716 research outputs found
Integration by parts on the law of the reflecting Brownian motion
We prove an integration by parts formula on the law of the reflecting
Brownian motion in the positive half line, where is a standard
Brownian motion. In other terms, we consider a perturbation of of the form
with smooth deterministic function and
and we differentiate the law of at . This
infinitesimal perturbation changes drastically the set of zeros of for any
. As a consequence, the formula we obtain contains an infinite
dimensional generalized functional in the sense of Schwartz, defined in terms
of Hida's renormalization of the squared derivative of and in terms of the
local time of at 0. We also compute the divergence on the Wiener space of a
class of vector fields not taking values in the Cameron-Martin space.Comment: 32 page
Fluctuations for a conservative interface model on a wall
We consider an effective interface model on a hard wall in (1+1) dimensions,
with conservation of the area between the interface and the wall. We prove that
the equilibrium fluctuations of the height variable converge in law to the
solution of a SPDE with reflection and conservation of the space average. The
proof is based on recent results obtained with L. Ambrosio and G. Savare on
stability properties of Markov processes with log-concave invariant measures
Conservative stochastic Cahn--Hilliard equation with reflection
We consider a stochastic partial differential equation with reflection at 0
and with the constraint of conservation of the space average. The equation is
driven by the derivative in space of a space--time white noise and contains a
double Laplacian in the drift. Due to the lack of the maximum principle for the
double Laplacian, the standard techniques based on the penalization method do
not yield existence of a solution. We propose a method based on infinite
dimensional integration by parts formulae, obtaining existence and uniqueness
of a strong solution for all continuous nonnegative initial conditions and
detailed information on the associated invariant measure and Dirichlet form.Comment: Published in at http://dx.doi.org/10.1214/009117906000000773 the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Approximate maximizers of intricacy functionals
G. Edelman, O. Sporns, and G. Tononi introduced in theoretical biology the
neural complexity of a family of random variables. This functional is a special
case of intricacy, i.e., an average of the mutual information of subsystems
whose weights have good mathematical properties. Moreover, its maximum value
grows at a definite speed with the size of the system.
In this work, we compute exactly this speed of growth by building
"approximate maximizers" subject to an entropy condition. These approximate
maximizers work simultaneously for all intricacies. We also establish some
properties of arbitrary approximate maximizers, in particular the existence of
a threshold in the size of subsystems of approximate maximizers: most smaller
subsystems are almost equidistributed, most larger subsystems determine the
full system.
The main ideas are a random construction of almost maximizers with a high
statistical symmetry and the consideration of entropy profiles, i.e., the
average entropies of sub-systems of a given size. The latter gives rise to
interesting questions of probability and information theory
A probabilistic study of neural complexity
G. Edelman, O. Sporns, and G. Tononi have introduced the neural complexity of
a family of random variables, defining it as a specific average of mutual
information over subfamilies. We show that their choice of weights satisfies
two natural properties, namely exchangeability and additivity, and we call any
functional satisfying these two properties an intricacy. We classify all
intricacies in terms of probability laws on the unit interval and study the
growth rate of maximal intricacies when the size of the system goes to
infinity. For systems of a fixed size, we show that maximizers have small
support and exchangeable systems have small intricacy. In particular,
maximizing intricacy leads to spontaneous symmetry breaking and failure of
uniqueness.Comment: minor edit
A Renewal version of the Sanov theorem
Large deviations for the local time of a process are investigated,
where for and are i.i.d.\ random
variables on a Polish space, is the -th arrival time of a renewal
process depending on . No moment conditions are assumed on the arrival
times of the renewal process.Comment: 13 page
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