X_{i}\left(t+1\right)\right),$ then go to the next step.

    2. (b)

Set $i:=i+1.$ If $i+1\leq N$ go back to step 4.a, otherwise go to the next
step.

  5. 5.

Read
$X\left(t\right)=\left(X_{1}\left(t\right),..,X_{N}\left(t\right)\right).$ If
$X\left(t+1\right)=X\left(t\right)$ stop, otherwise go to the next step.

  6. 6.

Set $i:=1$ and go to the next step.

    1. (a)

Set $j=1$ and go to the next step.

    2. (b)

Read the $p_{i,j}\left(t\right)$ entry of the vector
$\mathbf{p}\left(t\right)\mathbf{.}$ Sample a random variable $U$ uniformly
distributed on $\left[0,1\right].$ If $U\leq p_{i,j}\left(t\right)$ then set
$\omega_{i,j}\left(t+1\right):=\omega_{i,j}\left(t\right)$ if
$\omega_{i,j}\left(t\right)=1,$ otherwise set
$\omega_{i,j}\left(t+1\right):=1-\omega_{i,j}\left(t\right).$ If
$U>p_{i}\left(t\right)$ then set
$\omega_{i,j}\left(t+1\right):=1-\omega_{i,j}\left(t\right)$ if
$\omega_{i,j}\left(t\right)=1,$ otherwise set
$\omega_{i,j}\left(t+1\right):=\omega_{i,j}\left(t\right).$ Form the vector

$\omega\left(t+1\right):=\left(\omega_{1,1}\left(t+1\right),..,\omega_{1,N}\left(t+1\right),..,\omega_{i,1}\left(t+1\right),..,\omega_{i,j}\left(t+1\right)\right)$
(18)

and go to the next step.

    3. (c)

Set $j:=j+1.$ If $j+1\leq N$ go back to step 6.b, otherwise go to the next
step.

    4. (d)

Set $i:=i+1.$ If $i+1\leq N$ go back to step 6.a, otherwise go to the next
step.

  7. 7.

set
$t:=t+1,X\left(0\right):=X\left(t+1\right),\omega\left(0\right):=\omega\left(t+1\right)$
and go to step 3.

In terms of stochastic process the system evolution can be described as
follows.

Let $\Omega\ni\omega\longmapsto P\left(\omega\right)\in
St\left(\mathbf{V}\right)$ the stochastic matrix-valued function on
$\mathbb{R}^{\mathbf{V}}$ such that, for any $\omega\in\Omega,$

$P_{v,u}\left(\omega\right):=\frac{\sum_{e\in
E_{v}^{-}\left(\omega\right)}\delta_{e,\left(u,v\right)}r_{e}}{\sum_{e\in
E_{v}^{-}\left(\omega\right)}r_{e}}=\frac{r_{u,v}\mathbf{1}_{\mathcal{N}^{-}\left(v,\omega\right)}\left(u\right)}{\sum_{u\in\mathcal{N}^{-}\left(v,\omega\right)}r_{u,v}}=\frac{r_{u,v}\omega_{u,v}}{\sum_{u\in\mathbf{V}}r_{u,v}\omega_{u,v}}\
,\;u,v\in\mathbf{V\ .}$ (19)

###### Remark 3

We remark that, given $\omega\in\Omega,v\in\mathbf{V},$ by (19)
$u\in\mathcal{N}^{-}\left(v,\omega\right)$ iff $P_{v,u}\left(\omega\right)>0.$
Therefore, denoting by
$\overline{G}\left(\omega\right):=G\left(P\left(\omega\right)\right)$ the
graph associated to $P\left(\omega\right),$ this is the spanning graph of
$\overline{E}\left(\omega\right):=\left\\{e\in\mathbf{E}:e=\left(u,v\right)\
if\ \left(v,u\right)\in E\left(\omega\right)\right\\}.$

Considering $\Xi:=\left[0,1\right]^{\mathbf{V}}\subset\mathbb{R}^{\mathbf{V}}$
endowed with the norm
$\left\|X\right\|:=\left\|X\right\|_{\infty}=\sup_{v\in\mathbf{V}}\left|X_{v}\right|,$
let $\left(\mathfrak{X},\mathcal{F}\right)$ be the measurable space such that
$\mathfrak{X}:=\Xi\times\Omega$ and, if $\mathbf{V}$ is a finite set,
$\mathcal{F}:=\mathcal{B}\left(\Xi\right)\otimes\mathcal{P}\left(\Omega\right).$

For any $\omega\in\Omega,P\left(\omega\right)\in BL\left(\Xi\right),$
therefore we set

$\mathfrak{X}\ni\left(X,\omega\right)\longmapsto\mathcal{T}_{v}\left(X,\omega\right):=\sum_{u\in\mathbf{V}}P_{v,u}\left(\omega\right)X_{u}\in\left[0,1\right]\
,$ (20)

and consider the measurable map

$\mathfrak{X}\ni\left(X,\omega\right)\longmapsto\mathcal{T}\left(X,\omega\right):=\left\\{\mathcal{T}_{v}\left(X,\omega\right)\right\\}_{v\in\mathbf{V}}\in\Xi\
.$ (21)

Defining, by (15), the probability kernel from
$\left(\Xi,\mathcal{B}\left(\Xi\right)\right)$ to
$\left(\Omega,\mathcal{P}\left(\Omega\right)\right)$

$\displaystyle\mathfrak{X}\ni\left(X,\omega\right)\longmapsto\Pi\left(\omega|X\right)$
$\displaystyle:=\prod\limits_{e\in\mathbf{E}}\left[\delta_{\omega_{e},1}p\left(\left|\Delta_{e}X\right|\right)+\delta_{\omega_{e},0}\left(1-p\left(\left|\Delta_{e}X\right|\right)\right)\right]$
(22)
$\displaystyle=\prod\limits_{e\in\mathbf{E}}\left[\omega_{e}p\left(\left|\Delta_{e}X\right|\right)+\left(1-\omega_{e}\right)\left(1-p\left(\left|\Delta_{e}X\right|\right)\right)\right]\in\left[0,1\right]\
,$

we introduce the positive linear operator on $BM\left(\mathfrak{X}\right)$
such that

$BM\left(\mathfrak{X}\right)\ni\varphi\longmapsto\mathfrak{T}\varphi\left(X,\omega\right):=\sum_{\omega^{\prime}\in\Omega}\varphi\left(\mathcal{T}\left(X,\omega\right),\omega^{\prime}\right)\Pi\left(\omega^{\prime}|X\right)\in
BM\left(\mathfrak{X}\right)\ .$ (23)

Let $\mathbb{P}_{0}$ be the probability distribution on
$\left(\mathfrak{X}^{\mathbb{Z}_{+}},\mathfrak{C}\right),$ where
$\mathfrak{C}:=\mathcal{C}\left(\Xi\right)\otimes\mathcal{C}\left(\Omega\right),$
describing the homogeneous discrete time Markov process started at
$\left(X^{0},\omega^{0}\right)$ defined by the one-step transition probability
kernel associated to $\mathfrak{T}.$ We denote by
$\left\\{\chi_{t}\right\\}_{t\geq 0}$ the random process on
$\left(\mathfrak{X}^{\mathbb{Z}_{+}},\mathfrak{C},\mathbb{P}_{0}\right)$ such
that, $\forall t\geq 0,$

$\mathfrak{X}^{\mathbb{Z}_{+}}\ni\mathbf{x}\longmapsto\chi_{t}\left(\mathbf{x}\right)=\left(X\left(t\right),\omega\left(t\right)\right)\in\mathfrak{X}$
(24)

and by $\left\\{\mathfrak{F}_{t}\right\\}_{t\geq 0},$ with
$\mathfrak{F}_{t}:=\bigvee\limits_{s=0}^{t}\chi_{s}^{-1}\left(\mathcal{B}\left(\Xi\right)\otimes\mathcal{P}\left(\Omega\right)\right),$
the associated natural filtration. Therefore, denoting by $\mathbb{E}_{0}$ the
expectation value w.r.t. $\mathbb{P}_{0},$ for any bounded measurable function
$\varphi$ on $\mathfrak{X,}$

$\mathbb{E}_{0}\left[\varphi\circ\chi_{t+1}|\mathfrak{F}_{t}\right]=\mathbb{E}_{0}\left[\varphi\circ\chi_{t+1}|\chi_{t}\right]=\left(\mathfrak{T}\varphi\right)\left(\chi_{t}\right)\;\mathbb{P}_{0}-a.s.\
.$ (25)

Notice that, by (20),
$\mathfrak{T}:C\left(\mathfrak{X},\mathbb{R}\right)\circlearrowleft,$ that is
$\left\\{\chi_{t}\right\\}_{t\geq 0}$ is a Feller process.

Setting
$\pi_{\omega}:\mathfrak{X}\longmapsto\Omega,\pi_{X}:\mathfrak{X}\longmapsto\Xi,$
we denote by $\left\\{\mathfrak{w}_{t}\right\\}_{t\geq
0},\left\\{\mathfrak{x}_{t}\right\\}_{t\geq 0}$ the random processes on
$\left(\mathfrak{X}^{\mathbb{Z}_{+}},\mathfrak{C},\mathbb{P}_{0}\right)$ such
that, $\forall t\geq 0,$

$\mathfrak{X}^{\mathbb{Z}_{+}}\ni\mathbf{x}\longmapsto\mathfrak{w}_{t}\left(\mathbf{x}\right):=\pi_{\omega}\circ\chi_{t}\left(\mathbf{x}\right)=\omega\left(t\right)\in\Omega$
(26)

and

$\mathfrak{X}^{\mathbb{Z}_{+}}\ni\mathbf{x}\longmapsto\mathfrak{x}_{t}\left(\mathbf{x}\right):=\pi_{X}\circ\chi_{t}\left(\mathbf{x}\right)=X\left(t\right)\in\Xi\
.$ (27)

Hence, $\left\\{\chi_{t}\right\\}_{t\geq 0}$ can be represented as
$\left\\{\left(\mathfrak{x}_{t},\mathfrak{w}_{t}\right)\right\\}_{t\geq 0}.$
We also set $\left\\{\mathfrak{F}_{t}^{\omega}\right\\}_{t\geq 0},$ with
$\mathfrak{F}_{t}^{\omega}:=\bigvee\limits_{s=0}^{t}\mathfrak{w}_{s}^{-1}\left(\mathcal{P}\left(\Omega\right)\right),$
and $\left\\{\mathfrak{F}_{t}^{X}\right\\}_{t\geq 0},$ with
$\mathfrak{F}_{t}^{X}:=\bigvee\limits_{s=0}^{t}\mathfrak{x}_{s}^{-1}\left(\mathcal{B}\left(\Xi\right)\right).$

###### Remark 4

Notice that neither $\left\\{\mathfrak{w}_{t}\right\\}_{t\geq 0}$ nor
$\left\\{\mathfrak{x}_{t}\right\\}_{t\geq 0}$ are Markov processes. Indeed, by
(15), for any $t\geq 0,\mathfrak{w}_{t+1}$ is independent of
$\mathfrak{w}_{t}.$ Moreover, since $\forall t\geq 0,\mathfrak{F}_{t}^{X}$ and
$\mathfrak{F}_{t}^{\omega}$ are sub$\sigma$algebras of $\mathfrak{F}_{t},$ for
any $B\in\mathcal{P}\left(\Omega\right),$

