论文标题
在$ \ mathbb r _+^d $上的仿射递归上
On the affine recursion on $\mathbb R_+^d$
论文作者
论文摘要
我们修复了$ d \ geq 2 $，并表示$ \ Mathcal s $ $ d \ times d $矩阵的半组，具有非负条目。我们考虑一个序列$（a_n，b_n）_ {n \ geq 1} $ i。我。 d。随机变量具有$ \ Mathcal s \ times \ Mathbb r _+^d $中的随机变量，并研究Markov链$（x_n）_ {n \ geq 0} $ on $ \ mathbb r _+^d $定义的渐近行为_ {n \ geq 0} $ \ [ \ forall n \ geq 0，\ qquad x_ {n+1} = a_ {n+1} x_n+b_ {n+1}， \]其中$ x_0 $是固定的随机变量。我们假设矩阵$ a_n $的lyapunov指数等于$ 0 $，并在相当普遍的假设下证明，存在$（\ mathbb r^+）^d $ on $（\ mathbb r^+）^d $上的唯一（无限）rad rad，这对于链条$（x_n）_ {n \ geq 0} $是不变的。 $λ$的存在依赖于T.D.C.的最新工作关于随机矩阵产物规范的波动。它的独立性是一般财产的结果，称为“局部订单”，大约在20年前，由M. Babillot，博士Bougerol等人埃利（Bougerol et L. Elie）博士强调。
We fix $d \geq 2$ and denote $\mathcal S$ the semi-group of $d \times d$ matrices with non negative entries. We consider a sequence $(A_n, B_n)_{n \geq 1} $ of i. i. d. random variables with values in $\mathcal S\times \mathbb R_+^d$ and study the asymptotic behavior of the Markov chain $(X_n)_{n \geq 0}$ on $ \mathbb R_+^d$ defined by: \[ \forall n \geq 0, \qquad X_{n+1}=A_{n+1}X_n+B_{n+1}, \] where $X_0$ is a fixed random variable. We assume that the Lyapunov exponent of the matrices $A_n$ equals $0$ and prove, under quite general hypotheses, that there exists a unique (infinite) Radon measure $λ$ on $(\mathbb R^+)^d$ which is invariant for the chain $(X_n)_{n \geq 0}$. The existence of $λ$ relies on a recent work by T.D.C. Pham about fluctuations of the norm of product of random matrices . Its unicity is a consequence of a general property, called "local contractivity", highlighted about 20 years ago by M. Babillot, Ph. Bougerol et L. Elie in the case of the one dimensional affine recursion .
