Space-time boundaries for random walks obtained from diffuse measures
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.
scientific article; zbMATH DE number 590936
scientific article