Searcharxiv⌕ Search

arXiv subjects

Mark Freidlin

Publications and source records attributed to Mark Freidlin.

At least 19 recordsLinked to original sources
arXiv2024

Metastability in Parabolic Equations and Diffusion Processes with a Small Parameter

We study diffusion processes in $\mathbb{R}^d$ that leave invariant a finite collection of manifolds (surfaces or points) in $\mathbb{R}^d$ and small perturbations of such processes. Assuming certain ergodic properties at and near the invariant surfaces, we describe the rate at which the process gets attracted to or repelled from the surface, based on the local behavior of the coefficients. For processes that include, additionally, a small non-degenerate perturbation, we describe the metastable behavior. Namely, by allowing the time scale to depend on the size of the perturbation, we observe different asymptotic distributions of the process at different time scales. Stated in PDE terms, the results provide the asymptotics, at different time scales, for the solution of the parabolic Cauchy problem when the operator that degenerates on a collection of surfaces is perturbed by a small non-degenerate term. This asymptotic behavior switches at a finite number of time scales that are calculated and does not depend on the perturbation.

math.PR↗
arXiv2022

Perturbation of Systems with a First Integral: Motion on the Reeb Graph

We consider the long-time behavior of systems close to a system with a smooth first integral. Under certain assumptions, the limiting behavior, to some extent, turns out to be universal: it is determined by the first integral, the deterministic perturbation, and the initial point. Furthermore, it is the same for a broad class of noises. In particular, the long-time behavior of a deterministic system can, in a sense, be stochastic and stochastic systems can have a reduced stochasticity. The limiting distribution is calculated explicitly.

math.PR↗
arXiv2021

Perturbations of Parabolic Equations and Diffusion Processes with Degeneration: Boundary Problems, Metastability, and Homogenization

We study diffusion processes that are stopped or reflected on the boundary of a domain. The generator of the process is assumed to contain two parts: the main part that degenerates on the boundary in a direction orthogonal to the boundary and a small non-degenerate perturbation. The behavior of such processes determines the stabilization of solutions to the corresponding parabolic equations with a small parameter. Metastability effects arise in this case: the asymptotics of solutions, as the size of the perturbation tends to zero, depends on the time scale. Initial-boundary value problems with both the Dirichlet and the Neumann boundary conditions are considered. We also consider periodic homogenization for operators with degeneration.

math.AP↗
arXiv2021

Asymptotics in the Dirichlet Problem for Second Order Elliptic Equations with Degeneration on the Boundary

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.

math.AP↗
arXiv2020

Long Time Influence of Small Perturbations and Motion on the Simplex of Invariant Probability Measures

A general approach to a broad class of asymptotic problems related to long-time influence of small perturbations, of both deterministic and stochastic type, is presented in the paper. The main characteristic of this influence is a limiting motion on the simplex of invariant probability measures of the non-perturbed system in an appropriate time scale. The main tools we use in the paper are limit theorems for large deviations, modified averaging principle, and diffusion approximation.

math.PR↗
arXiv2020

Averaging in the case of multiple invariant measures for the fast system

We consider the averaging principle for deterministic or stochastic systems with a fast stochastic component (family of continuous-time Markov chains depending on the state of the system as a parameter). We show that, due to bifurcations in the simplex of invariant probability measures of the chains, the limiting system should be considered on a graph or on an open book with certain gluing conditions in the vertices of the graph (or on the bifurcation surface).

math.PR↗
arXiv2019

Diffusion in the presence of cells with semi-permeable membranes

We consider processes that coincide with a given diffusion process except on the boundaries of a finite collection of domains. The behavior on each of the boundaries is asymmetric: the process is much more likely to enter the interior of the domain than to enter the interior of its complement, with a small parameter controlling the trapping mechanism. We describe the limiting behavior of the processes. In particular, if the parameters controlling the boundary behavior have different orders of magnitude for different domains or if the domains are nested, metastable distributions between the trapping regions are described.

math.PR↗
arXiv2018

Boris R. Vainberg (on his 80th birthday)

Boris R. Vainberg was born on March 17, 1938, in Moscow. His father was a Lead Engineer in an aviation design institute. His mother was a homemaker. From early age, Boris was attracted to mathematics and spent much of his time at home and in school working through collections of practice problems for the Moscow Mathematical Olympiad. His first mathematical library consisted of the books he received as one of the prize-winners of these olympiads.

