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Stanislav Molchanov

Publications and source records attributed to Stanislav Molchanov.

At least 37 records · Page 2Linked to original sources
arXiv2017

Stable limit laws and structure of the scaling function for reaction-diffusion in random environment

We prove the emergence of stable fluctuations for reaction-diffusion in random environment with Weibull tails. This completes our work around the quenched to annealed transition phenomenon in this context of reaction diffusion. In [9], we had already considered the model treated here and had studied fully the regimes where the law of large numbers is satisfied and where the fluctuations are Gaussian, but we had left open the regime of stable fluctuations. Our work is based on a spectral approach centered on the classical theory of rank-one perturbations. It illustrates the gradual emergence of the role of the higher peaks of the environments. This approach also allows us to give the delicate exact asymptotics of the normalizing constants needed in the stable limit law.

math.PR↗
arXiv2017

Global stability in a nonlocal reaction-diffusion equation

We study stability of stationary solutions for a class of non-local semilinear parabolic equations. To this end, we prove the Feynman--Kac formula for a Lévy processes with time-dependent potentials and arbitrary initial condition. We propose sufficient conditions for asymptotic stability of the zero solution, and use them to the study of the spatial logistic equation arising in population ecology. For this equation, we find conditions which imply that its positive stationary solution is asymptotically stable. We consider also the case when the initial condition is given by a random field.

math.AP↗
arXiv2016

A multi-class extension of the mean field Bolker-Pacala population model

We extend our earlier mean field approximation of the Bolker-Pacala model of population dynamics by dividing the population into N classes, using a mean field approximation for each class but also allowing migration between classes as well as possibly suppressive influence of the population of one class over another class. For N at least 2, we obtain one symmetric non-trivial equilibrium for the system and give global limit theorems. For N=2, we calculate all equilibrium solutions, which, under additional conditions, include multiple non-trivial equilibria. Lastly, we prove geometric ergodicity regardless of the number of classes when there is no population suppression across the classes.

math.PR↗
arXiv2016

Resolvent bounds for jump generators and ground state asymptotics for nonlocal Schrödinger operators

The paper deals with jump generators with a convolution kernel. Assuming that the kernel decays either exponentially or polynomially we prove a number of lower and upper bounds for the resolvent of such operators. We consider two applications of these results. First we obtain pointwise estimates for principal eigenfunction of jump generators perturbed by a compactly supported potential (so-called nonlocal Schrödinger operators). Then we consider the Cauchy problem for the corresponding inhomogeneous evolution equations and study the behaviour of its solutions.

math-ph↗
arXiv2016

On ground state of non local Schrodinger operators

We study a ground state of a non local Schrodinger operator associated with an evolution equation for the density of population in the stochastic contact model in continuum with inhomogeneous mortality rates. We found a new effect in this model, when even in the high dimensional case the existence of a small positive perturbation of a special form (so-called, small paradise) implies the appearance of the ground state. We consider the problem in the Banach space of bounded continuous functions $C_b (R^d)$ and in the Hilbert space $L^2(R^d)$.

math-ph↗
arXiv2015

Heavy tails and one-dimensional localization

We address the fundamental questions concerning the operator \begin{eqnarray*} H^{θ_0}ψ(x)=-ψ"(x)+V(x,ω)ψ(x),\,ψ(0)\cosθ_0-ψ'(0)\sinθ_0=0. \end{eqnarray*} where the random potential $V$ has a variety of forms. In one example, it is composed of width one bumps of random heights where the square root of the heights are in the domain of attraction of a stable law with index $α\in(0,1)$ or in another it is composed of width one bumps of height one where the distance between bumps is in the domain of attraction of a stable law with index $α\in(0,1).$ We consider the existence of Lyapunov exponents, integrated density of states and the nature of the spectrum of the operator.

