Ergodic theorems for coupled random walks and other systems with locally interacting components

1981 ◽  
Vol 56 (4) ◽  
pp. 443-468 ◽  
Author(s):  
Thomas M. Liggett ◽  
Frank Spitzer
Keyword(s):  
Random Walks ◽  
Ergodic Theorems ◽  
Coupled Random Walks ◽  
Interacting Components

scholarly journals Fractional governing equations for coupled random walks

2012 ◽  
Vol 64 (10) ◽  
pp. 3021-3036 ◽  
Author(s):  
A. Jurlewicz ◽  
P. Kern ◽  
M.M. Meerschaert ◽  
H.-P. Scheffler
Keyword(s):  
Random Walks ◽  
Governing Equations ◽  
Coupled Random Walks

Ergodic theorems for dynamic random walks

1999 ◽  
Vol 329 (2) ◽  
pp. 153-158
Author(s):  
Nadine Guillotin-Plantard ◽  
Dominique Schneider
Keyword(s):  
Random Walks ◽  
Ergodic Theorems

scholarly journals Ergodic theorems for dynamic random walks

2003 ◽  
pp. 177-195
Author(s):  
Nadine Guillotin-Plantard ◽  
Dominique Schneider
Keyword(s):  
Random Walks ◽  
Ergodic Theorems

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.

scholarly journals Kazhdan projections, random walks and ergodic theorems

2019 ◽  
Vol 2019 (754) ◽  
pp. 49-86 ◽  
Author(s):  
Cornelia Druţu ◽  
Piotr W. Nowak
Keyword(s):  
Random Walks ◽  
Banach Spaces ◽  
Ergodic Theorems ◽  
Spectral Gaps ◽  
Norm Estimates ◽  
Markov Operators ◽  
Open Questions ◽  
Convergence Results ◽  
Uniformly Convex Banach Spaces ◽  
Property T

Abstract In this paper we investigate generalizations of Kazhdan’s property (T) to the setting of uniformly convex Banach spaces. We explain the interplay between the existence of spectral gaps and that of Kazhdan projections. Our methods employ Markov operators associated to a random walk on the group, for which we provide new norm estimates and convergence results. This construction exhibits useful properties and flexibility, and allows to view Kazhdan projections in Banach spaces as natural objects associated to random walks on groups. We give a number of applications of these results. In particular, we address several open questions. We give a direct comparison of properties (TE) and FE with Lafforgue’s reinforced Banach property (T); we obtain shrinking target theorems for orbits of Kazhdan groups; finally, answering a question of Willett and Yu we construct non-compact ghost projections for warped cones. In this last case we conjecture that such warped cones provide counterexamples to the coarse Baum–Connes conjecture.

Non-homogeneous Random Walks

2016 ◽  
Author(s):  
Mikhail Menshikov ◽  
Serguei Popov ◽  
Andrew Wade
Keyword(s):  
Random Walks
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