# Random walks with bounded first moment on finite-volume spaces

Timothée Bénard and Nicolas de Saxcé

###### Abstract

Let $G$ be a real Lie group, $\Lambda\leq G$ a lattice, and
$\Omega=G/\Lambda$. We study the equidistribution properties of the left
random walk on $\Omega$ induced by a probability measure $\mu$ on $G$. It is
assumed that $\mu$ has a finite first moment, and that the Zariski closure of
the group generated by the support of $\mu$ in the adjoint representation is
semisimple without compact factors. We show that for every starting point
$x\in\Omega$, the $\mu$-walk with origin $x$ has no escape of mass, and
equidistributes in Cesàro averages toward some homogeneous measure. This
extends several fundamental results due to Benoist-Quint and Eskin-Margulis
for walks with finite exponential moment.

###### Contents

  1. 1 Introduction
    1. 1.1 Main results
    2. 1.2 Dynamics of the proofs
  2. 2 Markov chains with a negative drift
    1. 2.1 Dominance condition and return times
    2. 2.2 Conservation of mass
    3. 2.3 Recurrence of empirical measures
  3. 3 Recurrence criterion for homogeneous chains
    1. 3.1 The controlled drift condition
    2. 3.2 Application to recurrence
  4. 4 Non-escape at infinity
    1. 4.1 The Benoist-Quint function on $\operatorname{\mathrm{SL}}_{d}(\mathbb{R})/\operatorname{\mathrm{SL}}_{d}(\mathbb{Z})$
    2. 4.2 General quotients $G/\Lambda$
  5. 5 Non-accumulation on invariant subspaces
    1. 5.1 Local drift
    2. 5.2 From local drift to global drift
  6. 6 Equidistribution
  7. 7 Conclusion

††footnotetext: The first author has received funding from the European
Research Council (ERC) under the European Union’s Horizon 2020 research and
innovation programme (grant agreement No. 803711).

##  1 Introduction

A homogeneous Markov chain on a finite-volume real homogeneous space is
formally incarnated by a triple $(G,\Lambda,\mu)$ where $G$ is a real Lie
group, $\Lambda$ a discrete subgroup of finite covolume in $G$, and $\mu$ a
Borel probability measure on $G$. The chain in question then corresponds to
the left $\mu$-random walk on the quotient $\Omega=G/\Lambda$. In other words,
the transition law at a point $x\in\Omega$ is given by the convolution
$\mu\ast\delta_{x}$, image of $\mu$ under $g\mapsto gx$.

In the last 20 years, the study of walks on finite-volume spaces has known
spectacular advances, which were achieved in analogy with Ratner’s theorems
describing the dynamics of Ad-unipotent flows on $\Omega$. The first milestone
was set by Eskin-Margulis who proved that the $n$-th step distribution of a
homogeneous chain essentially remains in a compact set [14]. A few years
later, Benoist-Quint managed to classify all the $\mu$-invariant probability
measures [3, 6], and extrapolated in [7] that almost every trajectory of the
random walk equidistributes in some finite-volume homogeneous subspace.
However, all these statements require stringent moment assumptions on the
measure $\mu$, such as _compact support_ or _finite exponential moment_ , the
latter meaning that for some $\alpha>0$, we have

$\int_{G}\lVert\operatorname{Ad}g\rVert^{\alpha}\,\mathrm{d}\mu(g)<\infty.$

In a recent paper, Eskin and Lindenstrauss [13] extended the techniques
developed by Benoist and Quint, and showed that their measure classification
was still valid in the case where $\mu$ only has a _finite first moment_. Our
goal is to weaken in the same way the moment assumptions in the Eskin-Margulis
recurrence theorem and in the Benoist-Quint equidistribution theorems. This
answers a question formulated by Benoist-Quint in the $10^{\text{th}}$ Takagi
Lectures [4, Question 2].

###  1.1 Main results

Let $G$ be a real Lie group, $\Lambda$ a lattice in $G$, and set
$\Omega=G/\Lambda$. We fix a Borel probability measure $\mu$ on $G$ and denote
by $\Gamma$ the semigroup generated by the support of $\mu$. The algebraic
group generated by its adjoint representation is denoted by
$H=\overline{\operatorname{Ad}\Gamma}^{Z}$, and we call $H^{nc}$ its non
compact part, defined as the smallest normal algebraic subgroup of $H$ such
that $H/H^{nc}$ is compact. All the theorems to follow are presented under the
next two assumptions.

  1. 1.

The non compact part $H^{nc}$ of $H$ is semisimple.

  2. 2.

The measure $\mu$ has a _finite first moment_ :

$\int_{G}\log\lVert\operatorname{Ad}g\rVert\,\mathrm{d}\mu(g)<\infty$

To state our results, we need to introduce the notion of homogeneous subspace.

######  Definition 1.1.

A closed subset $Y$ of $\Omega$ is _homogeneous_ if its stabilizer
$G_{Y}=\\{g\in G,\,gY=Y\\}$ acts transitively on $Y$. We add that $Y$ has
_finite volume_ if the action of $G_{Y}$ on $Y$ preserves a Borel probability
measure on $Y$. Such a measure is then unique and denoted by $\nu_{Y}$. If the
semigroup $\Gamma$ is included in $G_{Y}$ (and acts ergodically on
$(Y,\nu_{Y})$), we say that $Y$ is _$\Gamma$ -invariant_ (and
$\Gamma$-_ergodic_).

We denote by $\mathcal{S}_{\Omega}(\Gamma)$ the set of $\Gamma$-invariant
ergodic finite-volume closed homogeneous subsets of $\Omega$.

Our first result states that the $\mu$-walk on $\Omega$ essentially evolves in
a compact subset. It can be seen as an analog for random walks of the Dani-
Margulis recurrence theorem on Ad-unipotent flows [12].

######  Theorem A (Non-escape at infinity).

For every compact set $K\subset\Omega$ and every $\varepsilon>0$, there exists
a compact set $K^{\prime}\subset\Omega$ such that for every $x\in K$,

  1. (i)

for every $n\geq 0$, $(\mu^{*n}*\delta_{x})(K^{\prime})>1-\varepsilon$;

  2. (ii)

for $\mu^{\otimes\mathbb{N}^{*}}$-almost every instructions $(g_{i})_{i\geq
1}$,

$\liminf_{n\to+\infty}\frac{1}{n}\operatorname{card}\\{k\in\\{1,\dots,n\\}\ |\
g_{k}\dots g_{1}x\in K^{\prime}\\}>1-\varepsilon.$

Conclusion (i) means the mass of the $n$-th step distribution of the walk does
not escape at infinity. Conclusion (ii) expresses some positive recurrence of
the walk’s trajectories. The result actually holds under the slightly weaker
assumption that the image measure $(\operatorname{Ad}_{\mathfrak{s}})_{*}\mu$
of $\mu$ under the adjoint representation on the largest semisimple quotient
of $\mathfrak{g}$ without compact factors generates an algebraic group with
semisimple non-compact part (see Theorem A′ in Section 4). This theorem
generalizes [5] and [14], which assume that $\mu$ has a finite exponential
moment.

Our second result states that the $\mu$-walk on $\Omega$ does not accumulate
on a $\mu$-invariant homogeneous subspace, unless it is trapped inside it.

######  Theorem B (Unstability of invariant homogeneous subspaces).

Let $Y\in\mathcal{S}_{\Omega}(\Gamma)$ and consider a compact subset
$K\subseteq\Omega\smallsetminus Y$. For every $\varepsilon>0$, there exists a
neighborhood $O^{\prime}$ of $Y$ in $\Omega$ such that for all $x\in K$,

  1. (i)

for every $n\geq 0$, $(\mu^{*n}*\delta_{x})(O^{\prime})<\varepsilon$ ;

  2. (ii)

for $\mu^{\otimes\mathbb{N}^{*}}$-almost every instructions $(g_{i})_{i\geq
1}$,

$\limsup_{n\to+\infty}\frac{1}{n}\operatorname{card}\\{k\in\\{1,\dots,n\\}\ |\
g_{k}\dots g_{1}x\in O^{\prime}\\}<\varepsilon.$

For the purpose of studying the equidistribution of $\mu$-trajectories, we
prove a more general version (Theorem B′) where $Y$ is replaced by its orbit
$L_{0}Y$ under a compact subset $L_{0}$ of the centralizer of $H^{nc}$ in $G$.
Conclusion (ii) is obtained in [7] under the assumption that $\mu$ has a
finite exponential moment. If the space $\Omega=G/\Lambda$ is compact, item
(i) can be readily deduced from the arguments given in [6, Section 6], but it
appears to be new when $\Omega$ is not compact, even under exponential moments
assumptions.

Our third result states that each $\mu$-walk trajectory on $\Omega$
equidistributes in its closure toward some homogeneous measure. It can be seen
as an analog for random walks of Ratner’s equidistribution theorem [18] for
Ad-unipotent flows on $\Omega$. As observed in [6, 13], the assumption on $H$
must be strengthened in order to guarantee homogeneity. We ask that $H$ be
Zariski connected semisimple without compact factors, or equivalently
$H=H^{nc}$ semisimple.

######  Theorem C (Equidistribution in Cesàro-averages).

Suppose as well $H=H^{nc}$. For every $x\in\Omega$, we have:

  1. (i)

The orbit closure $Y=\overline{\Gamma\cdot x}\subset\Omega$ is a
$\Gamma$-invariant ergodic finite-volume closed homogeneous subset.

  2. (ii)

The sequence of measures
$(\frac{1}{n}\sum_{k=0}^{n-1}\mu^{*k}*\delta_{x})_{n\geq 1}$ converges to
$\nu_{Y}$ in the weak-$\ast$ topology.

  3. (iii)

For $\mu^{\otimes\mathbb{N}^{*}}$-almost every instructions $(g_{i})_{i\geq
1}$, the sequence of empirical measures
$(\frac{1}{n}\sum_{k=0}^{n-1}\delta_{g_{k}\dots g_{1}x})_{n\geq 1}$ converges
to $\nu_{Y}$ in the weak-$\ast$ topology.

Theorem C follows from [7] in the case where $\mu$ has a compact support,
which implies in particular that $\Gamma$ is compactly generated. It is a
well-known conjecture that the Cesàro-averages in the second item should not
be necessary. This will be proven in a follow-up paper [1] in the case where
two powers of $\mu$ are not mutually singular.

###  1.2 Dynamics of the proofs

The suppression of the exponential moment condition underlying Benoist-Quint’s
work requires most importantly to adapt the analytical core of their approach.
The abstract mecanism they rely on is inspired by Foster’s recurrence
criterion for walks with a negative drift. We recall it in Proposition 1.2
below. This result plays a fundamental role as it allows to convert increment
properties of the walk on $\Omega$ into asymptotic properties of its
distributions or trajectories. The first item is due to Eskin-Margulis [14,
Lemma 3.1] and the second to Benoist-Quint [7, Proposition 3.9].

######  Proposition 1.2 (Eskin-Margulis, Benoist-Quint).

Let $\Omega$ be a measurable space, $f:\Omega\rightarrow\mathbb{R}^{+}$ a
measurable function and $(X_{n})_{n\geq 0}$ a measurable Markov chain on
$\Omega$. Assume

  * $(\mathbf{EM})$

$\exists\alpha_{0}>0$,
$\,\sup_{x\in\Omega}\mathbb{E}_{x}\left(e^{\alpha_{0}|f(X_{1})-f(x)|}\right)<\infty$

  * $(\mathbf{D})$

$\exists R_{0},\lambda_{1}>0,$ $\forall x\in f^{-1}(R_{0},+\infty)$,
$\,\mathbb{E}_{x}(f(X_{1}))\leq f(x)-\lambda_{1}$

Then for all $\varepsilon>0$, there exists $R>0$ such that for $x\in\Omega$,

  1. (i)

$\forall n\geq n_{x},\quad\mathbb{P}_{x}[f(X_{n})>R]\,<\,\varepsilon$

  2. (ii)

$\mathbb{P}_{x}-\mbox{almost surely},$

$\limsup_{n\to+\infty}\frac{1}{n}\operatorname{card}\\{k\in\\{1,\dots,n\\}\ |\
f(X_{k})>R\\}<\varepsilon.$

It is certainly natural to wonder whether the moment conditions in Benoist-
Quint’s litterature may be loosened by weakening the requirement
($\mathbf{EM}$) above. A by-product of the present paper states that
Proposition 1.2 is still true if we replace ($\mathbf{EM}$) by the condition
that the increments of the walk are uniformly bounded in $L^{1+\eta}$ for some
$\eta>0$, more precisely by asking:

  * $(\mathbf{M}_{\eta})$

$\exists M>0$, $\forall x\in\Omega$,
$\,\mathbb{E}_{x}\left(|f(X_{1})-f(x)|^{1+\eta}\right)<M.$

However, carefully constructed (continuous) Markov chains on $\mathbb{R}^{+}$
show that _conclusions (i) and (ii) may not hold for walks with negative drift
and increments uniformly bounded in $L^{1}$_, in other words if
($\mathbf{EM}$) above is replaced by ($\mathbf{M}_{0}$) (see Section 2,
Examples 1 and 2). We need a new point of view, which reflects more refined
properties of homogeneous chains than the drift in expectation expressed by
$(\mathbf{D})$. Our crucial claim is that the height of a homogeneous walk
evolving in the cusps of a finite-volume space $\Omega=G/\Lambda$ must
decrease faster than a chain on $\mathbb{R}$ with i.i.d. increments of
negative mean. We formalize this concept by the notion of stochastic
dominance.

######  Definition 1.3.

Given real random variables $Z$ and $(Z_{i})_{i\in I}$, we say that
_$(Z_{i})_{i\in I}$ is stochastically dominated by $Z$_ if for every $i\in I$,
$t\in\mathbb{R}$

$\mathbb{P}(Z_{i}>t)\,\leq\,\mathbb{P}(Z>t).$

The abstract version of Proposition 1.2 we shall need is the following.

######  Theorem D (Recurrence of Markov chains with stochastic dominance).

Let $\Omega$ be a measurable space, $f:\Omega\rightarrow\mathbb{R}^{+}$ a
measurable function, and $(X_{n})_{n\geq 0}$ a measurable Markov chain on
$\Omega$.

Assume there exist a sublevel set $K:=\\{f\leq R_{0}\\}$ and integrable real
random variables $Z_{0}$, $Z_{1}$ with $\mathbb{E}[Z_{1}]:=-\lambda_{1}<0$
such that

  1. 1.

$(f(X_{1})-f(X_{0})|X_{0}=x)_{x\in K}$ is stochastically dominated by $Z_{0}$;

  2. 2.

$(f(X_{1})-f(X_{0})|X_{0}=x)_{x\in\Omega\smallsetminus K}$ is stochastically
dominated by $Z_{1}$.

Then for all $\varepsilon>0$, there exists $R>0$ such that for all
$x\in\Omega$,

  1. (i)

$\forall n\geq
1,\quad\mathbb{P}_{x}[f(X_{n})>R]\,<\,\varepsilon+\frac{f(x)}{n\lambda_{1}}.$

  2. (ii)

$\mathbb{P}_{x}-\mbox{almost surely},$

$\limsup_{n\to+\infty}\frac{1}{n}\operatorname{card}\\{k\in\\{1,\dots,n\\}\ |\
f(X_{k})>R\\}<\varepsilon.$

It is pleasant to note that conclusion (i) gives an _optimal bound_ on the
time $n$ at which the distribution $X_{n}$ starts to accumulate on the
sublevel set $\\{f\leq R\\}$ (see discussion below Proposition 2.4).

Once Theorem D is established, we use it to show that homogeneous random walks
do not escape at infinity, namely Theorem A. Theorem D will be applied to a
proper drift function $f:\Omega\to\mathbb{R}^{+}$ extracted from Benoist-
Quint’s paper [5]. Checking the stochastic condition of Theorem D will require
some work, mostly done in Section 3.

