29,434 research outputs found
The max-plus Martin boundary
We develop an idempotent version of probabilistic potential theory. The goal
is to describe the set of max-plus harmonic functions, which give the
stationary solutions of deterministic optimal control problems with additive
reward. The analogue of the Martin compactification is seen to be a
generalisation of the compactification of metric spaces using (generalised)
Busemann functions. We define an analogue of the minimal Martin boundary and
show that it can be identified with the set of limits of ``almost-geodesics'',
and also the set of (normalised) harmonic functions that are extremal in the
max-plus sense. Our main result is a max-plus analogue of the Martin
representation theorem, which represents harmonic functions by measures
supported on the minimal Martin boundary. We illustrate it by computing the
eigenvectors of a class of translation invariant Lax-Oleinik semigroups. In
this case, we relate the extremal eigenvectors to the Busemann points of a
normed space.Comment: 37 pages; 8 figures v1: December 20, 2004. v2: June 7, 2005. Section
12 adde
Integration and measures on the space of countable labelled graphs
In this paper we develop a rigorous foundation for the study of integration
and measures on the space of all graphs defined on a countable
labelled vertex set . We first study several interrelated -algebras
and a large family of probability measures on graph space. We then focus on a
"dyadic" Hamming distance function , which was
very useful in the study of differentiation on . The function
is shown to be a Haar measure-preserving
bijection from the subset of infinite graphs to the circle (with the
Haar/Lebesgue measure), thereby naturally identifying the two spaces. As a
consequence, we establish a "change of variables" formula that enables the
transfer of the Riemann-Lebesgue theory on to graph space
. This also complements previous work in which a theory of
Newton-Leibnitz differentiation was transferred from the real line to
for countable . Finally, we identify the Pontryagin dual of
, and characterize the positive definite functions on
.Comment: 15 pages, LaTe
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