$\displaystyle\mathbb{P}_{0}\left(\left\\{\mathfrak{w}_{t+1}\in
B\right\\}|\mathfrak{F}_{t}^{\omega}\right)$
$\displaystyle=\mathbb{E}_{0}\left[\mathbb{E}_{0}\left[\mathbf{1}_{B}\circ\pi_{\omega}\circ\chi_{t+1}|\mathfrak{F}_{t}\right]|\mathfrak{F}_{t}^{\omega}\right]$
(28)
$\displaystyle=\mathbb{E}_{0}\left[\mathbb{E}_{0}\left[\mathbf{1}_{B}\circ\pi_{\omega}\circ\chi_{t+1}|\chi_{t}\right]\mathfrak{F}_{t}^{\omega}\right]$
$\displaystyle=\mathbb{E}_{0}\left[\mathfrak{T}\left(\mathbf{1}_{B}\circ\pi_{\omega}\right)\left(\chi_{t}\right)|\mathfrak{F}_{t}^{\omega}\right]$
$\displaystyle=\mathbb{E}_{0}\left.\left[\sum_{\omega^{\prime}\in\Omega}\mathbf{1}_{B}\left(\omega^{\prime}\right)\Pi\left(\omega^{\prime}|\mathfrak{x}_{t}\right)\right|\mathfrak{F}_{t}^{\omega}\right]$
$\displaystyle=\mathbb{E}_{0}\left.\left[\sum_{\omega^{\prime}\in
B}\Pi\left(\omega^{\prime}|\mathfrak{x}_{t}\right)\right|\mathfrak{F}_{t}^{\omega}\right]\neq\mathbb{P}_{0}\left(\left\\{\mathfrak{w}_{t+1}\in
B\right\\}|\mathfrak{w}_{t}\right)$
$\displaystyle=\mathbb{P}_{0}\left\\{\mathfrak{w}_{t+1}\in
B\right\\}=\left(\mathfrak{T}^{t+1}\mathbf{1}_{B}\circ\pi_{\omega}\right)\left(X^{0},\omega^{0}\right)$

while, by (15), (23) and (25), $\forall\varphi\in
BM\left(\Xi,\mathbb{R}\right),$ since for any $t\geq 0,\mathfrak{F}_{t}^{X}$
is a sub$\sigma$algebra of $\mathfrak{F}_{t},$

$\displaystyle\mathbb{E}_{0}\left[\varphi\circ\mathfrak{x}_{t+1}|\mathfrak{F}_{t}^{X}\right]$
$\displaystyle=\mathbb{E}_{0}\left[\varphi\circ\pi_{X}\circ\chi_{t+1}|\mathfrak{F}_{t}^{X}\right]=\mathbb{E}_{0}\left[\mathbb{E}_{0}\left[\varphi\circ\pi_{X}\circ\chi_{t+1}|\mathfrak{F}_{t}\right]|\mathfrak{F}_{t}^{X}\right]$
(29)
$\displaystyle=\mathbb{E}_{0}\left[\mathbb{E}_{0}\left[\varphi\circ\pi_{X}\circ\chi_{t+1}|\chi_{t}\right]|\mathfrak{F}_{t}^{X}\right]=\mathbb{E}_{0}\left[\left(\mathfrak{T}\left(\varphi\circ\pi_{X}\right)\right)\left(\chi_{t}\right)|\mathfrak{F}_{t}^{X}\right]$
$\displaystyle=\sum_{\omega^{\prime}\in\Omega}\sum_{\mathfrak{w}_{t}\in\Omega}\varphi\left(\mathcal{T}\left(\mathfrak{x}_{t},\mathfrak{w}_{t}\right)\right)\Pi\left(\omega^{\prime}|\mathfrak{x}_{t}\right)\Pi\left(\mathfrak{w}_{t}|\mathfrak{x}_{t-1}\right)$
$\displaystyle=\sum_{\omega\in\Omega}\varphi\left(\mathcal{T}\left(\mathfrak{x}_{t},\omega\right)\right)\Pi\left(\omega|\mathfrak{x}_{t-1}\right)=\mathbb{E}_{0}\left[\varphi\circ\mathfrak{x}_{t+1}|\mathfrak{x}_{t},\mathfrak{x}_{t-1}\right]\;\mathbb{P}_{0}-a.s.\
.$

In particular, by (29), we get that $\left\\{\mathfrak{y}_{t}\right\\}_{t\geq
0}$ such that $\forall t\geq
0,\mathfrak{y}_{t}:=\left(\mathfrak{x}_{2t},\mathfrak{x}_{2t+1}\right)$ is a
homogeneous Markov process on
$\left(\mathfrak{X}^{\mathbb{Z}_{+}},\mathfrak{C},\mathbb{P}_{0}\right).$
Indeed, denoting by $\left\\{\mathfrak{F}_{t}^{\mathfrak{y}}\right\\}_{t\geq
0}$ the filtration generated by $\left\\{\mathfrak{y}_{t}\right\\}_{t\geq 0},$
since $\forall t\geq
0,\mathfrak{F}_{t}^{\mathfrak{y}}=\mathfrak{F}_{2t+1}^{X},$ for any bounded
measurable function $\varphi$ on $\Xi^{2},$

$\displaystyle\mathbb{E}\left[\varphi\circ\mathfrak{y}_{t+1}|\mathfrak{F}_{t}^{\mathfrak{y}}\right]$
$\displaystyle=\mathbb{E}\left[\varphi\left(\mathfrak{x}_{2t+2},\mathfrak{x}_{2t+3}\right)|\mathfrak{F}_{2t+1}^{X}\right]=\mathbb{E}\left[\varphi\left(\mathfrak{x}_{2t+2},\mathfrak{x}_{2t+3}\right)|\mathfrak{x}_{2t+1},\mathfrak{x}_{2t}\right]$
(30)
$\displaystyle=\mathbb{E}\left[\varphi\circ\mathfrak{y}_{t+1}|\mathfrak{y}_{t}\right]\
.$

Therefore, the transition operator associated to
$\left\\{\mathfrak{y}_{t}\right\\}_{t\geq 0}$ is

$\left(\mathbf{T}\varphi\right)\left(X_{1},X_{2}\right):=\sum_{\omega,\omega^{\prime}\in\Omega}\varphi\left(\mathcal{T}\left(X_{2},\omega\right),\mathcal{T}\left(\mathcal{T}\left(X_{2},\omega\right),\omega^{\prime}\right)\right)\Pi\left(\omega|X_{1}\right)\Pi\left(\omega^{\prime}|X_{2}\right)$
(31)

and, setting

$\Xi^{2}\ni\left(X_{1},X_{2}\right)\longmapsto\pi_{i}\left(X_{1},X_{2}\right):=X_{1}\delta_{i,1}+X_{2}\delta_{i,2}\in\Xi\
,\ i=1,2\ ,$ (32)

for any bounded measurable function $\varphi$ on $\Xi,$ we have

$\displaystyle\mathbb{E}_{0}\left[\varphi\circ\mathfrak{x}_{t+1}|\mathfrak{x}_{t},\mathfrak{x}_{t-1}\right]$
$\displaystyle=\mathbb{E}_{0}\left[\left(\varphi\circ\pi_{1}\right)\left(\mathfrak{y}_{t+1}\right)|\mathfrak{y}_{t}\right]$
(33)
$\displaystyle=\mathbb{E}_{0}\left[\mathbf{T}\left(\varphi\circ\pi_{1}\right)\left(\mathfrak{y}_{t}\right)\right]\;\mathbb{P}_{0}-a.s.\;.$

###  2.1 Consensus

If $X^{0}\in\Xi$ is such that $\forall
v\in\mathbf{V},X_{v}^{0}=x\in\left[0,1\right],$ then by (12)
$X\left(t\right)=X^{0},\forall t\geq 0.$ Hence these configuration, called
_consensus_ configurations, are stationary for the system evolution.

We denote by

$\mathcal{I}:=\bigcup\limits_{x\in\left[0,1\right]}\mathcal{I}_{x}$ (34)

where

$\mathcal{I}_{x}:=\left\\{X\in\Xi:X_{v}=x\ ,\ \forall v\in\mathbf{V}\right\\}$
(35)

and by $\mathbb{M}:\Xi\longrightarrow\Xi$ the _consensus projection map_ ,
that is the map associating to each belief configuration $X$ the consensus
configuration $\mathbb{M}X$ such that $\forall
v\in\mathbf{V},\left(\mathbb{M}X\right)_{v}:=\frac{1}{\left|\mathbf{V}\right|}\sum_{u\in\mathbf{V}}X_{u}.$
It is easy to see that $\mathbb{M}$ is a projection operator on $\mathcal{I},$
moreover an orthogonal projection if $\Xi$ is endowed with the Euclidean
structure $\left\langle\cdot,\cdot\right\rangle$ of $\mathbb{R}^{\mathbf{V}}.$
Indeed, $\forall X\in\Xi,\mathbb{M}^{2}X=\mathbb{M}X.$ Therefore, $\forall
X\in\Xi,$ we set

$dist\left(\mathcal{I},X\right):=\inf_{Y\in\mathcal{I}}\left\|X-Y\right\|\leq\left[\mathbb{I}-\mathbb{M}\right]X\
,$ (36)

where we denote by $\mathbb{I}$ the identity operator on
$\mathbb{R}^{\mathbf{V}}.$

Consequently, since if
$X=\mathbb{M}X,\forall\left(u,v\right)\in\mathbf{E},X_{u}-X_{v}=0,$ we can
modify the algorithm 2 erasing the line 5 and adding the line

  1. 3.e

If
$\sum_{i=1}^{N}\sum_{j=1}^{N}\left(1-\delta_{i,j}\right)p_{i,j}\left(t\right)=N\left(N-1\right)$
stop, otherwise proceed to the next step.

###  2.2 Invariant measures for $\mathfrak{T}$ and $\mathbf{T}$

Setting $\mathcal{X}:=\left(\mathbb{I}-\mathbb{M}\right)\Xi,$ we can represent
$\Xi$ as $\mathcal{I\oplus X}.$ Moreover, for any
$\omega\in\Omega,\mathcal{I}$ is invariant under
$\mathcal{T}\left(\cdot,\omega\right),$ since $X\in\Xi,$ by (21) we get
$\mathcal{T}\left(\mathbb{M}X,\omega\right)=\mathbb{M}X.$ Therefore

$\mathcal{T}\left(X,\omega\right)=\mathcal{T}\left(\mathbb{M}X+\left(\mathbb{I-M}\right)X,\omega\right)=\mathbb{M}X+\mathcal{T}\left(\left(\mathbb{I-M}\right)X,\omega\right)\
.$ (37)

Moreover, by (22), for any
$\omega\in\Omega,\Pi\left(\omega|X\right)=\Pi\left(\omega|\left(\mathbb{I-M}\right)X\right).$
Hence, denoting by $\delta_{\bar{1}}^{\omega}$ the Dirac measure at $\bar{1},$
by the definition of $p$ and by (15), given
$X\in\mathcal{I},\forall\omega\in\Omega,\Pi\left(\omega|X\right)=\prod\limits_{e\in\mathbf{E}}\delta_{\omega_{e},1}=\delta_{\bar{1}}^{\omega}.$

Denoting by $\delta^{X}$ the probability measure on
$\left(\Xi,B\left(\Xi\right)\right)$ concentrated on the beliefs configuration
$X\in\Xi,$ let $\delta_{\mathcal{I}}^{X}$ be the probability measure on
$\left(\Xi,\mathcal{B}\left(\Xi\right)\right)$ putting mass $1$ on the
configuration $X\in\mathcal{I}.$ It is easy to see that the probability
measure $\delta_{\mathcal{I}}^{X}\otimes\delta_{\bar{1}}^{\omega}$ on
$\left(\mathfrak{X},\mathcal{F}\right)$ is invariant for the evolution given
by $\mathfrak{T}.$ Indeed, if $X\in\mathcal{I},$ by (34) and (35), there
exists $x\in\left[0,1\right]$ such that $X\in\mathcal{I}_{x}.$ Hence, by (20),
for any $\omega\in\Omega,\mathcal{T}\left(X,\omega\right)=X.$ Therefore, given
any bounded measurable function $\varphi$ on $\mathfrak{X},$ by (21),
$\forall\omega,\omega^{\prime}\in\Omega,\delta_{\mathcal{I}}^{X}\left[\varphi\left(\mathcal{T}\left(\cdot,\omega\right),\omega^{\prime}\right)\right]=\delta_{\mathcal{I}}^{X}\left[\varphi\left(\cdot,\omega^{\prime}\right)\right].$
Thus, by (23),

