Tomasz Komorowski

Institute of Mathematics · Institute of Mathematics

We consider the long time, large scale behavior of the Wigner transform $W_\eps(t,x,k)$ of the wave function corresponding to a discrete wave equation on a 1-d integer lattice, with a weak multiplicative noise. This model has been introduced in Basile, Bernardin, and Olla to describe a system of interacting linear oscillators with a weak noise that...

We consider a stationary solution of the Poisson equation (λ + Lω)φΛ(x;ω) = -∂*b(x;ω), where λ>0 and Lω is a random, discrete, elliptic operator given by Lωu(x):= ∂* [a(x;ω)∂u(x)], x€Z. Here ∂f(x) := f(x + 1) - f(x) and ∂* f(x) := f(x - 1) - f(x) for an arbitrary function f:Z→R. The coefficients {(a(x;ω),b(x;ω)),x eZ} form a stationary random field...

We consider solutions of the Schrödinger equation with a weak time-dependent random potential. It is shown that when the two-point correlation function of the potential is rapidly decaying, then the Fourier transform [^(z)]e(t,x){\hat\zeta_\epsilon(t,\xi)} of the appropriately scaled solution converges point-wise in ξ to a stochastic complex Gaussi...

Suppose that $\{X_t,\,t\ge0\}$ is a non-stationary Markov process, taking values in a Polish metric space $E$. We prove the law of large numbers and central limit theorem for an additive functional of the form $\int_0^T\psi(X_s)ds$, provided that the dual transition probability semigroup, defined on measures, is strongly contractive in an appropria...

In this note we investigate the asymptotic behavior of the solutions of the heat equation with random, fast oscillating potential $$\begin{array}{rcl} \partial_tu_{\varepsilon}(t,x)&=&\dfrac12\Delta_xu_{\varepsilon}(t,x)+{\varepsilon}^{-\gamma}V\left(\dfrac{x}{{\varepsilon}}\right)u_{\varepsilon}(t,x),\,(t,x)\in(0,+\infty)\times{\mathbb R}^d, \\ u_...

We consider fluctuations of the solution W ε (t, x, k) of the Wigner equation which describes energy evolution of a solution of the Schrödinger equation with a random white noise in time potential. The expectation of W ε (t, x, k) converges as ε → 0 to \({\bar{W}(t,x,k)}\) which satisfies the radiative transport equation. We prove that when the ini...

Consider a Markov chain $\{X_n\}_{n\ge 0}$ with an ergodic probability measure $\pi$. Let $\Psi$ a function on the state space of the chain, with $\alpha$-tails with respect to $\pi$, $\alpha\in (0,2)$. We find sufficient conditions on the probability transition to prove convergence in law of $N^{1/\alpha}\sum_n^N \Psi(X_n)$ to a $\alpha$-stable la...

In this paper we consider an additive functional of an observable $V(x)$ of a Markov jump process. We assume that the law of the expected jump time $t(x)$ under the invariant probability measure $\pi$ of the skeleton chain belongs to the domain of attraction of a subordinator. Then, the scaled limit of the functional is a Mittag-Leffler proces, pro...

In this note we consider a passive tracer model describing particle dispersion in a turbulent flow. The trajectory of the particle is given by the solution of an ordinary differential equation , , where is a divergence-free, random vector field that is spatially homogeneous and isotropic. We show that trajectories of the tracer display superdiffusi...

We formulate a criterion for the existence and uniqueness of an invariant measure for a Markov process taking values in a Polish phase space. In addition, weak-$^*$ ergodicity, that is, the weak convergence of the ergodic averages of the laws of the process starting from any initial distribution, is established. The principal assumptions are the ex...

We consider a nearest neighbor, symmetric random walk on a homogeneous, ergodic random lattice Zd. The jump rates of the walk are independent, identically Bernoulli distributed random variables indexed by the bonds of the lattice. A standard result from the homogenization theory, see [A. De Masi, P.A. Ferrari, S. Goldstein, W.D. Wick, An invariance...

Suppose that the trajectory of a particle x(t; x, k) is a solution of the Newton equation \( \ddot x\left( {t;x{\text{,}}k} \right) = \delta ^{1/2} F\left( {x\left( {t;x{\text{,}}k} \right),\dot x\left( {t;x{\text{,}}k} \right)} \right),{\mathbf{ }}x\left( {\text{0}} \right) = x,\dot x\left( 0 \right) = k \) , where F(x, k) is a spatially homogeneo...

We show that the effective diffusivity of a random diffusion with a drift is a continuous function of the drift coefficient. In fact, in the case of a homogeneous and isotropic random environment the function is C∞ smooth outside the origin. We provide a one-dimensional example which shows that the diffusivity coefficient need not be differentiable...

In this paper we consider the motion of a tracer in a flow that is locally self-similar and whose correlations decay at infinity but at the rate that does not guarantee that the flow does not have "memory eect". We show that when the field is Gaussian the appropriately regularized scaling limit of the trajectory is a super-diusive fractional Browni...

We consider the solution of the equation r(t) = W(r(t)), r(0) = r 0 > 0 where W(⋅) is a fractional Brownian motion (f.B.m.) with the Hurst exponent α∈ (0,1). We show that for almost all realizations of W(⋅) the trajectory reaches in finite time the nearest equilibrium point (i.e. zero of the f.B.m.) either to the right or to the left of r 0, depend...

