PR ] 5 A pr 2 00 4 TREE-INDEXED PROCESSES

This article examines a recent body of work on stochastic processes indexed by a tree. Emphasis is on the application of this new framework to existing probability models. Proofs are largely omitted, with references provided.

[1]  David Thomas,et al.  The Art in Computer Programming , 2001 .

[2]  Quansheng Liu.  Sur certaines martingales de Mandelbrot , 1999 .

[3]  L. Carleson.  Selected Problems on Exceptional Sets , 1998 .

[4]  R. Pemantle.  Sharpness of Second Moment Criteria for Branching and Tree-Indexed Processes , 2004, math/0404090.

[5]  Y. Peres.  Intersection-equivalence of Brownian paths and certain branching processes , 1996 .

[6]  Thomas S. Salisbury,et al.  ENERGY, AND INTERSECTIONS OF MARKOV CHAINS* , 1996 .

[7]  E. Waymire,et al.  A CASCADE DECOMPOSITION THEORY WITH APPLICATIONS TO MARKOV AND EXCHANGEABLE CASCADES , 1996 .

[8]  R. Pemantle,et al.  Galton-Watson Trees with the Same Mean Have the Same Polar Sets , 1995, math/0404053.

[9]  Russell Lyons,et al.  Ergodic theory on Galton—Watson trees: speed of random walk and dimension of harmonic measure , 1995, Ergodic Theory and Dynamical Systems.

[10]  R. Pemantle,et al.  Domination Between Trees and Application to an Explosion Problem , 2004, math/0404044.

[11]  Yuval Peres,et al.  Tree-indexed random walks on groups and first passage percolation , 1994 .

[12]  Yuval Peres,et al.  Markov chains indexed by trees , 1994 .

[13]  R. Pemantle,et al.  Martin capacity for Markov chains and random walks in varying dimensions , 1994 .

[14]  R. Pemantle,et al.  Critical Random Walk in Random Environment on Trees of Exponential Growth , 2004, math/0404049.

[15]  J. Newman,et al.  Rates of Weed Spread in Spatially Heterogeneous Environments , 1993 .

[16]  Edward C. Waymire,et al.  A statistical analysis of mesoscale rainfall as a random cascade , 1993 .

[17]  R. Pemantle,et al.  Random walk in a random environment and rst-passage percolation on trees , 2004, math/0404045.

[18]  Russell Lyons,et al.  Random Walks, Capacity and Percolation on Trees , 1992 .

[19]  Steven N. Evans.  Polar and Nonpolar Sets for a Tree Indexed Process , 1992 .

[20]  J. Pitman,et al.  Size-biased sampling of Poisson point processes and excursions , 1992 .

[21]  Y. Peres,et al.  Random walks on a tree and capacity in the interval , 1992 .

[22]  Yuval Peres,et al.  A Correlation Inequality for Tree-Indexed Markov Chains , 1992 .

[23]  A. Sidorenko,et al.  Inequalities for functionals generated by bipartite graphs , 1991 .

[24]  R. Lyons.  Random Walks and Percolation on Trees , 1990 .

[25]  F. M. Dekking,et al.  On the structure of Mandelbrot's percolation process and other random cantor sets , 1990 .

[26]  J. Kahane,et al.  Décomposition des mesures selon la dimension , 1990 .

[27]  R. Lyons.  The Ising model and percolation on trees and tree-like graphs , 1989 .

[28]  B. Mohar,et al.  A Survey on Spectra of infinite Graphs , 1989 .

[29]  S. Krantz.  Fractal geometry , 1989 .

[30]  Thomas S. Salisbury,et al.  Capacity and energy for multiparameter Markov processes , 1989 .

[31]  D. Aldous.  Probability Approximations via the Poisson Clumping Heuristic , 1988 .

[32]  D. Aldous,et al.  A diffusion limit for a class of randomly-growing binary trees , 1988 .

[33]  B. Derrida,et al.  Polymers on disordered trees, spin glasses, and traveling waves , 1988 .

[34]  Rick Durrett,et al.  Connectivity properties of Mandelbrot's percolation process , 1988 .

[35]  J. Hawkes.  SOME RANDOM SERIES OF FUNCTIONS second edition (Cambridge Studies in Advanced Mathematics 5) , 1988 .

[36]  R. Mauldin,et al.  Exact Hausdorff dimension in random recursive constructions. , 1987, Proceedings of the National Academy of Sciences of the United States of America.

[37]  J. Laurie Snell,et al.  Random Walks and Electrical Networks , 1984 .

[38]  Curtis T. McMullen,et al.  The Hausdorff dimension of general Sierpiński carpets , 1984, Nagoya Mathematical Journal.

[39]  Alan Hastings,et al.  Dispersal strategies in patchy environments , 1984 .

[40]  J. Hawkes,et al.  Trees Generated by a Simple Branching Process , 1981 .

[41]  Stanley Sawyer,et al.  Isotropic random walks in a tree , 1978 .

[42]  M. Bramson.  Minimal displacement of branching random walk , 1978 .

[43]  H. Kesten.  Branching brownian motion with absorption , 1978 .

[44]  J. Biggins.  Chernoff's theorem in the branching random walk , 1977, Journal of Applied Probability.

[45]  Eugene Seneta,et al.  I. J. Bienayme Statistical Theory Anticipated. , 1976 .

[46]  J. Kingman.  The First Birth Problem for an Age-dependent Branching Process , 1975 .

[47]  J. Hammersley.  Postulates for Subadditive Processes , 1974 .

[48]  A. Joffe,et al.  Random variables, trees, and branching random walks☆ , 1973 .

[49]  D. Freedman,et al.  Random distribution functions , 1963 .

[50]  Two . dimensional Brownian Motion and Harmonic Functions , 2022 .