Recurrence on Affine Grassmannians

TL;DR

This paper investigates the dynamics of the affine group acting on affine subspaces, establishing conditions for the existence of stationary measures based on Lyapunov exponents, with implications for symmetric measures.

Contribution

It provides a precise criterion linking Lyapunov exponents to the existence of stationary measures on affine Grassmannians.

Findings

Stationary measure exists iff the (k+1)th Lyapunov exponent is negative.

For symmetric measures, stationary measures exist if and only if 2k ≥ d.

The result characterizes the action of affine groups on affine subspaces in terms of Lyapunov exponents.

Abstract

We study the action of the affine group $G$ of $\mathbb{R}^d$ on the space $X_{k,\,d}$ of $k$-dimensional affine subspaces. Given a compactly-supported Zariski dense probability measure $\mu$ on $G$, we show that $X_{k,\,d}$ supports a $\mu$-stationary measure $\nu$ if and only if the $(k\!+\!1)^{\rm th}$-Lyapunov exponent of $\mu$ is strictly negative. In particular, when $\mu$ is symmetric, $\nu$ exists if and only if $2k\geq d$.

Equations

$${\mu }^{\ast n}\ast {\delta }_x(C)\ge 1-\epsilon .$$

$${\lambda }_1+\dots +{\lambda }_p=\underset{n\to \infty }{\lim }\frac{1}{n}{\int }_G\log ||[||{\Lambda }^{p}g]d{\mu }^{\ast n}(g).$$

$${\lambda }_1={\int }_{{\mathbb{R}}^{\ast }⋉\mathbb{R}}\log |[|a]d\mu (a,u)$$

$$\frac{1}{n}{\sum }_{j=1}^n{\mu }^{\ast j}\ast {\delta }_x\underset{n\to \infty }{\to }0.$$

$$\frac{1}{n}{\sum }_{j=1}^n{\mu }^{\ast j}\ast {\delta }_x\underset{n\to \infty }{\to }\nu .$$

$$\mathcal{A}=\{(w,1)|w\in {\mathbb{R}}^d\}.$$

$$X_{k,d}={\mathrm{Gr}}_{k+1}(d+1)\setminus {\mathrm{Gr}}_{k+1}(d),$$

$$V:={\Lambda }^{k+1}{\mathbb{R}}^{d+1}\mathrm{and}W:={\Lambda }^{k+1}{\mathbb{R}}^d.$$

$$\phi :{\mathrm{Gr}}_{k+1}(d+1)⟶\mathbb{P}([)V];U⟼{\Lambda }^{k+1}U.$$

$$\phi :X_{k,d}\hookrightarrow X_{V,W}:=\mathbb{P}([)V]\setminus \mathbb{P}([)W].$$

$${\lambda }_{1,W}={\lambda }_1+\dots +{\lambda }_{k+1}\mathrm{and}{\lambda }_{1,W^{\prime }}={\lambda }_1+\dots +{\lambda }_k$$

$$Pu\le au+b.$$

$$P_{\mu }f(x)={\int }_{G}f(gx)d\mu (g).$$

$$X_{V,W}=\{[w,w^{\prime }]|w\in W,w^{\prime }\in W_s\setminus \{0\}\}.$$

$$u_{\delta }:X_{V,W}⟶{\mathbb{R}}_{+};[w,w^{\prime }]⟼\frac{||[||w{]}^{\delta }}{||[||w^{\prime }{]}^{\delta }}.$$

$$P_{\mu }^{n_0}{u}_{\delta }\le a{u}_{\delta }+b.$$

$$g=\begin{pmatrix}a_{g}&c_{g}\\ 0&d_{g}\end{pmatrix}\;\mbox{\rm with $a_{g}\in\mathrm{GL}(W),\,d_{g}\in\mathrm{GL}(W_{s}),\,c_{g}\in\mathcal{L}(W_{s},\,W)$.}$$

$${\lambda }_1-\epsilon$$

$${\lambda }_1^{\prime }-\epsilon$$

$$P_{\mu }^{n_0}{u}_{\delta }(x)=u_1(x{)}^{\delta }{\int }_G\frac{u_1(gx{)}^{\delta }}{u_1(x{)}^{\delta }}d{\mu }^{\ast n_0}(g).$$

$$c^{-1}=\frac{4}{n_0({\lambda }_1^{\prime }-{\lambda }_1)}{\int }_G||[||c_g]||[||a_g^{-1}]d{\mu }^{\ast n_0}(g).$$

$$K=\{[w,w^{\prime }]|w\in W,w^{\prime }\in W_s,||[||w^{\prime }]\ge c||[||w]\}.$$

$$\frac{u_1(gx)}{u_1(x)}=\frac{||[||a_{g}w+c_g{w}^{\prime }]}{||[||w]}\frac{||[||w^{\prime }]}{||[||d_g{w}^{\prime }]}\le \underset{g\in \mathrm{Supp}{\mu }^{\ast n_0}}{\sup }||[||d_g^{-1}](||[||a_g]+c||[||c_g]).$$

$$\frac{u_1(gx{)}^{\delta }}{u_1(x{)}^{\delta }}=e^{\delta \log \frac{u_1(gx)}{u_1(x)}}\le 1+\delta \log \frac{u_1(gx)}{u_1(x)}+{\delta }^2{M}_{n_0}.$$

$$\log \frac{u_1(gx)}{u_1(x)}$$

$$\le \log \frac{||[||a_{g}w]}{||[||w]}-\log \frac{||[||d_g{w}^{\prime }]}{||[||w^{\prime }]}+\frac{||[||c_g{w}^{\prime }]}{||[||a_{g}w]}$$

$$\le \log \frac{||[||a_{g}w]}{||[||w]}-\log \frac{||[||d_g{w}^{\prime }]}{||[||w^{\prime }]}+||[||c_g]||[||a_g^{-1}]c.$$

$${\int }_G\log \frac{u_1(gx)}{u_1(x)}d{\mu }^{\ast n_0}(g)\le n_0({\lambda }_1-{\lambda }_1^{\prime }+2\epsilon )+\frac{n_0({\lambda }_1^{\prime }-{\lambda }_1)}{4}\le \frac{n_0({\lambda }_1-{\lambda }_1^{\prime })}{2}.$$

$${\int }_G\frac{u_1(gx{)}^{\delta }}{u_1(x{)}^{\delta }}d{\mu }^{\ast n_0}(g)\le 1-\delta \kappa +{\delta }^2{M}_{n_0}.$$

$$P_{\mu }^{n_0}{u}_{\delta }(x)\le a_{n_0,\delta }{u}_{\delta }(x)+b_{n_0,\delta },$$

$${\nu }_1⊠{\nu }_2:={\int }_B{\nu }_{1,b}\otimes {\nu }_{2,b}d\beta (b)$$

$$b_1\cdots b_n=\left(\begin{array}{cc}{a}_n& c_n\\ 0& d_n\end{array}\right).$$

Full text

1. 1 Introduction

2. 1.1 Recurrence and Lyapunov Exponents

3. 1.2 Action on the Affine Grassmannians

4. 1.3 Action on $X_{V,W}$

5. 1.4 Strategy of the Proof

6. 2 Recurrence on affine Grassmannians

7. 3 Uniform Recurrence When $\lambda_{1}<\lambda_{1}^{\prime}$

8. 3.1 The Contraction Hypothesis

9. 3.2 Finding a Contracted Function

10. 4 Non-Recurrence in Law When $\lambda_{1}\geq\lambda_{1}^{\prime}$

11. 4.1 The Limit Measures

12. 4.2 No Stationary Measures on $X_{V,W}$

13. 4.3 Using the Limit Measures

14. 4.4 Using the Cartan Projection

15. 5 Uniqueness of the Stationary Measure

16. 5.1 No Stationary Measures on $Y_{V,W}$

17. 5.2 Proof of Uniqueness

18. 5.3 Limit of Means of Transition Probabilities

19. A Limit Laws on Reductive Groups

20. A.1 Cartan Decomposition

21. A.2 Limit Laws

22. A.3 Opposition Involution

23. A.4 Stationary Measures on Projective Spaces

1 Introduction

1.1 Recurrence and Lyapunov Exponents

†† Key words: Affine group, Grassmannian, random walk, recurrence, stationary probability.

