On the affine random walk on the torus

Let $\mu$ be a borelian probability measure on $\mathbf{G}:=\mathrm{SL}_d(\mathbb{Z}) \ltimes \mathbb{T}^d$. Define, for $x\in \mathbb{T}^d$, a random walk starting at $x$ denoting for $n\in \mathbb{N}$, $$\left\{\begin{array}{rcl} X_0 &=&x\\ X_{n+1} &=& a_{n+1} X_n + b_{n+1} \end{array}\right.$$ where $((a_n,b_n))\in \mathbf{G}^\mathbb{N}$ is an iid sequence of law $\mu$. Then, we denote by $\mathbb{P}_x$ the measure on $(\mathbb{T}^d)^\mathbb{N}$ that is the image of $\mu^{\otimes \mathbb{N}}$ by the map $\left((g_n) \mapsto (x,g_1 x, g_2 g_1 x, \dots , g_n \dots g_1 x, \dots)\right)$ and for any $\varphi \in \mathrm{L}^1((\mathbb{T}^d)^\mathbb{N}, \mathbb{P}_x)$, we set $\mathbb{E}_x \varphi((X_n)) = \int \varphi((X_n)) \mathrm{d}\mathbb{P}_x((X_n))$. Bourgain, Furmann, Lindenstrauss and Mozes studied this random walk when $\mu$ is concentrated on $\mathrm{SL}_d(\mathbb{Z}) \ltimes\{0\}$ and this allowed us to study, for any h\"older-continuous function $f$ on the torus, the sequence $(f(X_n))$ when $x$ is not too well approximable by rational points. In this article, we are interested in the case where $\mu$ is not concentrated on $\mathrm{SL}_d(\mathbb{Z}) \ltimes \mathbb{Q}^d/\mathbb{Z}^d$ and we prove that, under assumptions on the group spanned by the support of $\mu$, the Lebesgue's measure $\nu$ on the torus is the only stationary probability measure and that for any h\"older-continuous function $f$ on the torus, $\mathbb{E}_x f(X_n)$ converges exponentially fast to $\int f\mathrm{d}\nu$. Then, we use this to prove the law of large numbers, a non-concentration inequality, the functional central limit theorem and it's almost-sure version for the sequence $(f(X_n))$. In the appendix, we state a non-concentration inequality for products of random matrices without any irreducibility assumption

Environmental Protection, Producer Insolvency and Lender Liability

The present paper deals with some legal issues surrounding environmental protection, namely those issues concerning the liability of the different firms and individuals directly or indirectly involved in the generation of environment damaging accidents. We consider in particular the potential effects of extending a firm's liability in case of an environmental disaster to its lenders and financiers when the cost of this liability is too large in relation to the firm's assets. Such extended liability regimes exist or are considered in many countries. The most important case is the 1980/85 Comprehensive Environmental Response, Compensation and Liability Act (CERCLA) in the USA that led to an extensive jurisprudence over the last fifteen years. Nous traitons ici du cadre légal de la responsabilité directe des entreprises lors de désastres environnementaux et de l'extension de cette responsabilité aux prêteurs en cas de faillite de l'entreprise. De tels régimes existent ou sont à l'étude dans plusieurs pays. Le cas le plus connu est le Comprehensive Environmental Response, Compensation and Liability Act (CERCLA) de 1980/85 aux États-Unis et nous analysons les principaux cas de jurisprudence auxquels cette loi a donné lieu depuis 15 ans.Environment, Extended Lender Liability, CERCLA, Environnement, responsabilité, CERCLA

Exact Sparse Matrix-Vector Multiplication on GPU's and Multicore Architectures

We propose different implementations of the sparse matrix--dense vector multiplication (\spmv{}) for finite fields and rings \Zb/m\Zb. We take advantage of graphic card processors (GPU) and multi-core architectures. Our aim is to improve the speed of \spmv{} in the \linbox library, and henceforth the speed of its black box algorithms. Besides, we use this and a new parallelization of the sigma-basis algorithm in a parallel block Wiedemann rank implementation over finite fields