The proof of Theorem B is inspired by [6, Section 6] which obtains item (ii)
when $\mu$ has a finite exponential moment. However, several important changes
are needed. The approach of [6] consists in showing that the first return
random walk induced on a large compact subset $Q$ drifts away from $Y\cap Q$
in expectation. It is not possible to extract information on the original
$n$-th step distribution $\mu^{n}\ast\delta_{x}$ from this induced random
walk, so conclusion (i) cannot be obtained with such a strategy. Moreover,
checking that the induced walk does satisfy the new conditions of stochastic
dominance formulated in Theorem D would raise other significant difficulties.

Our solution instead is to construct a global drift function for every closed
invariant finite-volume homogeneous subset $Y$; this is done by gluing
together the Benoist-Quint drift function with a function that drifts away
from $Y$ on a compact subset. We believe this new technique could be useful to
construct drift functions in other contexts.

Finally, the equidistribution of each $\mu$-trajectory in its closure stated
in Theorem C results from a combination of Theorems A, B, the Eskin-
Lindenstrauss classification of stationary probability measures [13, Theorem
1.3], and the general strategy of [7]. The proof roughly goes as follows.
Consider a starting point $x\in\Omega$, a typical sequence of instructions
$(g_{i})_{i\geq 1}$ and write for $n\geq 1$,

$\nu_{n}=\frac{1}{n}\sum_{k=0}^{n-1}\delta_{g_{k}\dots g_{1}x}.$

Breiman’s law of large numbers and Theorem A imply that any weak-$\ast$ limit
$\nu$ of $(\nu_{n})_{n\geq 1}$ is a $\mu$-stationary probability measure. By
Theorem B, $\nu$ does not give mass to the $\Gamma$-invariant homogeneous
subspaces which do not contain $x$. The Eskin-Lindenstrauss classification of
stationary measures implies that $\nu=\nu_{Y}$ for some $Y\in
S_{\Omega}(\Gamma)$ and necessarily $Y:=\overline{\Gamma.x}$. This yields the
equidistribution statement.

Structure of the paper

Section 2 is dedicated to Theorem D. We first explain in greater detail the
role of the dominance hypothesis and then prove the result, subdivided into
two propositions (Propositions 2.4 and 2.6). We also check that the
conclusions still hold for walks with a negative drift and increments bounded
in $L^{1+\eta}$.

Section 3 converts Theorem D into a more handleable recurrence criterion for
homogeneous random walks. It will be used in Sections 3 and 4 to show that
homogeneous walks accumulate neither at infinity, nor around any proper
invariant homogeneous subspace.

Section 4 deals with Theorem A. We check that the Benoist-Quint drift function
satisfies the controlled drift assumption of Section 3.

Section 5 deals with Theorem B. The main part of the argument is the
construction of a new drift function for invariant homogeneous closed subsets.
It is obtained by gluing the Benoist-Quint function with some function which
drifts away from the homogeneous subspace on a compact set.

Section 6 contains the proof of Theorem C. It combines Theorems A, B and the
classification of stationary measures of Eskin-Lindenstrauss.

Section 7 concludes the paper with some possible further directions of
research.

Acknowledgements. We are indebted to Yves Benoist and Jean-François Quint for
several helpful discussions on the subject. It is a pleasure to thank them
here for sharing their knowledge, and for organizing the _Arbeitsgemeinschaft_
on rigidity of stationary measures at the Mathematisches Forschungsinstitut
Oberwolfach, where we learned about these problems. We also thank Jean-
Philippe Dru for his help in finding reference [19].

##  2 Markov chains with a negative drift

The goal of this section is to prove Theorem D stating that Markov chains on a
stratified state space and satisfying some stochastic dominance condition have
_neither escape of mass nor escape of empirical measures_. We also note that
these conclusions are still valid if the chain is assumed to have a negative
drift and increments uniformly bounded in $L^{1+\eta}$ for some $\eta>0$.
Counterexamples are given if $\eta=0$.

For the whole section, we fix a measurable space $\Omega$, a non-negative
measurable function $f:\Omega\rightarrow\mathbb{R}^{+}$, and a Markov chain
$(X_{n})_{n\geq 0}$ on $\Omega$. Given a point $x\in\Omega$, the notations
$\mathbb{E}_{x}$ and $\mathbb{P}_{x}$ will refer to the expectation and
probability conditional to $X_{0}=x$.

###  2.1 Dominance condition and return times

We first give some perspective to the stochastic dominance assumptions in
Theorem D. For future reference, these assumptions will be denoted by
$(\mathbf{SD})$. Recall for clarity:

  * $(\mathbf{SD})$

There exist a sublevel set $K:=\\{f\leq R_{0}\\}$ and integrable real random
variables $Z_{0}$, $Z_{1}$ with $\mathbb{E}[Z_{1}]:=-\lambda_{1}<0$ such that
every $x\in\Omega$, $t\in\mathbb{R}$,

$\mathbb{P}_{x}(f(X_{1})-f(x)>t)\,\,\leq\,\,\mathbb{P}(Z_{0}>t)\mathbbm{1}_{K}(x)\,+\,\mathbb{P}(Z_{1}>t)\mathbbm{1}_{\Omega\smallsetminus
K}(x).$

We begin by observing that $(\mathbf{SD})$ is a very natural condition as it
is satisfied for Markov chains with a well-behaved negative drift. This
criterion is expressed in Lemma 2.2 below. It will further be adapted to
homogeneous chains in Section 3. The statement uses the notion of standard
realisation of a random variable which we first recall.

######  Definition 2.1 (Standard realisation).

Let $Z$ be a real random variable. The random variable on the probability
space $([0,1],\mathscr{B}([0,1]),\text{leb}_{|[0,1]})$ defined by

$\begin{array}[]{lccl}Z^{\prime}:&[0,1]&\rightarrow&\mathbb{R}\\\
&s&\mapsto&\max\\{t\in\mathbb{R},\,\,\mathbb{P}(Z\geq t)\geq s\\}\end{array}$

has the same law as $Z$. We call it the _standard realisation_ of $Z$.

######  Remark.

This notion allows a second formulation of stochastic dominance: $Z_{1}$ is
dominated by $Z_{2}$ if and only if their standard realisations
$Z_{1}^{\prime},Z_{2}^{\prime}$ satisfy $Z_{1}^{\prime}\leq Z_{2}^{\prime}$
everywhere on $[0,1]$.

######  Lemma 2.2 (Criterion for stochastic dominance).

Let $(\Omega,f,(X_{n}))$ as above. Assume there exist a probability space
$(E,\mathbb{P})$, an integrable random variable $Z:E\rightarrow\mathbb{R}_{+}$
and constants $R_{0},\lambda,\alpha>0$ such that

  1. 1.

$\forall x\in\Omega$,

$f(x)>R_{0}\implies\mathbb{P}_{x}(f(X_{1})\leq f(x)-\lambda)\geq 1-\alpha$

  2. 2.

$\forall x\in\Omega$, $t\in\mathbb{R}_{+}$,

$\mathbb{P}_{x}(f(X_{1})-f(x)>t)\leq\mathbb{P}(Z>t)$

  3. 3.

Denoting by $Z^{\prime}$ the standard realisation of $Z$ we have

$\mathbb{E}(Z^{\prime}\mathbbm{1}_{[0,\alpha]})<\lambda(1-\alpha)$

Then $(\Omega,f,(X_{n}))$ satisfies the dominance condition $(\mathbf{SD})$
with constants $(R_{0},\lambda_{1})$ where
$\lambda_{1}:=\lambda(1-\alpha)-\mathbb{E}(Z^{\prime}\mathbbm{1}_{[0,\alpha]})$.

###### Proof.

Given $x\in\Omega$, we denote by $Y_{x}$ the variable $f(X_{1})-f(x)$ varying
under $\mathbb{P}_{x}$. Assumption _2._ of the lemma yields that for any
$x\in\Omega$, $Y_{x}$ is stochastically dominated by $Z$. In particular, we
may choose $Z_{0}=Z$.

We now define $Z_{1}$. Let $x\in\Omega\smallsetminus K$ and
$Y^{\prime}_{x},Z^{\prime}:[0,1]\rightarrow\mathbb{R}$ be the standard
realisations of $Y_{x},Z$ defined above. By assumption 1), we have
$Y^{\prime}_{x}\leq-\lambda$ on $(\alpha,1]$, hence we can write everywhere on
$[0,1]$

$Y^{\prime}_{x}\leq
Z^{\prime}\mathbbm{1}_{[0,\alpha]}-\lambda\mathbbm{1}_{(\alpha,1]}$

The right-hand side defines a random variable $Z_{1}$ independently of the
point $x\in\Omega\smallsetminus K$ chosen earlier. Moreover, assumption _3._
yields

$\mathbb{E}(Z_{1})\leq\mathbb{E}(Z^{\prime}\mathbbm{1}_{[0,\alpha]})-\lambda(1-\alpha)<0$

which concludes the proof. ∎

######  Remark.

The idea behind Lemma 2.2 can easily be understood through the graphs of the
repartition functions, as in Figure 1 below. Recall that the repartition
function of a real random variable $X$ is defined by
$F_{X}(t)=\mathbb{P}(X\leq t)$ for $t\in\mathbb{R}$.

$t$$1$$1-\alpha$$0$$-\lambda$$F_{f(X_{1})-f(x)}(t)$$F_{Z_{1}}(t)$$F_{Z}(t)$
Figure 1: Stochastic domination by $Z_{1}$ with $\mathbb{E}[Z_{1}]<0$.

One can readily compute the expectation of $Z_{1}$ from the above picture:

$\displaystyle\mathbb{E}[Z_{1}]$
$\displaystyle=\int_{\mathbb{R}}t\,\mathrm{d}F_{Z_{1}}(t)$
$\displaystyle=-(1-\alpha)\lambda+\int\mathbbm{1}_{\\{F_{Z}(t)\geq
1-\alpha\\}}\mathrm{d}F_{Z}(t)$
$\displaystyle=-(1-\alpha)\lambda+\mathbb{E}[\mathbbm{1}_{[0,\alpha]}Z^{\prime}]=-\lambda_{1}<0.$

We now turn to the first implications of condition $(\mathbf{SD})$. Its
crucial input is that it allows to _bound the tail probabilities of the return
time to the sublevel set $K$_ independently of the starting point $x\in K$,
and so that the resulting sequence of bounds is summable. This will be the key
ingredient to show that the mass of the $n$-th step distribution and the
empirical measures do not escape at infinity. Such bounds are also available
if the walk has a negative drift and increments uniformly bounded in
$L^{1+\eta}$ for some $\eta>0$, so the non-escape estimates are equally valid
in this case. This latter observation will not be used in the other sections,
but seems general enough to be of independent interest. It also extends the
results based on exponential moment conditions used in [14, 7].

######  Lemma 2.3 (Return times).

Suppose $(\Omega,f,(X_{n}))$ satisfy condition $(\mathbf{SD})$, or conditions
$(\mathbf{D})$ and $(\mathbf{M}_{\eta})$ for some $\eta>0$. Denote by

$\tau=\inf\\{n\geq 1\ |\ X_{n}\in K\\}$

the first return time to $K$. Then

$\sum_{n\geq 1}\sup_{x\in K}\mathbb{P}_{x}(\tau\geq n)<\infty.$

###### Proof.

Case 1: $(\mathbf{SD})$ is satisfied.

Denote by $(Y_{n})$ the Markov chain on $\mathbb{R}$ with i.i.d. increments
given by $Z_{1}$ and set

$\tau^{(Y)}=\\{\inf n\geq 1,\,Y_{n}\leq R_{0}\\}$

the first return time of $(Y_{n})$ to $(-\infty,R_{0}]$. Using the dominance
condition $(\mathbf{SD})$, one sees by induction on $n$ that for every
$x\in\Omega$ such that $f(x)>R_{0}$, we have

$\mathbb{P}_{x}(\tau\geq n)\geq\mathbb{P}_{f(x)}(\tau^{(Y)}\geq n).$

Now for $n\geq 2$, $x\in K$, we can write

$\displaystyle\mathbb{P}_{x}(\tau\geq n)$
$\displaystyle=\int_{(R,+\infty)}\mathbb{P}_{X_{1}}(\tau\geq
n-1)\,d\mathbb{P}_{x}(X_{1})$
$\displaystyle\leq\int_{(R,+\infty)}\mathbb{P}_{f(X_{1})}(\tau^{(Y)}\geq
n-1)\,d\mathbb{P}_{x}(X_{1})$
$\displaystyle\leq\int_{(R,+\infty)}\mathbb{P}_{R+Z_{0}}(\tau^{(Y)}\geq
n-1)\,d\mathbb{P}(Z_{0})$

where the last inequality is obtained using assumption $(\mathbf{SD})$ and the
fact that $t\mapsto\mathbb{P}_{t}(\tau^{(Y)}\geq n-1)$ is increasing on
$(R,+\infty)$. Observing that the right-hand side no longer depends on $x$ and
summing over $n$ we get

$\displaystyle\sum_{n\geq 2}\sup_{x\in K}\mathbb{P}_{x}(\tau\geq n)$
$\displaystyle\leq\int_{(R,+\infty)}\sum_{n\geq
2}\mathbb{P}_{R+Z_{0}}(\tau^{(Y)}\geq n-1)\,d\mathbb{P}(Z_{0})$
$\displaystyle=\int_{(R,+\infty)}\mathbb{E}_{R+Z_{0}}(\tau^{(Y)})\,d\mathbb{P}(Z_{0}).$

However, Foster’s criterion [17, Proposition 11.3.2] gives for every $t>R$,
the bound $\mathbb{E}_{t}(\tau^{(Y)})\leq(t-R)/\lambda_{1}$. Hence we may
conclude that

$\displaystyle\sum_{n\geq 2}\sup_{x\in K}\mathbb{P}_{x}(\tau\geq n)$
$\displaystyle\leq\frac{1}{\lambda_{1}}\int_{(R,+\infty)}Z_{0}\,d\mathbb{P}(Z_{0})$
$\displaystyle<\infty.$

Case 2: $(\mathbf{D})$ and $(\mathbf{M_{\eta}})$ are satisfied from some
$\eta>0$.

We can assume $\eta\in(0,1)$. Since $\mathbb{P}_{x}(\tau\geq
n)\leq\frac{1}{n^{1+\eta}}\mathbb{E}_{x}[\tau^{1+\eta}]$, it is enough to
prove that the family of return times $(\tau\,|\,X_{0}=x)_{x\in K}$ is
uniformly bounded in $L^{1+\eta}$. More generally, we show that for every
$x\in\Omega$,

$\displaystyle\mathbb{E}_{x}(\tau^{1+\eta})\leq\left(\frac{f(x)}{\lambda_{1}}\right)^{1+\eta}+M\frac{f(x)}{\lambda_{1}}+M_{1}$
(1)

where
$M_{1}=1+4\left((M^{\frac{1}{1+\eta}}+R_{0})^{1+\eta}+\frac{M(M+R_{0})}{\lambda_{1}}\right)$.

We first bound the return time for a starting point oustide of $K$. On the one
hand, notice that we already know by Foster’s criterion that for every
$x\in\Omega\smallsetminus K$,
$\mathbb{E}_{x}(\tau)\leq\frac{f(x)}{\lambda_{1}}$. In particular, Jensen’s
inequality gives

$\displaystyle\mathbb{E}_{x}(\tau^{\eta})\leq\left(\frac{f(x)}{\lambda_{1}}\right)^{\eta}.$
(2)

On the other hand, the combination of $(\mathbf{D})$ and $(\mathbf{M_{\eta}})$
yields a strong negative drift for $f(X_{1})^{1+\eta}$ as we now explain.
Observe that the inequality

$(1+t)^{1+\eta}\leq 1+(1+\eta)t+\lvert t\rvert^{1+\eta}$

where $-1\leq t\leq+\infty$, applied to $t=\frac{f(X_{1})-f(x)}{f(x)}$ gives

$f(X_{1})^{1+\eta}\leq
f(x)^{1+\eta}+(1+\eta)\left(f(X_{1})-f(x)\right)f(x)^{\eta}+\lvert
f(X_{1})-f(x)\rvert^{1+\eta},$

Taking expectation, we obtain for every $x\in\Omega\smallsetminus K$,

$\displaystyle\mathbb{E}_{x}(f(X_{1})^{1+\eta})\leq
f(x)^{1+\eta}-(1+\eta)\lambda_{1}f(x)^{\eta}+M.$ (3)

Inequalities (2) and (3) allow to apply a general result of Tweedie [19,
Theorem 3(iii)] (with $A=K$, $\psi(k)=(1+\eta)\lambda^{1+\eta}_{1}k^{\eta}-M$,
$g(x)=f(x)^{1+\eta}$, $g_{1}(x)=(1+\eta)\lambda_{1}f(x)^{\eta}-M$) to obtain
for all $x\in\Omega\smallsetminus K$,

$\mathbb{E}_{x}(\tau^{1+\eta})\leq\left(\frac{f(x)}{\lambda_{1}}\right)^{1+\eta}+M\mathbb{E}_{x}(\tau)$

so by Foster’s criterion again, for $x\in\Omega\smallsetminus K$,

$\mathbb{E}_{x}(\tau^{1+\eta})\leq\left(\frac{f(x)}{\lambda_{1}}\right)^{1+\eta}+M\frac{f(x)}{\lambda_{1}}.$

The general bound announced in (1) for every $x$ in $\Omega$ follows using the
moment condition $(\mathbf{M_{\eta}})$.