AMS-MSC : 22E40, 60J20.

Consider a locally compact group $G$ acting continuously on a locally compact second countable space $X$ and $\mu$ a probability measure on $G$. The associated random walk on $X$ is the Markov chain over $X$ defined by the transition probabilities $P_{x}=\mu*\delta_{x}$ for all $x\in X$. Our aim is to study the recurrence properties of such a random walk. We will not focus here on the almost sure recurrence as in [5] and [9] but on the recurrence in law as in [3], [6] and[10].

Definition 1.1**.**

The random walk on $X$ is recurrent in law at a point $x\in X$ if for all $\varepsilon>0$, there exists a compact set $C\subset X$ and $n_{0}\in\mathbb{N}^{*}$ such that for all $n\geq n_{0}$:

[TABLE]

The random walk on $X$ is uniformly recurrent in law if the same compact set $C$ can be chosen for all the starting points $x$. A probability measure $\nu$ on $X$ is said to be $\mu$-stationary or $\mu$-invariant if one has $\mu*\nu=\nu.$

Those definitions are tightly linked. Indeed, there exists a $\mu$-stationary probability measure on $X$ if and only if the random walk on $X$ is recurrent in law at some point $x\in X$ (see Lemma 3.3 for one implication).

In this paper, $G$ will always be a real algebraic group acting algebraically on a real algebraic variety $X$; the measure $\mu$ will be compactly supported and Zariski dense, which means that its support spans a Zariski dense subgroup in $G$.

When $G$ is a reductive group and $X=G/H$ is an algebraic homogeneous space, it is proven in [3] that there exists a $\mu$-stationary probability measure on $X$ if and only if $X$ is compact. The aim of our article is to focus on situations where the algebraic group $G$ is not reductive. In particular, in Corollary 1.5, we will exhibit examples of non-compact homogeneous spaces on which there always exists a $\mu$-stationary probability measure.

The key tool in our analysis will be to link the recurrence properties of these random walks to the Lyapunov exponents of $\mu$. The definition of these Lyapunov exponents depends on the choice of a linear action of $G$ on $\mathbb{R}^{d}$.

Definition 1.2**.**

Given a linear action of $G$ on $\mathbb{R}^{d}$, the Lyapunov exponents of $\mu$ are the real numbers $\lambda_{1},\,\ldots,\,\lambda_{d}$ such that, for all $1\leq p\leq d$, we have

[TABLE]

The sequence of Lyapunov exponents is always decreasing: $\lambda_{1}\geq\ldots\geq\lambda_{d}$ (see [14, Prop 1.2]). More properties of these exponents are given in [16], [14]; their use in the context of reductive groups is detailed in [11], [13], [12] and [4].

1.2 Action on the Affine Grassmannians

We assume now that $G$ is either the affine group $G=\mathrm{GL}(d,\mathbb{R})\ltimes\mathbb{R}^{d}$ or the special affine group $G=\mathrm{SL}(d,\mathbb{R})\ltimes\mathbb{R}^{d}$. For $1\leq p\leq d$, we denote by $\lambda_{p}$ the $p^{\rm th}$-Lyapunov exponent corresponding to the linear action of $G$ on $\mathbb{R}^{d}$. For instance, in dimension $d=1$, one has

[TABLE]

where $g=(a,u)\in\mathbb{R}^{*}\ltimes\mathbb{R}$. For any $d\geq 1$, Bougerol and Picard have shown in [6] that there exists a $\mu$-stationary probability measure on $\mathbb{R}^{d}$ if and only if the first Lyapunov exponent of $\mu$ is strictly negative: $\lambda_{1}<0$.

The main result of this paper is the following Theorem 1.3, which extends this equivalence to the affine Grassmannians $X_{k,\,d}$ where $0\leq k<d$. By definition the affine Grassmannian $X_{k,\,d}$ is the space of $k$-dimensional affine subspaces of $\mathbb{R}^{d}$. The group $G$ acts transitively on $X_{k,\,d}$.

Theorem 1.3**.**

*Let $G$ be the affine group or the special affine group of $\mathbb{R}^{d}$, let $\mu$ be a Zariski dense probability measure with compact support on $G$ and let $0\leq k<d$.

a) If $\lambda_{k+1}\geq 0$, then the random walk on $X_{k,\,d}$ is nowhere recurrent in law, there exists no $\mu$-stationary probability measure on $X_{k,\,d}$, and for all $x$ in $X_{k,\,d}$ the sequence of means of transition probabilities weakly converges to [math]:*

[TABLE]

b) If $\lambda_{k+1}<0$, then the random walk on $X_{k,\,d}$ is uniformly recurrent in law, there exists a unique $\mu$-stationary probability measure $\nu$ on $X_{k,\,d}$, and for all $x$ in $X_{k,\,d}$ the sequence of means of transition probabilities weakly converges to $\nu$:

[TABLE]

The result of Bougerol and Picard in [6] covers the $k=0$ case. In fact, their proof uses only the weaker assumption that $\mu$ has a finite first moment and that its support does not preserve any proper affine subspace of $\mathbb{R}^{d}$.

The following Corollary, which is particularly noteworthy insofar as it does not mention Lyapunov exponents, is deduced from Theorem 1.3.

Corollary 1.4**.**

Assume $\mu$ is symmetric. Then there exists a $\mu$-stationary probability measure $\nu$ on $X_{k,\,d}$ if and only if $2k\geq d$. In this case, $\nu$ is unique.

Proof.

Since $\mu$ is symmetric, for all $1\leq p\leq d$, the Lyapunov exponents satisfy the equalities $\lambda_{p}=-\lambda_{d+1-p}$. Moreover, since $\mu$ is Zariski dense in $G$, it follows from the Guivarc’h-Raugi simplicity theorem that the sequence of Lyapunov exponents is strictly decreasing: $\lambda_{1}>\cdots>\lambda_{d}$ (see [4, Corol. 10.15]). Therefore one has the equivalence $\lambda_{k+1}<0\Longleftrightarrow 2k\geq d$. ∎

Corollary 1.5**.**

Let $d\geq 2$. When $G$ is the special affine group and $k=d-1$, there exists a unique $\mu$-stationary probability measure on $X_{k,\,d}$.

Proof.

In this case, the sum of the Lyapunov exponents is zero. Hence, the simplicity of the Lyapunov exponents implies $\lambda_{d}<0$. ∎

For instance, when $G$ is the special affine group of $\mathbb{R}^{2}$, the random walk on the space of affine lines of $\mathbb{R}^{2}$ is always uniformly recurrent in law while the random walk on the space of points of $\mathbb{R}^{2}$ is nowhere recurrent in law.