$\displaystyle\delta_{\mathcal{I}}^{X}\otimes\delta_{\bar{1}}^{\omega}\left[\mathfrak{T}\varphi\right]$
$\displaystyle=\delta_{\bar{1}}^{\omega}\left[\delta_{\mathcal{I}}^{X}\left[\sum_{\omega^{\prime}\in\Omega}\varphi\left(\mathcal{T}\left(\cdot,\omega\right),\omega^{\prime}\right)\Pi\left(\omega^{\prime}|\cdot\right)\right]\right]$
(38)
$\displaystyle=\delta_{\bar{1}}^{\omega}\left[\sum_{\omega^{\prime}\in\Omega}\delta_{\mathcal{I}}^{X}\left[\varphi\left(\mathcal{T}\left(\cdot,\omega\right),\omega^{\prime}\right)\Pi\left(\omega^{\prime}|\cdot\right)\right]\right]$
$\displaystyle=\delta_{\bar{1}}^{\omega}\left[\sum_{\omega^{\prime}\in\Omega}\delta_{\mathcal{I}}^{X}\left[\varphi\left(\cdot,\omega^{\prime}\right)\prod\limits_{e\in\mathbf{E}}\delta_{\omega_{e}^{\prime},1}\right]\right]$
$\displaystyle=\delta_{\mathcal{I}}^{X}\otimes\delta_{\bar{1}}^{\omega}\left[\sum_{\omega^{\prime}\in\Omega}\varphi\left(\cdot,\omega^{\prime}\right)\prod\limits_{e\in\mathbf{E}}\delta_{\omega_{e}^{\prime},1}\right]$
$\displaystyle=\delta_{\mathcal{I}}^{X}\otimes\delta_{\bar{1}}^{\omega}\left[\varphi\left(\cdot,\bar{1}\right)\right]=\delta_{\mathcal{I}}^{X}\otimes\delta_{\bar{1}}^{\omega}\left[\varphi\right]\
.$

Thus the set $\mathfrak{I}_{\mathfrak{T}}$ of invariant probability measures
under $\mathfrak{T}$ is the weak limit of convex combinations of elements of
the set
$\left\\{\delta_{\bar{1}}^{\omega}\otimes\delta_{\mathcal{I}}^{X}\right\\}_{X\in\mathcal{I}}\subset\mathfrak{P}\left(\mathfrak{X},\mathcal{F}\right).$

Since for any $X_{2}\in\mathcal{I}$ and any
$\mu\in\mathfrak{P}\left(\Xi,\mathcal{B}\left(\Xi\right)\right),$ from (31) it
follows that

$\displaystyle\mu\otimes\delta_{\mathcal{I}}^{X_{2}}\left[\mathbf{T}\varphi\right]$
$\displaystyle=\int\mu\left(dX_{1}\right)\delta_{\mathcal{I}}^{X_{2}}\left[\sum_{\omega,\omega^{\prime}\in\Omega}\varphi\left(\mathcal{T}\left(\cdot,\omega\right),\mathcal{T}\left(\mathcal{T}\left(\cdot,\omega\right),\omega^{\prime}\right)\right)\Pi\left(\omega|X_{1}\right)\Pi\left(\omega^{\prime}|\cdot\right)\right]$
(39)
$\displaystyle=\int\mu\left(dX_{1}\right)\sum_{\omega,\omega^{\prime}\in\Omega}\varphi\left(X_{2},,\mathcal{T}\left(X_{2},\omega^{\prime}\right)\right)\Pi\left(\omega|X_{1}\right)\prod\limits_{e\in\mathbf{E}}\delta_{\omega_{e}^{\prime},1}$
$\displaystyle=\int\mu\left(dX_{1}\right)\sum_{\omega\in\Omega}\varphi\left(X_{2},,\mathcal{T}\left(X_{2},\bar{1}\right)\right)\Pi\left(\omega|X_{1}\right)$
$\displaystyle=\int\mu\left(dX_{1}\right)\sum_{\omega\in\Omega}\varphi\left(X_{2},X_{2}\right)\Pi\left(\omega|X_{1}\right)=\varphi\left(X_{2},X_{2}\right)\
,$

we have that the set $\mathfrak{I}_{\mathbf{T}}$ of invariant probability
measures under $\mathbf{T}$ is the weak limit of convex combinations of
elements of the set
$\left\\{\delta_{\mathcal{I}}^{\left(X,X\right)}\right\\}_{X\in\mathcal{I}}\subset\mathfrak{P}\left(\Xi^{2},\mathcal{B}\left(\Xi^{2}\right)\right),$
where
$\delta_{\mathcal{I}}^{\left(X,X\right)}:=\delta_{\mathcal{I}}^{X}\otimes\delta_{\mathcal{I}}^{X}.$

###  2.3 Emergence of consensus

###### Definition 5

Given $E\subseteq\mathbf{E},$ consider the spanning graph
$G\left(E\right)=\left(\mathbf{V},E\right).$ We call _pivots_ the elements $w$
of $\mathbf{V}$ such that $\mathcal{N}^{+}\left(w\right)=\mathbf{V}$ and
denote their collection by $\mathbf{P}\left(E\right).$ Moreover, for any
$\omega\in\Omega,$ we set
$\mathbf{P}\left(\omega\right):=\mathbf{P}\left(E\left(\omega\right)\right)$
and define
$\Omega_{\mathbf{P}}:=\left\\{\omega\in\Omega:\mathbf{P}\left(\omega\right)\neq\varnothing\right\\}.$

Let us denote by $\gamma$ the r.v.111Notice that $1-\gamma$ is the coefficient
of ergodicity [Se] of the transition probability matrix of the Markov chain on
$\mathbf{V}^{2}$ whose components are two independent versions of the Markov
chain defined by the transition probability matrix
$\left\\{P_{u,v}\left(\omega\right)\right\\}_{u,v\in\mathbf{V}}.$

$\Omega\ni\omega\longmapsto\gamma\left(\omega\right):=\min_{u,v\in\mathbf{V}\
:\ u\neq
v}\sum_{w,z\in\mathbf{V}}P_{u,w}\left(\omega\right)P_{v,z}\left(\omega\right)\wedge
P_{u,z}\left(\omega\right)P_{v,w}\left(\omega\right)\in[0,1)$ (40)

and by $\Gamma$ the r.v.

$\Xi\ni
X\longmapsto\Gamma\left(X\right):=\mathbb{E}\left[\gamma|X\right]=\sum_{\omega\in\Omega}\Pi\left(\omega|X\right)\gamma\left(\omega\right)\in[0,1)\
.$ (41)

###### Lemma 6

Given $\omega\in\Omega,\gamma\left(\omega\right)>0$ if and only if
$\mathbf{P}\left(\omega\right)$ is not empty.

Proof. Let $\omega\in\Omega$ be such that
$\mathbf{P}\left(\omega\right)\neq\varnothing.$ Denoting by
$u=u\left(\omega\right),v=v\left(\omega\right)$ the elements of $\mathbf{V}$
such that

$\displaystyle\sum_{w,z\in\mathbf{V}}P_{u,w}\left(\omega\right)P_{v,z}\left(\omega\right)\wedge
P_{u,z}\left(\omega\right)P_{v,w}\left(\omega\right)=$ (42)
$\displaystyle\min_{u^{\prime},u^{\prime\prime}\in\mathbf{V}}\sum_{w,z\in\mathbf{V}}P_{u^{\prime},w}\left(\omega\right)P_{u^{\prime\prime},z}\left(\omega\right)\wedge
P_{u^{\prime},z}\left(\omega\right)P_{u^{\prime\prime},w}\left(\omega\right)\
,$

for any $\bar{w}\in\mathbf{P}\left(\omega\right),$ we have

$\displaystyle\gamma\left(\omega\right)=\sum_{w,z\in\mathbf{V}}P_{u,w}\left(\omega\right)P_{v,z}\left(\omega\right)\wedge
P_{u,z}\left(\omega\right)P_{v,w}\left(\omega\right)=\sum_{w\in\mathbf{V}}P_{u,w}\left(\omega\right)P_{v,w}\left(\omega\right)+$
(43) $\displaystyle\sum_{w,z\in\mathbf{V}\ :\ w\neq
z}P_{u,w}\left(\omega\right)P_{v,z}\left(\omega\right)\wedge
P_{u,z}\left(\omega\right)P_{v,w}\left(\omega\right)\geq\left(P_{u,\bar{w}}\left(\omega\right)\wedge
P_{v,\bar{w}}\left(\omega\right)\right)^{2}>0\ .$

Conversely by (19), $\gamma\left(\omega\right)>0$ iff, for any
$u,v\in\mathbf{V}$ such that $u\neq
v,\mathcal{N}^{-}\left(u,\omega\right)\cap\mathcal{N}^{-}\left(v,\omega\right)\neq\varnothing,$
which is equivalent to say that $\gamma\left(\omega\right)>0$ implies that
there exists at least one $\bar{w}=\bar{w}\left(\omega\right)$ in $\mathbf{V}$
such that, by (50), $\mathcal{N}^{+}\left(\bar{w},\omega\right)=\mathbf{V},$
or, in other words, by Definition 5, that $\mathbf{P}\left(\omega\right)$ is
not empty.

In the next section we will prove the following result.

###### Theorem 7

The agents system reaches consensus for any realization of the initial value
of the noise $\omega_{0}\in\Omega$ and any initial configuration
$X^{0}\in\left\\{X\in\Xi:\Gamma\left(X\right)>0\right\\}.$

More precisely, for any
$X^{0}\in\left\\{X\in\Xi:\Gamma\left(X\right)>0\right\\}$ and any
$\omega_{0}\in\Omega,$ the sequence of probability measures
$\left\\{\mu_{0}^{t}\right\\}_{t\geq 0}$ on
$\left(\Xi,\mathcal{B}\left(\Xi\right)\right)$ such that

$\mathcal{B}\left(\Xi\right)\ni
A\longmapsto\mu_{0}^{t}\left(A\right):=\mathbb{P}_{0}\left\\{\mathbf{x}\in\mathfrak{X}^{\mathbb{Z}_{+}}:\pi_{X}\circ\chi_{t}\left(\mathbf{x}\right)\in
A\right\\}\in\left[0,1\right]\ ,$ (44)

converges to a probability measure $\mu_{0}^{\infty}$ supported on
$\mathcal{I}.$

###### Remark 8

We stress that this result give no information on the common value of the
beliefs when consensus is reached.

Given $X\in\Xi,$ let

$W\left(X\right):=\max_{u,v\in\mathbf{V}}\left|X_{u}-X_{v}\right|\ .$ (45)

Since

$W\left(\left[\mathbb{I}-\mathbb{M}\right]X\right)=W\left(X\right)\ ,$ (46)

$W$ is a seminorm on $\mathbb{R}^{\mathbf{V}}$ and therefore induces a norm on
$\mathbb{W}:=\mathbb{R}^{\mathbf{V}}/Ran\mathbb{M}.$

Hence, because $\mathbb{M}\mathcal{I}=\mathcal{I},$ for any $Y\in\mathcal{I},$
we have

$\left\|X-Y\right\|=W\left(X-Y\right)=W\left(\left[\mathbb{I}-\mathbb{M}\right]\left(X-Y\right)\right)=W\left(X\right)\
,$ (47)

which implies

$dist\left(\mathcal{I},X\right)=W\left(X\right)\ .$ (48)

For any $t\geq 0,$ let $W\left(t\right):=W\left(X\left(t\right)\right).$ In
the following we will prove that the random sequence
$\left\\{W\left(t\right)\right\\}_{t\geq 0}$ converges to zero w.p.1 w.r.t.
the noise, hence proving Theorem 7.

