arXiv subjects

Roland Prohaska

Publications and source records attributed to Roland Prohaska.

5 recordsLinked to original sources

arXiv2023

Expanding measures: Random walks and rigidity on homogeneous spaces

Let $G$ be a real Lie group, $Λ<G$ a lattice and $H<G$ a connected semisimple subgroup without compact factors and with finite center. We define the notion of $H$-expanding measures $μ$ on $H$ and, applying recent work of Eskin-Lindenstrauss, prove that $μ$-stationary probability measures on $G/Λ$ are homogeneous. Transferring a construction by Benoist-Quint and drawing on ideas of Eskin-Mirzakhani-Mohammadi, we construct Lyapunov/Margulis functions to show that $H$-expanding random walks on $G/Λ$ satisfy a recurrence condition and that homogeneous subspaces are repelling. Combined with a countability result, this allows us to prove equidistribution of trajectories in $G/Λ$ for $H$-expanding random walks and to obtain orbit closure descriptions. Finally, elaborating on an idea of Simmons-Weiss, we deduce Birkhoff genericity of a class of measures with respect to some diagonal flows and extend their applications to Diophantine approximation on similarity fractals to a non-conformal and weighted setting.

math.DS↗

arXiv2021

Spread Out Random Walks on Homogeneous Spaces

A measure on a locally compact group is called spread out if one of its convolution powers is not singular with respect to Haar measure. Using Markov chain theory, we conduct a detailed analysis of random walks on homogeneous spaces with spread out increment distribution. For finite volume spaces, we arrive at a complete picture of the asymptotics of the $n$-step distributions: They equidistribute towards Haar measure, often exponentially fast and locally uniformly in the starting position. In addition, many classical limit theorems are shown to hold. In the infinite volume case, we prove recurrence and a ratio limit theorem for symmetric spread out random walks on homogeneous spaces of at most quadratic growth. This settles one direction in a long-standing conjecture.

math.DS↗

arXiv2021

On Convergence of Random Walks on Moduli Space

The purpose of this note is to establish convergence of random walks on the moduli space of Abelian differentials on compact Riemann surfaces in two different modes: convergence of the $n$-step distributions from almost every starting point in an affine invariant submanifold towards the associated affine invariant measure, and almost sure pathwise equidistribution towards the affine invariant measure on the $SL_2(\mathbb{R})$-orbit closure of an arbitrary starting point. These are analogues to previous results for random walks on homogeneous spaces.

math.DS↗

arXiv2020

Markov Random Walks on Homogeneous Spaces and Diophantine Approximation on Fractals

In a first part, using the recent measure classification results of Eskin--Lindenstrauss, we give a criterion to ensure a.s. equidistribution of empirical measures of an i.i.d. random walk on a homogeneous space $G/Γ$. Employing renewal and joint equidistribution arguments, this result is generalized in the second part to random walks with Markovian dependence. Finally, following a strategy of Simmons--Weiss, we apply these results to Diophantine approximation problems on fractals and show that almost every point with respect to Hausdorff measure on a graph directed self-similar set is of generic type, so in particular, well approximable.

math.DS↗

arXiv2019

Aspects of Convergence of Random Walks on Finite Volume Homogeneous Spaces

We investigate three aspects of weak* convergence of the $n$-step distributions of random walks on finite volume homogeneous spaces $G/\Gamma$ of semisimple real Lie groups. First, we look into the obvious obstruction to the upgrade from Cesaro to non-averaged convergence: periodicity. We give examples where it occurs and conditions under which it does not. In a second part, we prove convergence towards Haar measure with exponential speed from almost every starting point. Finally, we establish a strong uniformity property for the Cesaro convergence towards Haar measure for uniquely ergodic random walks.

math.DS↗