1.3 Action on $X_{V,W}$

By embedding the affine Grassmannian $X_{k,\,d}$ of $\mathbb{R}^{d}$ in the projective space of a suitable exterior power $V$ of $\mathbb{R}^{d+1}$, we will deduce Theorem 1.3 from the following Theorem 1.6:

We first need two definitions. An algebraic group $G$ is Zariski connected if it is connected for the Zariski topology. A linear action of $G$ on a vector space $W$ is proximal if there exists a rank $1$ linear endomorphism $\pi$ of $W$ which is a limit of a sequence $\lambda_{n}\gamma_{n}$ with $\lambda_{n}>0$ and $\gamma_{n}$ in $\Gamma$.

Theorem 1.6**.**

*Let $V$ be a finite-dimensional real vector space, $G$ a Zariski connected algebraic subgroup of $\mathrm{GL}(V)$, $W$ a $G$-invariant subspace of $V$ such that

$(H1)$ $G$ acts irreducibly and proximally on $W$ and on $W^{\prime}:=V/W$.

$(H2)$ The representations of $G$ in $W$ and $W^{\prime}$ are not equivalent.

$(H3)$ $W$ has no $G$-invariant complementary subspace in $V$.

Let $X_{V,W}:=\mathbb{P}([)V]\smallsetminus\mathbb{P}([)W]$, let $\mu$ be a Zariski dense probability measure with compact support on $G$ and let $\lambda_{1}=\lambda_{1,W}$ and $\lambda^{\prime}_{1}=\lambda_{1,W^{\prime}}$ be the first Lyapunov exponents of $\mu$ in $W$ and $W^{\prime}$ respectively.

a) If $\lambda_{1}\geq\lambda^{\prime}_{1}$, then the random walk on $X_{V,W}$ is nowhere recurrent in law, there exists no $\mu$-stationary probability measure on $X_{V,W}$, and for all $x$ in $X_{V,W}$ one has the weak convergence $\textstyle\frac{1}{n}\sum_{j=1}^{n}\mu^{*j}*\delta_{x}\xrightarrow[n\rightarrow\infty]{}0.$

b) If $\lambda_{1}<\lambda^{\prime}_{1}$, then the random walk on $X_{V,W}$ is uniformly recurrent in law, there exists a unique $\mu$-stationary probability measure $\nu$ on $X_{V,W}$, and for all $x$ in $X_{V,W}$, one has the weak convergence $\textstyle\frac{1}{n}\sum_{j=1}^{n}\mu^{*j}*\delta_{x}\xrightarrow[n\rightarrow\infty]{}\nu.$*

1.4 Strategy of the Proof

In Chapter 2, we explain how to embed the affine Grassmannian $X_{k,d}$ in the variety $\mathbb{P}([)\Lambda^{k+1}\mathbb{R}^{d+1}]\smallsetminus\mathbb{P}([)\Lambda^{k+1}\mathbb{R}^{d}]$ and we deduce Theorem 1.3 from Theorem 1.6.

The last three chapters will deal with the proof of Theorem 1.6.

In Chapter 3, we prove the uniform recurrence in law when $\lambda_{1}<\lambda^{\prime}_{1}$ (Corollary 3.6). The crux of the proof is the construction of a proper function on $X_{V,W}$ which is contracted by the averaging operator (Proposition 3.5).

In Chapter 4, we prove the non-recurrence in law when $\lambda_{1}\geq\lambda^{\prime}_{1}$ (Proposition 4.4). The key point is the study of the ratio of the norms in $W$ and in $W^{\prime}$ of a random product $b_{1}\cdots b_{n}$. On the one hand, the existence of a $\mu$-stationary probability measure on $X_{V,W}$ would imply that these ratios are bounded (Lemma 4.5). On the other hand, when $\lambda_{1}\geq\lambda^{\prime}_{1}$, the Law of Large Numbers and the Law of Iterated Logarithms for these products prevent these ratios from being bounded (Lemma 4.6).

In Chapter 5, we prove the uniqueness of the $\mu$-stationary measure on $X_{V,W}$ (Proposition 5.3). Indeed, using the joining measure (Corollary 4.2) of two distinct $\mu$-stationary probability measures on $X_{V,W}$, we construct (Lemma 5.1) a $\mu$-stationary measure $\overline{\nu}$ on the space $\mathbb{P}([)W\oplus W^{\prime}]\smallsetminus(\mathbb{P}([)W]\cup\mathbb{P}([)W^{\prime}])$. This contradicts the classification of stationary measures in [3] since this space does not contain compact $G$-orbits (Lemma 5.2). The weak convergence of the sequence of means of transition probabilities follows easily (Corollary 5.4).

In Appendix A, we collect known facts on random walks on reductive groups.

In this paper, all the vector spaces will be finite dimensional real vector spaces, all the measures will be Borel measures and we will not distinguish between a real algebraic group and its group of real points.

2 Recurrence on affine Grassmannians

We explain first how to deduce Theorem 1.6 from Theorem 1.3

We use the notation of Theorem 1.6. The group $G$ is the affine group or the special affine group of $\mathbb{R}^{d}$, the space $X_{k,\,d}$ is the affine Grassmannian of $\mathbb{R}^{d}$, the probability measure $\mu$ on $G$ is Zariski dense and compactly supported.

Let us construct $G$-vector spaces $W\subset V$ to which we will apply Theorem 1.6. We identify the affine space $\mathbb{R}^{d}$ with the affine hyperplane of $\mathbb{R}^{d+1}=\mathbb{R}^{d}\oplus\mathbb{R}$:

[TABLE]

The group $G$ is then a subgroup of $\mathrm{GL}(d+1,\,\mathbb{R})$, which stabilizes $\mathcal{A}$, and we have

[TABLE]

where $\mathrm{Gr}_{k+1}(d\!+\!1)$ and $\mathrm{Gr}_{k+1}(d)$ are the Grassmannians of $(k\!+\!1)$-dimensional vector subspaces of $\mathbb{R}^{d+1}$ and of $\mathbb{R}^{d}$ respectively. Now, let

[TABLE]

The group $G$ acts linearly on the vector space $V$ and leaves invariant its vector subspace $W$. The Plücker map

[TABLE]

is an embedding of the Grassmannian variety in the projective space of $V$. It induces a $G$-equivariant injection

[TABLE]

Proposition 2.1**.**

*With the above notations,

a) Hypotheses $(H1)$, $(H2)$, $(H3)$ hold for these $V$, $W$ and $W^{\prime}=V/W$.

b) The $G$-equivariant inclusion $X_{k,\,d}\hookrightarrow X_{V,W}$ has closed image.

c) We have the equality $\lambda_{k+1}=\lambda_{1,\,W}-\lambda_{1,\,W^{\prime}}$.*

Proof of Proposition 2.1.

a) $(H1)$ : The representation of ${\rm SL}(d,\mathbb{R})$ in $W=\Lambda^{k+1}\mathbb{R}^{d}$ is irreducible by [7, Chap. 8.13.1.4]. This representation is proximal since the image in ${\rm GL}(W)$ of a diagonal element of $G$ with positive distinct eigenvalues is a proximal element of ${\rm GL}(W)$. The same is true for the representation in $W^{\prime}\simeq\Lambda^{k}\mathbb{R}^{d}$.

$(H2)$ : The fact that the representations of ${\rm SL}(d,\mathbb{R})$ in $W$ and $W^{\prime}$ are not equivalent is also proven in [7, Chap. 8.13.1.4].