###### Proposition 9

The sequence $\left\\{W\left(t\right)\right\\}_{t\geq 0}$ is non-increasing
hence bounded. Moreover, $\left\\{W\left(t\right)\right\\}_{t\geq 0}$ is a
non-negative $L^{1}$-supermartingale w.r.t.
$\left\\{\mathfrak{F}_{t}^{X}\right\\}_{t\geq 0},$ therefore
$\mathbb{P}_{0}$-a.s. convergent to a
$L^{1}\left(\mathfrak{X},\mathcal{F},\mathbb{P}_{0}\right)$ r.v. which we
denote by $W.$

Proof. By (12), given $u,v\in\mathbf{V}$ such that $u\neq v,$ for $t\geq 0,$

$\displaystyle X_{u}\left(t+1\right)-X_{v}\left(t+1\right)$
$\displaystyle=\left(X_{u}\left(t+1\right)-X_{u}\left(t\right)\right)-\left(X_{v}\left(t+1\right)-X_{v}\left(t\right)\right)+X_{u}\left(t\right)-X_{v}\left(t\right)$
(49) $\displaystyle=X_{u}\left(t\right)-X_{v}\left(t\right)+\frac{\sum_{e\in
E_{u}^{-}\left(t\right)}r_{e}\mathbf{1}_{\left(u,w\right)}\left(e\right)\left[X_{w}\left(t\right)-X_{u}\left(t\right)\right]}{\sum_{e\in
E_{u}^{-}\left(t\right)}r_{e}}$ $\displaystyle-\frac{\sum_{e^{\prime}\in
E_{v}^{-}\left(t\right)}r_{e^{\prime}}\mathbf{1}_{\left(v,z\right)}\left(e^{\prime}\right)\left[X_{z}\left(t\right)-X_{v}\left(t\right)\right]}{\sum_{e^{\prime}\in
E_{v}^{-}\left(t\right)}r_{e^{\prime}}}$ $\displaystyle=\frac{\sum_{e\in
E_{u}^{-}\left(t\right)}r_{e}\mathbf{1}_{\left(u,w\right)}\left(e\right)}{\sum_{e\in
E_{u}^{-}\left(t\right)}r_{e}}X_{w}\left(t\right)-\frac{\sum_{e^{\prime}\in
E_{v}^{-}\left(t\right)}r_{e^{\prime}}\mathbf{1}_{\left(v,z\right)}\left(e^{\prime}\right)}{\sum_{e^{\prime}\in
E_{v}^{-}\left(t\right)}r_{e^{\prime}}}X_{z}\left(t\right)\ .$

By (19), setting

$P_{u,v}\left(t\right):=P_{u,v}\left(\omega\left(t\right)\right)=\frac{\sum_{e\in
E_{v}^{-}\left(t\right)}\delta_{e,\left(v,u\right)}r_{e}}{\sum_{e\in
E_{u}^{-}\left(t\right)}r_{e}}=\frac{r_{v,u}\mathbf{1}_{\mathcal{N}^{-}\left(u,t\right)}\left(v\right)}{\sum_{v\in\mathcal{N}^{-}\left(u,t\right)}r_{v,u}}$
(50)

we can rewrite the previous expression as

$X_{u}\left(t+1\right)-X_{v}\left(t+1\right)=\sum_{w\in\mathbf{V}}P_{u,w}\left(t\right)X_{w}\left(t\right)-\sum_{z\in\mathbf{V}}P_{v,z}\left(t\right)X_{z}\left(t\right)\
.$ (51)

Since, $\forall t\geq 0,\sum_{v\in\mathbf{V}}P_{u,v}\left(t\right)=1,$ we have

$X_{u}\left(t+1\right)-X_{v}\left(t+1\right)=\sum_{w,z\in\mathbf{V}}P_{u,w}\left(t\right)P_{v,z}\left(t\right)\left[X_{w}\left(t\right)-X_{z}\left(t\right)\right]$
(52)

and, since
$\left[X_{w}\left(t\right)-X_{z}\left(t\right)\right]=-\left[X_{z}\left(t\right)-X_{w}\left(t\right)\right]\
,$we obtain

$X_{u}\left(t+1\right)-X_{v}\left(t+1\right)=\frac{1}{2}\sum_{w,z\in\mathbf{V}}\left\\{P_{u,w}\left(t\right)P_{v,z}\left(t\right)-P_{u,z}\left(t\right)P_{v,w}\left(t\right)\right\\}\left[X_{w}\left(t\right)-X_{z}\left(t\right)\right]\
.$ (53)

Hence

$\left|X_{u}\left(t+1\right)-X_{v}\left(t+1\right)\right|\leq\frac{1}{2}\sum_{w,z\in\mathbf{V}}\left|P_{u,w}\left(t\right)P_{v,z}\left(t\right)-P_{u,z}\left(t\right)P_{v,w}\left(t\right)\right|\left|X_{w}\left(t\right)-X_{z}\left(t\right)\right|\
.$ (54)

Since $\forall a,b\in\mathbb{R},a\wedge b=\frac{a+b-\left|a-b\right|}{2},$

$\displaystyle\left|X_{u}\left(t+1\right)-X_{v}\left(t+1\right)\right|$
$\displaystyle\leq\sum_{w,z\in\mathbf{V}}\left\\{\frac{P_{u,w}\left(t\right)P_{v,z}\left(t\right)+P_{u,z}\left(t\right)P_{v,w}\left(t\right)}{2}\right.$
(55) $\displaystyle\left.-P_{u,w}\left(t\right)P_{v,z}\left(t\right)\wedge
P_{u,z}\left(t\right)P_{v,w}\left(t\right)\right\\}\left|X_{w}\left(t\right)-X_{z}\left(t\right)\right|$
$\displaystyle\leq\sum_{w,z\in\mathbf{V}}\left\\{\frac{P_{u,w}\left(t\right)P_{v,z}\left(t\right)+P_{u,z}\left(t\right)P_{v,w}\left(t\right)}{2}\right.$
$\displaystyle\left.-P_{u,w}\left(t\right)P_{v,z}\left(t\right)\wedge
P_{u,z}\left(t\right)P_{v,w}\left(t\right)\right\\}\max_{w,z\in\mathbf{V}}\left|X_{w}\left(t\right)-X_{z}\left(t\right)\right|$
$\displaystyle\leq\left\\{1-\sum_{w,z\in\mathbf{V}}P_{u,w}\left(t\right)P_{v,z}\left(t\right)\wedge
P_{u,z}\left(t\right)P_{v,w}\left(t\right)\right\\}\max_{w,z\in\mathbf{V}}\left|X_{w}\left(t\right)-X_{z}\left(t\right)\right|\
.$

Therefore, choosing $u,v\in\mathbf{V}$ such that

$\left|X_{u}\left(t+1\right)-X_{v}\left(t+1\right)\right|=\max_{w,z\in\mathbf{V}}\left|X_{w}\left(t+1\right)-X_{z}\left(t+1\right)\right|\
,$ (56)

by (40) we get

$\displaystyle W\left(t+1\right)$
$\displaystyle\leq\left\\{1-\min_{u,v\in\mathbf{V}\ :\ u\neq
v}\sum_{w,z\in\mathbf{V}}P_{u,w}\left(t\right)P_{v,z}\left(t\right)\wedge
P_{u,z}\left(t\right)P_{v,w}\left(t\right)\right\\}W\left(t\right)$ (57)
$\displaystyle=\left(1-\gamma\left(\omega\left(t\right)\right)\right)W\left(t\right)\
;$

hence, $\forall t\geq 0,W\left(t\right)\leq W\left(0\right)\leq 1.$

Thus, representing the random sequence
$\left\\{W\left(t\right)\right\\}_{t\geq 0}$ as
$\left\\{W\circ\mathfrak{x}_{t}\right\\}_{t\geq 0},$ from (57) we get

$\displaystyle\mathbb{E}_{0}\left[W\left(t+1\right)|\mathfrak{F}_{t}^{X}\right]=\mathbb{E}_{0}\left[W\circ\mathfrak{x}_{t+1}|\mathfrak{F}_{t}^{X}\right]=\mathbb{E}_{0}\left[\mathbb{E}_{0}\left[W\circ\pi_{X}\circ\chi_{t+1}|\mathfrak{F}_{t}\right]|\mathfrak{F}_{t}^{X}\right]$
(58)
$\displaystyle\leq\mathbb{E}_{0}\left[\left[\mathbb{E}_{0}\left[\left[\left\\{1-\gamma\circ\pi_{\omega}\right\\}W\circ\pi_{X}\right|\chi_{t}\right]\right|\mathfrak{F}_{t}^{X}\right]$
$\displaystyle=\mathbb{E}_{0}\left[\left[\mathbb{E}_{0}\left\\{1-\gamma\left(\pi_{\omega}\circ\chi_{t}\right)\right\\}W\circ\mathfrak{x}_{t}\right|\mathfrak{F}_{t}^{X}\right]$
$\displaystyle=\left\\{1-\mathbb{E}_{0}\left[\left[\gamma\left(\pi_{\omega}\circ\chi_{t}\right)\right|\mathfrak{F}_{t}^{X}\right]\right\\}W\left(t\right)\leq
W\left(t\right)\ ,$

that is $\left\\{W\left(t\right)\right\\}_{t\geq 0}$ is a
$L^{1}$-supermartingale w.r.t. $\left\\{\mathfrak{F}_{t}^{X}\right\\}_{t\geq
0}.$

####  2.3.1 Asymptotic estimate of
$\mathbb{E}_{0}\left[W\left(t\right)\right]$

###### Lemma 10

The sequence
$\left\\{\mathbb{E}_{0}\left[\gamma|\mathfrak{F}_{t}^{X}\right]\right\\}_{t\geq
0}$ is predictable w.r.t. the filtration
$\left\\{\mathfrak{F}_{t}^{X}\right\\}_{t\geq 0}.$

Proof. For any $t\geq 1,$ by (19),

$\displaystyle\mathbb{E}_{0}\left[\gamma|\mathfrak{F}_{t}^{X}\right]=\mathbb{E}_{0}\left[\left[\gamma\left(\pi_{\omega}\circ\chi_{t}\right)\right|\mathfrak{F}_{t}^{X}\right]=\mathbb{E}_{0}\left[\left[\gamma\left(\mathfrak{w}_{t}\right)\right|\mathfrak{x}_{t-1}\right]=$
(59)
$\displaystyle\sum_{\omega\in\Omega}\Pi\left(\omega|\mathfrak{x}_{t-1}\right)\min_{u,v\in\mathbf{V}\
:\ u\neq
v}\sum_{w,z\in\mathbf{V}}P_{u,w}\left(\omega\right)P_{v,z}\left(\omega\right)\wedge
P_{u,z}\left(\omega\right)P_{v,w}\left(\omega\right)=\Gamma\left(\mathfrak{x}_{t-1}\right)\
.$

Let us set

$\displaystyle\mathfrak{X}\ni\left(X,\omega\right)\longmapsto\Pi\left(\omega|W\left(X\right)\right)$
$\displaystyle:=\prod\limits_{e\in\mathbf{E}}\left[\delta_{\omega_{e},1}p\left(W\left(X\right)\right)+\delta_{\omega_{e},0}\left(1-p\left(W\left(X\right)\right)\right)\right]$
(60)
$\displaystyle=\prod\limits_{e\in\mathbf{E}}\left[\omega_{e}p\left(W\left(X\right)\right)+\left(1-\omega_{e}\right)\left(1-p\left(W\left(X\right)\right)\right)\right]\in\left[0,1\right]\
.$