∎

###  2.2 Conservation of mass

We show conclusion (i) in Theorem D: random walks on $\Omega$ satisfying the
stochastic dominance assumption $(\mathbf{SD})$, or having a negative drift
and increments uniformly bounded in $L^{1+\eta}$, have no escape of mass.

######  Proposition 2.4.

Suppose $(\Omega,f,(X_{n}))$ satisfy condition $(\mathbf{SD})$, or conditions
$(\mathbf{D})$ and $(\mathbf{M}_{\eta})$ for some $\eta>0$. Then for all
$\varepsilon>0$, there exists $R>0$ such that for $x\in\Omega$, $n\geq 1$,

$\mathbb{P}_{x}[f(X_{n})>R]\,<\,\varepsilon+\frac{f(x)}{n\lambda_{1}}$

######  Remark.

Choosing $\varepsilon$ very small, we see that the first time $n\geq 1$ such
that $\mathbb{P}_{x}[f(X_{n})>R]<1$ is bounded above by
$\frac{f(x)}{\lambda_{1}(1-\varepsilon)}\simeq\frac{f(x)}{\lambda_{1}}$. When
$f(x)\to+\infty$, this bound is essentially optimal, as one can see by
considering a deterministic walk by translation of $-\lambda_{1}$, for which
the return time is equal to $\frac{f(x)-R}{\lambda_{1}}$.

The key is to use Lemma 2.3 to show the following renewal estimate.

######  Lemma 2.5 (Renewal estimate).

Suppose $(\Omega,f,(X_{n}))$ satisfy condition $(\mathbf{SD})$, or conditions
$(\mathbf{D})$ and $(\mathbf{M}_{\eta})$ for some $\eta>0$. Then for any
$\alpha>0$, there exists $\ell\geq 0$, such that for all $x\in K$,
$n\geq\ell$,

$\mathbb{P}_{x}(\exists i\in\\{n-\ell,\dots,n\\}:\,X_{i}\in K)>1-\alpha.$

In words, this lemma states that for any interval of integers
$I\subseteq\mathbb{N}$ of large length $\ell$, the walk starting from an
arbitrary point in $K$ has a very good chance to come back to $K$ during $I$.

###### Proof.

Let $\alpha>0$. According to Lemma 2.3, we may choose $\ell\geq 0$ such that

$\sum_{n\geq\ell}\sup_{x\in K}\mathbb{P}_{x}(\tau>n)<\alpha.$

Let us check that $\ell$ satisfies the conclusions of the lemma. Given
$n\geq\ell$, it is convenient to set $I_{n}=\\{n-\ell,\dots,n\\}$. Assume that
$A_{n}\in\mathbb{R}^{+}$ is a constant such that for $x\in K$,
$k\in\\{\ell,\dots,n\\}$, we have

$\mathbb{P}_{x}(\forall i\in I_{k},\,X_{i}\notin K)\leq A_{n}.$

We can write

$\displaystyle\mathbb{P}_{x}(\forall i\in I_{n+1},\,X_{i}\notin K)\,$
$\displaystyle\leq\,\mathbb{P}_{x}(\forall i\in I_{n+1},\,X_{i}\notin
K;\tau\leq n)+\mathbb{P}_{x}(\tau>n)$
$\displaystyle\leq\sum_{k=1}^{n-\ell}\,\mathbb{P}_{x}(\forall i\in
I_{n+1},\,X_{i}\notin K;\tau=k)+\mathbb{P}_{x}(\tau>n)$
$\displaystyle\leq\sum_{k=1}^{n-\ell}\,A_{n}\mathbb{P}_{x}(\tau=k)+\mathbb{P}_{x}(\tau>n)$
$\displaystyle\leq A_{n}+\mathbb{P}_{x}(\tau>n).$

Hence, defining a sequence $(A_{n})_{n\geq\ell}$ by

$\left\\{\begin{array}[]{l}A_{\ell}=0\\\ \forall n\geq\ell,\
A_{n+1}=A_{n}+\sup_{x\in K}\mathbb{P}_{x}(\tau>n)\end{array}\right.$

we can conclude that for all $n\geq\ell$,

$\mathbb{P}_{x}(\forall i\in I_{n},\,X_{i}\notin K)\leq A_{n}<\alpha.$

∎

Let us now prove the conservation of mass anounced above.

###### Proof of Proposition 2.4.

We first deal with the case where $x\in K$. We denote by $\ell\geq 0$ the
constant given by Lemma 2.5 for $\alpha=\varepsilon/2$, then choose the
constant $R>0$ large enough so that for every $x\in K$ and every $n\leq\ell$,

$\mathbb{P}_{x}(f(X_{n})>R)<\varepsilon/2.$

By Lemma 2.5, this yields for $x\in K$, and arbitrary $n\geq\ell$

$\displaystyle\mathbb{P}_{x}(f(X_{n})>R)$
$\displaystyle<\varepsilon/2+\mathbb{P}_{x}(f(X_{n})>R;\,\,\exists
i\in\\{n-\ell,\dots,n\\},X_{i}\in K)$ $\displaystyle<\varepsilon.$

To deal with the complementary case $x\in\Omega\smallsetminus K$, denote by
$\tau$ the first return time to $K$ and notice that for $n\geq 0$,

$\displaystyle\mathbb{P}_{x}(f(X_{n})>R)$
$\displaystyle<\mathbb{P}_{x}(f(X_{n})>R;\,\tau\leq n)+\mathbb{P}_{x}(\tau>n)$
$\displaystyle<\varepsilon+\frac{\mathbb{E}_{x}[\tau]}{n}.$

In view of Foster’s criterion [17, Proposition 11.3.2], we have
$\mathbb{E}_{x}[\tau]\leq\frac{f(x)}{\lambda_{1}}$ as soon as $f(x)>R_{0}$.
Hence, we obtain

$\displaystyle\mathbb{P}_{x}\left(f(X_{n})>R\right)<\varepsilon+\frac{f(x)}{n\lambda_{1}}$

which finishes the proof of Proposition 2.4 ∎

We conclude Section 2.2 by a counterexample to Proposition 2.4 when the walk
$(\Omega,f,X_{n})$ is only assumed to satisfy $(\mathbf{D})$ and
$(\mathbf{M}_{0})$.

######  Example 1 (Escape of mass).

Let $[0,\frac{1}{2}]\rightarrow[0,1),x\mapsto\alpha_{x}$ be a continuous map
such that $\alpha^{-1}(0)=0$. We consider the Markov chain $(X_{n})_{n\geq 0}$
on $\Omega=[0,+\infty)$ whose transition probability measures are given by

$\left\\{\begin{array}[]{ll}\mu_{0}=\delta_{0}\\\
\mu_{x}=(1-\alpha_{x})\delta_{x}+\alpha_{x}\delta_{\alpha^{-1}_{x}}&\text{if
$0<x\leq 1/2$}\\\ \mu_{x}=(2x-1)\delta_{0}+(2-2x)\mu_{\frac{1}{2}}&\text{if
$1/2\leq x\leq 1$}\\\ \mu_{x}=\delta_{x-1}&\text{if $x\geq 1$.}\\\
\end{array}\right.$

Note that $\mu_{x}$ depends continously on $x$. Letting $f=\text{Id}$, the
random walk $(X_{n})_{n\geq 0}$ satisfies

$\sup_{x\in\Omega}\mathbb{E}_{x}[|f(X_{1})-f(x)|]\leq
1\quad\text{and}\quad\sup_{x\in(1,+\infty)}\mathbb{E}_{x}[f(X_{1})-f(x)]=-1.$

However, given a fixed $x_{0}\in(0,1/2]$, one can choose the family
$(\alpha_{x})_{x\in[0,1/2]}$ so that for every $R>0$, there exists $n\geq 0$
such that

$\displaystyle\mathbb{P}_{x_{0}}(f(X_{n})>R)\geq 1/2.$ (4)

To see this, first observe that a trajectory with origin
$x_{0}\in(0,\frac{1}{2}]$ stays at $x_{0}$ for some time, then jumps out of
$[0,1]$ and comes back to it with constant increments of $-1$ until reaching a
point $x_{1}\in[0,1]$. Then, if $x_{1}\in(0,\frac{1}{2}]$, the process repeats
itself to give some point $x_{2}\in[0,1]$, and so on. Arguing step by step, we
can choose the coefficients $\alpha_{x}$ such that the return points
$x_{0},x_{1},x_{2},\dots$ form an infinite sequence in $(0,\frac{1}{2}]$,
satisfy $x_{i}>x_{i+1}$, $(x_{i})_{i\geq 0}\to 0$, and so that the sequence
$(\alpha_{x_{i}})_{i\geq 0}$ decreases to $0$ fast enough to have for every
$i\geq 1$, some $n_{i}\geq 1$ for which

$\mathbb{P}_{x_{0}}(X_{n_{i}}=x_{i})\geq 1-1/i.$

Let us now check the property (4) announced above. Fix $R>0$. For $i,k\geq 0$,

$\displaystyle\mathbb{P}_{x_{0}}(X_{n_{i}+k}\leq R)$
$\displaystyle=\mathbb{P}_{x_{0}}(X_{n_{i}+k}\leq
R;X_{n_{i}}=x_{i})+\mathbb{P}_{x_{0}}(X_{n_{i}+k}\leq R;X_{n_{i}}\neq x_{i})$
$\displaystyle\leq\mathbb{P}_{x_{i}}(X_{k}\leq R)+1/i.$ (5)

Assume $i$ to be large enough so that $\alpha_{x_{i}}^{-1}>R$ and set
$k_{i}=\lfloor\alpha^{-1}_{x_{i}}-R\rfloor$ the greatest integer smaller than
$\alpha^{-1}_{x_{i}}-R$. We can then write

$\displaystyle\mathbb{P}_{x_{i}}(X_{k_{i}}\leq R)$
$\displaystyle=\mathbb{P}_{x_{i}}(X_{1},\dots,X_{k_{i}}=x_{i})$
$\displaystyle=(1-\alpha_{x_{i}})^{k_{i}}$
$\displaystyle=e^{k_{i}\log(1-\alpha_{x_{i}})}\underset{i\to+\infty}{\longrightarrow}e^{-1}.$
(6)

Equations (1) and (1) give that for large $i$,

$\mathbb{P}_{x_{0}}(X_{n_{i}+k_{i}}\leq R)\leq 1/2$

as announced in (4).

###  2.3 Recurrence of empirical measures

We show conclusion (ii) in Theorem D: random walks on $\Omega$ satisfying the
stochastic dominance assumption ($\mathbf{SD}$), or having a negative drift
and increments uniformly bounded in $L^{1+\eta}$, must have positive recurrent
trajectories.

######  Proposition 2.6.

Suppose $(\Omega,f,(X_{n}))$ satisfy condition $(\mathbf{SD})$, or conditions
$(\mathbf{D})$ and $(\mathbf{M}_{\eta})$ for some $\eta>0$. For every
$\varepsilon>0$, there exists $R>0$ such that for every $x\in\Omega$,
$\mathbb{P}_{x}$-almost every $(X_{n})$,

$\limsup_{n\to+\infty}\frac{1}{n}\operatorname{card}\\{k\in\\{1,\dots,n\\}\ |\
f(X_{k})>R\\}<\varepsilon.$

###### Proof.

We start with some notation. Let $\tau_{i}$ be the $i$-th return time in $K$.
More precisely, $\tau_{0}=0$ by convention, and for all $i\geq 1$,

$\tau_{i}=\inf\\{n>\tau_{i-1}\ |\ X_{n}\in K\\}.$

We also let $Z_{i}=\tau_{i}-\tau_{i-1}$ be the length of the $i$-th excursion
and

$Z^{ext}_{i}=\operatorname{card}\\{n\in\\{\tau_{i-1},\dots,\tau_{i}-1\\}\ |\
f(X_{n})>R\\}$

the time spent outside of $\\{f\leq R\\}$ during the excursion.

To prove Proposition 2.6, it is sufficient to show that if $R>0$ is large
enough, then for any $x\in K$, $\mathbb{P}_{x}$-almost surely, one has for
large $n\geq 0$,

$\displaystyle\sum_{i=1}^{n}Z^{ext}_{i}<\varepsilon\sum_{i=1}^{n-1}Z_{i}.$ (7)

We explain how to obtain (7). The assumption $(\mathbf{SD})$, or the
conditions $(\mathbf{D})$ and $(\mathbf{M}_{\eta})$, both guarantee that

$\displaystyle\sup_{x\in K}\mathbb{P}_{x}(Z^{ext}_{1}\neq
0)\underset{R\to+\infty}{\longrightarrow}0.$ (8)

Indeed, we can write for $x\in K$,

$\displaystyle\mathbb{P}_{x}(Z^{ext}_{1}\neq 0)$
$\displaystyle\leq\mathbb{P}_{x}(Z^{ext}_{1}\neq 0;\,\tau\leq
N)+\mathbb{P}_{x}(\tau\geq N)$ $\displaystyle\leq\mathbb{P}_{x}(\exists i\leq
N,f(X_{i})>R;\,\tau\leq N)+\mathbb{P}_{x}(\tau\geq N)$
$\displaystyle\leq\sum_{i\leq
N}\mathbb{P}_{x}(f(X_{i})>R)+\mathbb{P}_{x}(\tau\geq N)$
$\displaystyle\leq\frac{N(R_{0}+NM)}{R}+\sup_{y\in K}\mathbb{P}_{y}(\tau\geq
N)$

where $M>0$ is a fixed uniform $L^{1}$-bound on the positive increment of the
walk, i.e. such that $\mathbb{E}_{x}[\max\left(0,f(X_{1})-f(x)\right)]<M$. The
right-hand side of the inequality does not depend on $x\in K$. In view of
Lemma 2.3, we can choose $N$ large, then $R$ even larger, to have the right-
hand side arbitrarily close to $0$, whence (8).

Observing that
$\mathbb{E}_{x}(Z^{ext}_{1})\leq\mathbb{E}_{x}(\tau\mathbbm{1}_{Z^{ext}_{1}\neq
0})$ and using the uniform integrability of the variables
$(\tau\,|\,X_{0}=x)_{x\in K}$ given by Lemma 2.3, we deduce from (8) that

$\sup_{x\in
K}\mathbb{E}_{x}(Z^{ext}_{1})\underset{R\to+\infty}{\longrightarrow}0.$

In particular, we can choose $R$ so that $\sup_{x\in
K}\mathbb{E}_{x}(Z^{ext}_{1})<\varepsilon/2$. Fix $x\in K$. Denote by
$\mathcal{F}_{i}$ the sub $\sigma$-algebra of $\Omega$ generated by
$(X_{0},\dots,X_{\tau_{i}})$. The random variables $Z^{ext}_{i}$, $i\geq 1$,
statisfy for $n\geq 1$,

$\mathbb{P}_{x}(Z^{ext}_{i}\geq n\,|\,\mathcal{F}_{i-1})\leq\sup_{y\in
K}\mathbb{P}_{y}(\tau\geq n).$

As the right-hand side of the inequality is summable by Lemma 2.3, we may
apply Kolmogorov’s law of large numbers for dependent variables [9, Theorem
A.6] and obtain that $\mathbb{P}_{x}$-almost surely,

$\frac{1}{n}\sum_{i=1}^{n}Z^{ext}_{i}=\frac{1}{n}\sum_{i=1}^{n}\mathbb{E}_{x}[Z^{ext}_{i}\,|\,\mathcal{F}_{i-1}]+o(1).$

Moreover, by our choice for $R$, we have
$\mathbb{E}_{x}[Z^{ext}_{i}\,|\,\mathcal{F}_{i-1}]<\varepsilon/2$. We infer
that for large $n\geq 0$,

$\frac{1}{n}\sum_{i=1}^{n}Z^{ext}_{i}<\varepsilon\,\frac{1}{n}\sum_{i=1}^{n-1}Z_{i}$

keeping in mind that by definition $Z_{i}\geq 1$. This concludes the proof of
inequality (7), whence the proposition. ∎

We conclude with a counterexample to Proposition 2.4 when the walk
$(\Omega,f,X_{n})$ is only assumed to satisfy $(\mathbf{D})$ and
$(\mathbf{M}_{0})$.