$(H3)$ : Since the representation of ${\rm SL}(d,\mathbb{R})$ in $W$ and $W^{\prime}$ are irreducible and are not equivalent, by Schur’s lemma, the only ${\rm SL}(d,\mathbb{R})$-invariant complementary subspace of $W$ in $V\simeq W\oplus W^{\prime}$ is $W^{\prime}$. But $W^{\prime}$ is not invariant by the translations of $G$.

b) The image $\varphi(X_{k,d})$ is closed in $X_{V,W}$ since $\varphi^{-1}(\mathbb{P}([)W])=\mathrm{Gr}_{k+1}(d)$.

c) This equality is the difference of the equalities

[TABLE]

which follow from the very definition (1.1) of the Lyapunov exponents. ∎

Proof of Theorem 1.6 $\Longrightarrow$ Theorem 1.3.

We use Proposition 2.1.

If $\lambda_{k+1}\geq 0$, then we can apply Theorem 1.6 in the case where $\lambda_{1,\,W}\geq\lambda_{1,\,W^{\prime}}$, and there can be no $\mu$-stationary probability measure on $X_{k,\,d}$.

Conversely, if $\lambda_{k+1}<0$, we are in the case where $\lambda_{1,\,W}<\lambda_{1,\,W^{\prime}}$. Since $X_{k,\,d}$ is a $G$-invariant closed subset of $X_{V,W}$, we obtain uniform recurrence in law on $X_{k,\,d}$. Lemma 3.3 then ensures the existence of a $\mu$-stationary probability measure on $X_{k,\,d}$, which is thus the unique $\mu$-stationary probability measure on $X_{V,W}$. ∎

3 Uniform Recurrence When $\lambda_{1}<\lambda_{1}^{\prime}$

The goal of this Chapter is to show that the random walk on $X_{V,W}$ is uniformly recurrent in law when $\lambda_{1}<\lambda_{1}^{\prime}$ (Corollary 3.6).

3.1 The Contraction Hypothesis

We recall in this section the uniform contraction hypothesis and why this condition implies the uniform recurrence in law.

The setting is very general (see [15], [10] or [2] for more details). Let $X$ be a locally compact second-countable space and $P$ a Markov-Feller operator on $X$.

Definition 3.1**.**

The operator $P$ satisfies the uniform contraction hypothesis (UCH) if there exists a proper map $u:X\rightarrow[0,\,\infty[$ and two constants $0<a<1$ and $b>0$ such that, over $X$,

[TABLE]

We recall that a map is proper if the inverse image of every compact set is relatively compact. The definition of recurrence in law extends to Markov chains on $X$. Uniform recurrence in law is fundamentally linked with (UCH):

Proposition 3.2**.**

If $P$ satisfies (UCH), then the associated Markov chain on $X$ is uniformly recurrent in law.

Proof.

See [15, Thm 15.0.1], [10, Lem 3.1] or [2, Lem 2.1]. ∎

Lemma 3.3**.**

If $P$ is recurrent in law at point $x\in X$, there exists a $P$-invariant probability measure on $X$.

Proof.

By the Banach-Alaoglu Theorem, the sequence of means of transition probabilities $\nu_{n}=\frac{1}{n}\sum_{j=1}^{n}P^{j}_{x}$ has at least one accumulation point $\nu_{\infty}$ for the weak-$*$ topology. This finite measure $\nu_{\infty}$ is $P$-invariant. Since $P$ is recurrent in law at $x$, there is no escape of mass and $\nu_{\infty}$ is a probability measure. ∎

The following lemma is a useful tool to check (UCH).

Lemma 3.4**.**

Let $n\geq 1$. If $P^{n}$ satisfies $(UCH)$ then $P$ satisfies $(UCH)$ too.

Proof.

Let $u$ be the proper map and $a$,$b$ the constants such that $P^{n}u\leq au+b$ over $X$. Let $\alpha_{k}=a^{-k/n}$ for $0\leq k\leq n-1$, $a^{\prime}=a^{1/n}$, $b^{\prime}=\frac{a^{\prime}}{a}b$. Then the proper map $u^{\prime}:X\rightarrow\mathbb{R}_{+}$ defined by $u^{\prime}=\sum_{k=0}^{n-1}\alpha_{k}P^{k}u$ satisfies the inequality $Pu^{\prime}\leq a^{\prime}u^{\prime}+b^{\prime}$ on $X$, and thus $P$ satisfies(UCH). ∎

3.2 Finding a Contracted Function

In this section, we use again the notations and assumptions of Theorem 1.6. We will prove that the averaging operator satisfies the uniform contraction hypothesis.

We recall that $W\subset V$ are real vector spaces, $G$ is a Zariski connected algebraic subgroup of $\mathrm{GL}(V)$ preserving $W$ and satisfying $(H1)$, $(H2)$, $(H3)$. We identify the quotient $W^{\prime}=V/W$ with a complementary subspace $W_{s}$ of $W$ in $V$. Note that this subspace $W_{s}$ is not $G$-invariant. We recall also that $\mu$ is a Zariski dense probability measure on $G$ with compact support and that $\lambda_{1}$ and $\lambda^{\prime}_{1}$ are the first Lyapunov exponents of $\mu$ in $W$ and $W^{\prime}$, and that we are studying the associated random walk on the $G$-space $X_{V,W}:=\mathbb{P}([)V]\smallsetminus\mathbb{P}([)W]$.

The corresponding Markov operator $P_{\mu}:\mathcal{C}^{0}([)X_{V,W}]\longrightarrow\mathcal{C}^{0}([)X_{V,W}]$ is given by

[TABLE]

Proposition 3.5**.**

*Same notations and assumptions as in Theorem 1.6.

If $\lambda_{1}<\lambda^{\prime}_{1}$, then the Markov operator $P_{\mu}$ satisfies (UCH).*

Proof.

The space $X_{V,W}$ can be seen as the set

[TABLE]

Choose a norm on $V$, and, for $\delta>0$, consider the functions

[TABLE]

These functions are proper and well-defined. We want to find $\delta>0$, $a\in]0,\,1[$, $b>0$, $n_{0}\in\mathbb{N}^{*}$ such that, over $X_{V,W}$, one has the inequality

[TABLE]

Since $W$ is $G$-invariant, we can write $g\in G$ as

[TABLE]

Let $0<\varepsilon<\frac{\lambda^{\prime}_{1}-\lambda_{1}}{8}$. Then, by a lemma due to Furstenberg (cf. [4, Thm 4.28], [11]) since $G$ acts irreducibly on $W$ and $W^{\prime}$ there exists $n_{0}\in\mathbb{N}^{*}$ such that for all $n\geq n_{0}$, for all non-zero $w\in W,\,w^{\prime}\in W_{s}$, the following inequalities hold:

[TABLE]

For $\delta>0$ and $x=[w,\,w^{\prime}]\in X_{V,W}$, one computes

[TABLE]

We will give an upper bound for the right-hand integral for all $x$ in the complementary set of some compact $K$ in $X$ ; since map $P_{\mu}^{n_{0}}u_{\delta}$ is bounded on the compact set $K$, this will give inequality (3.2). Let $c>0$ be the constant defined by

[TABLE]

Let $K$ be the compact subset of $X_{V,W}$ given by

[TABLE]

For $\mu^{*n_{0}}$-almost every $g\in G$, for all $x\in X\smallsetminus K$, the following ratio is bounded:

[TABLE]

Therefore, we can find some constant $M_{n_{0}}>0$ such that for all $\delta>0$, for all $x\in X\smallsetminus K$, for $\mu^{*n_{0}}$-almost every $g\in G$, we can write

[TABLE]

For all $x\in X\smallsetminus K$, for $\mu^{*n_{0}}$-almost every $g\in G$, the following upper bound holds:

[TABLE]

Using inequalities (3.4), (3.5) and the definition of $c$, we get the inequality

[TABLE]

Let $\kappa=\frac{n_{0}(\lambda^{\prime}_{1}-\lambda_{1})}{2}>0$. We then get the upper bound, for all $x\in X\smallsetminus K$,

[TABLE]

Choose $\delta>0$ such that $a_{n_{0},\,\delta}:=1-\delta\kappa+\delta^{2}M_{n_{0}}$ is strictly between [math] and $1$. Therefore, since $K$ is compact, there exists a constant $b_{n_{0},\,\delta}$ such that for all $x\in X$:

[TABLE]

and, by Lemma 3.4, the operator $P_{\mu}$ satisfies (UCH). ∎

Corollary 3.6**.**

*Same notations and assumptions as in Theorem 1.6.

