We consider fast oscillating random perturbations of dynamical systems in regions where one can introduce action-angle-type coordinates. In an appropriate time scale, the evolution of first ...integrals, under the assumption that the set of resonance tori is small enough, is approximated by a diffusion process. If action-angle coordinates can be introduced only piece-wise, the limiting diffusion process should be considered on an open-book space. Such a process can be described by differential operators, one in each page, supplemented by some gluing conditions at the binding of the open book.
This third edition has been substantially revised and updated, with fresh chapters and augmented bibliographical references. It remains a very detailed and profound mathematical treatment of the ...long-term behavior of randomly perturbed dynamical systems.
This book discusses some aspects of the theory of partial differential equations from the viewpoint of probability theory. It is intended not only for specialists in partial differential equations or ...probability theory but also for specialists in asymptotic methods and in functional analysis. It is also of interest to physicists who use functional integrals in their research. The work contains results that have not previously appeared in book form, including research contributions of the author.
The diffusion process in a region
governed by the operator
inside the region and undergoing instantaneous co-normal reflection at the boundary is considered. We show that the slow component of this ...process converges to a diffusion process on a certain graph corresponding to the problem. This allows to find the main term of the asymptotics for the solution of the corresponding Neumann problem in
G
. The operator
is, up to the factor
ε
− 1
, the result of small perturbation of the operator
. Our approach works for other operators (diffusion processes) in any dimension if the process corresponding to the non-perturbed operator has a first integral, and the
ε
-process is non-degenerate on non-singular level sets of this first integral.
Asymptotic problems for classical dynamical systems, stochastic processes, and PDEs can lead to stochastic processes and differential equations on spaces with singularities. We consider the averaging ...principle for systems with conservation laws perturbed by small noise, where, after a change of time scale, the limiting slow motion is a diffusion process on a space which is called in topology an open book: the space consisting of a number of
n-dimensional manifold pieces (pages) that are glued together, sometimes several at a time, at the “binding”, which is made up of manifolds of lower dimension. A diffusion process on such a space is determined by differential operators governing the process inside the pages, and gluing conditions, which determine its behavior after hitting the binding.
We prove weak convergence of measures in the function space that correspond to the slow-motion process in our averaging problem, and calculate the characteristics of the limiting process.
We study small perturbations of the Dirichlet problems for second order elliptic equations that degenerate on the boundary. The limit of the solution, as the perturbation tends to zero, is ...calculated. The result is based on a certain asymptotic self-similarity near the boundary, which holds in the generic case. In the last section, we briefly consider the stabilization of solutions to the corresponding parabolic equations with a small parameter. Metastability effects arise in this case: the asymptotics of the solution depends on the time scale. Initial-boundary value problem with the Neumann boundary condition is discussed in the last section as well.
This book discusses some aspects of the theory of partial differential equations from the viewpoint of probability theory. It is intended not only for specialists in partial differential equations or ...probability theory but also for specialists in asymptotic methods and in functional analysis. It is also of interest to physicists who use functional integrals in their research. The work contains results that have not previously appeared in book form, including research contributions of the author.
For a wide class of dynamical systems perturbed by a random noise, we describe the deterministic component of the long-time evolution of the perturbed system. In particular, for any initial point and ...for a given timescale, the metastable state can be defined. Stochastic resonance is the result of the change of the metastable state if a relatively small and slowly changing deterministic perturbation is added to the system. If this perturbation is periodic, then under certain assumption, the system will perform a motion which is close to a large amplitude oscillation with the same period or with a period proportional to the period of determinisitic perturbation. All these effects are manifestations of the laws of the large deviations for the perturbed system.
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