######  Example 2 (Escape of empirical measures).

Given a real number $r\in\mathbb{R}$, we denote by $\lfloor r\rfloor$ the
largest integer $k$ such that $k\leq r$. Fix a sequence $(x_{i})_{n\geq
3}\in(0,1)^{\mathbb{N}_{\geq 3}}$ decreasing to $0$, and consider the Markov
chain $(X_{n})_{n\geq 0}$ on $\Omega=[0,+\infty)$ whose transition probability
measures are given by

$\left\\{\begin{array}[]{ll}\mu_{0}=\delta_{0}&\\\
\mu_{x_{i}}=(1-\frac{1}{i\log i})\delta_{x_{i+1}}+\frac{1}{i\log
i}\delta_{x_{i+1}+\lfloor i\sqrt{\log i}\rfloor}&\text{ for $i\geq 3$}\\\
\mu_{x}=\delta_{x-1}&\text{ if $x>1$}\\\ x\mapsto\mu_{x}\ \text{affine on each
segment}\ [x_{i+1},x_{i}].&\end{array}\right.$

The chain $(X_{n})_{n\geq 0}$ has a continuous family of transition
probability measures and satisfies, for $f=\text{Id}$,

$\sup_{x\in\Omega}\mathbb{E}_{x}[|f(X_{1})-f(x)|]\leq
3\quad\text{and}\quad\sup_{x\in(1,+\infty)}\mathbb{E}_{x}[f(X_{1})-f(x)]=-1.$

However, for $\mathbb{P}_{x_{3}}$-almost every trajectory $(X_{n})_{n\geq 0}$,
for every $\varepsilon,R>0$, there exists $n_{0}\geq 1$ such that

$\displaystyle\frac{1}{n_{0}}\operatorname{card}\\{k\in\\{1,\dots,n_{0}\\}\ |\
f(X_{k})\leq R\\}<\varepsilon.$ (9)

To see this, denote by $\tau_{j}$ the $j$-th return time to $[0,1]$. In other
words, $\tau_{0}=0$ by convention, and for all $j\geq 1$,

$\tau_{j}=\inf\\{n>\tau_{j-1}\ |\ X_{n}\in[0,1]\\}.$

The random variables $(\tau_{j}-\tau_{j-1})_{j\geq 1}$ varying under
$\mathbb{P}_{x_{3}}$ are independent of respective laws

$\displaystyle(1-\frac{1}{(j+2)\log(j+2)})\delta_{1}+\frac{1}{(j+2)\log(j+2)}\delta_{\lfloor(j+2)\sqrt{\log(j+2)}\rfloor+1}$
($j\geq 1$)

As the sequence $(\frac{1}{n\log n})_{n\geq 3}$ is not summable, the converse
of the Borel-Cantelli lemma yields that $\mathbb{P}_{x_{3}}$-almost surely we
have $\limsup_{j\to+\infty}\frac{\tau_{j}-\tau_{j-1}}{j}=+\infty$. In
particular, given $\varepsilon,R>0$, there exists $j_{0}\geq 1$ such that

$j_{0}<\frac{\varepsilon}{R+1}\,\tau_{j_{0}}.$

Setting $n_{0}=\tau_{j_{0}}$, we get

$\operatorname{card}\\{k\in\\{1,\dots,n_{0}\\}\ |\ X_{k}\leq
1\\}<\frac{\varepsilon}{R+1}n_{0}.$

Finally, observing that for any $k\geq 0$ such that $X_{k}\leq R$ there exists
a time $j\in\\{k,\dots,k+\lfloor R\rfloor\\}$ for which $X_{j}\in[0,1]$, we
conclude

$\operatorname{card}\\{k\in\\{1,\dots,n_{0}\\}\ |\ X_{k}\leq R\\}<\varepsilon
n_{0}$

as announced in (9).

##  3 Recurrence criterion for homogeneous chains

We apply the results of the previous section to the particular case of
homogeneous random walks and obtain a handy criterion to check that the mass
or the empirical measures of the walk do not accumulate near a given subset
(Proposition 3.3 below). This criterion will be applied in two different
contexts, first to prove non-escape at infinity, and later to derive non-
accumulation on invariant homogeneous subspaces.

###  3.1 The controlled drift condition

In the whole section we fix $\Omega$ a locally compact topological space,
$\Gamma$ a topological semigroup acting continously on $\Omega$, and $\mu$ a
Borel probability measure on $\Gamma$. We call $\mu$-walk on $\Omega$ the
Markov chain on $\Omega$ whose law of transition at $x\in\Omega$ is
$\mu\ast\delta_{x}$, image of $\mu$ under $g\mapsto gx$. We denote by
$\overline{\Omega}=\Omega\cup\\{\infty\\}$ the Alexandroff compactification of
$\Omega$, and write $\mathscr{P}(\overline{\Omega})$ the collection of its
subsets. Given
$\mathcal{S}\subseteq\mathscr{P}(\overline{\Omega})\times\mathscr{P}(\overline{\Omega})$,
we have a decomposition $\mathcal{S}=\cup_{Y\in
p_{1}(\mathcal{S})}\\{Y\\}\times\mathcal{S}_{Y}$ where $p_{1}$ is the first
coordinate projection map and
$\mathcal{S}_{Y}\subseteq\mathscr{P}(\overline{\Omega})$ is the fibre above
$Y$ in $\mathcal{S}$. Finally, given a finite-dimensional real vector space
$V$, we set
$\operatorname{\mathrm{SL}}^{\pm}(V)=\\{g\in\operatorname{\mathrm{GL}}(V)\ |\
\det g=\pm 1\\}$.

######  Definition 3.1.

Let
$\mathcal{S}\subseteq\mathscr{P}(\overline{\Omega})\times\mathscr{P}(\overline{\Omega})$,
$\mathcal{A}\subseteq\mathscr{P}(\Omega)$. We say that the $\mu$-walk has a
_controlled drift away from $\mathcal{S}$ on $\mathcal{A}$_ if:

  * •

There exists a linear representation
$\Phi:\Gamma\rightarrow\operatorname{\mathrm{SL}}^{\pm}(V)$ on a finite-
dimensional normed real vector space $V$ such that

$\int_{G}\log\|\Phi(g)\|\,d\mu(g)<+\infty.$

  * •

There exist constants $\lambda,C>0$ such that for any $\alpha>0$, there exists
$N\geq 1$ such that for $n\geq N$, for any $Y\in p_{1}(\mathcal{S})$, any
compact subset $K\subseteq\Omega$ such that $K\cap Y=\varnothing$, any
$Q\in\mathcal{A}$, there is $Y^{\prime}\in\mathcal{S}_{Y}$, and a measurable
function $f:\Omega\to[0,+\infty]$ which is bounded on $K$, whose upper level
sets are neighborhoods of $Y^{\prime}$, and which satisfies

    1. (i)

$\exists\,T\in[0,+\infty)$, $\forall x\in Q\cap f^{-1}[T,+\infty)$,

$\displaystyle\mu^{*n}(g\in G\ |\ f(gx)\leq f(x)-n\lambda)\geq 1-\alpha$
(probable decrease)

    2. (ii)

$\forall x\in f^{-1}[0,+\infty)$, $\forall g\in\Gamma$,

$\displaystyle f(gx)-f(x)\leq C\log\lVert\Phi(g)\rVert.$ (control of
variations)

######  Remark.

1) Strictly speaking, the definition expresses a drift away from some
$Y^{\prime}\in\mathcal{S}_{Y}$ which could have nothing to do with $Y$.
However, in all the applications below, we will always have that $Y\subseteq
Y^{\prime}$ for all $Y^{\prime}\in\mathcal{S}_{Y}$, so in particular, the walk
does drift away from $Y$ too. We choose to sum it up using the terminology of
drift away from $\mathcal{S}$.

3) The condition (ii) implies that $f^{-1}[0,+\infty)$ is $\Gamma$-invariant.

2) If $\mathcal{A}=\\{\Omega\\}$ then we simply say that the $\mu$-walk has a
_controlled drift away from $\mathcal{S}$_. If additionally,
$\mathcal{S}=\\{\infty,\infty\\}$, we talk about _controlled drift away from
infinity_. Note that $f$ is proper in this case: its sublevel sets have
compact closure in $\Omega$.

4) The map $\Phi$ will later be referred to as the _control function_ for the
drift. Indeed, its role is to bound the growth of $f$ in the unlikely event
its value goes up along the walk.

To get more familiar with the notion of controlled drift, we start by
exploring some stability properties. Given
$\mathcal{S}_{0},\mathcal{S}_{1}\subseteq\mathscr{P}(\overline{\Omega})\times\mathscr{P}(\overline{\Omega})$,
we denote by $\mathcal{S}_{0}\vee\mathcal{S}_{1}$ the set of pairs
$(Y,Y^{\prime}_{0}\cup Y^{\prime}_{1})$ where $Y\in p_{1}(\mathcal{S}_{0})\cap
p_{1}(\mathcal{S}_{1})$, $Y^{\prime}_{0}\in\mathcal{S}_{0,Y}$,
$Y^{\prime}_{1}\in\mathcal{S}_{1,Y}$.

######  Lemma 3.2.

  * a)

If the $\mu$-walk on $\Omega$ has a controlled drift away from both
$\mathcal{S}_{0},\mathcal{S}_{1}$, then it has a controlled drift away from
$\mathcal{S}_{0}\vee\mathcal{S}_{1}$.

  * b)

If the $\mu$-walk on $\Omega$ has a controlled drift away from infinity and a
controlled drift away from $\mathcal{S}$ on compact sets of $\Omega$, then it
has a controlled drift away from $\mathcal{S}$.

We explain _a)_. The proof of the second assertion is much more involved and
postponed to Section 5 where it will be used.

###### Proof of Lemma 3.2 a).

We respectively index by $0$ and $1$ the notations referring to
$\mathcal{S}_{0}$, $\mathcal{S}_{1}$. Let $C_{i},\lambda_{i}$ be the constants
given by the controlled drift assumption 3.1 for $\mathcal{S}_{i}$. Set
$C=\max(C_{0},C_{1})$, $\lambda=\min(\lambda_{0},\lambda_{1})$. Let
$\alpha>0$, then $N_{i}$ as in 3.1. Set $N=\max(N_{0},N_{1})$, $n\geq N$. Let
$Y\in p_{1}(\mathcal{S}_{0})\cap p_{1}(\mathcal{S}_{1})$ and consider a
compact set $K\subset\Omega\smallsetminus Y$. Let $Y^{\prime}_{i},f_{i},T_{i}$
be given by 3.1. We set $Y^{\prime}=Y^{\prime}_{0}\cup Y^{\prime}_{1}$,

$f=\max(f_{0},f_{1})$

and claim that the map $f$ satisfies all the requirements in Definition 3.1
(with $2\alpha$ instead of $\alpha$). Indeed, all the conditions are obvious
except the one entitled “probable decrease”. To check it, we choose
$T\geq\max(T_{0},T_{1})$ and large enough so that

$\mu^{*n}(g\in\Gamma\,|\,T_{0}+C_{0}\Phi_{0}(g)\leq T-n\lambda)\geq 1-\alpha$

and the same holds when we replace $0$ by $1$. Now let $x\in
f^{-1}[T,+\infty)$. If $f_{1}(x)=f(x)$, then

$\mu^{*n}(g\in\Gamma\,|\,f_{1}(gx)\leq f(x)-n\lambda_{1})\geq 1-\alpha$

On the other hand, distinguishing the cases $f_{0}(x)<T_{0}$ and $f_{0}(x)\geq
T_{0}$, the condition on $T$ guarantees that

$\mu^{*n}(g\in\Gamma\,|\,f_{0}(gx)\leq f(x)-n\lambda)\geq 1-\alpha$

so combining these inequalities

$\mu^{*n}(g\in\Gamma\,|\,f(gx)\leq f(x)-n\lambda)\geq 1-2\alpha.$

The same holds if $f_{0}(x)=f(x)$ by symmetry.

∎

###  3.2 Application to recurrence

The proposition below states that a walk that drifts away from $\mathcal{S}$
does not accumulate on $\mathcal{S}$. In particular, when
$\mathcal{S}=\\{\infty,\infty\\}$, the conclusion exactly means that the walk
has neither escape of mass nor escape of empirical measures at infinity.

######  Proposition 3.3.

Let $(\Omega,G,\mu)$ be as above and
$\mathcal{S}\subseteq\mathscr{P}(\overline{\Omega})\times\mathscr{P}(\overline{\Omega})$.
Assume that the $\mu$-walk on $\Omega$ has a controlled drift away from
$\mathcal{S}$. Then for every $Y\in p_{1}(\mathcal{S})$, every compact set
$K\subseteq\Omega\smallsetminus Y$, there is $Y^{\prime}\in\mathcal{S}_{Y}$
such that for every $\varepsilon>0$, for some neighborhood $O^{\prime}$ of
$Y^{\prime}$, for all $x\in K$,

  1. (i)

for every $n\geq 0$, $(\mu^{*n}*\delta_{x})(O^{\prime})<\varepsilon$;

  2. (ii)

for $\mu^{\otimes\mathbb{N}^{*}}$-almost every instructions $(g_{i})_{i\geq
1}$,

$\limsup_{n\to+\infty}\frac{1}{n}\operatorname{card}\\{k\in\\{1,\dots,n\\}\ |\
g_{k}\dots g_{1}x\in O^{\prime}\\}<\varepsilon.$

The proof can be summarized as follows. If $\alpha>0$ is small enough, $n\geq
N$ is large enough, then for every $Y\in p_{1}(\mathcal{S})$, every compact
set $K\subseteq\Omega\smallsetminus Y$, the function $f$ given by the drift
property 3.1 satisfies the assumptions of Lemma 2.2 for the $\mu^{*n}$-random
walk on $f^{-1}[0,+\infty)$ for some $n\in\mathbb{N}^{*}$, whence the
stochastic dominance condition $(\mathbf{SD})$. Theorem D then applies and
leads to non-accumulation results for the $\mu^{\ast n}$-walk on
$f^{-1}[0,+\infty)$ which can be extended to the $\mu$-walk.

We begin with a lemma that will allow to check condition 3. in Lemma 2.2.

######  Lemma 3.4.

Let $\Phi:\Gamma\rightarrow\operatorname{\mathrm{SL}}^{\pm}(V)$ be a linear
representation of $\Gamma$ on a finite-dimensional normed vector space $V$
such that

$\int_{G}\log\|\Phi(g)\|\,d\mu(g)<+\infty$

Then for all $\eta>0$, there exists $\alpha>0,N^{\prime}\geq 0$ such that for
all $n\geq N^{\prime}$, if $Z_{n}^{\prime}$ denotes the standard realisation
of $Z_{n}=\log\lVert\Phi(g)\rVert$ for $\mu^{*n}$, then

$\mathbb{E}[Z_{n}^{\prime}\mathbbm{1}_{[0,\alpha]}]\leq n\eta.$

###### Proof.

The law of large numbers [9, Lemma 4.27] guarantees that there exists a
constant $\ell\geq 0$ such that the sequence of real random variables
$(\frac{1}{n}Z_{n})_{n\geq 1}$ converges to $\ell$ in $L^{1}$. This
convergence also holds for $(\frac{1}{n}Z^{\prime}_{n})_{n\geq 1}$ as it has
the same law. Now let $\alpha>0$, $n\geq 1$.