math.HO↗
arXiv2018

Front propagation for reaction-diffusion equations in composite structures

We consider asymptotic problems concerning the motion of interface separating the regions of large and small values of the solution of a reaction-diffusion equation in the media consisting of domains with different characteristics (composites). Under certain conditions, the motion can be described by the Huygens principle in the appropriate Finsler (e.g., Riemannian) metric. In general, the motion of the interface has, in a sense, non-local nature. In particular, the interface may move by jumps. We are mostly concerned with the nonlinear term that is of KPP type. The results are based on limit theorems for large deviations.

math.PR↗
arXiv2017

On diffusions in media with pockets of large diffusivity

We consider diffusion processes in media with pockets of large diffusivity. The asymptotic behavior of such processes is described when the diffusion coefficients in the pockets tend to infinity. The limiting process is identified as a diffusion on the space where each of the pockets is treated as a single point, and certain conditions on the behavior of the process on the boundary of the pockets are imposed. Calculation of various probabilities and expectations related to the limiting process leads to new initial-boundary (and boundary) problems for the corresponding parabolic (and elliptic) PDEs.

math.PR↗
arXiv2016

Metastable distributions of Markov chains with rare transitions

In this paper we consider Markov chains with transition rates that depend on a small parameter $\varepsilon$. Under a mild assumption on the asymptotics of these transition rates, we describe the behavior of the chain at various $\varepsilon$-dependent time scales, i.e., we calculate the metastable distributions.

math.PR↗
arXiv2016

On the Smoluchowski-Kramers approximation for SPDEs and its interplay with large deviations and long time behavior

We discuss here the validity of the small mass limit (the so-called Smoluchowski-Kramers approximation) on a fixed time interval for a class of semi-linear stochastic wave equations, both in the case of the presence of a constant friction term and in the case of the presence of a constant magnetic field. We also consider the small mass limit in an infinite time interval and we see how the approximation is stable in terms of the invariant measure and of the large deviation estimates and the exit problem from a bounded domain of the space of square integrable functions.

math.PR↗
arXiv2015

On stochastic perturbations of slowly changing dynamical systems

In this paper we consider a diffusion process obtained as a small random perturbation of a dynamical system attracted to a stable equilibrium point. The drift and the diffusive perturbation are assumed to evolve slowly in time. We describe the asymptotics of the time it takes the process to exit a given domain and the limiting distribution of the exit point.

math.PR↗
arXiv2015

On the behavior of diffusion processes with traps

We consider processes that coincide with a given diffusion process outside a finite collection of domains. In each of the domains, there is, additionally, a large drift directed towards the interior of the domain. We describe the limiting behavior of the processes as the magnitude of the drift tends to infinity, and thus the domains become trapping, with the time to exit the domains being exponentially large. In particular, in exponential time scales, metastable distributions between the trapping regions are considered.

math.PR↗
arXiv2015

Large Deviations for the Langevin equation with strong damping

We study large deviations in the Langevin dynamics, with damping of order $\e^{-1}$ and noise of order $1$, as $\e\downarrow 0$. The damping coefficient is assumed to be state dependent. We proceed first with a change of time and then, we use a weak convergence approach to large deviations and their equivalent formulation in terms of the Laplace principle, to determine the good action functional. Some applications of these results to the exit problem from a domain and to the wave front propagation for a suitable class of reaction diffusion equations are considered.

math.PR↗
arXiv2014

SPDEs on narrow domains and on graphs: an asymptotic approach

We introduce here a class of stochastic partial differential equations defined on a graph and we show how they are obtained as the limit of suitable stochastic partial equations defined in a narrow channel, as the width of the channel goes to zero. To our knowledge, this is the first time an SPDE on a graph is studied.

math.PR↗
arXiv2014

On stochastic perturbations of dynamical systems with a "rough" symmetry. Hierarchy of Markov chains

We consider long-time behavior of dynamical systems perturbed by a small noise. Under certain conditions, a slow component of such a motion, which is most important for long- time evolution, can be described as a motion on the cone of invariant measures of the non- perturbed system. The case of a finite number of extreme points of the cone is considered in this paper. As is known, in the generic case, the long-time evolution can be described by a hierarchy of cycles defined by the action functional for corresponding stochastic processes. This, in particular, allows to study metastable distributions and such effects as stochastic resonance. If the system has some symmetry in the logarithmic asymptotics of transition probabilities (rough symmetry),the hierarchy of cycles should be replaced by a hierarchy of Markov chains and their invariant measures.

math.PR↗