If $\lambda_{1}<\lambda^{\prime}_{1}$, then the random walk on $X$ is uniformly recurrent in law.*

Proof.

This is a direct consequence of Proposition 3.5: since $P_{\mu}$ satisfies (UCH), we only need to apply Proposition 3.2. ∎

4 Non-Recurrence in Law When $\lambda_{1}\geq\lambda_{1}^{\prime}$

The goal of this Chapter is to show that the random walk on $X_{V,W}$ is nowhere recurrent in law when $\lambda_{1}\geq\lambda_{1}^{\prime}$ (Proposition 4.4).

4.1 The Limit Measures

We recall in this section the definition and the properties of the limit probability measures associated to a stationary measure.

The setting is very general. Let $G$ be a locally compact group acting on a second countable locally compact space $X$ and $\mu$ be a probability measure on $G$. Let $B$ be the product space $B=G^{\mathbb{N}^{*}}$ and $\beta$ be the product measure $\beta=\mu^{\otimes\mathbb{N}^{*}}$. The following lemma is due to Furstenberg. See [1, Lem 3.2] or [4, Lemma 2.17].

Lemma 4.1**.**

Let $\nu$ be a $\mu$-stationary probability measure on $X$. For $\beta$-almost every $b\in B$, the sequence $(b_{1}\cdots b_{n})_{*}\nu$ of probability measures on $X$ has a limit $\nu_{b}$, which we will call limit probability. Moreover, we have $\nu=\int_{B}\nu_{b}\operatorname{d}\!{\beta}(b).$

The following construction will be useful in Chapter 5. See [1, Cor 3.5] for a proof.

Corollary 4.2**.**

Let $\nu_{1}$ and $\nu_{2}$ be two $\mu$-stationary probability measures on $X$. Then the probability measure on $X\times X$

[TABLE]

is $\mu$-stationary. It is called the joining measure of $\nu_{1}$ and $\nu_{2}$.

This corollary will be used in combination with the following basic lemma.

Lemma 4.3**.**

Let $m_{1}$, $m_{2}$ be probability measures on a topological space $X$ and let $\Delta_{X}:=\{(x,x)\mid x\in X\}$ be the diagonal of $X$.

If $m_{1}\otimes m_{2}(\Delta_{X})=1$, then $m_{1}$ and $m_{2}$ are identical Dirac measures.

Proof.

By assumption, we have $m_{1}\otimes m_{2}(\Delta_{X})=\int_{X}m_{1}(\{x\})\operatorname{d}\!{m}_{2}(x)=1.$ Hence, for $m_{2}$-almost every $x\in X$, we have $m_{1}(\{x\})=1$, which implies that measures $m_{1}$ and $m_{2}$ are identical Dirac measures. ∎

4.2 No Stationary Measures on $X_{V,W}$

In this section, we again use the same notations and assumptions as in Theorem 1.6. We will prove that the space $X_{V,W}$ supports no $\mu$-stationary measures.

Recall that $W\subset V$ are real vector spaces, $G$ is a Zariski connected algebraic subgroup of $\mathrm{GL}(V)$ preserving $W$ and satisfying $(H1)$, $(H2)$, $(H3)$, Also recall that $\mu$ is a Zariski dense probability measure on $G$ with compact support, that $\lambda_{1}$ and $\lambda^{\prime}_{1}$ are the first Lyapunov exponents of $\mu$ in $W$ and in $W^{\prime}:=V/W$, and that we are studying the associated random walk on the $G$-space $X_{V,W}:=\mathbb{P}([)V]\smallsetminus\mathbb{P}([)W]$.

Proposition 4.4**.**

*Same notations and assumptions as in Theorem 1.6.

If $\lambda_{1}\geq\lambda^{\prime}_{1}$, then the random walk on $X_{V,W}$ is nowhere recurrent in law, and there exists no $\mu$-stationary probability measure on $X_{V,W}$.*

Proof.

By Lemma 3.3 the first assertion follows from the second one. This second assertion is a consequence of the following Lemmas 4.5 and 4.6. ∎

Let $B=G^{\mathbb{N}^{*}}$ and $\beta=\mu^{\otimes\mathbb{N}^{*}}$. For $b=(b_{1},b_{2},\ldots)$ in $B$ we write as in (3.3):

[TABLE]

Lemma 4.5**.**

Same notations and assumptions as in Theorem 1.6. If there exists a $\mu$-stationary probability measure on $X_{V,W}$, then for $\beta$-almost every $b\in B$, we have

[TABLE]

The proof of Lemma 4.5 will be given in Section 4.3. It relies on the properties of the limit probability measures $\nu_{b}$.

Lemma 4.6**.**

Same notations and assumptions as in Theorem 1.6. If $\lambda_{1}\geq\lambda^{\prime}_{1}$, then for $\beta$-almost every $b\in B$, one has

[TABLE]

The proof of Lemma 4.6 will be given in Section 4.4. It relies on the law of large numbers and on the law of the iterated logarithm for the random variables $\log\|a_{n}\|\!-\!\log\|d_{n}\|$.

4.3 Using the Limit Measures

The aim of this section is to prove Lemma 4.5.

We will need the following analog of [4, Prop. 3.7] for a non-irreducible action.

Lemma 4.7**.**

Same notations and assumptions as in Theorem 1.6. Let $\nu$ be a $\mu$-stationary probability measure on $\mathbb{P}([)V]$ such that $\nu(\mathbb{P}([)W])=0$. Then for every proper subspace $U$ of $V$, we have $\nu(\mathbb{P}([)U])=0.$

Proof.

Assume there exists a proper subspace $U$ of $V$ such that $\nu(\mathbb{P}([)U])>0$. Let $r_{0}$ be the minimal dimension of such a subspace $U$. If $U_{1}$ and $U_{2}$ are two distinct vector subspaces of dimension $r_{0}$, one has the equality

[TABLE]

Let $\alpha:=\sup\{\nu(\mathbb{P}([)U])\,|\,U\subset V,\,\dim U=r_{0}\}>0\;\;$ and consider the set

[TABLE]

This set is finite and non-empty. By $\mu$-stationarity of $\nu$, for $\mu$-almost every $g\in G$, we have $g^{-1}F=F$. Therefore, since $\mu$ is Zariski dense in $G$, this set $F$ is $G$-invariant. Since $G$ is Zariski connected, all the subspaces $U$ belonging to $F$ are $G$-invariant. But by $(H1)$, $(H2)$ and $(H3)$, the only proper $G$-invariant subspace of $V$ is $W$. This is contradictory since, by assumption, we have $\nu(\mathbb{P}([)W])=0$. ∎

Proof of Lemma 4.5.

