This volume is comprised of two parts: the first contains articles by S. N. Evans, F. Ledrappier, and Figa-Talomanaca. These articles arose from a Centre de Recherches de Mathematiques (CRM) seminar entitled, 'Topics in Probability on Lie Groups: Boundary Theory'. Evans gives a synthesis of his pre-1992 work on Gaussian measures on vector spaces over a local field. Ledrappier uses the freegroup on $d$ generators as a paradigm for results on the asymptotic properties of random walks and harmonic measures on the Martin boundary.These articles are followed by a case study by Figa-Talamanca using Gelfand pairs to study a diffusion on a compact ultrametric space. The second part of the book is an appendix to the book ""Compactifications of Symmetric Spaces (Birkhauser)"" by Y. Guivarc'h and J. C. Taylor. This appendix consists of an article by each author and presents the contents of this book in a more algebraic way. L. Ji and J.-P. Anker simplifies some of their results on the asymptotics of the Green function that were used to compute Martin boundaries. And Taylor gives a self-contained account of Martin boundary theory for manifolds using the theory of second order strictly elliptic partial differential operators.
Heat kernel and Green function estimates on noncompact symmetric spaces. II by J.-P. Anker and L. Ji Local fields, Gaussian measures, and Brownian motions by S. N. Evans An application of Gelfand pairs to a problem of diffusion in compact ultrametric spaces by A. Figa-Talamanca Compactifications of symmetric spaces and positive eigenfunctions of the Laplacian by Y. Guivarc'h, J. C. Taylor, and L. Ji Some asymptotic properties of random walks on free groups by F. Ledrappier The Martin compactification associated with a second order strictly elliptic partial differential operator on a manifold $\textbf M$ by J. C. Taylor.