$\displaystyle\frac{1}{n}\mathbb{E}[Z_{n}^{\prime}\mathbbm{1}_{[0,\alpha]}]$
$\displaystyle\leq\alpha\ell+\mathbb{E}[\lvert\frac{1}{n}Z^{\prime}_{n}-\ell\rvert]$
$\displaystyle\leq\alpha(\ell+1)$

if $n$ is large enough. The result follows if we specify
$\alpha=\frac{\eta}{\ell+1}$ above. ∎

We now apply the results of Section 2 with the drift function $f$ constructed
above to derive Theorem A′.

######  Lemma 3.5.

Assume that the $\mu$-walk on $\Omega$ has a controlled drift away from
$\mathcal{S}$. Then there exists $\alpha_{0}>0$, $n_{0}\geq N$ such that for
any $Y\in p_{1}(S)$, any compact $K\subseteq\Omega\smallsetminus Y$, any map
$f$ as in Definition 3.1, the $\mu^{\ast n_{0}}$-walk on $f^{-1}[0,+\infty)$
satisfies condition $(\mathbf{SD})$ with drift function $f$.

###### Proof.

Let $\lambda,C>0$ be given by Definition 3.1. Choose a small
$\alpha_{0}\in(0,1/2)$ as in Lemma 3.4 for $\eta=\frac{\lambda}{2C}$. Fix an
integer $n_{0}\geq\max(N,N^{\prime})$ where $N,N^{\prime}$ are the constants
given respectively by Definition 3.1 and Lemma 3.4 for our choice of $\alpha$.
Consider $Y\in p_{1}(\mathcal{S})$, a compact set
$K\subseteq\Omega\smallsetminus Y$, and a map $f$ as in Definition 3.1.

We show that the $\mu^{\ast n_{0}}$-random walk on $f^{-1}[0,+\infty)$
satisfies the stochastic dominance condition $(\mathbf{SD})$. To do so, we
only need to check that the assumptions of Lemma 2.2 do hold in the present
context.

  1. 1.

By Definition 3.1, there exists $T>0$ such that for $x\in f^{-1}[T,+\infty)$,

$\mu^{*n_{0}}(g\in\Gamma\ |\ f(gx)\leq f(x)-n_{0}\lambda)\geq 1-\alpha_{0}.$

  2. 2.

For every $x\in f^{-1}[0,+\infty)$, $g\in\Gamma$, we have $f(gx)-f(x)\leq
C\log\|\Phi(g)\|$. In particular, denoting by $Z$ the variable
$C\log\|\Phi(g)\|$ where $g$ varies with respect to $\mu^{\ast n_{0}}$, the
law of $f(gx)-f(x)$ for $\mu^{\ast n_{0}}$ is stochastically dominated by $Z$.

  3. 3.

Lemma 3.4 and our choice of constants imply that, writing $Z^{\prime}$ the
standard realisation of $Z$,

$\mathbb{E}[Z^{\prime}\mathbbm{1}_{[0,\alpha_{0}]}]\leq\frac{n_{0}\lambda}{2}.$

Hence the conditions of Lemma 2.2 are satisfied, which implies that
$(\mathbf{SD})$ holds for the $\mu^{\ast n_{0}}$-walk on $f^{-1}[0,+\infty)$.
∎

###### Proof of Proposition 3.3.

Fix $\alpha_{0}>0$, $n_{0}\geq N$ as in Lemma 3.5, $Y\in p_{1}(\Omega)$, a
compact $K\subseteq\Omega\smallsetminus Y$, and let
$Y^{\prime}\in\mathcal{S}_{Y}$, $f:\Omega\rightarrow[0,+\infty]$ be given by
the drift condition 3.1. Let us check the conclusions of the proposition with
$O^{\prime}=\\{f>M\\}$ for some sufficiently large $M$. Lemma 3.5 combined
with Theorem D already yields the result for the $\mu^{*n_{0}}$-walk: for any
$\varepsilon>0$, there is $R>0$ such that for all $x\in K$,

  1. (i)

for every $q\geq 0$, $\mu^{\ast qn_{0}}(g\in\Gamma\ |\ f(gx)>R)<\varepsilon$;

  2. (ii)

for $\mu^{\otimes\mathbb{N}^{*}}$-almost every trajectory $(g_{i})_{i\geq 1}$,

$\limsup_{n\to+\infty}\frac{1}{q}\operatorname{card}\\{k\in\\{1,\dots,q\\}\ |\
f(g_{kn_{0}}\dots g_{1}x)>R\\}<\varepsilon.$

To conclude, we just check that (i) and (ii) also hold for $\mu$ (i.e. if
$n_{0}=1$) up to increasing $R$ and doubling $\varepsilon$.

For (i), choose $R_{1}>0$ such that for every $1\leq m<n_{0}$,

$\mu^{\ast m}(g\in G\ |\ \log\lVert\Phi(g)\rVert>R_{1})\leq\varepsilon.$

Let $R_{2}=R+CR_{1}$ and $n\geq 0$. Writing $n=qn_{0}+m$ for some integers
$q,m\geq 0$ with $m<n_{0}$, we obtain for every $x\in K$

$\displaystyle\mu^{\ast n}(g\in G\ |\ f(gx)>R_{2})$
$\displaystyle\leq\mu^{\ast qn_{0}}(g\in G\ |\ f(gx)>R)$
$\displaystyle\qquad+\mu^{\ast m}(g\in G\ |\ \log\lVert\Phi(g)\rVert>R_{1})$
$\displaystyle<2\varepsilon.$

For (ii), the argument is very similar. Given $R_{3}>0$, set for $q\geq 1$

$Y_{q}=\mathbbm{1}_{\\{\exists n\in\\{qn_{0}+1,\dots,qn_{0}+n_{0}\\}:\
\log\lVert\Phi(g_{n}\dots g_{qn_{0}+1})\rVert>R_{3}\\}}.$

The random variables $Y_{q}$ are independent, identically distributed, and
satisfy $\mathbb{E}[Y_{q}]<\eta$ if $R_{3}$ is chosen large enough. Therefore,
by the classical law of large numbers,

$\limsup_{q\to+\infty}\frac{1}{q}\operatorname{card}\\{k\in\\{1,\dots,q\\}\ |\
\exists\,kn_{0}<n\leq n_{0}(k+1):\log\lVert\Phi(g_{n}\dots
g_{kn_{0}+1})\rVert>R_{3}\\}<\varepsilon.$

Setting $R_{4}=R+CR_{3}$, we get for $\mu^{\otimes\mathbb{N}^{\ast}}$-almost
every $(g_{i})_{i\geq 1}\in G^{\mathbb{N}^{\ast}}$,

$\limsup_{n\to+\infty}\frac{1}{n}\operatorname{card}\\{k\in\\{1,\dots,n\\}\ |\
f(g_{k}\dots g_{1}x)>R_{4}\\}<2\varepsilon$

which finishes the proof of the proposition. ∎

##  4 Non-escape at infinity

We now apply the results of the previous section to show that random walks on
finite-volume homogeneous spaces have neither escape of mass nor escape of
empirical measures at infinity. In what follows, $G$ denotes a real Lie group,
$\mathfrak{g}$ its Lie algebra, $\mathfrak{r}$ its amenable radical,
$\mathfrak{s}=\mathfrak{g}/\mathfrak{r}$ the largest quotient of
$\mathfrak{g}$ which is semisimple without compact factors, and we denote by
$\operatorname{Ad}_{\mathfrak{s}}:G\to\operatorname{Aut}\mathfrak{s}$ the
adjoint action of $G$ on $\mathfrak{s}$. Given a probability measure $\mu$ on
$G$, we let $H_{\mathfrak{s}}\subset\operatorname{Aut}\mathfrak{s}$ be the
real algebraic group generated by the support of
$(\text{Ad}_{\mathfrak{s}})_{\ast}\mu$ and call $H^{nc}_{\mathfrak{s}}$ its
non compact part, i.e. the smallest normal algebraic subgroup of
$H_{\mathfrak{s}}$ such that $H_{\mathfrak{s}}/H^{nc}_{\mathfrak{s}}$ is
compact.

Our goal is to show the following theorem, which is due to Benoist and Quint
[5, Theorem 1.1] in the case where $\mu$ has a finite exponential moment on
$\mathfrak{s}$.

######  Theorem A′.

_(Recurrence of semisimple random walks)_ Let $G$ be a real Lie group,
$\Lambda$ a lattice in G, and $\Omega=G/\Lambda$. Assume that $\mu$ is a
probability measure on $G$ such that

  1. 1.

$\mu$ has a finite first moment on $\mathfrak{s}$:
$\int_{G}\log\lVert\operatorname{Ad}_{\mathfrak{s}}g\rVert\ d\mu(g)<+\infty$ ;

  2. 2.

$H^{nc}_{\mathfrak{s}}$ is semisimple.

Then, for every compact set $K\subset\Omega$ and every $\varepsilon>0$, there
exists a compact set $K^{\prime}\subset\Omega$ such that for every $x\in K$,

  1. (i)

for every $n\geq 0$, $(\mu^{*n}*\delta_{x})(K^{\prime})>1-\varepsilon$;

  2. (ii)

for $\mu^{\otimes\mathbb{N}^{*}}$-almost every instructions $(g_{i})_{i\geq
1}$,

$\liminf_{n\to+\infty}\frac{1}{n}\operatorname{card}\\{k\in\\{1,\dots,n\\}\ |\
g_{k}\dots g_{1}x\in K^{\prime}\\}>1-\varepsilon.$

######  Remark.

In particular, every Radon $\mu$-stationary measure on $\Omega$ must have
finite mass.

By Proposition 3.3, the proof of Theorem A′ reduces to showing that the
$\mu$-walk on $\Omega$ has a controlled drift away from infinity. We first
check that this is the case if
$\Omega=\operatorname{\mathrm{SL}}_{d}(\mathbb{R})/\operatorname{\mathrm{SL}}_{d}(\mathbb{Z})$
and then deduce the result for general quotients $\Omega=G/\Lambda$.

###  4.1 The Benoist-Quint function on
$\operatorname{\mathrm{SL}}_{d}(\mathbb{R})/\operatorname{\mathrm{SL}}_{d}(\mathbb{Z})$

In this paragraph, we prove Theorem A′ in the particular case where
$G=\operatorname{\mathrm{SL}}_{d}(\mathbb{R})$,
$\Lambda=\operatorname{\mathrm{SL}}_{d}(\mathbb{Z})$ for some $d\geq 2$. The
drift functions we use were introduced by Benoist and Quint in [5], although
they were rather defined in an exponential form. We begin by recalling their
construction.

Fix a probability measure $\mu$ on
$\operatorname{\mathrm{SL}}_{d}(\mathbb{R})$, and let $H$ be the algebraic
group generated by the support of $\mu$. We assume that the non compact part
$H^{nc}$ of $H$ is semisimple, or equivalently that $H$ is reductive with
compact center. Consider the representation of $H$ on the exterior algebra
$V=\wedge^{*}\mathbb{R}^{d}$ and decompose it into its isotypical components
for the induced action of $H^{nc}$:

$V=\oplus_{j\in J}V^{(j)}.$

Alternatively, denoting by $H^{\circ}$ the identity component of $H$ in the
Zariski topology, each $V^{(j)}$ is obtained by summing together all the
irreducible subrepresentations of $H^{\circ}$ on $V$ with a given highest
weight. The action of $H$ on $V$ then induces an action of the group of
connected components $F=H/H^{\circ}$ on the set $\\{V^{(j)},j\in J\\}$ which
stabilizes the subspace $V^{H^{nc}}$ of $H^{nc}$-invariant vectors. By the law
of large numbers [9, Theorem 10.9, Corollary 10.12], we can associate to each
representation $V^{(j)}$ a Lyapunov exponent $\lambda_{1}^{(j)}\in\mathbb{R}$,
that quantifies the exponential growth rate of the walk on $V^{(j)}$ and only
depends on the $F$-orbit of $V^{(j)}$. Moreover, by semisimplicity of
$H^{nc}$, we have $\lambda^{(j)}_{1}\geq 0$, with $\lambda_{1}^{(j)}=0$ if and
and only if $V^{(j)}=V^{H^{nc}}$ [9, Theorem 10.9 (f)].

Fix $H^{c}\subseteq H$ a maximal compact subgroup of $H$ and some
$H^{c}$-invariant Euclidean norm on $\mathbb{R}^{d}$. This induces a natural
$H^{c}$-invariant Euclidean norm on the exterior algebra $V$. Given $v\in V$,
write $v=\sum_{j\in J}v^{(j)}$ according to the above decomposition of $V$,
and define

$\lvert v\rvert=\max_{j\in J}\lVert
v^{(j)}\rVert^{\frac{1}{\lambda_{1}^{(j)}}}$

with the convention that $\frac{1}{0}=+\infty$ and

$\lVert v^{(j)}\rVert^{+\infty}=\left\\{\begin{array}[]{ll}0&\mbox{if}\ \lVert
v^{(j)}\rVert<1\\\ +\infty&\mbox{if}\ \lVert v^{(j)}\rVert\geq
1.\end{array}\right.$

The following lemma expresses that the $\mu$-walk on $V$ uniformly expands the
quasi-norm of vectors $v$ whose $H^{nc}$-invariant component is not too
weighty.

######  Lemma 4.1.

Given $k\geq 1$, $\lambda\in(0,1)$, $\alpha>0$, there exists $n_{0}\geq 0$
such that for every $n\geq n_{0}$, for every finite family $v_{1},\dots,v_{k}$
of non-zero vectors in $\wedge^{*}\mathbb{R}^{d}$ satisfying $0<\lvert
v_{\ell}\rvert<+\infty$ for each $\ell$, one has

$\mu^{*n}\left(g\in H\ \left|\ \min_{1\leq\ell\leq k}\log\frac{\lvert
gv_{\ell}\rvert}{\lvert v_{\ell}\rvert}\geq n\lambda\right.\right)\geq
1-\alpha.$

###### Proof.

Up to replacing $\alpha$ by $\alpha/k$, we may assume that $k=1$. For $v\in V$
such that $0<\lvert v\rvert<+\infty$, write $v=\sum_{j\in J}v^{(j)},$ and let

$J_{v}=\\{j\in J\ |\ \lvert v^{(j)}\rvert>0\\}.$

Then, for every $g\in H$, one can bound

$\displaystyle\min_{j\in J_{v}}\log\frac{\lvert gv^{(j)}\rvert}{\lvert
v^{(j)}\rvert}\leq\log\frac{\lvert gv\rvert}{\lvert v\rvert}\leq\max_{j\in
J_{v}}\log\frac{\lvert gv^{(j)}\rvert}{\lvert v^{(j)}\rvert}.$ (10)

For the left inequality, just consider $j$ such that $\lvert v\rvert=\lvert
v^{(j)}\rvert$, and for the right inequality, choose $j$ for which $\lvert
gv\rvert=\lvert gv^{(j)}\rvert$.

By the law of large numbers for product of random matrices [9, Theorem 4.28
(b)], on each representation $V^{(j)}$ such that $\lambda_{1}^{(j)}>0$, the
sequence of functions

$(g_{i})_{i\geq 1}\mapsto\frac{1}{n}\log\frac{\lvert g_{n}\dots
g_{1}v^{(j)}\rvert}{\lvert v^{(j)}\rvert}$

converges to $1$ in $L^{1}(G^{\mathbb{N}^{*}},\mu^{\otimes\mathbb{N}^{*}})$,
uniformly over all non-zero $v^{(j)}\in V^{(j)}$. Combining this fact and the
left inequality of (10) yields the lemma. ∎

To define the Benoist-Quint function on the set
$\Omega=\operatorname{\mathrm{SL}}_{d}(\mathbb{R})/\operatorname{\mathrm{SL}}_{d}(\mathbb{Z})$
of unimodular lattices of $\mathbb{R}^{d}$, we need to enhance some convexity
property of the above quasi-norm on $V$. For that, we make the following
definition.

