arXiv subjects

Yves Guivarc'H

Publications and source records attributed to Yves Guivarc'H.

4 recordsLinked to original sources

arXiv2017

On spectral gap properties and extreme value theory for multivariate affine stochastic recursions

We consider a general multivariate affine stochastic recursion and the associated Markov chain on $\mathbb R^{d}$. We assume a natural geometric condition which implies existence of an unbounded stationary solution and we show that the large values of the associated stationary process follow extreme value properties of classical type, with a non trivial extremal index. We develop some explicit consequences such as convergence to Fr{é}chet's law or to an exponential law, as well as convergence to a stable law. The proof is based on a spectral gap property for the action of associated positive operators on spaces of regular functions with slow growth, and on the clustering properties of large values in the recursion.

math.PR↗

arXiv2015

Spectral gap properties for linear random walks and Pareto's asymptotics for affine stochastic recursions

Let $V=\mathbb R^d$ be the Euclidean $d$-dimensional space, $μ$ (resp $λ$) a probability measure on the linear (resp affine) group $G=G L (V)$ (resp $H= \Aff (V))$ and assume that $μ$ is the projection of $λ$ on $G$. We study asymptotic properties of the iterated convolutions $μ^n *δ\_{v}$ (resp $λ^n*δ\_{v})$ if $v\in V$, i.e asymptotics of the random walk on $V$ defined by $μ$ (resp $λ$), if the subsemigroup $T\subset G$ (resp.\ $Σ\subset H$) generated by the support of $μ$ (resp $λ$) is "large". We show spectral gap properties for the convolution operator defined by $μ$ on spaces of homogeneous functions of degree $s\geq 0$ on $V$, which satisfy H{ö}lder type conditions. As a consequence of our analysis we get precise asymptotics for the potential kernel $Σ\_{0}^{\infty} μ^k * δ\_{v}$, which imply its asymptotic homogeneity. Under natural conditions the $H$-space $V$ is a $λ$-boundary; then we use the above results and radial Fourier Analysis on $V\setminus \{0\}$ to show that the unique $λ$-stationary measure $ρ$ on $V$ is "homogeneous at infinity" with respect to dilations $v\rightarrow t v$ (for $t\textgreater{}0$), with a tail measure depending essentially of $μ$ and $Σ$. Our proofs are based on the simplicity of the dominant Lyapunov exponent for certain products of Markov-dependent random matrices, on the use of renewal theorems for "tame" Markov walks, and on the dynamical properties of a conditional $λ$-boundary dual to $V$.

math.PR↗

arXiv2005

Group-theoretic compactification of Bruhat-Tits buildings

Let GF denote the rational points of a semisimple group G over a non-archimedean local field F, with Bruhat-Tits building X. This paper contains five main results. We prove a convergence theorem for sequences of parahoric subgroups of GF in the Chabauty topology, which enables to compactify the vertices of X. We obtain a structure theorem showing that the Bruhat-Tits buildings of the Levi factors all lie in the boundary of the compactification. Then we obtain an identification theorem with the polyhedral compactification (previously defined in analogy with the case of symmetric spaces). We finally prove two parametrization theorems extending the BruhatTits dictionary between maximal compact subgroups and vertices of X: one is about Zariski connected amenable subgroups, and the other is about subgroups with distal adjoint action.

math.GR↗

arXiv2004

Semigroup actions on tori and stationary measures on projective spaces

Let $Γ$ be a sub-semigroup of $G=GL(d,\mathbb R),$ $d>1.$ We assume that the action of $Γ$ on $\R^d$ is strongly irreducible and that $Γ$ contains a proximal and expanding element. We describe contraction properties of the dynamics of $Γ$ on $\R^d$ at infinity. This amounts to the consideration of the action of $Γ$ on some compact homogeneous spaces of $G,$ which are extensions of the projective space $\pr^{d-1}.$ In the case where $Γ$ is a sub-semigroup of $GL(d,\R)\cap M(d,\Z)$ and $Γ$ has the above properties, we deduce that the $Γ$-orbits on $\T^d=\R^d\slash\Z^d$ are finite or dense.

math.DS↗