We assume also that there exists a $\mu$-stationary probability measure $\nu$ on $X_{V,W}$. In order to prove (4.3), it is enough to check that for $\beta$-almost every $b\in B$, for all accumulation points $\pi$ in ${\rm End}(V)$ of the sequence $p_{n}:=\frac{b_{1}\cdots b_{n}}{||[||b_{1}\cdots b_{n}]}$, the image of $\pi$ is not included in $W$ :

[TABLE]

Lemma 4.7 shows that $\nu(\mathbb{P}([)\operatorname{Ker}\pi])=0$, hence the image probability measure $\pi_{*}\nu$ is well-defined and the sequence ${p_{n}}_{*}\nu$ weakly converges to $\pi_{*}\nu$. By Lemma 4.1 this sequence ${p_{n}}_{*}\nu$ also weakly converges to $\nu_{b}$, and therefore we have

[TABLE]

Therefore, for $\beta$-almost all $b$ in $B$, one has, for all accumulation point $\pi$,

[TABLE]

Since $\nu(\mathbb{P}([)W])=0$, one also has, for $\beta$-almost all $b$ in $B$,

[TABLE]

and hence the images $\operatorname{Im}\pi$ are not contained in $W$. This proves (4.5). ∎

4.4 Using the Cartan Projection

The aim of this section is to prove Lemma 4.6.

Let $\rho$ be the natural projection

[TABLE]

The image group $\overline{G}:=\rho(G)$ is a reductive subgroup of $\mathrm{GL}(W)\times\mathrm{GL}(W^{\prime})$. The image measure $\overline{\mu}:=\rho_{*}\mu$ is a Zariski dense probability measure on $\overline{G}$.

The proof of Lemma 4.6 will use the notations of Appendix A with the reductive group $\overline{G}$ and its probability measure $\overline{\mu}$. In particular, $\mathfrak{[}g]$ is the Lie algebra of $\overline{G}$, $\mathfrak{[}a]$ is the Lie algebra of a maximal split torus of $\overline{G}$, $\kappa$ is the Cartan projection, $\sigma_{\overline{\mu}}$ is the Lyapunov vector, $\Phi_{\overline{\mu}}$ is the covariance $2$-tensor, $\mathfrak{[}a]_{\overline{\mu}}$ is its linear span, and $K_{\overline{\mu}}$ is the unit ball of $\mathfrak{[}a]_{\overline{\mu}}$.

We will also use the following two lemmas. We set $r=\dim W$ and $r^{\prime}=\dim W^{\prime}$.

Lemma 4.8**.**

The highest weights $\chi$ and $\chi^{\prime}$ of the representations of $\overline{G}$ in $W$ and $W^{\prime}$ are distinct.

Proof.

Since $\overline{G}$ is Zariski connected, Condition $(H1)$ tells us that $W$ and $W^{\prime}$ are irreducible representations of $\mathfrak{[}g]$ and that their highest weight spaces are one dimensional. Condition $(H2)$ tells us that these representations of $\mathfrak{[}g]$ are not equivalent. Therefore as in [7, Chap 8.6.3], the highest weights $\chi$ and $\chi^{\prime}$ must be distinct. ∎

Lemma 4.9**.**

The center $Z$ of $\overline{G}$ is equal to $Z=\{(\alpha I_{r},\beta I_{r^{\prime}})\in\overline{G}\,|\,\alpha,\,\beta\in\mathbb{R}^{*}\}.$

Proof.

By Schur’s lemma, the commutant of $\overline{G}$ in ${\rm End}(W)$ is a division algebra. Since the representation of $\overline{G}$ in $W$ is proximal, this commutant is the field $\mathbb{R}$ of scalar matrices. Therefore $Z$ acts on $W$ (and also on $W^{\prime}$) by scalar matrices. ∎

Proof of Lemma 4.6.

Fix norms on $W$ and $W^{\prime}$ as in Lemma A.1, so that, for any element $g=(a,d)$ in $\overline{G}$ with $a\in\mathrm{GL}(W)$, $d\in\mathrm{GL}(W^{\prime})$, one has

[TABLE]

In particular, the first Lyapunov exponents in $W$ and $W^{\prime}$ are given by

[TABLE]

Let $\overline{B}=\overline{G}^{\mathbb{N}^{*}}$ and $\overline{\beta}=\overline{\mu}^{\otimes\mathbb{N}^{*}}$. For $b=(b_{1},b_{2},\ldots)\in\overline{B}$, we write $b_{1}\cdots b_{n}=(a_{n},\,d_{n}).$ We distinguish three cases:

First case : $\lambda_{1}>\lambda^{\prime}_{1}$.   In this case one has $(\chi-\chi^{\prime})(\sigma_{\overline{\mu}})>0$. According to (4.7) and the Law of Large Numbers A.4, for $\overline{\beta}$-almost every $b\in\overline{B}$, we have

[TABLE]

Second case : $\lambda_{1}=\lambda^{\prime}_{1}$ and $(\chi-\chi^{\prime})(\mathfrak{[}a]_{\overline{\mu}})\neq 0$.  In this case, one has $(\chi-\chi^{\prime})(\sigma_{\overline{\mu}})=0$ and there exists $x$ in the unit ball $K_{\overline{\mu}}$ of $\mathfrak{[}a]_{\overline{\mu}}$ such that $(\chi-\chi^{\prime})(x)>0.$ According to the Law of the Iterated Logarithm A.4, for $\overline{\beta}$-almost every $b\in\overline{B}$, there exists an increasing sequence of integers $n_{i}$ such that

[TABLE]

and therefore such that

[TABLE]

Third case : $\lambda_{1}=\lambda^{\prime}_{1}$ and $(\chi-\chi^{\prime})(\mathfrak{[}a]_{\overline{\mu}})=0$.  Let

[TABLE]

Since the group $\overline{G}$ is reductive, by Lemma 4.9, the subgroup $S$ is semisimple. Let $\mathfrak{[}s]$ be the Lie algebra of $S$. By [4, Thm 13.19], we have $\mathfrak{[}a]\cap\mathfrak{[}s]\subset\mathfrak{[}a]_{\overline{\mu}}$, and thus also

[TABLE]

We introduce the group morphism $\delta$ defined by:

[TABLE]

For every $g=(a,d)$ in $\overline{G}$, we can write $g=sz$ with $s\in S$ and $z\in Z$. Using Equations (4.7), (4.8) and the equality $\kappa(g)=\kappa(s)+\kappa(z)$, we compute,

[TABLE]

We want to describe the behavior of the random variable $T_{n}=\log\left(||[||a_{n}]/||[||d_{n}]\right)$ on $\overline{B}$ where as above $(a_{n},\,d_{n})=b_{1}\cdots b_{n}$. Using Equation (4.9), we see that

[TABLE]

is the sum of $n$ real-valued independent and identically distributed random variables $\delta(b_{i})$. Note that the law of the variable $\delta(b_{1})$ has compact support. Since $\lambda_{1}=\lambda^{\prime}_{1}$, we have $\mathbb{E}(\delta(b_{1}))=\frac{1}{n}\mathbb{E}(T_{n})\xrightarrow[n\rightarrow\infty]{}0$. Thus the variable $\delta(b_{1})$ is centered.