Since $\Omega$ is a poset w.r.t. the partial order relation:
$\omega\leq\omega^{\prime}$ if, $\forall
e\in\mathbf{E},\omega_{e}\leq\omega_{e}^{\prime},$ we have

###### Lemma 11

For any
$X\in\Xi,\Pi\left(\cdot|X\right)\overset{st}{\geq}\Pi\left(\cdot|W\left(X\right)\right).$
Moreover, for any $t\geq
0,\Pi\left(\cdot|W\left(t+1\right)\right)\overset{st}{\geq}\Pi\left(\cdot|W\left(t\right)\right).$

Proof. Let us consider first the statement
$\Pi\left(\cdot|X\right)\overset{st}{\geq}\Pi\left(\cdot|W\left(X\right)\right).$
For $X\in\mathcal{I},$ by (22) and (60) $\Pi\left(\cdot|X\right)$ and
$\Pi\left(\cdot|W\left(X\right)\right)$ coincide. Let now $X\in\mathcal{X}.$
By (22) and (60) $\Pi\left(\cdot|\cdot\right)$ is irreducible, then to prove
$\Pi\left(\cdot|X\right)\overset{st}{\geq}\Pi\left(\cdot|W\left(X\right)\right)$
is enough to prove that the Holley inequality is satisfied, namely

$\Pi\left(\omega\vee\omega^{\prime}|X\right)\Pi\left(\omega\wedge\omega^{\prime}|W\left(X\right)\right)\geq\Pi\left(\omega|X\right)\Pi\left(\omega^{\prime}|W\left(X\right)\right)\;,\quad\omega,\omega^{\prime}\in\Omega\
,$ (61)

where $\omega\vee\omega^{\prime}\in\Omega$ is such that $\forall
e\in\mathbf{E},\left(\omega\vee\omega^{\prime}\right)_{e}=\omega_{e}\vee\omega_{e}^{\prime}$
and $\omega\wedge\omega^{\prime}\in\Omega$ is such that $\forall
e\in\mathbf{E},\left(\omega\wedge\omega^{\prime}\right)_{e}=\omega_{e}\wedge\omega_{e}^{\prime}.$
This is equivalent to prove that, for any $e,f\in\mathbf{E},$

$\Pi\left(\omega^{\left\\{e\right\\}}|X\right)\Pi\left(\omega_{\left\\{e\right\\}}|W\left(X\right)\right)\geq\Pi\left(\omega^{\left\\{e\right\\}}|W\left(X\right)\right)\Pi\left(\omega_{\left\\{e\right\\}}|X\right)$
(62)

and

$\Pi\left(\omega^{\left\\{ef\right\\}}|X\right)\Pi\left(\omega_{\left\\{ef\right\\}}|W\left(X\right)\right)\geq\Pi\left(\omega_{\left\\{f\right\\}}^{\left\\{e\right\\}}|W\left(X\right)\right)\Pi\left(\omega_{\left\\{e\right\\}}^{\left\\{f\right\\}}|X\right)\
,$ (63)

where, for any $E\subset\mathbf{E},\omega^{E}\in\Omega$ is such that $\forall
e\in\mathbf{E},\omega_{e}^{E}:=\omega_{e}\mathbf{1}_{E^{c}}\left(e\right)+\mathbf{1}_{E}\left(e\right)$
and $\omega_{E}\in\Omega$ is such that $\forall
e\in\mathbf{E},\left(\omega_{E}\right)_{e}:=\omega_{e}\mathbf{1}_{E^{c}}\left(e\right)$
(see e.g. [Gr] Theorem 2.3). But, by (22) and (60), $\Pi\left(\cdot|X\right)$
and $\Pi\left(\cdot|W\left(X\right)\right)$ are product measures, then (62)
becomes

$\Pi\left(\omega_{e}=1|X\right)\Pi\left(\omega_{e}=0|W\left(X\right)\right)\geq\Pi\left(\omega_{e}=1|W\left(X\right)\right)\Pi\left(\omega_{e}=0|X\right)\
,$

which can be rewritten as

$p\left(\left|\Delta_{e}X\right|\right)\left(1-p\left(W\left(X\right)\right)\right)\geq
p\left(W\left(X\right)\right)\left(1-p\left(\left|\Delta_{e}X\right|\right)\right)$
(64)

and (63) becomes

$\displaystyle\Pi\left(\omega_{e}=1|X\right)\Pi\left(\omega_{f}=1|X\right)\Pi\left(\omega_{e}=0|W\left(X\right)\right)\Pi\left(\omega_{f}=0|W\left(X\right)\right)\geq$
(65)
$\displaystyle\Pi\left(\omega_{e}=1|W\left(X\right)\right)\Pi\left(\omega_{f}=0|W\left(X\right)\right)\Pi\left(\omega_{e}=0|X\right)\Pi\left(\omega_{f}=1|X\right)$

which is again (64). Since by (45), for any
$e\in\mathbf{E,}W\left(X\right)\geq\Delta_{e}X$ and since
$p:\left[0,1\right]\circlearrowleft$ is non increasing, we have, for any
$e\in\mathbf{E},p\left(\left|\Delta_{e}X\right|\right)\geq
p\left(W\left(X\right)\right)$ and consequently
$\left(1-p\left(W\left(X\right)\right)\right)\geq\left(1-p\left(\left|\Delta_{e}X\right|\right)\right)$
which proves (64).

The proof of the statement
$\Pi\left(\cdot|W\left(t+1\right)\right)\overset{st}{\geq}\Pi\left(\cdot|W\left(t\right)\right),t\geq
0,$ follow the same lines of the proof of
$\Pi\left(\cdot|X\right)\overset{st}{\geq}\Pi\left(\cdot|W\left(X\right)\right)$
since, by (57), $W\left(t+1\right)\leq W\left(t\right),$ which implies
$p\left(W\left(t+1\right)\right)\geq p\left(W\left(t\right)\right).$

##### Proof of Theorem 7

Since $\gamma$ is an non-decreasing function, by (41) and by the previous
lemma we have that

$\Gamma\left(X\right)\geq\sum_{\omega\in\Omega}\mathbb{P}\left(\omega|W\left(X\right)\right)\gamma\left(\omega\right)=\Gamma\left(W\left(X\right)\right)$
(66)

and, for any $t\geq
0,\Gamma\left(W\left(t+1\right)\right)\geq\Gamma\left(W\left(t\right)\right).$
Then, by (58) and Lemma 10 we get

$\displaystyle\mathbb{E}_{0}\left[W\left(t\right)\right]$
$\displaystyle=\mathbb{E}_{0}\left[\mathbb{E}_{0}\left[W\circ\pi_{X}\circ\chi_{t}|\mathfrak{F}_{t-1}^{X}\right]\right]\leq\mathbb{E}_{0}\left[\left(1-\mathbb{E}_{0}\left[\gamma|\mathfrak{F}_{t-1}^{X}\right]\right)W\circ\pi_{X}\circ\chi_{t-1}\right]$
(67)
$\displaystyle=\mathbb{E}_{0}\left[\left(1-\Gamma\left(X\left(t-2\right)\right)\right)W\left(t-1\right)\right]\leq\mathbb{E}_{0}\left[\left(1-\Gamma\left(W\left(t-2\right)\right)\right)W\left(t-1\right)\right]$
$\displaystyle=\mathbb{E}_{0}\left[\left(1-\Gamma\left(W\circ\pi_{X}\circ\chi_{t-2}\right)\right)\mathbb{E}_{0}\left[W\circ\pi_{X}\circ\chi_{t-1}|\mathfrak{F}_{t-2}^{X}\right]\right]$
$\displaystyle\leq\mathbb{E}_{0}\left[\left(1-\Gamma\left(W\circ\pi_{X}\circ\chi_{t-2}\right)\right)\left(1-\mathbb{E}_{0}\left[\gamma|\mathfrak{F}_{t-2}^{X}\right]\right)W\circ\pi_{X}\circ\chi_{t-2}\right]$
$\displaystyle=\mathbb{E}_{0}\left[\left(1-\Gamma\left(W\left(t-2\right)\right)\right)\left(1-\Gamma\left(X\left(t-2\right)\right)\right)W\left(t-2\right)\right]$
$\displaystyle\leq\mathbb{E}_{0}\left[\left(1-\Gamma\left(W\left(t-2\right)\right)\right)^{2}W\left(t-2\right)\right]\leq\mathbb{E}_{0}\left[\left(1-\Gamma\left(W\left(t-3\right)\right)\right)^{2}W\left(t-2\right)\right]\
.$

Iterating this inequality, after $k$ steps, with $k\leq t,$ we obtain

$\mathbb{E}_{0}\left[W\left(t\right)\right]\leq\mathbb{E}_{0}\left[\left(1-\Gamma\left(W\left(t-k\right)\right)\right)^{k-1}W\left(t-k+1\right)\right]$
(68)

which, by (57) implies

$\mathbb{E}_{0}\left[W\left(t\right)\right]\leq\mathbb{E}_{0}\left[W\left(1\right)\right]\left(1-\Gamma\left(W\left(X^{0}\right)\right)\right)^{t-1}\leq
W\left(X^{0}\right)\left(1-\Gamma\left(W\left(X^{0}\right)\right)\right)^{t}\
.$ (69)

Therefore, for any $X^{0}\in\left\\{X\in\Xi:\Gamma\left(X\right)>0\right\\},$
since $W\left(X^{0}\right)$ and for any $\varepsilon>0$ the Markov inequality
implies

$\mathbb{P}_{0}\left\\{W\left(t\right)>\varepsilon\right\\}\leq\frac{\mathbb{E}_{0}\left[W\left(t\right)\right]}{\varepsilon}\leq\varepsilon^{-1}\left(1-\Gamma\left(W\left(X^{0}\right)\right)\right)^{t}\
,$ (70)

by the Borel-Cantelli Lemma $\left\\{W\left(t\right)\right\\}_{t\geq 0}$
converges to zero $\mathbb{P}_{0}$-a.s., that
is$\mu_{0}^{\infty}:=\lim_{t\rightarrow\infty}\mathbb{P}_{0}\left\\{X\left(t\right)\in\cdot\right\\}$
is supported on $\mathcal{I}.$

###  2.4 Convergence to the stationary measure of
$\left\\{\chi_{t}\right\\}_{t\in\mathbb{Z}_{+}}$ and
$\left\\{\mathfrak{y}_{t}\right\\}_{t\in\mathbb{Z}_{+}}$

We can rephrase (69) and therefore the content of Theorem 7 in terms of
exponential (more precisely geometric since $t\in\mathbb{Z}_{+}$) convergence
to an element of the set of the invariant measures of the Markov chains
defined by the transition operators $\mathfrak{T}$ and $\mathbf{T}.$ More
precisely, for any $\varepsilon>0$ and $t>0,$ given
$\chi^{0}=\left(X^{0},\omega^{0}\right)\in\left\\{X\in\Xi:\Gamma\left(X\right)>0\right\\}\times\Omega,$
by (36)
$\left\\{\left\|\left(\mathbb{I}-\mathbb{M}\right)\mathfrak{x}_{t}\right\|>\varepsilon\right\\}\subseteq\left\\{W\left(t\right)>\varepsilon\right\\}.$
Hence, by Theorem 7, $\left\\{\mathfrak{v}_{t}\right\\}_{t\in\mathbb{Z}_{+}}$
converges to zero $\mathbb{P}_{0}-a.s.$ and, by (71),
$\left\\{\mathfrak{w}_{t}\right\\}_{t\in\mathbb{Z}_{+}}$ converges to
$\bar{1}\mathbb{P}_{0}-a.s..$ But, since, by (23), for any
$Y\in\mathcal{I},\omega\in\Omega,\mathfrak{T}\varphi\left(Y,\omega\right)=\varphi\left(Y,\bar{1}\right),\left\\{\mathfrak{u}_{t}\right\\}_{t\in\mathbb{Z}_{+}}$
converges $\mathbb{P}_{0}-a.s.$ to an element of $\mathcal{I}$ which we denote
by $X^{\infty}.$

