418 research outputs found
Passive tracer in a flow corresponding to a two dimensional stochastic Navier Stokes equations
In this paper we prove the law of large numbers and central limit theorem for
trajectories of a particle carried by a two dimensional Eulerian velocity
field. The field is given by a solution of a stochastic Navier--Stokes system
with a non-degenerate noise. The spectral gap property, with respect to
Wasserstein metric, for such a system has been shown in [9]. In the present
paper we show that a similar property holds for the environment process
corresponding to the Lagrangian observations of the velocity. In consequence we
conclude the law of large numbers and the central limit theorem for the tracer.
The proof of the central limit theorem relies on the martingale approximation
of the trajectory process
Asymptotics of the solutions of the stochastic lattice wave equation
We consider the long time limit theorems for the solutions of a discrete wave
equation with a weak stochastic forcing. The multiplicative noise conserves the
energy and the momentum. We obtain a time-inhomogeneous Ornstein-Uhlenbeck
equation for the limit wave function that holds both for square integrable and
statistically homogeneous initial data. The limit is understood in the
point-wise sense in the former case, and in the weak sense in the latter. On
the other hand, the weak limit for square integrable initial data is
deterministic
Long time, large scale limit of the Wigner transform for a system of linear oscillators in one dimension
We consider the long time, large scale behavior of the Wigner transform
W_\eps(t,x,k) of the wave function corresponding to a discrete wave equation
on a 1-d integer lattice, with a weak multiplicative noise. This model has been
introduced in Basile, Bernardin, and Olla to describe a system of interacting
linear oscillators with a weak noise that conserves locally the kinetic energy
and the momentum. The kinetic limit for the Wigner transform has been shown in
Basile, Olla, and Spohn. In the present paper we prove that in the unpinned
case there exists such that for any the
weak limit of W_\eps(t/\eps^{3/2\gamma},x/\eps^{\gamma},k), as \eps\ll1,
satisfies a one dimensional fractional heat equation with . In the pinned case an analogous
result can be claimed for W_\eps(t/\eps^{2\gamma},x/\eps^{\gamma},k) but the
limit satisfies then the usual heat equation
From a kinetic equation to a diffusion under an anomalous scaling
A linear Boltzmann equation is interpreted as the forward equation for the
probability density of a Markov process (K(t), i(t), Y(t)), where (K(t), i(t))
is an autonomous reversible jump process, with waiting times between two jumps
with finite expectation value but infinite variance, and Y(t) is an additive
functional of K(t). We prove that under an anomalous rescaling Y converges in
distribution to a two-dimensional Brownian motion. As a consequence, the
appropriately rescaled solution of the Boltzmann equation converges to a
diffusion equation
Central Limit Theorem and recurrence for random walks in bistochastic random environments
We prove the annealed Central Limit Theorem for random walks in bistochastic
random environments on with zero local drift. The proof is based on a
"dynamicist's interpretation" of the system, and requires a much weaker
condition than the customary uniform ellipticity. Moreover, recurrence is
derived for .Comment: 13 pages; to appear in the special issue of J. Math. Phys. on
"Statistical Mechanics on Random Structures
Cut Points and Diffusions in Random Environment
In this article we investigate the asymptotic behavior of a new class of
multi-dimensional diffusions in random environment. We introduce cut times in
the spirit of the work done by Bolthausen, Sznitman and Zeitouni, see [4], in
the discrete setting providing a decoupling effect in the process. This allows
us to take advantage of an ergodic structure to derive a strong law of large
numbers with possibly vanishing limiting velocity and a central limit theorem
under the quenched measure.Comment: 44 pages; accepted for publication in "Journal of Theoretical
Probability
Nonequilibrium dynamics of a stochastic model of anomalous heat transport
We study the dynamics of covariances in a chain of harmonic oscillators with
conservative noise in contact with two stochastic Langevin heat baths. The
noise amounts to random collisions between nearest-neighbour oscillators that
exchange their momenta. In a recent paper, [S Lepri et al. J. Phys. A: Math.
Theor. 42 (2009) 025001], we have studied the stationary state of this system
with fixed boundary conditions, finding analytical exact expressions for the
temperature profile and the heat current in the thermodynamic (continuum)
limit. In this paper we extend the analysis to the evolution of the covariance
matrix and to generic boundary conditions. Our main purpose is to construct a
hydrodynamic description of the relaxation to the stationary state, starting
from the exact equations governing the evolution of the correlation matrix. We
identify and adiabatically eliminate the fast variables, arriving at a
continuity equation for the temperature profile T(y,t), complemented by an
ordinary equation that accounts for the evolution in the bulk. Altogether, we
find that the evolution of T(y,t) is the result of fractional diffusion.Comment: Submitted to Journal of Physics A, Mathematical and Theoretica
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