589,883 research outputs found
Convergence to equilibrium for finite Markov processes, with application to the Random Energy Model
We estimate the distance in total variation between the law of a finite state
Markov process at time t, starting from a given initial measure, and its unique
invariant measure. We derive upper bounds for the time to reach the
equilibrium. As an example of application we consider a special case of finite
state Markov process in random environment: the Metropolis dynamics of the
Random Energy Model. We also study the process of the environment as seen from
the process
On ergodic two-armed bandits
A device has two arms with unknown deterministic payoffs and the aim is to
asymptotically identify the best one without spending too much time on the
other. The Narendra algorithm offers a stochastic procedure to this end. We
show under weak ergodic assumptions on these deterministic payoffs that the
procedure eventually chooses the best arm (i.e., with greatest Cesaro limit)
with probability one for appropriate step sequences of the algorithm. In the
case of i.i.d. payoffs, this implies a "quenched" version of the "annealed"
result of Lamberton, Pag\`{e}s and Tarr\`{e}s [Ann. Appl. Probab. 14 (2004)
1424--1454] by the law of iterated logarithm, thus generalizing it. More
precisely, if ,
, are the deterministic reward sequences we would get if we
played at time , we obtain infallibility with the same assumption on
nonincreasing step sequences on the payoffs as in Lamberton, Pag\`{e}s and
Tarr\`{e}s [Ann. Appl. Probab. 14 (2004) 1424--1454], replacing the i.i.d.
assumption by the hypothesis that the empirical averages
and converge, as
tends to infinity, respectively, to and , with rate at
least , for some . We also show a
fallibility result, that is, convergence with positive probability to the
choice of the wrong arm, which implies the corresponding result of Lamberton,
Pag\`{e}s and Tarr\`{e}s [Ann. Appl. Probab. 14 (2004) 1424--1454] in the
i.i.d. case.Comment: Published in at http://dx.doi.org/10.1214/10-AAP751 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Asymptotic expansions of Laplace integrals for quantum state tomography
Bayesian estimation of a mixed quantum state can be approximated via maximum
likelihood (MaxLike) estimation when the likelihood function is sharp around
its maximum. Such approximations rely on asymptotic expansions of
multi-dimensional Laplace integrals. When this maximum is on the boundary of
the integration domain, as it is the case when the MaxLike quantum state is not
full rank, such expansions are not standard. We provide here such expansions,
even when this maximum does not belong to the smooth part of the boundary, as
it is the case when the rank deficiency exceeds two. These expansions provide,
aside the MaxLike estimate of the quantum state, confidence intervals for any
observable. They confirm the formula proposed and used without precise
mathematical justifications by the authors in an article recently published in
Physical Review A.Comment: 17 pages, submitte
Boundary stabilization and control of wave equations by means of a general multiplier method
We describe a general multiplier method to obtain boundary stabilization of
the wave equation by means of a (linear or quasi-linear) Neumann feedback. This
also enables us to get Dirichlet boundary control of the wave equation. This
method leads to new geometrical cases concerning the "active" part of the
boundary where the feedback (or control) is applied. Due to mixed boundary
conditions, the Neumann feedback case generate singularities. Under a simple
geometrical condition concerning the orientation of the boundary, we obtain a
stabilization result in linear or quasi-linear cases
Liapunov Multipliers and Decay of Correlations in Dynamical Systems
The essential decorrelation rate of a hyperbolic dynamical system is the
decay rate of time-correlations one expects to see stably for typical
observables once resonances are projected out. We define and illustrate these
notions and study the conjecture that for observables in , the essential
decorrelation rate is never faster than what is dictated by the {\em smallest}
unstable Liapunov multiplier
Scale Invariant Cosmology
An attempt is made here to extend to the microscopic domain the scale
invariant character of gravitation - which amounts to consider expansion as
applying to any physical scale. Surprisingly, this hypothesis does not prevent
the redshift from being obtained. It leads to strong restrictions concerning
the choice between the presently available cosmological models and to new
considerations about the notion of time. Moreover, there is no horizon problem
and resorting to inflation is not necessary.Comment: TeX, 20 page
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