Given $X\in\Xi,$ let us set $X=\left(U,V\right)$ such that
$U:=\mathbb{M}U,V:=\left(\mathbb{I}-\mathbb{M}\right)X$ and consider the
random processes $\left\\{\mathfrak{u}_{t}\right\\}_{t\in\mathbb{Z}_{+}}$ and
$\left\\{\mathfrak{u}_{t}\right\\}_{t\in\mathbb{Z}_{+}}$ such that $\forall
t\geq 0,\mathfrak{u}_{t}:=\mathbb{M}\mathfrak{x}_{t}$ and
$\mathfrak{v}_{t}:=\left(\mathbb{I}-\mathbb{M}\right)\mathfrak{x}_{t}.$ From
(22),(23) and (37), for any bounded measurable function $\varphi$ on
$\Xi^{2}\times\Omega,$

$\mathfrak{T}\varphi\left(U,V,\omega\right)=\sum_{\omega^{\prime}\in\Omega}\varphi\left(U+\mathbb{M}\mathcal{T}\left(V,\omega\right),\left(\mathbb{I}-\mathbb{M}\right)\mathcal{T}\left(V,\omega\right),\omega^{\prime}\right)\Pi\left(\omega^{\prime}|V\right)\
.$ (71)

Hence, $\left\\{\mathfrak{z}_{t}\right\\}_{t\in\mathbb{Z}_{+}}$ such that,
$\forall t\geq
0,\mathfrak{z}_{t}:=\left(\mathfrak{v}_{t},\mathfrak{w}_{t}\right),$ is an
homogeneous Markov process.

Let us introduce on $\Omega$ the metric

$\Omega\times\Omega\ni\left(\omega,\omega^{\prime}\right)\longmapsto\mathbf{d}\left(\omega,\omega^{\prime}\right):=\frac{1}{\left|\mathbf{E}\right|}\sum_{e\in\mathbf{E}}\left(1-\delta_{\omega_{e},\omega_{e}^{\prime}}\right)\in\left[0,1\right]\
.$ (72)

###### Lemma 12

From (45), for any $\left(X^{0},\omega^{0}\right)\in\mathfrak{X}$ and $t\geq
1,$ we have

$\mathbb{E}\left[\mathbf{d}\left(\mathfrak{w}_{t},\bar{1}\right)|\left(X^{0},\omega^{0}\right)\right]\leq\mathbb{E}\left[\left(1-p\left(W\circ\mathfrak{x}_{t-1}\right)\right)|\left(X^{0},\omega^{0}\right)\right]\
.$ (73)

Proof. For any $\omega\in\Omega,$ we get

$\mathbf{d}\left(\omega,\bar{1}\right)=\frac{1}{\left|\mathbf{E}\right|}\sum_{e\in\mathbf{E}}\left(1-\delta_{\omega_{e},1}\right)=\frac{1}{\left|\mathbf{E}\right|}\sum_{e\in\mathbf{E}}\left(1-\omega_{e}\right)\
.$ (74)

Hence, by the Markov property, from (23) and (22) we have

$\displaystyle\mathbb{E}_{0}\left[\mathbf{d}\left(\mathfrak{w}_{t},\bar{1}\right)\right]$
$\displaystyle=\mathbb{E}_{0}\left[\mathbb{E}\left[\mathbf{d}\left(\mathfrak{w}_{t},\bar{1}\right)|\left(\mathfrak{x}_{t-1},\mathfrak{w}_{t-1}\right)\right]\right]$
(75)
$\displaystyle=\mathbb{E}_{0}\left[\sum_{\omega^{\prime}\in\Omega}\frac{1}{\left|\mathbf{E}\right|}\sum_{e\in\mathbf{E}}\left(1-\omega_{e}^{\prime}\right)\Pi\left(\omega^{\prime}|\mathfrak{x}_{t-1}\right)\right]$
$\displaystyle=\frac{1}{\left|\mathbf{E}\right|}\sum_{e\in\mathbf{E}}\mathbb{E}_{0}\left[\sum_{\omega^{\prime}\in\Omega}\left(1-\omega_{e}^{\prime}\right)\left(\omega_{e}^{\prime}p\left(\left|\Delta_{e}\mathfrak{x}_{t-1}\right|\right)\times\right.\right.$
$\displaystyle\left.\left.\times\left(1-\omega_{e}^{\prime}\right)\left(1-p\left(\left|\Delta_{e}\mathfrak{x}_{t-1}\right|\right)\right)\right)\right]$
$\displaystyle=\frac{1}{\left|\mathbf{E}\right|}\sum_{e\in\mathbf{E}}\mathbb{E}_{0}\left[\left(1-p\left(\left|\Delta_{e}\mathfrak{v}_{t-1}\right|\right)\right)\right]$
$\displaystyle\leq\frac{1}{\left|\mathbf{E}\right|}\sum_{e\in\mathbf{E}}\mathbb{E}_{0}\left[\left(1-p\left(W\left(t-1\right)\right)\right)\right]$
$\displaystyle\leq\mathbb{E}_{0}\left[\left(1-p\left(W\left(t-1\right)\right)\right)\right]\
.$

Let $\Xi_{\mathcal{I}}:=\Xi\backslash\mathcal{I},$ Given a bounded measurable
function $\varphi$ on $\Xi_{\mathcal{I}}\times\Omega\subset\mathfrak{X,}$ let

$\displaystyle\left\|\nabla_{V}\varphi\right\|_{1}$
$\displaystyle:=\sup_{\left(V^{\prime},\omega\right)\in\Xi_{\mathcal{I}}\times\Omega}\sum_{v\in\mathbf{V}}\left|\frac{\partial}{\partial
V_{v}}\varphi\right|\left(V^{\prime},\omega\right)\ ,$ (76)
$\displaystyle\left\|\varphi\right\|_{\Omega}$
$\displaystyle:=\sup_{V\in\Xi_{\mathcal{I}}}\sup_{\omega,\omega^{\prime}\in\Omega\
:\
\omega\neq\omega^{\prime}}\frac{\left|\varphi\left(V,\omega\right)-\varphi\left(V,\omega^{\prime}\right)\right|}{\mathbf{d}\left(\omega,\omega^{\prime}\right)}$
(77)

and consider the Banach space $\mathbb{L}$ of measurable functions $\varphi$
on $\Xi_{\mathcal{I}}\times\Omega,$ with norm

$\left\|\varphi\right\|_{L}:=\sup_{\left(V,\omega\right)\in\Xi_{\mathcal{I}}\times\Omega}\left|\varphi\left(V,\omega\right)\right|+\left\|\nabla_{V}\varphi\right\|_{1}+\left\|\varphi\right\|_{\Omega}\
.$ (78)

Since for any $X\in\Xi,$

$\displaystyle\left\|\left(\mathbb{I}-\mathbb{M}\right)X\right\|$
$\displaystyle=\sup_{v\in\mathbf{V}}\left|X_{v}-\left(\mathbb{M}X\right)_{v}\right|=\sup_{v\in\mathbf{V}}\left|X_{v}-\frac{1}{\left|\mathbf{V}\right|}\sum_{u\in\mathbf{V}}X_{u}\right|$
(79)
$\displaystyle=\sup_{v\in\mathbf{V}}\left|\left(1-\frac{1}{\left|\mathbf{V}\right|}\right)X_{v}-\frac{1}{\left|\mathbf{V}\right|}\sum_{u\in\mathbf{V}\
:\ u\neq v}X_{u}\right|$
$\displaystyle=\sup_{v\in\mathbf{V}}\left|\frac{\left|\mathbf{V}\right|-1}{\left|\mathbf{V}\right|}X_{v}-\frac{1}{\left|\mathbf{V}\right|}\sum_{u\in\mathbf{V}\
:\ u\neq v}X_{u}\right|$
$\displaystyle=\sup_{v\in\mathbf{V}}\left|\frac{1}{\left|\mathbf{V}\right|}\sum_{u\in\mathbf{V}\
:\ u\neq v}\left(X_{v}-X_{u}\right)\right|\leq W\left(X\right)\ ,$

then, for $\varphi\in\mathbb{L},$ by (79), we have

$\displaystyle\left|\varphi\left(\left(\mathbb{I}-\mathbb{M}\right)X,\omega\right)-\varphi\left(0,\omega\right)\right|$
$\displaystyle=\left|\int_{0}^{1}ds\sum_{v\in\mathbf{V}}\left(\frac{\partial}{\partial
V_{v}}\varphi\right)\left(s\left(\mathbb{I}-\mathbb{M}\right)X,\omega\right)\left(\left(\mathbb{I}-\mathbb{M}\right)X\right)_{v}\right|$
(80)
$\displaystyle=\left|\int_{0}^{1}ds\sum_{v\in\mathbf{V}}\left(\frac{\partial}{\partial
V_{v}}\varphi\right)\left(s\left(X-\mathbb{M}X\right),\omega\right)\frac{1}{\left|\mathbf{V}\right|}\sum_{u\in\mathbf{V}\
:\ u\neq v}\left(X_{v}-X_{u}\right)\right|$
$\displaystyle\leq\left\|\nabla_{V}\varphi\right\|_{1}W\left(X\right)\ .$

###### Proposition 13

Starting from an initial state
$\chi^{0}=\left(X^{0},\omega^{0}\right)\in\left\\{X\in\Xi:\Gamma\left(X\right)>0\right\\}\times\Omega\subset\mathfrak{X},$
the Markov chain $\left\\{\chi_{t}\right\\}_{t\geq 0}$ weakly converges to the
degenerate random vector
$\left(X^{\infty},\bar{1}\right)\in\mathcal{I}\times\Omega,$ where
$X^{\infty}$ is the $\mathbb{P}_{0}-a.s.$ limit of the random process
$\left\\{\mathfrak{u}_{t}\right\\}_{t\in\mathbb{Z}_{+}}.$ Moreover, if $p$ is
concave function and $\lim_{x\downarrow 0}\frac{1-p\left(x\right)}{x}$ exists,
the rate of convergence is exponential.

Proof. Given $\varphi\in\mathbb{L},$ by (80) and (75) we have

$\displaystyle\left|\mathbb{E}_{0}\left[\varphi\left(\mathfrak{v}_{t},\mathfrak{w}_{t}\right)\right]-\varphi\left(0,\bar{1}\right)\right|\leq\mathbb{E}_{0}\left[\left|\varphi\left(\mathfrak{v}_{t},\mathfrak{w}_{t}\right)-\varphi\left(0,\bar{1}\right)\right|\right]$
(81)
$\displaystyle\leq\mathbb{E}_{0}\left[\left|\varphi\left(\mathfrak{v}_{t},\mathfrak{w}_{t}\right)-\varphi\left(0,\mathfrak{w}_{t}\right)\right|\right]+\mathbb{E}_{0}\left[\left|\varphi\left(0,\mathfrak{w}_{t}\right)-\varphi\left(0,\bar{1}\right)\right|\right]$
$\displaystyle\leq\left\|\nabla_{X}\varphi\right\|_{1}\mathbb{E}_{0}\left[W\left(t\right)\right]+\left\|\varphi\right\|_{\Omega}\mathbb{E}_{0}\left[\left(1-p\left(W\left(t-1\right)\right)\right)\right]$
$\displaystyle\leq\left\|\varphi\right\|_{L}\left(\mathbb{E}_{0}\left[W\left(t\right)\right]\vee\mathbb{E}_{0}\left[\left(1-p\left(W\left(t-1\right)\right)\right)\right]\right)$

which tends to zero in the limit $t\rightarrow\infty$ by Theorem 7. Clearly,
if $p$ is concave, $1-p$ is convex, hence, by (69),

$\displaystyle\mathbb{E}_{0}\left[\left(1-p\left(W\left(t-1\right)\right)\right)\right]$
$\displaystyle\leq\left(1-p\left(\mathbb{E}_{0}\left[W\left(t-1\right)\right]\right)\right)$
(82) $\displaystyle\leq
1-p\left(W\left(X^{0}\right)\left(1-\Gamma\left(X^{0}\right)\right)^{t-1}\right)\
,$

therefore, if $\lim_{x\downarrow 0}\frac{1-p\left(x\right)}{x}$ is finite the
rate of convergence is exponential.