######  Definition 4.2 (Corrected quasi-norm).

Given a large parameter $A>0$ and $i\in\\{1,\dots,d-1\\}$, define a corrected
version $\lvert\cdot\rvert_{A}$ of the quasi-norm $\lvert\cdot\rvert$ by

$\forall v\in\wedge^{i}\mathbb{R}^{d},\quad\lvert v\rvert_{A}=\left\lvert
e^{Ai(d-i)}v\right\rvert$

We can also view $\|.\|$, $\lvert\cdot\rvert$ and $\lvert\cdot\rvert_{A}$ as
functions on discrete subgroups of $\mathbb{R}^{d}$, by identifying
$\Delta=\mathbb{Z}a_{1}\oplus\dots\oplus\mathbb{Z}a_{i}$ with $\pm
a_{1}\wedge\dots\wedge a_{i}\in\wedge^{i}\mathbb{R}^{d}$. This allows to
define a function $\phi_{A}$ with values in $[-\infty,+\infty]$ by setting:

$\phi_{A}(\Delta)=\log\frac{1}{\lvert\Delta\rvert_{A}}\ \ \mbox{if}\
\text{rank\,}\Delta\notin\\{0,d\\}\quad\mbox{and}\quad\phi_{A}(\Delta)=0\\\ \
\mbox{otherwise.}$

######  Remark.

A discrete subgroup $\Delta\subseteq\mathbb{R}^{d}$ of rank
$i\in\\{1,\dots,d-1\\}$ satisfies $\phi_{A}(\Delta)=+\infty$ if and only if
$\Delta\in V^{H^{nc}}$ and $\lVert\Delta\rVert<e^{-Ai(d-i)}$. Since $H$ acts
by isometries on $V^{H^{nc}}$, it follows that the set
$\\{\phi_{A}=+\infty\\}$ is $H$-invariant.

######  Definition 4.3 (Benoist-Quint function).

For $A>0$, define a function $f_{A}$ on $\Omega$ with values in $[0,+\infty]$
by

$f_{A}(x)=\max_{\Delta\leq x}\phi_{A}(\Delta).$

A few comments are in order. First of all, the notation $\Delta\leq x$ means
that $\Delta$ varies in the set of subgroups of the lattice $x$. As $\Delta$
can be chosen of rank $0$ or $d$, our conventions for $\phi_{A}$ impose that
$f_{A}\geq 0$.

Note also that the maximum defining $f_{A}$ is well defined. Indeed, one can
observe that if $\Delta\leq x$ satisfies $\phi_{A}(\Delta)>0$ then
$\text{rank}(\Delta)\notin\\{0,d\\}$ and $\|\Delta^{(j)}\|<1$ for each $j\in
J$. Since $\wedge^{*}x$ is a discrete subset of $V=\wedge^{*}\mathbb{R}^{d}$,
the set of subgroups $\Delta\leq x$ such that $\phi_{A}(\Delta)>0$ is finite.

Finally, observe that the set $\Omega_{A}:=\\{f_{A}<+\infty\\}$ is
$H$-invariant, because this is the case for $\\{\phi_{A}=+\infty\\}$.
Moreover, for every compact subset $K\subset\Omega$, the norm $\|\Delta\|$ of
non-trivial subgroups of lattices $x\in K$ is uniformly bounded away from $0$
as $x$ varies in $K$. This implies that $K\subset\Omega_{A}$ for large enough
$A$.

######  Proposition 4.4 (Drift for the Benoist-Quint function on
$\operatorname{\mathrm{SL}}_{d}$).

Let $\mu$ be a probability measure on
$G=\operatorname{\mathrm{SL}}_{d}(\mathbb{R})$ and $H$ the algebraic subgroup
generated by $\mu$. Assume that $\mu$ has a finite moment of order $1$, and
that $H^{nc}$ is semisimple.

  1. (i)

$\forall\lambda\in(0,1),\ \forall\alpha>0,\,\exists N\geq 0,\,\forall n\geq
N,\ \exists A_{0},A_{1}>0,\,\forall A\geq A_{1}$, $\forall x\in
f^{-1}_{A}(A_{0},+\infty)$,

$\mu^{*n}(g\in G\ |\ f_{A}(gx)\leq f_{A}(x)-n\lambda)\geq 1-\alpha$

  2. (ii)

$\exists C>0$, $\forall A>0$, $\forall x\in\phi^{-1}_{A}(\mathbb{R})$,
$\forall g\in H$,

$\phi_{A}(gx)-\phi_{A}(x)\leq C\log\lVert g\rVert$

which implies the same inequality with $f_{A}$ instead of $\phi_{A}$.

The proof of this proposition is based on a convexity property of $\phi_{A}$,
Lemma 4.5 below, which is a reformulation of [5, Lemma 4.2]. The first
statement should be understood as follows: if two primitive subgroups
$\Delta,\Delta^{\prime}\leq x$ take large values under $\phi_{A}$, then one
can obtain an even larger value by considering the intersection
$\Delta\cap\Delta^{\prime}$ or the sum $\Delta+\Delta^{\prime}$.

######  Lemma 4.5 (Weak submodularity property [5]).

For every $A_{0}>0$, there exists $A_{1}>0$, such that for all $A\geq A_{1}$,
for every lattice $x\in\Omega$, all primitive subgroups
$\Delta,\Delta^{\prime}\leq x$ satisfying $\Delta\subsetneq\Delta^{\prime}$,
$\Delta^{\prime}\subsetneq\Delta$ and
$\phi_{A}(\Delta),\phi_{A}(\Delta^{\prime})>0$,

$\min\left(\phi_{A}(\Delta),\phi_{A}(\Delta^{\prime})\right)+A_{0}<\max\left(\phi_{A}(\Delta\cap\Delta^{\prime}),\phi_{A}(\Delta+\Delta^{\prime})\right).$

In particular, given $x\in\Omega_{A}$ such that $f_{A}(x)>A_{0}$, for each
$i\in\\{1,\dots,d-1\\}$, there exists at most one primitive subgroup
$\Delta_{i}\leq x$ of rank $i$ such that

$\phi_{A}(\Delta_{i})\geq f_{A}(x)-A_{0}$

###### Proof.

We check that [5, Lemma 4.2] does imply Lemma 4.5. We first explain why our
notations correspond to those of [5]. Fix a Cartan subspace $\mathfrak{a}$ in
the Lie algebra of $H$ compatible with our choice of maximal compact subgroup
$H^{c}$, and an open Weyl chamber $\mathfrak{a}^{++}\subseteq\mathfrak{a}$.
Each isotypical component $V^{(j)}$ corresponds to a highest weight
$\lambda^{(j)}\in\mathfrak{a}^{*}$, and by [9, Lemma 8.18] one has

$\lambda_{1}^{(j)}=\langle\lambda^{(j)},\sigma_{\mu}\rangle$ (11)

where $\sigma_{\mu}\in\mathfrak{a}^{++}$ is the Lyapunov vector of $\mu$,
given by the law of large numbers [9, Theorem 10.9].

Now set $\delta_{\lambda}=\langle\lambda,\sigma_{\mu}\rangle$ for
$\lambda\in\\{\lambda^{(j)},j\in J\\}$ and $\delta_{i}=i(d-i)$,
$i\in\\{1,\dots,d-1\\}$. Then, for every $A>0$ and every discrete subgroup
$\Delta$ in $\mathbb{R}^{d}$ of rank $1\leq i<d$, the function $\phi_{A}$ can
be expressed from the function $\varphi_{\varepsilon_{0}}$ used in [5] by

$\phi_{A}(\Delta)=\log\varphi_{\varepsilon_{0}}(\Delta),\quad\mbox{where}\
\varepsilon_{0}=e^{-A}.$

Let us now check that [5, Lemma 4.2] does imply the weak submodularity
property. Fix $A_{2}>0$ such that for $A\geq A_{2}$, any $\Delta$, the
inequality $\phi_{A}(\Delta)>0$ implies $\|\Delta\|<1$. Let $A\geq A_{2}$, and
consider $x\in\Omega$, as well as two primitive subgroups
$\Delta\neq\Delta^{\prime}\leq x$ which are not included in one another and
such that $\phi_{A}(\Delta),\phi_{A}(\Delta^{\prime})>0$. Let
$(u_{1},\dots,u_{r})$ be a basis for $\Delta\cap\Delta^{\prime}$ and complete
it into basis $(u_{1},\dots,u_{r},v_{1},\dots,v_{s})$,
$(u_{1},\dots,u_{r},w_{1},\dots,w_{t})$ of $\Delta$ and $\Delta^{\prime}$. The
assumption $\Delta\subsetneq\Delta^{\prime}$,
$\Delta^{\prime}\subsetneq\Delta$ guarantees that $s,t\geq 1$. Note also that
$(u_{1},\dots,u_{r},v_{1},\dots,v_{s},w_{1},\dots,w_{t})$ is a basis for
$\Delta+\Delta^{\prime}$.

We now apply [5, Lemma 4.2] to $u=(u_{i})$, $v=(v_{j})$, $w=(w_{k})$. Note
that case $iv)$ cannot occur as the conditions $A\geq A_{2}$ and
$\phi_{A}(\Delta),\phi_{A}(\Delta^{\prime})>0$ impose that if $r=0$ and
$s+t=d$, then $\|\Delta+\Delta^{\prime}\|=\|v\wedge w\|<1$ which is absurd for
a sublattice of a unimodular lattice. Hence we are left with cases $i)$,
$ii)$, $iii)$. As long as $A>C_{1}$ where $C_{1}>1$ is some constant depending
only on $H$, they all express the inequality

$\min\left(\phi_{A}(\Delta),\phi_{A}(\Delta^{\prime})\right)+R(A)<\max\left(\phi_{A}(\Delta\cap\Delta^{\prime}),\phi_{A}(\Delta+\Delta^{\prime})\right)$

with $R(A)=\frac{1}{2}(\max_{\lambda\in P^{+}}\delta_{\lambda})^{-1}(A-\log
C_{1})$. To conclude, for any $A_{0}>0$ choose $A_{1}>0$ such that
$A>A_{2},C_{1}$ and $R(A_{1})>A_{0}$ to get the announced result.

For the second statement, assume for a contradiction that
$\Delta^{\prime}_{i}\leq\Delta$ is another primitive subgroup of rank $i$
satisfying $\phi_{A}(\Delta^{\prime}_{i})\geq f_{A}(x)-A_{0}$. The above
inequality applied to $\Delta_{i}$ and $\Delta^{\prime}_{i}$ implies

$f_{A}(x)\leq\min\left(\phi_{A}(\Delta_{i}),\phi_{A}(\Delta^{\prime}_{i})\right)+A_{0}<\max\left(\phi_{A}(\Delta_{i}\cap\Delta^{\prime}_{i}),\phi_{A}(\Delta_{i}+\Delta^{\prime}_{i})\right)\leq
f_{A}(x)$

which is absurd. ∎

We are now ready to prove Proposition 4.4.

###### Proof of Proposition 4.4.

Item (ii) is an immediate consequence of our definitions, so we only prove
(i).

Let $\lambda\in(0,1)$ and $\alpha>0$. By Lemma 4.1, there exists $N\geq 0$
such that for all $n\geq N$, for all $v_{1},\dots,v_{d}$ in $V$ with $0<\lvert
v_{i}\rvert<+\infty$,

$\mu^{*n}\left(g\in H\ \left|\ \min_{1\leq i\leq k}\log\frac{\lvert
gv_{i}\rvert}{\lvert v_{i}\rvert}\geq n\lambda\right.\right)\geq
1-\frac{\alpha}{2}.$ (12)

Fix such $n\geq N$, let $C>0$ be such that (ii) holds, and choose a constant
$A_{0}>0$ large enough so that

$\mu^{*n}\left(g\in H\ \left|\
C\log\|g\|\leq\frac{1}{2}A_{0}\right.\right)\geq 1-\frac{\alpha}{2}.$

Let $A_{1}>0$ be the parameter given by Lemma 4.5, and $A\geq A_{1}$. Then let
$x\in\Omega_{A}$ such that $f_{A}(x)>A_{0}$ and $\mathscr{D}:=\\{\Delta\leq x\
|\ \Delta\text{ primitive },\,\phi_{A}(\Delta)\geq f_{A}(x)-A_{0}\\}$. The
second statement of Lemma 4.5 guarantees that $\text{card}\mathscr{D}\leq d$.
In particular, by our choice for $n$, and observing that
$\frac{|g\Delta|}{|\Delta|}=\frac{|g\Delta|_{A}}{|\Delta|_{A}}$,

$\mu^{*n}\left(g\in H\ |\
\forall\Delta\in\mathscr{D},\,\phi_{A}(g\Delta)\leq\phi_{A}(\Delta)-n\lambda\right)\geq
1-\frac{\alpha}{2}.$

Moreover, as $|\phi_{A}(gx)-\phi_{A}(x)|\leq C\log\|g\|$, we have

$\mu^{*n}(g\in
H\,|\,f_{A}(gx)=\max_{\Delta\in\mathscr{D}}\phi_{A}(g\Delta))\geq
1-\frac{\alpha}{2}$

which concludes the proof.

∎

###  4.2 General quotients $G/\Lambda$

This paragraph concludes Section 4 by the proof of Theorem A′ for general
quotients $G/\Lambda$. We follow the argument of [5, Section 6].

######  Proposition 4.6.

Let $(G,\Lambda,\Omega,\mu)$ be as in Theorem A′. Then the $\mu$-walk on
$\Omega$ has a controlled drift away from infinity.

###### Proof sketch.

First case: $G$ is a semisimple $\mathbb{Q}$-group and
$\Lambda=G(\mathbb{Z})$.
If $G\hookrightarrow\operatorname{\mathrm{SL}}_{N}$ is an embedding defined
over $\mathbb{Q}$, it induces a proper map

$G/\Lambda\hookrightarrow\operatorname{\mathrm{SL}}_{N}(\mathbb{R})/\operatorname{\mathrm{SL}}_{N}(\mathbb{Z}),$

Hence the controlled drift away from infinity in
$\operatorname{\mathrm{SL}}_{N}(\mathbb{R})/\operatorname{\mathrm{SL}}_{N}(\mathbb{Z})$
given by Proposition 4.4 induces by restriction the result on $G/\Lambda$.

Second case: $G=\operatorname{Aut}\mathfrak{g}$ with $\mathfrak{g}$ semisimple
of real rank $1$.
Here, the construction does not use the Benoist-Quint function. We recall the
argument given in [5, Section 6]. By the results of Garland and Raghunathan
[15] on reduction theory for lattices in semisimple Lie groups of real rank
$1$ (see also [5, Proof of Lemma 6.3, 2nd case]), there exists a
representation $V$ of $G$ and a finite family of vectors $v_{1},\dots,v_{r}$
in $V$ such that

  1. (i)

each $\Lambda$-orbit $\Lambda v_{i}$ is discrete in $V$;

  2. (ii)

for each $i=1,\dots,r$, the orbit $G\cdot v_{i}$ is compact in $\mathbb{P}(V)$
and contains no $H^{nc}$-invariant element;

  3. (iii)

a sequence $(g_{n}\Lambda)_{n\geq 0}$ goes to infinity in $G/\Lambda$ if and
only if

$\lim_{n\to\infty}\left(\min_{1\leq i\leq r}\min_{\gamma\in\Lambda}\lVert
g_{n}\gamma v_{i}\rVert\right)=0;$

  4. (iv)

for every $A_{0}\geq 0$, there exists $\delta>0$ such that for every $g\in G$,
if

$\min_{1\leq i\leq r}\min_{\gamma\in\Lambda}\lVert g\gamma v_{i}\rVert=\lVert
g\gamma_{0}v_{i_{0}}\rVert\leq\delta,$

then for every $\gamma v_{i}\neq\gamma_{0}v_{i_{0}}$, $\lVert
g\gamma_{0}v_{i_{0}}\rVert\leq e^{-A_{0}}\lVert g\gamma v_{i}\rVert$.

The function

$f(g\Lambda)=\max_{1\leq i\leq r}\max_{\gamma\in\Lambda}\log\frac{1}{\lVert
g\gamma v_{i}\rVert}$

is therefore proper on $G/\Lambda$. One can easily adapt the proof of
Proposition 4.4 to show that it has the properties of “probable decrease” and
“control of variation” required for the controlled drift 3.1 away from
infinity.