If this random variable $\delta(b_{1})$ were almost surely [math], it would mean that for $\overline{\mu}$-almost every $g=(a,\,d)\in\overline{G}$, we have $\delta(g)=0$. Since $\overline{\mu}$ is Zariski dense in $\overline{G}$, this would imply $\delta(G)=0$, or, equivalently,

[TABLE]

where $\mathfrak{[}z]\subset\mathfrak{[}a]$ is the Lie algebra of $Z$. Equalities (4.8) and (4.10) would tell us that the highest weights $\chi$ and $\chi^{\prime}$ were equal. This would contradict Lemma 4.8.

Therefore this centered variable $\delta(b_{1})$ is not almost surely [math]. Thus the classical recurrence properties of real random walks (cf e.g. [8, Thm 3.38]) tell us that $\sup_{n\geq 1}T_{n}=\infty$ almost surely.

In each of these three cases, we have checked (4.4). ∎

5 Uniqueness of the Stationary Measure

The main aim of this chapter is to prove the uniqueness of the stationary measure on $X_{V,W}$ (Proposition 5.3).

5.1 No Stationary Measures on $Y_{V,W}$

The proof of uniqueness will rely on the following Lemma 5.1.

We keep the notations and assumptions of Theorem 1.6. Let $p$ be the projection

[TABLE]

and let $Y_{V,W}$ be the $G$-invariant subvariety of $X_{V,W}^{2}$

[TABLE]

Lemma 5.1**.**

*Same notations and assumptions as in Theorem 1.6.

There is no $\mu$-stationary probability measure $\widetilde{\nu}$ on $Y_{V,W}$.*

Proof.

Suppose that such a measure $\widetilde{\nu}$ does exist. Consider again the natural projection $\rho:G\longrightarrow\mathrm{GL}(W)\times\mathrm{GL}(W^{\prime})$ introduced in (4.6). Let $\overline{G}:=\rho(G)$ be the image of $G$ by $\rho$, a reductive subgroup of $\mathrm{GL}(W)\times\mathrm{GL}(W^{\prime})$, and let $\overline{\mu}:=\rho_{*}\mu$ be the image of $\mu$ by $\rho$, a Zariski dense probability measure on $\overline{G}$. Now consider the map

[TABLE]

where $\overline{Y}:=\mathbb{P}([)W\oplus W^{\prime}]\smallsetminus(\mathbb{P}([)W]\cup\mathbb{P}([)W^{\prime}])$. Let $\overline{\nu}=f_{*}\widetilde{\nu}$ be the probability measure on $\overline{Y}$ that is the image of $\widetilde{\nu}$ by $\rho$. Since the map $f$ is equivariant, the probability measure $\overline{\nu}$ is $\overline{\mu}$-stationary. According to Proposition A.6 such a measure $\overline{\nu}$ is supported by a compact $\overline{G}$-orbit in $\overline{Y}$. This contradicts the following Lemma 5.2. ∎

Lemma 5.2**.**

There are no compact $\overline{G}$-orbits in $\overline{Y}$.

Proof.

Such a compact orbit would be of the form $\overline{G}/\overline{H}$, where $\overline{H}$ is an algebraic subgroup of $\overline{G}$ containing a conjugate of the group $\overline{A}\,\overline{N}$ with $\overline{A}$ a maximal split subtorus of $\overline{G}$ and $\overline{N}$ a maximal unipotent subgroup normalized by $\overline{A}$. Since $W$ and $W^{\prime}$ are proximal irreducible representations of $\overline{G}$, there is only one $\overline{N}$-invariant line $\mathbb{R}v$ in $W$ and one $\mathbb{R}v^{\prime}$ in $W^{\prime}$. Hence the $\overline{N}$-invariant lines in $W\oplus W^{\prime}$ are included in the plane $\mathbb{R}v\oplus\mathbb{R}v^{\prime}$. Since, by Lemma 4.8, the highest weights $\chi$ and $\chi^{\prime}$ of $W$ and $W^{\prime}$ are distinct, the lines $\mathbb{R}v$ and $\mathbb{R}v^{\prime}$ are the only $\overline{A}$-invariant lines in $\mathbb{R}v\oplus\mathbb{R}v^{\prime}$. Therefore, a compact $\overline{G}$-orbit in $\mathbb{P}([)W\oplus W^{\prime}]$ is contained in $\mathbb{P}([)W]\cup\mathbb{P}([)W^{\prime}]$. ∎

5.2 Proof of Uniqueness

We can now show the uniqueness of the $\mu$-stationary probability measure $\nu$ on $X_{V,W}$. The same proof will tell us that its limit probability measures $\nu_{b}$ are Dirac measures.

Proposition 5.3**.**

*Same notations and assumptions as in Theorem 1.6.

If $\lambda_{1}<\lambda^{\prime}_{1}$, the $\mu$-stationary probability measure $\nu$ on $X_{V,W}$ is unique.

Moreover, the limit measures $\nu_{b}$ are $\beta$-almost surely Dirac measures.*

Proof of Proposition 5.3.

Let $\nu_{1}$ and $\nu_{2}$ be two $\mu$-stationary probability measures on $X_{V,W}$. By Corollary 4.2 the joining measure $\nu_{1}\boxtimes\nu_{2}$ on $X_{V,W}^{2}$ is $\mu$-stationary.

Let us show that its support is contained in the subvariety

[TABLE]

where $p:X_{V,W}\rightarrow\mathbb{P}([)W^{\prime}]$ is again the canonical projection. Since the action of $G$ on $W^{\prime}$ is irreducible and proximal, there exists a unique $\mu$-stationary measure $\nu^{\prime}_{0}$ on $\mathbb{P}([)W^{\prime}]$ called the Furstenberg measure. Its limit probability measures $\nu^{\prime}_{0,\,b}$ are $\beta$-almost surely Dirac measures $\delta_{\xi_{b}}$ for some $\xi_{b}\in\mathbb{P}([)W^{\prime}]$. See [4, Prop. 3.7] for more detail on the Furstenberg measure. Since $\nu^{\prime}_{0}$ is unique, we have the equalities

[TABLE]

Therefore, for $\beta$-almost every $b\in B$, we have

[TABLE]

and hence

[TABLE]

By the very definition (4.1) of the joining measure, integrating this equality gives

[TABLE]

By definition, this set $Z_{V,W}$ is the union $Z_{V,W}=Y_{V,W}\cup\Delta_{X_{V,W}}$. By Lemma 5.1, the $G$-variety $Y_{V,W}$ does not support $\mu$-stationary measures. Therefore the joining measure $\nu_{1}\boxtimes\nu_{2}$ is supported on the diagonal $\Delta_{X_{V,W}}$. Hence, for $\beta$ almost every $b$ in $B$, the measure $\nu_{1,b}\otimes\nu_{2,b}$ is also supported on the diagonal: $\nu_{1,b}\otimes\nu_{2,b}(\Delta_{X_{V,W}})=1.$ Therefore, by Lemma 4.3, the limit probability measures $\nu_{1,b}$ and $\nu_{2,b}$ are both equal to the same Dirac measures. Hence, by Lemma 4.1, one has $\nu_{1}=\nu_{2}$. ∎

5.3 Limit of Means of Transition Probabilities

In this section we prove that the sequence of means of the transition probabilities $\mu^{*n}*\delta_{x}$ on $X_{V,W}$ always has a limit.