Similar conclusions hold for the Markov chain
$\left\\{\mathfrak{y}_{t}\right\\}_{t\geq 0}.$ Indeed, by (31) and (37),
setting $X_{1}=\left(U_{1},V_{1}\right),X_{2}=\left(U_{2},V_{2}\right),$ for
any bounded measurable $\varphi:\Xi^{4}\times\Omega\rightarrow\mathbb{R},$

$\displaystyle\left(\mathbf{T}\varphi\right)\left(U_{1},V_{1},U_{2},V_{2}\right)$
$\displaystyle:=\sum_{\omega,\omega^{\prime}\in\Omega}\varphi\left(U_{2}+\mathbb{M}\mathcal{T}\left(V_{2},\omega\right),\left(\mathbb{I}-\mathbb{M}\right)\mathcal{T}\left(V_{2},\omega\right),\right.$
(83)
$\displaystyle\left.U_{2}+\mathbb{M}\mathcal{T}\left(\mathcal{T}\left(V_{2},\omega\right),\omega^{\prime}\right),\left(\mathbb{I}-\mathbb{M}\right)\mathcal{T}\left(\mathcal{T}\left(V_{2},\omega\right),\omega^{\prime}\right)\right)\times$
$\displaystyle\times\Pi\left(\omega|V_{1}\right)\Pi\left(\omega^{\prime}|V_{2}\right)\
.$

Hence, $\left\\{\mathfrak{Z}_{t}\right\\}_{t\geq 0}$ such that $\forall t\geq
0,\mathfrak{Z}_{t}:=\left(\mathfrak{v}_{2t},\mathfrak{v}_{2t+1}\right)$ is an
homogeneous Markov process.

Moreover, from (57), for any $t\geq 0$ we get

$\displaystyle\mathbb{E}_{0}\left[W\circ\mathfrak{x}_{t+1}|\mathfrak{F}_{t-1}^{X}\right]$
$\displaystyle=\mathbb{E}_{0}\left[\left.\mathbb{E}_{0}\left[W\circ\mathfrak{x}_{t+1}|\mathfrak{F}_{t}^{X}\right]\right|\mathfrak{F}_{t-1}^{X}\right]$
(84)
$\displaystyle=\mathbb{E}_{0}\left[\left(1-\Gamma\circ\mathfrak{x}_{t-1}\right)W\circ\mathfrak{x}_{t}|\mathfrak{F}_{t-1}^{X}\right]$
$\displaystyle=\mathbb{E}_{0}\left[W\circ\mathfrak{x}_{t}|\mathfrak{F}_{t-1}^{X}\right]\left(1-\Gamma\circ\mathfrak{x}_{t-1}\right)$
$\displaystyle\leq\left(1-\Gamma\circ\mathfrak{x}_{t-1}\right)\left(1-\Gamma\circ\mathfrak{x}_{t-2}\right)W\circ\mathfrak{x}_{t-1}$
$\displaystyle\leq\left(1-\Gamma\circ
W\circ\mathfrak{x}_{t-1}\right)\left(1-\Gamma\circ
W\circ\mathfrak{x}_{t-2}\right)W\circ\mathfrak{x}_{t-1}$
$\displaystyle\leq\left(1-\Gamma\circ
W\circ\mathfrak{x}_{t-2}\right)^{2}W\circ\mathfrak{x}_{t-1}\ .$

Hence, by (32), defining

$\displaystyle\Xi^{2}$
$\displaystyle\ni\left(X_{1},X_{2}\right)\longmapsto\bar{\Gamma}\left(X_{1},X_{2}\right):=\left(\Gamma\circ\pi_{1}\right)\left(X_{1},X_{2}\right)\in\left(0,1\right)\
,$ (85) $\displaystyle\Xi^{2}$
$\displaystyle\ni\left(X_{1},X_{2}\right)\longmapsto\bar{W}\left(X_{1},X_{2}\right):=\left(W\circ\pi_{1}\right)\left(X_{1},X_{2}\right)\in\left[0,1\right]\
,$ (86)

from (84) we have

$\mathbb{E}_{0}\left[\bar{W}\circ\mathfrak{y}_{t+1}|\mathfrak{F}_{t}^{\mathfrak{y}}\right]\leq\left(1-\bar{\Gamma}\circ\mathfrak{y}_{t}\right)^{2}\bar{W}\circ\mathfrak{y}_{t}\;,\
t\geq 0\ .$ (87)

Therefore, proceeding as in (67), by (87) we get

$\displaystyle\mathbb{E}_{0}\left[\bar{W}\circ\mathfrak{y}_{t}\right]$
$\displaystyle\leq\mathbb{E}_{0}\left[\left(1-\bar{\Gamma}\circ\mathfrak{y}_{t-1}\right)^{2}\bar{W}\circ\mathfrak{y}_{t-1}\right]$
(88)
$\displaystyle\leq\mathbb{E}_{0}\left[\left(1-\bar{\Gamma}\circ\bar{W}\circ\mathfrak{y}_{t-1}\right)^{2}\bar{W}\circ\mathfrak{y}_{t-1}\right]$
$\displaystyle\leq\mathbb{E}_{0}\left[\left(1-\bar{\Gamma}\circ\bar{W}\circ\mathfrak{y}_{t-2}\right)^{2}\bar{W}\circ\mathfrak{y}_{t-1}\right]\
.$

Thus, iterating,

$\mathbb{E}_{0}\left[\mathbf{T}^{t}\bar{W}\right]\leq\left(1-\bar{\Gamma}\circ\mathfrak{y}_{0}\right)^{2t}\bar{W}\circ\mathfrak{y}_{0}\
.$ (89)

Let us denote by $\mathcal{L}$ be the Banach space of bounded measurable
functions $\varphi$ on $\Xi_{\mathcal{I}}^{2}$ with norm

$\left\|\varphi\right\|_{\mathcal{L}}:=\sup_{\left(V_{1},V_{2}\right)\in\Xi_{\mathcal{I}}^{2}}\left|\varphi\left(V_{1},V_{2}\right)\right|+\left\|\nabla\varphi\right\|_{1}\
,$ (90)

where

$\left\|\nabla\varphi\right\|_{1}:=\sup_{\left(V^{\prime},V^{\prime\prime}\right)\in\Xi_{\mathcal{I}}^{2}}\sum_{v\in\mathbf{V}}\left[\left|\frac{\partial}{\partial\left(V_{1}\right)_{v}}\varphi\right|\left(V^{\prime},V^{\prime\prime}\right)+\left|\frac{\partial}{\partial\left(V_{2}\right)_{v}}\varphi\right|\left(V^{\prime},V^{\prime\prime}\right)\right]\
.$ (91)

###### Corollary 14

The Markov chain $\left\\{\mathfrak{y}_{t}\right\\}_{t\geq 0}$ started at
$\left(X^{0},X^{1}\right)\in\left\\{X\in\Xi:\Gamma\left(X\right)>0\right\\}\times\Xi$
converges weakly to the degenerate random vector
$\left(X^{\infty},X^{\infty}\right)\in\Xi^{2}$ with exponential rate.

Proof. Given $\varphi\in\mathcal{L},$ proceeding as in (80), by (58) and (89),
we have

$\displaystyle\left|\mathbb{E}_{0}\left[\varphi\left(\mathfrak{Z}_{t}\right)\right]-\varphi\left(0,0\right)\right|$
$\displaystyle\leq\mathbb{E}_{0}\left[\left|\varphi\left(\mathfrak{Z}_{t}\right)-\varphi\left(0,0\right)\right|\right]$
(92)
$\displaystyle\leq\left\|\varphi\right\|_{\mathcal{L}}\mathbb{E}_{0}\left[\bar{W}\circ\mathfrak{y}_{t}\right]\
.$

##  3 Large system evolution

Given $N\in\mathbb{N},$ let
$\mathbf{V}_{N}:=\left\\{1,..,N\right\\}\subset\mathbb{N}$ and denote by
$\mathbf{E}_{N}$ the subset of
$\mathbf{E}:=\left\\{\left(u,v\right)\in\mathbb{N}\times\mathbb{N}\right\\}$
such that
$\mathbf{E}_{N}:=\left\\{\left(u,v\right)\in\mathbf{V}_{N}\times\mathbf{V}_{N}\right\\}.$

In this section we set $\Xi:=\left[0,1\right]^{\mathbb{N}}$ and, denoting by
$X_{N}:=\left(X_{1},..,X_{N}\right)$ the element of
$\Xi_{N}:=\left[0,1\right]^{N}$ representing the restriction of the beliefs
configuration $X\in\Xi$ to $\mathbf{V}_{N},$ we endow $\Xi$ with the metric
induced by the norm
$\left\|X\right\|:=\sum_{N\in\mathbb{N}}2^{-N}\left\|X_{N}\right\|_{\infty}.$

Setting $\Omega:=\left\\{0,1\right\\}^{\mathbf{E}},$ we denote by $\omega_{N}$
the restriction of the configuration $\omega\in\Omega$ to
$\Omega_{N}:=\left\\{0,1\right\\}^{\mathbf{E}_{N}}$ and, by (6), if
$E:=E\left(\omega\right),$ we set $E_{N}:=E\cap\mathbf{E}_{N}.$

Then, for any $X\in\Xi,\overline{\Pi}\left(\cdot|X\right)$ denotes the random
field on $\left(\Omega,\mathcal{C}\left(\Omega\right)\right)$ such that, for
any cylinder event
$\mathbf{C}_{N}\left(\omega^{\prime}\right)=\left\\{\omega\in\Omega:\omega_{N}=\omega^{\prime}\right\\},N\geq
1,\omega^{\prime}\in\Omega_{N},$

$\sum_{\omega\in\Omega}\overline{\Pi}\left(\omega|X\right)\mathbf{1}_{\mathbf{C}_{N}\left(\omega^{\prime}\right)}\left(\omega\right)=\Pi\left(\omega^{\prime}|X_{N}\right)$
(93)

where $\Pi\left(\omega^{\prime}|X_{N}\right)$ is given by (22). Moreover, for
any $M\geq N,$

$\sum_{\omega\in\Omega_{M}}\Pi\left(\omega|X_{M}\right)\mathbf{1}_{\left\\{\omega\in\Omega_{M}\
:\
\omega_{N}=\omega^{\prime}\right\\}}\left(\omega\right)=\sum_{\omega\in\Omega}\overline{\Pi}\left(\omega|X\right)\mathbf{1}_{\mathbf{C}_{N}\left(\omega^{\prime}\right)}\left(\omega\right)=\Pi\left(\omega^{\prime}|X_{N}\right)$
(94)

Setting $\mathbf{V}:=\mathbb{N}$ for notational convenience, let
$\mathcal{K}\left(\mathbf{V}\right)$ be the set of the transition probability
kernels on $\left(\mathbf{V},\mathcal{P}\left(\mathbf{V}\right)\right).$ From
(19), given the $\mathcal{C}\left(\Omega\right)$-measurable function
$\Omega\ni\omega\longmapsto
P\left(\omega\right)\in\mathcal{K}\left(\mathbf{V}\right)$ such that
$\forall\omega\in\Omega,$