Third case: $G=\operatorname{Aut}\mathfrak{g}$ where $\mathfrak{g}$ is
semisimple without compact ideal.
Replacing $\Lambda$ by a finite-index subgroup if necessary, we may decompose
$\Omega=G/\Lambda$ into a finite product

$\Omega=\Omega_{1}\times\dots\times\Omega_{r},$

where $\Omega_{i}=G_{i}/\Lambda_{i}$ is the quotient of a semisimple group
$G_{i}$ without compact factors by an _irreducible_ lattice $\Lambda_{i}$. By
the Margulis Arithmeticity Theorem [16, Theorem 1.16, page 299], either
$G_{i}$ has real rank one, or $\Lambda_{i}$ is an arithmetic subgroup. In both
cases, we have already shown that the $\mu$-walk on $\Omega_{i}$ has a
controlled drift away from infinity. This is equivalent to saying that the
random walk on $\Omega$ has a controlled drift away from
$\mathcal{S}_{i}=(\infty,S_{i})$ where $S_{i}$ is the product of the
$(\Omega_{j})_{j\neq i}$ and $\infty_{i}$, the point at infinity in
$\Omega_{i}$. Proposition 3.2 then implies that the $\mu$-walk on $\Omega$
drifts away from $\mathcal{S}_{1}\vee\dots\vee\mathcal{S}_{r}$, i.e. from
$(\infty,\infty)$.

Fourth case: General case.
Let $\operatorname{Ad}_{\mathfrak{s}}:G\to\operatorname{Aut}\mathfrak{s}$ be
the adjoint action of $G$ on $\mathfrak{s}$. By [8, Lemma 6.4], the image
$\Lambda_{S}=\operatorname{Ad}_{\mathfrak{s}}\Lambda$ is a lattice in $S$, and
the induced map $p_{S}:\Omega\to\operatorname{Aut}\mathfrak{s}/\Lambda_{S}$ is
proper. By what precedes, the walk on
$\operatorname{Aut}\mathfrak{s}/\Lambda_{S}$ drifts away from infinity, and
denoting by $f^{(S)}$ the drift function involved, we obtain a drift function
for $G/\Lambda$ by setting

$f=f^{(S)}\circ p_{S}.$

∎

##  5 Non-accumulation on invariant subspaces

The goal of this section is to prove Theorem B′, stating that a random walk on
$G/\Lambda$ does not accumulate on a proper invariant subset unless it is
stuck inside it. The statement is slightly more technical than its naive
counterpart Theorem B of the introduction. It is indeed the version we need to
show that the equidistribution results of Benoist and Quint [7] hold under a
finite first moment assumption. The proof of Theorem B′ is based on the
construction of an appropriate drift function as in Definition 3.1.

######  Theorem B′.

_(Unstability of invariant homogeneous subspaces)_ Let $G$ be a real Lie
group, $\Lambda$ a lattice in $G$, and $\Omega=G/\Lambda$. Let $\mu$ be a
probability measure on $G$ and denote by $H\subseteq\emph{Aut}(\mathfrak{g})$
the real algebraic group generated by the support of
$\operatorname{Ad}_{*}\mu$. We assume that $\mu$ has a finite first moment and
that $H^{nc}$ is semisimple. Denote by $L$ the centralizer of $H^{nc}$ in $G$.
Let $Y\in\mathcal{S}_{\Omega}(\Gamma)$, and consider compact subsets
$L_{0}\subseteq L$, $K\subseteq\Omega\smallsetminus L_{0}Y$.

Then, for every $\varepsilon>0$, there exists a neighborhood $O^{\prime}$ of
$L_{0}Y$ in $\Omega$ such that for all $x\in K$,

  1. (i)

for every $n\geq 0$, $(\mu^{*n}*\delta_{x})(O^{\prime})<\varepsilon$;

  2. (ii)

for $\mu^{\otimes\mathbb{N}^{*}}$-almost every instructions $(g_{i})_{i\geq
1}$,

$\limsup_{n\to+\infty}\frac{1}{n}\operatorname{card}\\{k\in\\{1,\dots,n\\}\ |\
g_{k}\dots g_{1}x\in O^{\prime}\\}<\varepsilon.$

######  Remark.

1) More precisely, the notation $L$ refers to the subgroup of $G$ defined by
the Lie algebra $\mathfrak{\ell}=\\{v\in\mathfrak{g}\,|\,\forall a\in
H^{nc},\,a(v)=v\\}$. Note that this subgroup may not be closed in $G$.

2) In the context of Theorem B′, for any $Y\in S_{\Omega}(\Gamma)$,
$x\in\Omega\smallsetminus LY$, we have

  1. 1.

Every weak limit $\nu$ of $(\mu^{*n}*\delta_{x})_{n\geq 0}$ satisfies
$\nu(LY)=0$.

  2. 2.

For $\mu^{\otimes\mathbb{N}^{*}}$-almost every trajectory $(g_{n})_{n\geq 1}$,
every weak limit $\nu$ of the sequence of empirical measures
$(\frac{1}{n}\sum_{k=0}^{n-1}\delta_{g_{k}\dots g_{1}x})_{n\geq 1}$ satisfies

$\nu(LY)=0.$

To prove Theorem B′, we show that the $\mu$-walk on $\Omega$ drifts away from
$L$-neighborhoods of $Y$.

######  Proposition 5.1.

Let $(G,\Lambda,\Omega,\mu)$ be as in Theorem B′. Let $\mathcal{S}_{L}$ be the
collection of pairs $(Y,UY)$ where $Y\in S_{\Omega}(\Gamma)$ and $U\subseteq
L$ is a neighborhood of $1\in L$.

Then the $\mu$-walk on $\Omega$ has a controlled drift away from
$\mathcal{S}_{L}$.

It is easy to check that Proposition 5.1 implies Theorem B′.

###### Proof of Theorem B′.

For every $\ell\in L_{0}$, we have $K\subseteq\Omega\smallsetminus\ell Y$.
Proposition 5.1 implies that the $\mu$-walk on $\Omega$ has a controlled drift
away from the collection $\mathcal{S}$ of $\\{(\ell Y,U\ell Y)\\}$ where $U$
varies among the neighborhoods of $1$ in $L$. Choose an arbitrary $U_{\ell}$
such that the conclusions of Proposition 3.3 hold for the $\mu$-walk on
$\Omega$. As $L_{0}$ is compact, there exists a finite covering
$L_{0}\subseteq U_{\ell_{1}}\ell_{1}\cup\dots\cup U_{\ell_{m}}\ell_{m}$. Let
$\varepsilon>0$, and for $i=1,\dots,m$, choose a neighborhood $O^{\prime}_{i}$
of $U_{\ell_{i}}\ell_{i}Y$ as in Proposition 3.3 for the constant
$\varepsilon/m$. Then $O^{\prime}=\cup_{i=1}^{m}O^{\prime}_{i}$ is a
neighborhood of $L_{0}Y$ which satisfies the recurrence properties $(i)$ and
$(ii)$ announced in Theorem B′. ∎

The rest of the section is dedicated to the proof of Proposition 5.1. It can
be decomposed into two independent statements.

######  Lemma 5.2.

The $\mu$-walk on $\Omega$ has a controlled drift away from $\mathcal{S}_{L}$
on compact subsets of $\Omega$.

######  Lemma 3.2 b).

In the general context of Section 3, if a random walk has controlled drift
away from infinity and controlled drift from some
$\mathcal{S}\subseteq\mathscr{P}(\overline{\Omega})\times\mathscr{P}(\overline{\Omega})$
on compact subsets, then it has controlled drift away from $\mathcal{S}$.

###  5.1 Local drift

Let us prove Lemma 5.2. To construct a local drift function for some $UY$, we
exhibit a transverse direction to $UY$ which is expanded by the walk. The Lie
algebra $\mathfrak{g}$ of $G$ is endowed with an arbitrary norm, and given a
subspace $\mathfrak{t}\subseteq\mathfrak{g}$, we set
$B_{\mathfrak{t}}(0,\delta)=\\{v\in\mathfrak{t}\ |\ \|v\|<\delta\\}$.

######  Lemma 5.3 (Transverse coordinate).

Let $Y\in S_{\Omega}(\Gamma)$, denote by $\mathfrak{s}$ be the Lie algebra of
$G_{Y}=\operatorname{Stab}_{G}Y$, $\mathfrak{\ell}$ the Lie algebra of $L$,
and fix an $H$-invariant subspace $\mathfrak{t}\subseteq\mathfrak{g}$ such
that

$\mathfrak{g}=\mathfrak{t}\oplus(\mathfrak{\ell}+\mathfrak{s}).$

For any compact subset $M\subset\Omega$, there exists a neighborhood $U$ of
$1$ in $L$ and $\delta>0$ such that for every $x\in M$, there is at most one
$v\in B_{\mathfrak{t}}(0,\delta)$ for which $x\in e^{v}UY$.

###### Proof of Lemma 5.3.

Let $\mathfrak{\ell}^{\prime}\subset\mathfrak{\ell}$ be a subspace such that
$\mathfrak{\ell}+\mathfrak{s}=\mathfrak{\ell}^{\prime}\oplus\mathfrak{s}$, and
$O$ a relatively compact open neighborhood of $M$ in $\Omega$. There exist
small neighborhoods of the origin $V_{\mathfrak{t}}\subset\mathfrak{t}$,
$V_{\mathfrak{\ell}^{\prime}}\subset\mathfrak{\ell}^{\prime}$ and
$V_{\mathfrak{s}}\subset\mathfrak{s}$ such that

$\begin{array}[]{ccc}V_{\mathfrak{t}}\times
V_{\mathfrak{\ell}^{\prime}}\times(Y\cap O)&\to&\Omega\\\
(v_{\mathfrak{t}},v_{\mathfrak{\ell}^{\prime}},y)&\mapsto&e^{v_{\mathfrak{t}}}e^{v_{\mathfrak{\ell}^{\prime}}}y\end{array}$

induces a diffeomorphism on its image and
$e^{V_{\mathfrak{\ell}^{\prime}}}e^{V_{\mathfrak{s}}}$ contains a neighborhood
$U$ of $1$ in $L$.

Moreover, reducing those neighborhoods if necessary, we may assume that

$e^{V_{\mathfrak{s}}}e^{-V_{\mathfrak{\ell}^{\prime}}}e^{-V_{\mathfrak{t}}}M\subset
O.$

Given $x\in M$, assume that for some $v_{i}\in V_{\mathfrak{t}}$, $u_{i}\in
U_{0}$ and $y_{i}\in Y$, $i=1,2$,

$x=e^{v_{1}}u_{1}y_{1}=e^{v_{2}}u_{2}y_{2}.$

Writing $u_{i}=u_{i}^{\prime}s_{i}$, with $u_{i}^{\prime}\in
e^{V_{\mathfrak{\ell}^{\prime}}}$ and $s_{i}\in e^{V_{\mathfrak{s}}}$, we find
$e^{v_{1}}u_{1}^{\prime}(s_{1}y_{1})=e^{v_{2}}u_{2}^{\prime}(s_{2}y_{2})$ and
since $s_{i}y_{i}=s_{i}u_{i}^{-1}e^{-v_{i}}x\in Y\cap O$, this implies
$u_{1}^{\prime}=u_{2}^{\prime}$, $s_{1}y_{1}=s_{2}y_{2}$, and $v_{1}=v_{2}$. ∎

We now use the previous lemma to prove local drift away from some $UY$. Given
$Y\in S_{\Omega}(\Gamma)$, and $U\subseteq L$ a neighborhood of $1\in L$, we
introduce for every $x\in\Omega$, the set $\mathfrak{t}_{x}=\\{v\in
B_{\mathfrak{t}}(0,1)\ |\ x\in e^{v}UY\\}$ and define a function on $\Omega$
by

$f_{UY}(x)=\left\\{\begin{array}[]{ll}\max_{v\in\mathfrak{t}_{x}}\log\frac{1}{\lVert
v\rVert}&\mbox{if}\ \mathfrak{t}_{x}\neq\varnothing\\\
0&\mbox{otherwise}.\end{array}\right.$

The proof of Lemma 5.2 will follow from the observation that for any $Y\in
S_{\Omega}(\Gamma)$ and any compact set $Q\subseteq\Omega$, we can choose $U$
small enough so that the function $f_{UY}$ statistically decreases when the
walk passes through $Q$.

###### Proof of Lemma 5.2.

Let $\mathfrak{b}\subseteq\mathfrak{g}$ be the unique $H$-invariant subspace
such that $\mathfrak{g}=\mathfrak{b}\oplus\mathfrak{\ell}$. Since
$\mathfrak{b}$ intersects $\mathfrak{\ell}$ trivially, the law of large
numbers [9, Theorem 4.28] and Furstenberg’s theorem on the positivity of the
first Lyapunov exponent [9, Theorem 4.32] show that there exists $\lambda>0$
such that for every $\alpha>0$, there exists $N\geq 0$ such that for every
$n\geq N$, and every $v\in\mathfrak{b}\setminus\\{0\\}$,

$\mu^{*n}\left(g\in G\ \left|\ \log\frac{\lVert gv\rVert}{\lVert v\rVert}\geq
n\lambda\right.\right)\geq 1-\frac{\alpha}{3}.$ (13)

Fix $n\geq N$. Let $Y\in S_{\Omega}(\Gamma)$ and consider $K,Q\subseteq\Omega$
compact sets with $K\cap Y=\varnothing$. We can assume
$\mathfrak{\ell}+\mathfrak{s}\neq\mathfrak{g}$ otherwise $UY$ is open in
$\Omega$ for any open set $U\subseteq L$, and the drift statement follows
trivially by letting $f$ be infinite on $UY$ and null elsewhere. Consider a
compact set $M\subseteq\Omega$ such that

$\mu^{*n}(g\in G\ |\ gQ\subseteq M)\geq 1-\frac{\alpha}{3}$ (14)

and choose $\mathfrak{t}\subseteq\mathfrak{b}$, $U\subseteq L$, $\delta>0$ as
in Lemma 5.3, with $U$ open relatively compact in $L$, and so that
$K\cap\overline{U}Y=\varnothing$. Set $Y^{\prime}=UY$ and $f=f_{UY}$. We show
that $f$ satisfies the properties required for controlled drift. The fact that
$f$ is bounded on $K$ comes from the observation that $\overline{U}Y$ is
closed which, combined with the condition $K\cap\overline{U}Y=\varnothing$,
imposes that the distance between $K$ and $UY$ is bounded below by some
positive constant. The condition on upper-level sets is straightforward.

Let us check the condition (i) of 3.1. Set $T>0$ so large that

$\mu^{*n}(g\in G\ |\ \lVert\operatorname{Ad}g\rVert<e^{T}\delta)\geq
1-\frac{\alpha}{3}.$ (15)

Let $x\in Q$ such that $T\leq f(x)<+\infty$, i.e. $x\in e^{v}UY$ with
$0<\lVert v\rVert\leq e^{-T}$. Let $g\in G$ be any element satisfying
conditions (13), (14), and (15). By (15) one has $gx\in
e^{\operatorname{Ad}(g)v}UY$ with
$\lVert\operatorname{Ad}(g)v\rVert\leq\lVert\text{Ad}g\rVert\lVert
v\rVert<\delta$. As $gx\in M$ by (14), our choice for $(U,\delta)$ yields that
$\mathfrak{t}_{gx}=\\{\operatorname{Ad}(g)v\\}$, hence
$f(gx)=\log\frac{1}{\|\operatorname{Ad}(g)v\|}$. By (13),

$f(gx)\leq f(x)-n\lambda.$

To check (ii), observe that for $g\in\Gamma$, one has
$\lVert\operatorname{Ad}g^{-1}\rVert^{-1}\,\|v\|\leq\lVert\operatorname{Ad}gv\rVert$
and
$\lVert\operatorname{Ad}g^{-1}\rVert\leq\lVert\operatorname{Ad}g\rVert^{\kappa}$
for some constant $\kappa>0$ depending only on the normed vector space
$(\mathfrak{g},\|.\|)$. So $C=\kappa$ leads to the bound announced in the
lemma.

