Searcharxiv⌕ Search

arXiv subjects

Leonid Koralov

Publications and source records attributed to Leonid Koralov.

At least 19 recordsLinked to original sources
arXiv2026

A Criterion for Convergence of Diffusion Processes on Graphs

In this paper, we consider diffusion processes on a graph. On the $i$-th edge, each process is governed by an operator of the form $D_{v_i}D^+_{u_i}$, and the behavior at the vertices is determined by the gluing conditions. The paper characterizes the weak convergence of a family of such processes to a limiting process in terms of the behavior of the scale and speed functions $u_i^n$ and $v_i^n$ and the parameters of the gluing conditions. This may be viewed as a generalization of Freidlin and Wentzell's result on the weak convergence of a family of processes on a line.

math.PR↗
arXiv2026

Limiting Behavior of Randomly Perturbed Diffusions with Invariant Repelling Surfaces

We study small perturbations of diffusion processes in $\mathbb{R}^d$ that leave invariant a finite collection of hypersurfaces. Each surface is assumed to be repelling for the unperturbed process, and the unperturbed motion on each of the surfaces is assumed to be ergodic. These surfaces separate the space into a finite number of domains, each of which carries an invariant measure of the unperturbed process. We describe the asymptotics of the densities of the invariant measures near the invariant surfaces. We then describe the asymptotic behavior of the perturbed process: at different time scales (depending on the size of the perturbation), metastable distributions are described in terms of linear combinations of the ergodic invariant measures of the unperturbed system. The coefficients in the linear combination depend on the time scale but are shown not to depend on the perturbation.

math.PR↗
arXiv2026

Distribution of particles near the front in supercritical branching Brownian motion with compactly supported branching

We investigate the long-time behavior of a $d-$dimensional supercritical branching Brownian motion with a compactly supported branching potential. It is known that, for $\mathbf{v}\in \mathbb{R}^d$, all the moments of the normalized number of particles in a bounded domain centered at $\mathbf{v} t$ converge, as $t \rightarrow \infty$, provided that $\|\mathbf{v}\|$ is strictly less than the asymptotic speed of the front. The limiting distribution does not depend on $\mathbf{v}$. Using sharp asymptotics for the solutions of parabolic PDEs with compact potential, we prove that the normalized number of particles in a bounded time-dependent domain located near the front converges in distribution and with all the moments. The limit, however, now depends on the asymptotic location of the domain.

math.PR↗
arXiv2025

Spectral properties of operators and wave propagation in high-contrast media

The paper aims to study the spectral properties of elliptic operators with highly inhomogeneous coefficients and related issues concerning wave propagation in high-contrast media. A unified approach to solving problems in bounded domains with Dirichlet or Neumann boundary conditions, as well as in infinite periodic media, is proposed. For a small parameter $\varepsilon > 0$ characterizing the contrast of the components of the medium, the analyticity of the eigenvalues and eigenfunctions is established in a neighborhood of $\varepsilon = 0$. Effective operators corresponding to $\varepsilon = 0$ are described.

math.AP↗
arXiv2025

Energy growth for systems of coupled oscillators with partial damping

We consider two interacting particles on the circle. The particles are subject to stochastic forcing, which is modeled by white noise. In addition, one of the particles is subject to friction, which models energy dissipation due to the interaction with the environment. We show that, in the diffusive limit, the absolute value of the velocity of the other particle converges to the reflected Brownian motion. In other words, the interaction between the particles are asymptotically negligible in the scaling limit. The proof combines averaging for large energies with large deviation estimates for small energies.

math.PR↗
arXiv2024

Metastable Distributions of Semi-Markov Processes

In this paper, we consider semi-Markov processes whose transition times and transition probabilities depend on a small parameter $\varepsilon$. Understanding the asymptotic behavior of such processes is needed in order to study the asymptotics of various randomly perturbed dynamical and stochastic systems. The long-time behavior of a semi-Markov process $X^\varepsilon_t$ depends on how the point $(1/\varepsilon, t(\varepsilon))$ approaches infinity. We introduce the notion of complete asymptotic regularity (a certain asymptotic condition on transition probabilities and transition times), originally developed for parameter-dependent Markov chains, which ensures the existence of the metastable distribution for each initial point and a given time scale $t(\varepsilon)$. The result may be viewed as a generalization of the ergodic theorem to the case of parameter-dependent semi-Markov processes.

math.PR↗
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↗
arXiv2023

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

Asymptotic behavior of branching diffusion processes in periodic media

We study the asymptotic behavior of branching diffusion processes in periodic media. For a super-critical branching process, we distinguish two types of behavior for the normalized number of particles in a bounded domain, depending on the distance of the domain from the region where the bulk of the particles is located. At distances that grow linearly in time, we observe intermittency (i.e., the $k$-th moment dominates the $k$-th power of the first moment for some $k$), while, at distances that grow sub-linearly in time, we show that all the moments converge. A key ingredient in our analysis is a sharp estimate of the transition kernel for the branching process, valid up to linear in time distances from the location of the initial particle.

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

Multi-type branching processes with time dependent branching rates

Under mild non-degeneracy assumptions on branching rates in each generation, we provide a criterion for almost-sure extinction of a multi-type branching process with time-dependent branching rates. We also provide a criterion for the total number of particles (conditioned on survival and divided by the expectation of the resulting random variable) to approach an exponential random variable as time goes to infinity.

math.PR↗
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

A fractional kinetic process describing the intermediate time behaviour of cellular flows

This paper studies the intermediate time behaviour of a small random perturbation of a periodic cellular flow. Our main result shows that on time scales shorter than the diffusive time scale, the limiting behaviour of trajectories that start close enough to cell boundaries is a fractional kinetic process: A Brownian motion time changed by the local time of an independent Brownian motion. Our proof uses the Freidlin-Wentzell framework, and the key step is to establish an analogous averaging principle on shorter time scales. As a consequence of our main theorem, we obtain a homogenization result for the associated advection-diffusion equation. We show that on intermediate time scales the effective equation is a fractional time PDE that arises in modelling anomalous diffusion.

math.PR↗