50 research outputs found
Finitary isomorphisms of Brownian motions
Ornstein and Shields (Advances in Math., 10:143-146, 1973) proved that
Brownian motion reflected on a bounded region is an infinite entropy Bernoulli
flow and thus Ornstein theory yielded the existence of a measure-preserving
isomorphism between any two such Brownian motions. For fixed h >0, we construct
by elementary methods, isomorphisms with almost surely finite coding windows
between Brownian motions reflected on the intervals [0, qh] for all positive
rationals q.Comment: Published at https://doi.org/10.1214/19-AOP1412 in the Annals of
Probability by the Institute of Mathematical Statistic
Finitary isomorphisms of some infinite entropy Bernoulli flows
A consequence of Ornstein theory is that the infinite entropy flows associated with Poisson processes and continuous-time irreducible Markov chains on a finite number of states are isomorphic as measure-preserving systems. We give an elementary construction of such an isomorphism that has an additional finitariness property, subject to the additional conditions that the Markov chain has a uniform holding rate and a mixing skeleton
Factors of IID on Trees
Classical ergodic theory for integer-group actions uses entropy as a complete
invariant for isomorphism of IID (independent, identically distributed)
processes (a.k.a. product measures). This theory holds for amenable groups as
well. Despite recent spectacular progress of Bowen, the situation for
non-amenable groups, including free groups, is still largely mysterious. We
present some illustrative results and open questions on free groups, which are
particularly interesting in combinatorics, statistical physics, and
probability. Our results include bounds on minimum and maximum bisection for
random cubic graphs that improve on all past bounds.Comment: 18 pages, 1 figur
A monotone Sinai theorem
Sinai proved that a nonatomic ergodic measure-preserving system has any
Bernoulli shift of no greater entropy as a factor. Given a Bernoulli shift, we
show that any other Bernoulli shift that is of strictly less entropy and is
stochastically dominated by the original measure can be obtained as a monotone
factor; that is, the factor map has the property that for each point in the
domain, its image under the factor map is coordinatewise smaller than or equal
to the original point.Comment: Published at http://dx.doi.org/10.1214/14-AOP968 in the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org