3,895 research outputs found
On the long-time behavior of a perturbed conservative system with degeneracy
We consider in this work a model conservative system subject to dissipation
and Gaussian-type stochastic perturbations. The original conservative system
possesses a continuous set of steady states, and is thus degenerate. We
characterize the long-time limit of our model system as the perturbation
parameter tends to zero. The degeneracy in our model system carries features
found in some partial differential equations related, for example, to
turbulence problems.Comment: Revised version. We added a Section 6 on the connection with the
Euler-Arnold equation. To appear at Journal of Theoretical Probabilit
Hypoelliptic multiscale Langevin diffusions: Large deviations, invariant measures and small mass asymptotics
We consider a general class of non-gradient hypoelliptic Langevin diffusions
and study two related questions. The first one is large deviations for
hypoelliptic multiscale diffusions. The second one is small mass asymptotics of
the invariant measure corresponding to hypoelliptic Langevin operators and of
related hypoelliptic Poisson equations. The invariant measure corresponding to
the hypoelliptic problem and appropriate hypoelliptic Poisson equations enter
the large deviations rate function due to the multiscale effects. Based on the
small mass asymptotics we derive that the large deviations behavior of the
multiscale hypoelliptic diffusion is consistent with the large deviations
behavior of its overdamped counterpart. Additionally, we rigorously obtain an
asymptotic expansion of the solution to the related density of the invariant
measure and to hypoelliptic Poisson equations with respect to the mass
parameter, characterizing the order of convergence. The proof of convergence of
invariant measures is of independent interest, as it involves an improvement of
the hypocoercivity result for the kinetic Fokker-Planck equation. We do not
restrict attention to gradient drifts and our proof provides explicit
information on the dependence of the bounds of interest in terms of the mass
parameter
Random perturbations of dynamical systems with reflecting boundary and corresponding PDE with a small parameter
We study the asymptotic behavior of a diffusion process with small diffusion
in a domain . This process is reflected at with respect to a
co-normal direction pointing inside . Our asymptotic result is used to study
the long time behavior of the solution of the corresponding parabolic PDE with
Neumann boundary condition.Comment: 17 pages, 1 figure, comments are welcom
On diffusion in narrow random channels
We consider in this paper a solvable model for the motion of molecular
motors. Based on the averaging principle, we reduce the problem to a diffusion
process on a graph. We then calculate the effective speed of transportation of
these motors.Comment: 23 pages, 3 figures, comments are welcom
Large deviations and averaging for systems of slow–fast reaction–diffusion equations
We study a large deviation principle for a system of stochastic reaction--diffusion equations (SRDEs) with a separation of fast and slow components and small noise in the slow component. The derivation of the large deviation principle is based on the weak convergence method in infinite dimensions, which results in studying averaging for controlled SRDEs. By appropriate choice of the parameters, the fast process and the associated control that arises from the weak convergence method decouple from each other. We show that in this decoupling case one can use the weak convergence method to characterize the limiting process via a "viable pair" that captures the limiting controlled dynamics and the effective invariant measure simultaneously. The characterization of the limit of the controlled slow-fast processes in terms of viable pair enables us to obtain a variational representation of the large deviation action functional. Due to the infinite--dimensional nature of our set--up, the proof of tightness as well as the analysis of the limit process and in particular the proof of the large deviations lower bound is considerably more delicate here than in the finite--dimensional situation. Smoothness properties of optimal controls in infinite dimensions (a necessary step for the large deviations lower bound) need to be established. We emphasize that many issues that are present in the infinite dimensional case, are completely absent in finite dimensions.First author draf
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