Corollary 5.4**.**

*Same notations and assumptions as in Theorem. 1.6. Let $x\in X_{V,W}$.

a) When $\lambda_{1}\geq\lambda^{\prime}_{1}$, one has the weak convergence $\frac{1}{n}\sum_{j=1}^{n}\mu^{*j}*\delta_{x}\xrightarrow[n\rightarrow\infty]{}0.$

b) When $\lambda_{1}<\lambda^{\prime}_{1}$, one has the weak convergence $\frac{1}{n}\sum_{j=1}^{n}\mu^{*j}*\delta_{x}\xrightarrow[n\rightarrow\infty]{}\nu.$*

Proof of Corollary 5.4.

Every accumulation point of the sequence of probability measures $\frac{1}{n}\sum_{j=1}^{n}\mu^{*j}*\delta_{x}$ is a $\mu$-stationary finite measure.

When $\lambda_{1}\geq\lambda^{\prime}_{1}$, by Proposition 4.4, such a measure is necessarily [math].

When $\lambda_{1}<\lambda^{\prime}_{1}$, by Corollary 3.6, the corresponding Markov chain is recurrent in law; hence, no mass is lost and the accumulation points are thus $\mu$-stationary probability measures. By Proposition 5.3, there is only one such measure. ∎

Appendix A Limit Laws on Reductive Groups

In this appendix, we recall some facts about random walks on reductive groups, which are mainly detailed in [4].

A.1 Cartan Decomposition

Let $G$ be a Zariski connected real algebraic reductive group. Let $A$ be a maximal split subtorus of $G$, $\mathfrak{[}a]$ be the Lie algebra of $A$, $\mathfrak{[}a]_{+}\subset\mathfrak{[}a]$ be a Weyl chamber and $A_{+}=\exp\mathfrak{[}a]_{+}$. There exists a maximal compact subgroup $K$ of $G$ such that $G$ has a Cartan decomposition $G=KA_{+}K.$ The Cartan projection of $G$ is the unique map $\kappa:G\rightarrow\mathfrak{[}a]_{+}$ such that, for all $g\in G$,

[TABLE]

The Cartan projection is useful because of the following Lemma.

Lemma A.1**.**

([4, Lem. 6.33]) Let $G$ be a Zariski connected real algebraic reductive group, $\rho$ be an irreducible algebraic representation of $G$ in a real vector space $W$ and $\chi\in\mathfrak{[}a]^{*}$ be the highest weight of $W$. There exists a norm on $W$ such that, for all $g\in G$,

[TABLE]

A.2 Limit Laws

Let $\mu$ be a Zariski dense probability measure with compact support on $G$. Let $B=G^{\mathbb{N}^{*}}$ and $\beta=\mu^{\otimes\mathbb{N}^{*}}$. We now recall two limit laws for the Cartan projection.

Define the Lyapunov vector (see [4, Thm 10.9]) $\sigma_{\mu}\in\mathfrak{[}a]_{+}$ as the limit

[TABLE]

Theorem A.2**.**

(Law of Large Numbers, [4, Thm 10.9]) Let $G$ be a Zariski connected real reductive group and $\mu$ a Zariski dense probability measure with compact support on $G$. For $\beta$-almost every $b\in B$, we have the convergence

[TABLE]

Define the covariance 2-tensor (see [4, Prop.14.18]) $\Phi_{\mu}\in S^{2}\mathfrak{[}a]$ as the limit

[TABLE]

$\Phi_{\mu}$ is a symmetric $2$-tensor on $\mathfrak{[}a]$. We denote by $\mathfrak{[}a]_{\mu}\subset\mathfrak{[}a]$ the linear span of $\Phi_{\mu}$ which is the smallest subspace $\mathfrak{[}a]_{\mu}$ of $\mathfrak{[}a]$ such that we have $\Phi_{\mu}\in S^{2}\mathfrak{[}a]_{\mu}$. We can see $\Phi_{\mu}$ as an inner product over $\mathfrak{[}a]_{\mu}$. We then denote by $K_{\mu}$ the closed unit ball of $\mathfrak{[}a]_{\mu}$ for the metric corresponding to $\Phi_{\mu}$. When $G$ is semisimple, the covariance $2$-tensor $\Phi_{\mu}$ is non degenerate i.e. one has $\mathfrak{[}a]_{\mu}=\mathfrak{[}a]$.

Theorem A.3**.**

(Law of the Iterated Logarithm, [4, Thm 13.17]) Let $G$ be a Zariski connected real reductive group and $\mu$ a Zariski dense probability measure with compact support on $G$. Then, for $\beta$-almost every $b\in B$, the set of accumulation points of the sequence

[TABLE]

A.3 Opposition Involution

In Chapter 4 we need a variation of Theorems A.2 and A.3 where the order in the product of $b_{i}$’s is inverted.

Corollary A.4**.**

*Let $G$ be a Zariski connected real reductive group and $\mu$ a Zariski dense probability measure with compact support on $G$.

a) For $\beta$-almost every $b\in B$, we have the convergence*

[TABLE]

b) For $\beta$-almost every $b\in B$, the set of accumulation points of the sequence

[TABLE]

The proof will use the probability measure $\check{\mu}$ on $G$ which is the image of $\mu$ by the map $g\mapsto g^{-1}$. Recall that there exists a linear map $\iota:\mathfrak{[}a]\rightarrow\mathfrak{[}a]$ called the opposition involution (see [4, §8.2]) such that, for all $g\in G$, we have

[TABLE]

Lemma A.5**.**

The Lyapunov vector $\sigma_{\check{\mu}}$ of $\check{\mu}$, its covariance $2$-tensor $\Phi_{\check{\mu}}$ and the closed unit ball $K_{\check{\mu}}$ of the linear span $\mathfrak{[}a]_{\check{\mu}}$ of $\Phi_{\check{\mu}}$ are equal to:

[TABLE]

Proof.

These identities follow from (A.2), (A.3) and (A.4). ∎

Proof of Corollary A.4.

We apply the Law of Large Numbers A.2 to the probability measure $\check{\mu}$. For $\beta$-almost every $b\in B$, the sequence $\frac{1}{n}\kappa(b_{n}^{-1}\cdots b_{1}^{-1})$ converges to $\sigma_{\check{\mu}}$. Applying the opposition involution $\iota$, we find that, for $\beta$-almost every $b\in B$, the sequence $\frac{1}{n}\kappa(b_{1}\cdots b_{n})$ converges to $\iota(\sigma_{\check{\mu}})$ which is equal to $\sigma_{\mu}$ by Lemma A.5.

In the same manner, Theorem A.3 tells us that the set of accumulation points of the sequence $\frac{\kappa(b_{1}\cdots b_{n})-n\sigma_{\mu}}{\sqrt{2n\log\log n}}$ is exactly $\iota(K_{\check{\mu}})$ which is equal to $K_{\mu}$ by Lemma A.5. ∎

A.4 Stationary Measures on Projective Spaces

In Chapter 5, we use the classification of stationary measures in [3, Thm 1.7]:

Proposition A.6**.**

Let $V$ be a real vector space, $G\subset\mathrm{GL}(V)$ be a reductive algebraic subgroup of $\mathrm{GL}(V)$ and $\mu$ be a Zariski dense probability measure on $G$.

Then every $\mu$-stationary probability measure $\nu$ on $\mathbb{P}([)V]$ is supported by a compact $G$-orbit.

Conversely every compact $G$-orbit in $\mathbb{P}([)V]$ supports a unique $\mu$-stationary probability measure $\nu$.