$P_{v,u}\left(\omega\right)=\frac{\sum_{e\in E_{v}^{-}\cap
E\left(\omega\right)}r_{e}\delta_{e,\left(u,v\right)}}{\sum_{e\in
E_{v}^{-}\cap
E\left(\omega\right)}r_{e}}=:\mathfrak{p}_{v,u}\left(E\left(\omega\right)\right)\in\left[0,1\right]\;;\;v,u\in\mathbf{V}\
,$ (95)

we denote as in (21) and (20)
$\mathfrak{X}\ni\left(X,\omega\right)\longmapsto\overline{\mathcal{T}}_{\cdot}\left(X,\omega\right)\in\Xi$
such that, for any $v\in\mathbf{V}$ and any
$\left(X,\omega\right)\in\mathfrak{X},\overline{\mathcal{T}}_{v}\left(X,\omega\right):=\sum_{u\in\mathbf{V}}\mathfrak{p}_{v,u}\left(E\left(\omega\right)\right)X_{u}.$
Then, the operator

$BM\left(\mathfrak{X}\right)\ni\varphi\longmapsto\overline{\mathfrak{T}}\varphi\left(X,\omega\right):=\sum_{\omega^{\prime}\in\Omega}\varphi\left(\overline{\mathcal{T}}\left(X,\omega\right),\omega^{\prime}\right)\overline{\Pi}\left(\omega^{\prime}|X\right)\in
BM\left(\mathfrak{X}\right)$ (96)

represents the transition probability kernel of the homogeneous Markov chain
$\left\\{\chi_{t}\right\\}_{t\geq 0}$ on
$\left(\mathfrak{X}^{\mathbb{Z}_{+}},\mathfrak{C},\mathbb{P}_{0}\right)$ with
initial condition $\left(X^{0},\omega^{0}\right)\in\mathfrak{X}$ such that, by
(23), for any
$\omega^{\prime}\in\Omega_{N},B\in\mathcal{B}\left(\left[0,1\right]^{N}\right),$

$\displaystyle\mathbb{P}_{0}\left(\left\\{\chi_{t+1}\in\mathbf{C}_{N}\left(B\right)\times\mathbf{C}_{N}\left(\omega^{\prime}\right)\right\\}|\chi_{t}\right)=\left(\overline{\mathfrak{T}}\mathbf{1}_{\mathbf{C}_{N}\left(B\right)\times\mathbf{C}_{N}\left(\omega^{\prime}\right)}\right)\left(\chi_{t}\right)$
(97)
$\displaystyle=\Pi\left(\omega^{\prime}|X_{N}\left(t\right)\right)\mathbf{1}_{B}\left(\overline{\mathcal{T}}_{N}\left(X\left(t\right),\omega\left(t\right)\right)\right)\
.$

Consequently, the operator

$\mathfrak{\bar{L}}\left(\Xi^{2}\right)\ni\varphi\longmapsto\left(\overline{\mathbf{T}}\varphi\right)\left(X,Y\right):=\sum_{\omega,\omega^{\prime}\in\Omega}\varphi\left(\overline{\mathcal{T}}\left(Y,\omega\right),\overline{\mathcal{T}}\left(\overline{\mathcal{T}}\left(Y,\omega\right),\omega^{\prime}\right)\right)\overline{\Pi}\left(\omega|X\right)\overline{\Pi}\left(\omega^{\prime}|Y\right)\in\mathfrak{\bar{L}}\left(\Xi^{2}\right)\
,$ (98)

defined as in (31), represents the transition probability kernel of the
homogeneous Markov chain $\left\\{\mathfrak{y}_{t}\right\\}_{t\geq 0}$ on
$\left(\mathfrak{X}^{\mathbb{Z}_{+}},\mathfrak{C},\mathbb{P}_{0}\right)$ such
that, for any $B\in\mathcal{B}\left(\left[0,1\right]^{2N}\right),$

$\displaystyle\mathbb{P}_{0}\left(\left\\{\mathfrak{y}_{t+1}\in\mathbf{C}_{N}\left(B\right)\right\\}|\mathfrak{y}_{t}\right)=\left(\overline{\mathbf{T}}\mathbf{1}_{\mathbf{C}_{N}\left(B\right)}\right)\left(\mathfrak{y}_{t}\right)=$
(99)
$\displaystyle\sum_{\omega,\omega^{\prime}\in\Omega}\mathbf{1}_{B}\left(\overline{\mathcal{T}}_{N}\left(X\left(2t+1\right),\omega\right),\overline{\mathcal{T}}_{N}\left(\overline{\mathcal{T}}\left(X\left(2t+1\right),\omega\right),\omega^{\prime}\right)\right)\overline{\Pi}\left(\omega|X\left(2t\right)\right)\overline{\Pi}\left(\omega^{\prime}|X\left(2t+1\right)\right)=$
$\displaystyle\sum_{\omega,\omega^{\prime}\in\Omega}\mathbf{1}_{B}\left(\overline{\mathcal{T}}_{N}\left(X\left(2t+1\right),\omega\right),\overline{\mathcal{T}}_{N}\left(\overline{\mathcal{T}}\left(X\left(2t+1\right),\omega\right),\omega^{\prime}\right)\right)\overline{\Pi}\left(\omega|X\left(2t\right)\right)\overline{\Pi}\left(\omega^{\prime}|X\left(2t+1\right)\right)=$
$\displaystyle\sum_{\omega,\omega^{\prime}\in\Omega}\mathbf{1}_{B}\left(\mathcal{T}\left(X_{N}\left(2t+1\right),\omega\right),\mathcal{T}\left(\overline{\mathcal{T}}_{N}\left(X\left(2t+1\right),\omega\right),\omega^{\prime}\right)\right)\overline{\Pi}\left(\omega|X\left(2t\right)\right)\overline{\Pi}\left(\omega^{\prime}|X\left(2t+1\right)\right)=$
$\displaystyle\sum_{\omega,\omega^{\prime}\in\Omega}\mathbf{1}_{B}\left(\mathcal{T}\left(X_{N}\left(2t+1\right),\omega\right),\mathcal{T}\left(\mathcal{T}\left(X_{N}\left(2t+1\right),\omega\right),\omega^{\prime}\right)\right)\overline{\Pi}\left(\omega|X\left(2t\right)\right)\overline{\Pi}\left(\omega^{\prime}|X\left(2t+1\right)\right)\
.$

###### Remark 15

Notice that if the cardinality of the set
$R:=\left\\{\left(u,v\right)\in\mathbf{V}\times\mathbf{V}:r_{u,v}>0\right\\}$
is finite, there exists $M>N$ such that

$\mathbb{P}_{0}\left(\left\\{\chi_{t+1}\in\mathbf{C}_{N}\left(B\right)\times\mathbf{C}_{N}\left(\omega^{\prime}\right)\right\\}|\chi_{t}\right)=\Pi\left(\omega^{\prime}|X_{N}\left(t\right)\right)\mathbf{1}_{B}\left(\mathcal{T}\left(X_{N}\left(t\right),\omega_{M}\left(t\right)\right)\right)$
(100)

and

$\displaystyle\mathbb{P}_{0}\left(\left\\{\mathfrak{y}_{t+1}\in\mathbf{C}_{N}\left(B\right)\right\\}|\mathfrak{y}_{t}\right)=$
(101)
$\displaystyle\sum_{\omega_{M},\omega_{M}^{\prime}\in\Omega_{M}}\mathbf{1}_{B}\left(\mathcal{T}\left(X_{N}\left(2t+1\right),\omega\right),\mathcal{T}\left(\mathcal{T}\left(X_{N}\left(2t+1\right),\omega\right),\omega^{\prime}\right)\right)\times$
$\displaystyle\times\Pi\left(\omega_{M}|X_{M}\left(2t\right)\right)\Pi\left(\omega_{M}^{\prime}|X_{M}\left(2t+1\right)\right)=$
$\displaystyle\left(\mathbf{T1}_{\left\\{\left(X^{\left(1\right)},X^{\left(2\right)}\right)\in\Xi_{M}^{2}\
:\ \left(X_{N}^{\left(1\right)},X_{N}^{\left(2\right)}\right)\in
B\right\\}}\right)\left(\mathfrak{y}_{t}\right)\ .$

In the following we make this assumption.

###### Proposition 16

Let the initial datum $X^{0}\in\Xi$ be such that
$\alpha:=\inf_{N\in\mathbb{N}}\Gamma\left(X_{N}^{0}\right)>0.$ Then, for any
$\varphi\in C\left(\Xi\right)$ and any $X^{1}\in\Xi,$

$\lim_{t\rightarrow\infty}\left|\mathbb{E}\left[\varphi\circ\mathfrak{Z}_{t}|\mathfrak{Z}_{0}=\left(X^{0},X^{1}\right)\right]-\varphi\left(0,0\right)\right|=0\
.$ (102)

Proof. For any $\varphi\in C\left(\Xi\right)$ there exists a sequence
$\left\\{\varphi_{N}\right\\}_{N\in\mathbb{N}}$ such that, $\forall N\geq
1,\varphi_{N}$ is a continuous $\mathcal{B}\left(\Xi_{N}\right)$-measurable
function uniformly convergent to $\varphi$ [Ge]. Hence, given $\varepsilon>0,$
there exists $N_{\varepsilon}\geq 1$ such that for any
$N>N_{\varepsilon},\left\|\varphi-\varphi_{N}\right\|_{\infty}<\varepsilon.$
Moreover, denoting by
$\mathcal{I}_{N}:=\bigcup\limits_{x\in\left[0,1\right]}\left\\{X_{N}\in\Xi_{N}:\left(X_{N}\right)_{v}=x\
,\ \forall v\in\mathbf{V}_{N}\right\\}$ and
$\Xi_{\mathcal{I}}^{N}:=\Xi_{N}\backslash\mathcal{I}_{N},$ by the Stone-
Weierstass theorem there exists $\phi\in\mathcal{L}_{N}$ (with
$\mathcal{L}_{N}$ defined as $\mathcal{L}$ in the previous section) such that
$\left\|\phi_{N}-\varphi_{N}\right\|_{\infty}<\varepsilon.$ Then, the thesis
follows from Corollary 14. Indeed,

$\displaystyle\left|\mathbb{E}\left[\varphi\circ\mathfrak{y}_{t}|\mathfrak{y}_{0}=\left(X^{0},X^{1}\right)\right]-\varphi\left(0,0\right)\right|$
$\displaystyle\leq
2\varepsilon+\left|\mathbb{E}\left[\varphi_{N}\circ\mathfrak{y}_{t}|\mathfrak{y}_{0}=\left(X^{0},X^{1}\right)\right]-\varphi_{N}\left(0,0\right)\right|$
(103) $\displaystyle\leq
4\varepsilon+\left|\mathbb{E}\left[\phi_{N}\circ\mathfrak{y}_{t}|\mathfrak{y}_{0}=\left(X^{0},X^{1}\right)\right]-\phi_{N}\left(0,0\right)\right|$
$\displaystyle=4\varepsilon+\left|\delta^{\left(X_{M}^{0},X_{M}^{1}\right)}\mathbf{T}^{t}\left(\phi_{N}-\phi_{N}\left(0,0\right)\right)\right|\
,$

where $M\geq N.$ But,

$\delta^{\left(X_{M}^{0},X_{M}^{1}\right)}\mathbf{T}^{t}\left|\phi_{N}-\phi_{N}\left(0,0\right)\right|\leq\left\|\phi_{N}\right\|_{\mathcal{L}_{N}}\left(1-\alpha\right)^{2t}\
.$ (104)

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