On the homogeneity at infinity of the stationary probability for an affine random walk [unknown]

Y. Guivarc'h, E. Le Page
2015 Contemporary Mathematics   unpublished
We consider an affine random walk on R. We assume the existence of a stationary probability ν on R and we describe the shape at infinity of ν, if ν has unbounded support. We discuss the connections of the result with geometrical or probabilistic problems. I -Introduction Let G be the affine group of the line. For g ∈ G, x ∈ R, we write gx = a(g)x + b(g) with a(g) ∈ R * , b(g) ∈ R. Let µ be a probability on G. We denote by P the Markov operator on R defined by P ϕ(x) = ϕ(gx)µ(dg) where ϕ is a
more » ... dg) where ϕ is a bounded Borel function. Our hypothesis H µ is stated below and we observe that H µ (1) and H µ (2) imply that P has a unique stationary probability ν (see [16] ) ; if H µ (4) is also valid, then suppν is unbounded. Here we are interested in the "shape at infinity" of ν ; we will show that for some α > 0, the quantities |t| α ν[t, ∞) and |t| α ν(−∞, t] have limits at infinity, we discuss their positivity and we illustrate the possible uses of this result by two corollaries in two different contexts. This "homogeneity at infinity" of ν plays an essential role in extreme value theory (see [19] ), for random variables associated with the Markov chain X x n with kernel P on R. Also, for random walk in a random medium on Z (see [21] ) the slow diffusion property is closely related to this homogeneity (see [6] , [17] ). Furthermore the construction of ν given here provides a natural construction of a large class of heavy tailed measures which generates "anomalous" random walks on the additive group R. This class of measures appears now to be of great interest from the physical point of view (see [2] ). In the geometrical context of excursions of geodesic flows on manifolds of negative curvature the "logarithm law" is well known (see [22] , [18]), and we will discuss analogous properties for the Markov chain X x n . We assume that µ satisfies the following set of conditions H µ . H µ (1) : (|ℓn|a(g)|| + |(ℓn|b(g)||))µ(dg) < ∞. H µ (2) : For some α > 0 |a(g)| α µ(dg) = 1. H µ (3) : |a(g)| α ℓn|a(g)|µ(dg) < ∞, |b(g)| α µ(dg) < ∞. H µ (4) : The elements of suppµ have no common fixed point in R. H µ (5) : The set {ℓn|a(g)| ; g ∈ suppµ} generates a dense subgroup of R. Then we have the Theorem 1 Assume that µ satisfies H µ . Then 1) There exists c + ≥ 0, c − ≥ 0 such that lim t→∞ |t| α ν(t, ∞) = c + , lim t→−∞ |t| α ν(−∞, t) = c − . Moreover c = c + + c − > 0. 2) If µ{g ∈ G ; a(g) < 0} > 0, then c + = c − > 0.
doi:10.1090/conm/631/12600 fatcat:wyoe3nfnjzeqxgeb54j2ejnvc4