Space-time boundaries for random walks obtained from diffuse measures
type: Item
instance of: scholarly article
title: Space-time boundaries for random walks obtained from diffuse measures
zbMATH Open document ID: 0814.60066
DOI: 10.1007/BF02773675
published in: Israel Journal of Mathematics
publication date: 1995-06-18T00:00:00Z
review text: Let \(X\) be a discrete group, and let \(\{ \mu_ 0, \mu_ 1, \mu_ 2, \dots\}\) be a family of probability measures on \(X\). The set \(C = \biggl\{ (x,n) \in X \times N \text{ such that } \mu_ n* \mu_{n-1}* \cdots* \mu_ 0 \bigl( \{x\} \bigr) > 0 \biggr\}\) is called the space- time cone determined by \(\mu_ 0, \mu_ 1, \dots\) A space-time harmonic function on the cone \(C\) is a nonzero function \(f : C \to R^ +\) such that \[ f(x,n) = \sum_{y \in X} f(x + y, n + 1) \mu_{n+1} \bigl( \{y\} \bigr). \] The extreme rays of the set of space-time harmonic functions form the space-time boundary of the random walk associated to \(\mu_ 0, \mu_ 1, \dots\) by the natural way. One of the results for this construction is that the space-time boundary admits a natural identification with the convex hull of the support of the measure \(\mu\) (when \(\mu = \mu_ 1 = \mu_ 2 = \cdots)\). These and some others results are generalized to the case when \(X\) is a local compact Abelian group and \(\mu_ i\) are measures on \(X\). To this end the author proposes the new construction for the space-time boundary of the corresponding random walk.
reviewed by: N. S. Bratijchuk
Mathematics Subject Classification ID: 60G50
Mathematics Subject Classification ID: 60B15
zbMATH DE Number: 590936
zbMATH Keywords: harmonic functions
zbMATH Keywords: space-time boundary
zbMATH Keywords: random walk
author: David Handelman
MaRDI profile type: Publication
cites work: Q3900495
cites work: Angular momentum, convex Polyhedra and Algebraic Geometry
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cites work: Dimension Groups and Their Affine Representations
cites work: Q4118035
cites work: Rank functions and \(K_0\) of regular rings
cites work: Convexity properties of the moment mapping
cites work: Q3894702
cites work: Positive polynomials and product type actions of compact groups
cites work: Q3744397
cites work: Deciding eventual positivity of polynomials
cites work: Positive polynomials, convex integral polytopes, and a random walk problem
cites work: Extending traces on fixed point \(C^ *\) algebras under Xerox product type actions of compact Lie groups
cites work: Representing polynomials by positive linear functions on compact convex polyhedra
cites work: Iterated multiplication of characters of compact connected Lie groups
cites work: Effectiveness of an affine invariant for indecomposable integral polytopes
cites work: REPRESENTATION RINGS AS INVARIANTS FOR COMPACT GROUPS AND LIMIT RATIO THEOREMS FOR THEM
cites work: The Martin Boundary for Random Walk
cites work: Q4405392
cites work: Convex Analysis
cites work: Some Properties of Laplace Transforms of Measures
cites work: Q5338702
cites work: The K-functor (Grothendieck group) of the infinite symmetric group
description: scientific article; zbMATH DE number 590936