| 概要 | Let T be a rooted multi-type Galton-Watson (MGW) tree of finitely many
types with at least one offspring at each vertex and an offspring
distribution with exponential tails. The r-biased random walk X(t)
on T is the nearest neighbor random walk which, when at a vertex v
with d(v) offspring, moves closer to the root with probability
r/(r+d(v)) and to each of the offspring with probability 1/(r+d(v)).
This walk is transient if and only if 0<r<R, with R the Perron-Frobenius
eigenvalue for the (assumed) irreducible matrix of expected offspring
numbers. Following the approach of Peres and Zeitouni (2008), in a joint
work with Nike Sun we show that at the critical value r=R, for almost
every T, the process |X(nt)|/sqrt(n) converges in law as n goes to
infinity to a deterministic positive multiple of a reflected Brownian
motion. Our proof is based on a new explicit description of a reversing
measure for this walk from the point of view of the particle, a
construction which extends to the reversing measure for
a biased random walk with random environment (RWRE) on MGW trees,
again at a critical value of the bias.
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