math.PR↗
arXiv2015

On the Critical Behavior of a Homopolymers Model

Taking $P^0$ to be the measure induced by simple, symmetric nearest neighbor continuous time random walk on ${\bf{Z^d}}$ starting at $0$ with jump rate $2d$ define, for $β\ge 0,\,t>0,$ the Gibbs probability measure $P_{β,t}$ by specifying its density with respect to $P^0$ as \begin{eqnarray} \frac{dP_{β,t}}{dP^0}=Z_{β,t}(0)^{-1}e^{β\int_0^tδ_0(x_s)ds} \end{eqnarray} where $Z_{β,t}(0)\equiv E^0[e^{β\int_0^tδ_0(x_s)ds}].$ This Gibbs probability measure provides a simple model for a homopolymer with an attractive potential at the origin. In a previous paper \cite{CM07}, we showed that for dimension $d\ge3$ there is a phase transition in the behavior of these paths from diffusive behavior for $β$ below a critical parameter to positive recurrent behavior for $β$ above this critical value. This corresponds to a transition from a diffusive or stretched out phase to a globular phase for the polymer. The critical value was determined by means of the spectral properties of the operator $Δ+βδ_0$ where $Δ$ is the discrete Laplacian on ${\bf{Z^d}}.$ In this paper we give a description of the polymer at the critical value where the phase transition takes place. The behavior at the critical parameter is in some sense midway between the two phases and dimension dependent.

math.PR↗
arXiv2015

Spectral dimension and Bohr's formula for Schrodinger operators on unbounded fractal spaces

We establish an asymptotic formulas for the eigenvalue counting function of the Schrödinger operator $-Δ+V$ for some unbounded potentials $V$ on several types of unbounded fractal spaces. We give sufficient conditions for Bohr's formula to hold on metric measure spaces which admit a cellular decomposition, and then verify these conditions for fractafolds and fractal fields based on nested fractals. In particular, we partially answer a question of Fan, Khandker, and Strichartz regarding the spectral asymptotics of the harmonic oscillator potential on the infinite blow-up of a Sierpinski gasket.

math-ph↗
arXiv2013

A Mean Field Approximation of the Bolker-Pacala Population Model

We approximate the Bolker-Pacala model of population dynamics with the logistic Markov chain and analyze the latter. We find the asymptotics of the degenerated hypergeometric function and use these to prove a local CLT and large deviations result. We also state global limit theorems and obtain asymptotics for the first passage time to the boundary of a large interval.

math.PR↗
arXiv2013

Density behavior of spatial birth-and-death stochastic evolution of mutating genotypes under selection rates

We consider birth-and-death stochastic evolution of genotypes with different lengths. The genotypes might mutate that provides a stochastic changing of lengthes by a free diffusion law. The birth and death rates are length dependent which corresponds to a selection effect. We study an asymptotic behavior of a density for an infinite collection of genotypes. The cases of space homogeneous and space heterogeneous densities are considered.

q-bio.PE↗
arXiv2013

Lyapunov exponent of the random Schrödinger operator with short-range correlated noise potential

We study the influence of disorder on propagation of waves in one-dimensional structures. Transmission properties of the process governed by the Schrödinger equation with the white noise potential can be expressed through the Lyapunov exponent $γ$ which we determine explicitly as a function of the noise intensity σand the frequency ω. We find uniform two-parameter asymptotic expressions for $γ$ which allow us to evaluate $γ$ for different relations between σand ω. The value of the Lyapunov exponent is also obtained in the case of a short-range correlated noise, which is shown to be less than its white noise counterpart.

math-ph↗
arXiv2012

Branching diffusion in the super-critical regime

We investigate the long-time evolution of branching diffusion processes (starting with a single particle) in inhomogeneous media. The qualitative behavior of the processes depends on the intensity of the branching. We analyze the super-critical case, when the total number of particles growing exponentially with positive probability. We study the asymptotics of the number of particles in different regions of space and describe the growth of the region occupied by the particles.

math.PR↗
arXiv2011

The effect of disorder on the wave propagation in one-dimensional periodic optical systems

The influence of disorder on the transmission through periodic waveguides is studied. Using a canonical form of the transfer matrix we investigate dependence of the Lyapunov exponent $γ$ on the frequency $ν$ and magnitude of the disorder $σ$. It is shown that in the bulk of the bands $γ\sim σ^2$, while near the band edges it has the order $γ\sim σ^{2/3}$. This dependence is illustrated by numerical simulations.

math-ph↗
arXiv2009

Explicit parametrix and local limit theorems for some degenerate diffusion processes

For a class of degenerate diffusion processes of rank 2, i.e. when only Poisson brackets of order one are needed to span the whole space, we obtain a parametrix representation of the density from which we derive some explicit Gaussian controls that characterize the additional singularity induced by the degeneracy. We then give a local limit theorem with the usual convergence rate for an associated Markov chain approximation. The key point is that the "weak" degeneracy allows to exploit the techniques first introduced by Konakov and Molchanov and then developed by Konakov and Mammen that rely on Gaussian approximations.

