Bimodules over $\mathop {\rm VN}\nolimits (G)$, harmonic operators and the non-commutative Poisson boundary

[ACCESS RESTRICTED TO THE UNIVERSITY OF MISSOURI AT AUTHOR'S REQUEST.] Trace and extension theory lay the foundation for solving a plethora of boundary value problems. In developing this theory, one typically needs well-behaved extension operators from a specified domain to the entire Euclidean space. Historically, three extension operators have developed much of the theory in the setting of Lipschitz domains (and rougher domains); those due to A.P. Calderon, E.M. Stein, and P.W. Jones. In this dissertation, we generalize Stein's extension operator to weighted Sobolev spaces and Jones' extension operator to domains with partially vanishing traces. We then develop a rich trace/extension theory as a tool in solving a Poisson boundary value problem with Dirichlet boundary condition where the differential operator in question is of second order in divergence form with bounded coefficients satisfying the Legendre-Hadamard ellipticity condition.

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On transformations of Markov chains and Poisson boundary

Transactions of the American Mathematical Society ◽

10.1090/tran/7975 ◽

2019 ◽

Vol 373 (3) ◽

pp. 2207-2227

Author(s):

Iddo Ben-Ari ◽

Behrang Forghani

Keyword(s):

Markov Chains ◽

Poisson Boundary

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The Poisson boundary of $\operatorname{Out}(F_{N})$

Duke Mathematical Journal ◽

10.1215/00127094-3166308 ◽

2016 ◽

Vol 165 (2) ◽

pp. 341-369 ◽

Cited By ~ 2

Author(s):

Camille Horbez

Keyword(s):

Poisson Boundary

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An Abramov formula for stationary spaces of discrete groups

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2012.167 ◽

2013 ◽

Vol 34 (3) ◽

pp. 837-853 ◽

Cited By ~ 3

Author(s):

YAIR HARTMAN ◽

YURI LIMA ◽

OMER TAMUZ

Keyword(s):

Random Walk ◽

Probability Measure ◽

Finite Index ◽

Discrete Group ◽

Return Time ◽

Discrete Groups ◽

Poisson Boundary ◽

Index Subgroup ◽

Expected Return ◽

Finite Index Subgroup

AbstractLet $(G, \mu )$ be a discrete group equipped with a generating probability measure, and let $\Gamma $ be a finite index subgroup of $G$. A $\mu $-random walk on $G$, starting from the identity, returns to $\Gamma $ with probability one. Let $\theta $ be the hitting measure, or the distribution of the position in which the random walk first hits $\Gamma $. We prove that the Furstenberg entropy of a $(G, \mu )$-stationary space, with respect to the action of $(\Gamma , \theta )$, is equal to the Furstenberg entropy with respect to the action of $(G, \mu )$, times the index of $\Gamma $ in $G$. The index is shown to be equal to the expected return time to $\Gamma $. As a corollary, when applied to the Furstenberg–Poisson boundary of $(G, \mu )$, we prove that the random walk entropy of $(\Gamma , \theta )$ is equal to the random walk entropy of $(G, \mu )$, times the index of $\Gamma $ in $G$.

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