∎

###  5.2 From local drift to global drift

###### Proof of Lemma 3.2 b).

In this proof, the notations of 3.1 referring to the drift away from infinity
will be indexed by $0$, and those referring to the drift away from
$\mathcal{S}$ on compact subsets will be indexed by $1$. Now let
$(\Phi_{i},\lambda_{i},C_{i})_{i=0,1}$ be given by 3.1. Set $c>1$ such that
$c>\lambda_{0,\infty}$ and $\frac{c\lambda_{0}}{2}>2C_{1}\lambda_{1,\infty}$
where $\lambda_{i,\infty}\geq 0$ is the first Lyapunov exponent of
$\Phi_{i\ast}\mu$. Fix $\alpha>0$ and denote by $N_{0},N_{1}\geq 0$ the
associated constants in 3.1. Fix $n\in\mathbb{N}$ large enough so that
$n\geq\max(N_{0},N_{1})$ and

$\mu^{\ast n}(g\in G\,|\,\log\|\Phi_{0}(g)\|\leq cn)\geq 1-\alpha$ $\mu^{\ast
n}(g\in G\,|\,\log\|\Phi_{1}(g)\|\leq\frac{c\lambda_{0}}{2C_{1}}n)\geq
1-\alpha$

Let $Y\in S_{\Omega}(\Gamma)$, let $K\subseteq\Omega\smallsetminus Y$ be a
compact set. The controlled drift away from infinity associates to $K$ a
proper drift function $f_{0}:\Omega\rightarrow[0,+\infty]$ and a threshold
$T_{0}>0$ as in 3.1. Let $T^{\prime}>0$ be a large parameter which will be
specified later. Choose a compact subset $Q_{1}\subseteq\Omega$ containing
$K\cup\\{f_{0}\leq T^{\prime}\\}$, and consider the objects $Y^{\prime}\in
S_{Y}$, $f_{1}:\Omega\rightarrow[0,+\infty]$, $T_{1}>0$ associated to
$(K,Q_{1})$ as in 3.1 for the drift away from $\mathcal{S}$ on compact sets.

We are now all set up to define our drift function: for $x\in\Omega$, set

$f(x)=c_{x}f_{0}(x)+f_{1}(x)$

where

$c_{x}=\left\\{\begin{array}[]{ll}c&\mbox{if}\ f_{0}(x)\geq T^{\prime}\\\
c\cdot\frac{f_{0}(x)-T_{0}}{T^{\prime}-T_{0}}&\mbox{if}\ T_{0}\leq
f_{0}(x)\leq T^{\prime}\\\ 0&\mbox{if}\ f_{0}(x)\leq T_{0}.\end{array}\right.$

Let us check that $f$ satisfies the properties enumerated in Definition 3.1.
The only non-trivial conditions to check are those enumerated as (i) and (ii).

Probable decrease
We prove that if the parameter $T^{\prime}$ above is chosen sufficiently
large, then $f$ satisfies the property (i) in 3.1, with $T=cT^{\prime}+T_{1}$
and $\lambda=\frac{1}{2}\min(\lambda_{0},\lambda_{1})$. Let $x\in\Omega$ such
that $f(x)\in[T,+\infty)$. We study separately different cases, according to
the values of $f_{0}(x),f_{1}(x)$. Observe that the definition of $T$ imposes
that $f_{0}(x)>T^{\prime}$ or $f_{1}(x)>T_{1}$.

  1. (a)

$f_{0}(x)>T^{\prime}$

In this case, the conditions $\left\\{\begin{array}[]{l}c_{gx}f_{0}(gx)\leq
cf_{0}(x)-nc\lambda_{0}\\\
C_{1}\log\lVert\Phi_{1}(g)\rVert\leq\frac{nc\lambda_{0}}{2}\end{array}\right.$
are simultaneously satisfied with $\mu^{\ast n}$-probability at least
$1-2\alpha$ and ensure that

$f(gx)\leq f(x)-\frac{nc\lambda_{0}}{2}$

  2. (b)

$T_{0}<f_{0}(x)\leq T^{\prime}$ and $f_{1}(x)>T_{1}$
Then, with $\mu^{\ast n}$-probability at least $1-2\alpha$, one has both
$\left\\{\begin{array}[]{l}f_{0}(gx)\leq f_{0}(x)-n\lambda_{0}\\\
f_{1}(gx)\leq f_{1}(x)-n\lambda_{1}.\end{array}\right.$ This certainly implies
$c_{gx}\leq c_{x}$ and therefore

$f(gx)\leq f(x)-n\lambda_{1}.$

  3. (c)

$f_{0}(x)\leq T_{0}$ and $f_{1}(x)>T_{1}$
With $\mu^{\ast n}$-probability at least $1-2\alpha$, one has
$\left\\{\begin{array}[]{l}f_{0}(gx)\leq T_{0}+C_{0}cn\\\ f_{1}(gx)\leq
f_{1}(x)-n\lambda_{1}.\end{array}\right.$

This implies

$c_{gx}f_{0}(gx)\leq c\cdot\frac{C_{0}cn}{T^{\prime}-T_{0}}f_{0}(gx)\leq
c\cdot\frac{C_{0}cn}{T^{\prime}-T_{0}}(T_{0}+C_{0}cn)\leq\frac{n\lambda_{1}}{2}$

up to choosing $T^{\prime}$ large enough in the definition of $f$, and
therefore

$f(gx)\leq f(x)-\frac{n\lambda_{1}}{2}.$

Putting together (a), (b), (c), we obtain for every $x\in f^{-1}[T,+\infty)$,

$\mu^{*n}(g\in G\ |\ f(gx)\leq f(x)-n\lambda)\geq 1-2\alpha$

where $\lambda=\frac{1}{2}\min(\lambda_{0},\lambda_{1})>0$.

Control of variations
We show that if $T^{\prime}$ is chosen large enough in the definition of $f$,
then we have the bound (ii) of 3.1. with $C=3cC_{0}+C_{1}$ and
$\Phi=\Phi_{0}\oplus\Phi_{1}$. Let $x\in\Omega$, $g\in\Gamma$. We can write

$f(gx)-f(x)=\left(c_{gx}f_{0}(gx)-c_{x}f_{0}(x)\right)+\left(f_{1}(gx)-f_{1}(x)\right).$

Definition 3.1 guarantees that

$f_{1}(gx)-f_{1}(x)\leq C_{1}\log\|\Phi_{1}(g)\|$

so it remains to bound the first term. We can assume that $f_{0}(x)\leq
f_{0}(gx)$. Then,

$\displaystyle c_{gx}f_{0}(gx)-c_{x}f_{0}(x)$
$\displaystyle=c_{gx}\left(f_{0}(gx)-f_{0}(x)\right)+f_{0}(x)(c_{gx}-c_{x})$
$\displaystyle\leq c\,C_{0}\log\|\Phi_{0}(g)\|+f_{0}(x)(c_{gx}-c_{x})$

If $f_{0}(x)>T^{\prime}$, then $c_{x}=c_{gx}=c$ and we are done. Otherwise,
observing that

$c_{gx}-c_{x}\leq c\cdot\frac{f_{0}(gx)-f_{0}(x)}{T^{\prime}-T_{0}}\leq
c\cdot\frac{\,C_{0}\log\lVert\Phi_{0}(g)\rVert}{T^{\prime}-T_{0}}$

we obtain

$\displaystyle f_{0}(x)(c_{gx}-c_{x})$
$\displaystyle\leq\frac{T^{\prime}}{T^{\prime}-T_{0}}c\,C_{0}\log\lVert\Phi_{0}(g)\rVert$
$\displaystyle\leq 2c\,C_{0}\log\lVert\Phi_{0}(g)\rVert$

provided $T^{\prime}$ is chosen large enough in the definition of $f$. In any
case, we conclude that for $x\in f^{-1}[0,+\infty)$, $g\in\Gamma$,

$f(gx)-f(x)\leq 3cC_{0}\log\|\Phi_{0}(g)\|+C_{1}\log\|\Phi_{1}(g)\|.$

∎

##  6 Equidistribution

The goal of this section is to establish Theorem C. We shall need the
following lemma, obtained by Benoist and Quint [7, Proposition 2.1] in the
greater generality of $S$-adic Lie groups, but with the additional assumption
that $\Gamma$ is compactly generated. The proof is almost the same as in [7],
but we include it to explain in detail how to modify their argument to avoid
this assumption.

######  Lemma 6.1.

Let $G$ be a real Lie group, $\Lambda$ a lattice in $G$, $\Gamma$ a subgroup
of $G$ and denote by $L$ be the centralizer of $\Gamma$ in $G$. Assume that
$\overline{\operatorname{Ad}\Gamma}^{Z}$ is Zariski connected semisimple and
without compact factors. Then the set $\mathcal{S}_{\Omega}(\Gamma)$ is a
countable union of $L$-orbits.

###### Proof.

Fix a dense countable subset $D\subset\Gamma$. Given
$Y\in\mathcal{S}_{\Omega}(\Gamma)$, there exists a finite family of elements
$g_{1},\dots,g_{s}$ in $D$ generating a group $\Gamma_{f}$ such that
$\overline{\operatorname{Ad}\Gamma_{f}}^{Z}=\overline{\operatorname{Ad}\Gamma}^{Z}$
and that moreover $\Gamma_{f}$ acts transitively on the (finite) set of
connected components of $Y$. Let $g\in G$ be such that $g\Lambda\in Y$, and
write

$H=g^{-1}\Gamma_{f}G_{Y}^{\circ}g.$

Since $H\Lambda=g^{-1}Y$ has finite volume, $H\cap\Lambda$ is a lattice in
$H$, and $H^{\circ}\cap\Lambda$ is a lattice in $H^{\circ}$. Because $H$ and
$H^{\circ}$ are compactly generated, the groups $\Sigma=H\cap\Lambda$ and
$\Delta=H^{\circ}\cap\Lambda$ are both finitely generated, and since $\Lambda$
contains only countably many finitely generated subgroups, we may assume that
they are fixed. (The finitely generated group $\Gamma_{f}$ was introduced
precisely to ensure that $H$ be compactly generated, so we can use the
argument of [7].) Then, $H$ belongs to the set $\mathcal{T}(G,\Delta,\Sigma)$
of closed subgroups such that

  1. (i)

$\Sigma$ is a lattice in $H$;

  2. (ii)

$\Delta=\Sigma\cap H^{\circ}$ is a lattice in $H^{\circ}$;

  3. (iii)

there exists a subgroup $\Gamma^{\prime}$ in $G$ such that
$\overline{\operatorname{Ad}\Gamma^{\prime}}^{Z}$ is connected semisimple
without compact factors and acts ergodically on $H/\Sigma$.

By [7, Lemma 2.5], this set is countable, so we may assume that $H$ is fixed.
Let $L_{f}$ denote the centralizer of $\Gamma_{f}$ in $G$, and note that
$L_{f}^{\circ}=L^{\circ}$ as their Lie algebras are given by the sets of
invariant vectors of
$\overline{\operatorname{Ad}{\Gamma_{f}}}^{Z}=\overline{\operatorname{Ad}{\Gamma}}^{Z}$.
By [7, Lemma 2.2], the set of fixed points of $\Gamma_{f}$ in $G/H$ is a
countable union of $L_{f}^{\circ}$-orbits, and therefore also a countable
union of $L$-orbits. Since $Y=gH\Lambda$ and $gH$ is a fixed point of
$\Gamma_{f}$ in $G/H$, the lemma follows. ∎

###### Proof of Theorem C.

With Theorem A, Theorem B′, Lemma 6.1, and [13, Theorem 1.3] at hand, the
theorem follows from the argument given in [7, §2.3 and §4.1]; we include the
proof for completeness. We show the following assertion, which implies items
(i), (ii) and (iii): for every $x\in\Omega$, there exists a $\Gamma$-invariant
ergodic finite-volume homogeneous closed subset $Y$ containing $x$ such that
$\left(\frac{1}{n}\sum_{k=0}^{n-1}\delta_{g_{k}\dots g_{1}x}\right)_{n\geq 1}$
converges to $\nu_{Y}$.

Let $(g_{n})_{n\geq 1}$ be a $\mu^{\otimes\mathbb{N}^{*}}$-typical sequence of
instructions, and $\nu$ any weak limit of the sequence
$\left(\frac{1}{n}\sum_{k=0}^{n-1}\delta_{g_{k}\dots g_{1}x}\right)_{n\geq
1}$. By Theorem A, $\nu$ is a probability measure. Moreover, by Breiman’s law
of large numbers [10] (see also [7, Lemma 3.2]) the measure $\nu$ is
$\mu$-stationary. Consider a disintegration

$\nu=\int\nu_{\alpha}\mathrm{d}\mathbb{P}(\alpha)$

where each $\nu_{\alpha}$ is an ergodic $\mu$-stationary measure. By the
classification of stationary measures [13, Theorem 1.3], each $\nu_{\alpha}$
is equal to a homogeneous measure $\nu_{Y_{\alpha}}$ for some invariant
ergodic finite-volume homogeneous closed subset $Y_{\alpha}$. Using
Proposition 6.1, we may rewrite this integral as a countable sum

$\nu=\sum_{i\in\mathbb{N}}\nu_{i},$ (16)

where each $\nu_{i}$ is a $\mu$-stationary measure supported on $LY_{i}$, for
some $Y_{i}\in\mathcal{S}_{\Omega}(\Gamma)$. If $x\not\in LY$, Theorem B′
implies that $\nu(LY)=0$, so there must exist
$Y\in\mathcal{S}_{\Omega}(\Gamma)$ such that $x\in Y$. Choose such $Y$ of
minimal dimension. Replacing $G$ by $G_{Y}=\\{g\in G\ |\ gY=Y\\}$, the proof
boils down to the case where $\Omega\in S_{\Omega}(\Gamma)$, $x$ does not
belong to any proper $Z\in S_{\Omega}(\Gamma)$, and we need to show that
$\nu=\nu_{\Omega}$. This comes directly from a second application of the
decomposition (16) combined with Theorem B′. ∎

##  7 Conclusion

To conclude, we mention a few problems that were not discussed here, but might
lead to interesting continuations of our study.

Compact factors.

When the group $H=\overline{\operatorname{Ad}\Gamma}^{Z}$ is semisimple but is
allowed to have compact factors, it is still possible to describe all ergodic
$\mu$-stationary measures on $\Omega$. Some might not be homogeneous, but they
can nevertheless be written as an integral average of homogeneous measures
[13]. It should be possible to obtain a generalization of Theorem C to this
setting.

Drift function on $G$.

On a general quotient $G/\Lambda$, the Benoist-Quint drift function is
essentially constructed by embedding $G/\Lambda$ into some space of lattices
$\operatorname{\mathrm{SL}}_{d}(\mathbb{R})/\operatorname{\mathrm{SL}}_{d}(\mathbb{Z})$.
It would be desirable to have an intrinsic construction of such a function, at
least in the case where $G$ is a linear algebraic group defined over
$\mathbb{Q}$ and $\Lambda=G(\mathbb{Z})$ an arithmetic subgroup. For example,
this would lead to simple formulas for the optimal return time to a compact
set, similar to those available for
$\operatorname{\mathrm{SL}}_{d}(\mathbb{R})/\operatorname{\mathrm{SL}}_{d}(\mathbb{Z})$.

Unipotency assumption.

It was originally suggested in [14] that Theorem A could hold under the
assumption that $H=\overline{\operatorname{Ad}\Gamma}^{Z}$ is generated by
unipotents. As observed by Emmanuel Breuillard [11, Proposition 10.4], this is
not true for conclusion $(i)$ in general. But this might be the case for
$(ii)$, and $(i)$ would then hold for the Cesàro averages
$\frac{1}{n}\sum_{k=0}^{n-1}\mu^{*k}*\delta_{x}$. In a slightly different
direction, the theorem could be valid as stated if $H$ is assumed to be
perfect, i.e. if its Lie algebra $\mathfrak{h}$ satisfies
$\mathfrak{h}=[\mathfrak{h},\mathfrak{h}]$.

No moment assumption.

As suggested at the end of [2], one may hope that the results on homogeneous
random walks obtained here, and in particular Theorem A, are still valid
without any moment assumption.

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