186 research outputs found
A Multiscale Approach to Brownian Motors
The problem of Brownian motion in a periodic potential, under the influence
of external forcing, which is either random or periodic in time, is studied in
this paper. Multiscale techniques are used to derive general formulae for the
steady state particle current and the effective diffusion tensor. These
formulae are then applied to calculate the effective diffusion coefficient for
a Brownian particle in a periodic potential driven simultaneously by additive
Gaussian white and colored noise. Our theoretical findings are supported by
numerical simulations.Comment: 19 pages, 2 figures, to appear in Physics Letters A. Revised for
Periodic Homogenization for Hypoelliptic Diffusions
We study the long time behavior of an Ornstein-Uhlenbeck process under the
influence of a periodic drift. We prove that, under the standard diffusive
rescaling, the law of the particle position converges weakly to the law of a
Brownian motion whose covariance can be expressed in terms of the solution of a
Poisson equation. We also derive upper bounds on the convergence rate
Analysis of White Noise Limits for Stochastic Systems with Two Fast Relaxation Times
In this paper we present a rigorous asymptotic analysis for stochastic
systems with two fast relaxation times. The mathematical model analyzed in this
paper consists of a Langevin equation for the particle motion with
time-dependent force constructed through an infinite dimensional Gaussian noise
process. We study the limit as the particle relaxation time as well as the
correlation time of the noise tend to zero and we obtain the limiting equations
under appropriate assumptions on the Gaussian noise. We show that the limiting
equation depends on the relative magnitude of the two fast time scales of the
system. In particular, we prove that in the case where the two relaxation times
converge to zero at the same rate there is a drift correction, in addition to
the limiting It\^{o} integral, which is not of Stratonovich type. If, on the
other hand, the colored noise is smooth on the scale of particle relaxation
then the drift correction is the standard Stratonovich correction. If the noise
is rough on this scale then there is no drift correction. Strong (i.e.
pathwise) techniques are used for the proof of the convergence theorems.Comment: 35 pages, 0 figures, To appear in SIAM J. MM
Spectral methods for multiscale stochastic differential equations
This paper presents a new method for the solution of multiscale stochastic
differential equations at the diffusive time scale. In contrast to
averaging-based methods, e.g., the heterogeneous multiscale method (HMM) or the
equation-free method, which rely on Monte Carlo simulations, in this paper we
introduce a new numerical methodology that is based on a spectral method. In
particular, we use an expansion in Hermite functions to approximate the
solution of an appropriate Poisson equation, which is used in order to
calculate the coefficients of the homogenized equation. Spectral convergence is
proved under suitable assumptions. Numerical experiments corroborate the theory
and illustrate the performance of the method. A comparison with the HMM and an
application to singularly perturbed stochastic PDEs are also presented
Estimating eddy diffusivities from noisy Lagrangian observations
The problem of estimating the eddy diffusivity from Lagrangian observations
in the presence of measurement error is studied in this paper. We consider a
class of incompressible velocity fields for which is can be rigorously proved
that the small scale dynamics can be parameterised in terms of an eddy
diffusivity tensor. We show, by means of analysis and numerical experiments,
that subsampling of the data is necessary for the accurate estimation of the
eddy diffusivity. The optimal sampling rate depends on the detailed properties
of the velocity field. Furthermore, we show that averaging over the data only
marginally reduces the bias of the estimator due to the multiscale structure of
the problem, but that it does significantly reduce the effect of observation
error
Mean Field Limits for Interacting Diffusions in a Two-Scale Potential
In this paper we study the combined mean field and homogenization limits for
a system of weakly interacting diffusions moving in a two-scale, locally
periodic confining potential, of the form considered
in~\cite{DuncanPavliotis2016}. We show that, although the mean field and
homogenization limits commute for finite times, they do not, in general,
commute in the long time limit. In particular, the bifurcation diagrams for the
stationary states can be different depending on the order with which we take
the two limits. Furthermore, we construct the bifurcation diagram for the
stationary McKean-Vlasov equation in a two-scale potential, before passing to
the homogenization limit, and we analyze the effect of the multiple local
minima in the confining potential on the number and the stability of stationary
solutions
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