Transfer Operators, Induced Probability Spaces, and Random Walk Models

P. Jorgensen, F. Tian

2017, v.23, №2, 187-210

ABSTRACT

We study a family of discrete-time random-walk models. The starting
point is a fixed generalized transfer operator $R$ subject to a set
of axioms, and a given endomorphism in a compact Hausdorff space $X$.
Our setup includes a host of models from applied dynamical systems,
and it leads to general path-space probability realizations of the
initial transfer operator. The analytic data in our construction is
a pair $\left(h,\lambda\right)$, where $h$ is an $R$-harmonic function
on $X$, and $\lambda$ is a given positive measure on $X$ subject
to a certain invariance condition defined from $R$. With this we
show that there are then discrete-time random-walk realizations in
explicit path-space models; each associated to a probability measures
$\mathbb{P}$ on path-space, in such a way that the initial data allows
for spectral characterization: The initial endomorphism in $X$ lifts
to an automorphism in path-space with the probability measure $\mathbb{P}$
quasi-invariant with respect to a shift automorphism. The latter takes
the form of explicit multiresolutions in $L^{2}$ of $\mathbb{P}$
in the sense of Lax\tire Phillips scattering theory.

Keywords: unbounded operator, closable operator, spectral theory, discrete analysis, distribution of point-masses, probability space, stochastic processes, discrete time, path-space measure, endomorphism, harmonic functions

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