math.PR↗
arXiv2007

Geometric characterization of intermittency in the parabolic Anderson model

We consider the parabolic Anderson problem $\partial_tu=Δu+ξ(x)u$ on $\mathbb{R}_+\times\mathbb{Z}^d$ with localized initial condition $u(0,x)=δ_0(x)$ and random i.i.d. potential $ξ$. Under the assumption that the distribution of $ξ(0)$ has a double-exponential, or slightly heavier, tail, we prove the following geometric characterization of intermittency: with probability one, as $t\to\infty$, the overwhelming contribution to the total mass $\sum_xu(t,x)$ comes from a slowly increasing number of ``islands'' which are located far from each other. These ``islands'' are local regions of those high exceedances of the field $ξ$ in a box of side length $2t\log^2t$ for which the (local) principal Dirichlet eigenvalue of the random operator $Δ+ξ$ is close to the top of the spectrum in the box. We also prove that the shape of $ξ$ in these regions is nonrandom and that $u(t,\cdot)$ is close to the corresponding positive eigenfunction. This is the geometric picture suggested by localization theory for the Anderson Hamiltonian.

math.PR↗
arXiv2007

Transition asymptotics for reaction-diffusion in random media

We describe a universal transition mechanism characterizing the passage to an annealed behavior and to a regime where the fluctuations about this behavior are Gaussian, for the long time asymptotics of the empirical average of the expected value of the number of random walks which branch and annihilate on ${\mathbb Z}^d$, with stationary random rates. The random walks are independent, continuous time rate $2dκ$, simple, symmetric, with $κ\ge 0$. A random walk at $x\in{\mathbb Z}^d$, binary branches at rate $v_+(x)$, and annihilates at rate $v_-(x)$. The random environment $w$ has coordinates $w(x)=(v_-(x),v_+(x))$ which are i.i.d. We identify a natural way to describe the annealed-Gaussian transition mechanism under mild conditions on the rates. Indeed, we introduce the exponents $F_θ(t):=\frac{H_1((1+θ)t)-(1+θ)H_1(t)}θ$, and assume that $\frac{F_{2θ}(t)-F_θ(t)}{θ\log(κt+e)}\to\infty$ for $|θ|>0$ small enough, where $H_1(t):=\log < m(0,t)>$ and $ $ denotes the average of the expected value of the number of particles $m(0,t,w)$ at time $t$ and an environment of rates $w$, given that initially there was only one particle at 0. Then the empirical average of $m(x,t,w)$ over a box of side $L(t)$ has different behaviors: if $ L(t)\ge e^{\frac{1}{d} F_ε(t)}$ for some $ε>0$ and large enough $t$, a law of large numbers is satisfied; if $ L(t)\ge e^{\frac{1}{d} F_ε(2t)}$ for some $ε>0$ and large enough $t$, a CLT is satisfied. These statements are violated if the reversed inequalities are satisfied for some negative $ε$. Applications to potentials with Weibull, Frechet and double exponential tails are given.

math.PR↗
arXiv2007

On bounded solutions of the balanced generalized pantograph equation

The question about the existence and characterization of bounded solutions to linear functional-differential equations with both advanced and delayed arguments was posed in early 1970s by T. Kato in connection with the analysis of the pantograph equation, y'(x)=ay(qx)+by(x). In the present paper, we answer this question for the balanced generalized pantograph equation of the form -a_2 y''(x)+a_1 y'(x)+y(x)=int_0^infty y(qx) m(dq), where a_1 > or = 0, a_2 > or = 0, a_1^2+a_2^2>0, and m is a probability measure. Namely, setting K:=int_0^infty ln(q) m(dq), we prove that if K < or = 0 then the equation does not have nontrivial (i.e., nonconstant) bounded solutions, while if K>0 then such a solution exists. The result in the critical case, K=0, settles a long-standing problem. The proof exploits the link with the theory of Markov processes, in that any solution of the balanced pantograph equation is an L-harmonic function relative to the generator L of a certain diffusion process with "multiplication" jumps. The paper also includes three "elementary" proofs for the simple prototype equation y'(x)+y(x)=(1/2)y(qx)+(1/2)y(x/q), based on perturbation, analytical, and probabilistic techniques, respectively, which may appear useful in other situations as efficient exploratory tools.

math.PR↗