Consider a Markov chain $\{X_n\}_{n\ge 0}$ with an ergodic probability
measure $\pi$. Let $\Psi$ a function on the state space of the chain, with
$\alpha$-tails with respect to $\pi$, $\alpha\in (0,2)$. We find sufficient
conditions on the probability transition to prove convergence in law of
$N^{1/\alpha}\sum_n^N \Psi(X_n)$ to a $\alpha$-stable law. ``Martingale
approximation'' approach and ``coupling'' approach give two different sets of
conditions. We extend these results to continuous time Markov jump processes
$X_t$, whose skeleton chain satisfies our assumptions. If waiting time between
jumps has finite expectation, we prove convergence of $N^{-1/\alpha}\int_0^{Nt}
V(X_s) ds$ to a stable process. In the case of waiting times with infinite
average, we prove convergence to a Mittag-Leffler process.Comment: Accepted for the publication in Annals of Applied Probabilit
We formulate a criterion for the existence and uniqueness of an invariant
measure for a Markov process taking values in a Polish phase space. In
addition, weak-$^*$ ergodicity, that is, the weak convergence of the ergodic
averages of the laws of the process starting from any initial distribution, is
established. The principal assumptions are the existence of a lower bound for
the ergodic averages of the transition probability function and its local
uniform continuity. The latter is called the e-property. The general result is
applied to solutions of some stochastic evolution equations in Hilbert spaces.
As an example, we consider an evolution equation whose solution describes the
Lagrangian observations of the velocity field in the passive tracer model. The
weak-$^*$ mean ergodicity of the corresponding invariant measure is used to
derive the law of large numbers for the trajectory of a tracer.Comment: Published in at http://dx.doi.org/10.1214/09-AOP513 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
We establish the self-averaging properties of the Wigner transform of a mixture of states in the regime when the correlation length of the random medium is much longer than the wave length but much shorter than the propagation distance. The main ingredients in the proof are the error estimates for the semiclassical approximation of the Wigner transform by the solution of the Liouville equations, and the limit theorem for two-particle motion along the characteristics of the Liouville equations. The results are applied to a mathematical model of the time-reversal experiments for the acoustic waves, and self-averaging properties of the re-transmitted wave are proved.
We consider a one dimensional infinite chain of harmonic oscillators whose dynamics is perturbed by a stochastic term conserving energy and momentum. We prove that in the unpinned case the macroscopic evolution of the energy converges to a fractional diffusion governed by −|∆| 3/4 . For a pinned system we prove that its energy evolves diffusively, generalizing some results of [4].
We consider solutions of the Schrödinger equation with a weak time-dependent random potential. It is shown that when the two-point correlation function of the potential is rapidly decaying, then the Fourier transformζ ε (t, ξ) of the appropriately scaled solution converges point-wise in ξ to a stochastic complex Gaussian limit. On the other hand, when the two-point correlation function decays slowly, we show that the limit ofζ ε (t, ξ) has the formζ 0 (ξ ) exp(i B κ (t, ξ)) where B κ (t, ξ) is a fractional Brownian motion.
Suppose that {X t , t ≥ 0} is a non-stationary Markov process, taking values in a Polish metric space E. We prove the law of large numbers and central limit theorem for an additive functional of the form T 0 ψ(X s )ds, provided that the dual transition probability semigroup, defined on measures, is strongly contractive in an appropriate Wasserstein metric. Function ψ is assumed to be Lipschitz on E.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.