scholarly journals Spread out random walks on homogeneous spaces

2020 ◽  
pp. 1-35
Author(s):  
ROLAND PROHASKA
Keyword(s):  
Random Walks ◽  
Haar Measure ◽  
Limit Theorems ◽  
Homogeneous Spaces ◽  
Locally Compact Group ◽  
Quadratic Growth ◽  
Infinite Volume ◽  
Starting Position ◽  
Markov Chain Theory ◽  
Ratio Limit

Abstract A measure on a locally compact group is said to be 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.

Sharp ergodic theorems for group actions and strong ergodicity

1999 ◽  
Vol 19 (4) ◽  
pp. 1037-1061 ◽  
Author(s):  
ALEX FURMAN ◽  
YEHUDA SHALOM
Keyword(s):  
Probability Measure ◽  
Random Walks ◽  
Homogeneous Spaces ◽  
Locally Compact Group ◽  
Invariant Mean ◽  
Spectral Condition ◽  
Ergodic Theorems ◽  
Averaging Operator ◽  
The Mean ◽  
Probability Measure Space

Let $\mu$ be a probability measure on a locally compact group $G$, and suppose $G$ acts measurably on a probability measure space $(X,m)$, preserving the measure $m$. We study ergodic theoretic properties of the action along $\mu$-i.i.d. random walks on $G$. It is shown that under a (necessary) spectral assumption on the $\mu$-averaging operator on $L^2(X,m)$, almost surely the mean and the pointwise (Kakutani's) random ergodic theorems have roughly $n^{-1/2}$ rate of convergence. We also prove a central limit theorem for the pointwise convergence. Under a similar spectral condition on the diagonal $G$-action on $(X\times X,m\times m)$, an almost surely exponential rate of mixing along random walks is obtained.The imposed spectral condition is shown to be connected to a strengthening of the ergodicity property, namely, the uniqueness of $m$-integration as a $G$-invariant mean on $L^\infty(X,m)$. These related conditions, as well as the presented sharp ergodic theorems, never occur for amenable $G$. Nevertheless, we provide many natural examples, among them automorphism actions on tori and actions on Lie groups' homogeneous spaces, for which our results can be applied.

Ratio Limit Theorems for Random Walks and L�vy Processes

2005 ◽  
Vol 23 (4) ◽  
pp. 357-380
Author(s):  
Minzhi Zhao ◽  
Jiangang Ying
Keyword(s):  
Random Walks ◽  
Limit Theorems ◽  
Ratio Limit Theorems ◽  
Ratio Limit

scholarly journals Recurrence and ergodicity of random walks on linear groups and on homogeneous spaces

2011 ◽  
Vol 32 (4) ◽  
pp. 1313-1349 ◽  
Author(s):  
Y. GUIVARC’H ◽  
C. R. E. RAJA
Keyword(s):  
Random Walks ◽  
Closed Subgroup ◽  
Homogeneous Spaces ◽  
Local Fields ◽  
Quadratic Growth ◽  
Linear Groups ◽  
Compact Groups ◽  
Large Classes ◽  
Skew Products ◽  
Recurrent Random Walk

AbstractWe discuss recurrence and ergodicity properties of random walks and associated skew products for large classes of locally compact groups and homogeneous spaces. In particular, we show that a closed subgroup of a product of finitely many linear groups over local fields supports an adapted recurrent random walk if and only if it has at most quadratic growth. We give also a detailed analysis of ergodicity properties for special classes of random walks on homogeneous spaces and for associated homeomorphisms with infinite invariant measure. The structural properties of closed subgroups of linear groups over local fields and the properties of group actions with respect to certain Radon measures associated with random walks play an important role in the proofs.

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