We consider the motion of a particle in a two-dimensional spatially homogeneous mixing potential and show that its momentum converges to the Brownian motion on a circle. This complements the limit theorem of Kesten and Papanicolaou (1980) proved in dimensionsd≥3.

We consider the movement of a particle advected by a random flow of the form v + δF(x), with a constant drift, F(x), the fluctuation, given by a zero mean, stationary random field and δ 1 so that the drift dominates over the fluctuation. The two-point correlation matrix R(x) of the random field decays as |x|2α−2, as |x| → +∞ with α < 1. The Kubo f...

We consider the motion of a particle governed by a weakly random Hamiltonian flow. We identify temporal and spatial scales on which the particle trajectory converges to a spatial Brownian motion. The main technical issue in the proof is to obtain error estimates for the convergence of the solution of the stochastic acceleration problem to a momentu...

We study the asymptotic behavior of an inertial tracer particle in a random force field. We show that there exists a probability measure, under which the process describing the velocity and environment seen from the vantage point of the moving particle is stationary and ergodic. This measure is equivalent to the underlying probability for the Euler...

We consider a tracer particle performing a nearest neighbor random walk on in dimension d[greater-or-equal, slanted]3 with random jump rates. This kind of a walk models the motion of a charged particle under a constant external electric field. We assume that the jump rates admit only two values 0<[gamma]-<[gamma]+<+[infinity], representing the lowe...

We study the transport of a passive tracer particle in a steady strongly mixing flow with a nonzero mean velocity. We show that there exists a probability measure under which the particle Lagrangian velocity process is stationary. This measure is absolutely continuous with respect to the underlying probability measure for the Eulerian flow.

In this paper we rigorously establish the existence of the mobility coefficient for a tagged particle in a simple symmetric exclusion process with adsorption/desorption of particles, in a presence of an external force field interacting with the particle. The proof is obtained using a perturbative argument. In addition, we show that, for a constant...

The trajectories of a passive tracer in a turbulent flow satisfy the ordinary differential equation x(t)=V(t,x(t)), where V(t,x) is a stationary random field, the so-called Eulerian velocity. It is a nontrivial question to define the dynamics of the tracer in the case when the realizations of the Eulerian field are only spatially Hlder regular beca...

We consider the passive scalar transport in an incompressible random flow. Our basic result is a proof of the convergence of a certain numerical scheme for the computation of the eddy diffusivity tensor. The scheme leads to the formula for the diffusivity expressed in terms of an infinite series. We give a rigorous proof of the geometric bounds on...

We study the model of a motion of a passive tracer particle in a turbulent flow that is strongly mixing in time variable. In (8) we have shown that there exists a probability measure equivalent to the underlying physical probability under which the quasi-Lagrangian velocity process - i.e. the velocity of the flow observed from the vintage point of...

In the present article we consider a continuous time random walk on an anisotropic random,lattice. We show the existence of a steady state ¯„fi for the environment process (‡(t))t‚0 corresponding to the walk. This steady state has the property that the ergodic averages of (F(‡(t)))t‚0, where F is local (i.e. depends on finitely many bonds of the la...

We establish conditions for the frozen path approximation for turbulent transport in a class of nonmixing Gaussian flows with long-range correlation. We identify the regimes of fractional Brownian motion limit as well as the Brownian motion limit.

We establish the self-averaging properties of the Wigner transform of a mixture of states in the regime when the correlation length of the random medium is much longer than the wave length but much shorter than the propagation distance. The main ingredients in the proof are the error estimates for the semiclassical approximation of the Wigner trans...

We consider random walk in a random environment on d-dimensional integer lattice with a uniform local drift. The environment is described as a stationary field of random vectors , , |e|=1 taking values in the standard 2d-dimensional simplex with the support of the law of lying on one side of a certain hyperplane (the so-called non-nestling conditio...

We consider random walk in a random environment on d-dimensional integer lattice with a uniform local drift. The environment is described as a stationary field of random vectors , , |e|=1 taking values in the standard 2d-dimensional simplex with the support of the law of lying on one side of a certain hyperplane (the so-called non-nestling conditio...

We consider a class of two-fold stochastic random walks in a random environ-ment. The transition probability is given by an ergodic random field on Z d with two-fold stochastic realizations. The central limit theorem for this class of random walks has been claimed by Kozlov under certain strong mixing conditions (cf. [4], Theorem 3, p. 121). Howeve...

We consider a model of motion of a passive tracer partitcle under a random, non-steady (time dependent), incompressible velocity flow in a medium with positive molecular diffusivity. We show the existence of the effective diffusivity tensor for the flow provided that its relaxation time is sufficiently small. In contrast to the previous papers by C...

We establish the self-averaging properties of the Wigner transform of a mixture of states in the regime when the correlation length of the random medium is much longer than the wave length but much shorter than the propagation distance. The main ingredients in the proof are the error estimates for the semiclassical approximation of the Wigner trans...

In the present article we consider a motion of a passive tracer particle, whose trajectory satisfies the It stochastic differential equation d x(t) = V(t,x(t)) dt + Ö{2k}\sqrt {2\kappa} d w(t), where w() is a Brownian motion, V is a stationary Gaussian random field with incompressible realizations independent of w() and >0. We prove the superdiff...

We study a diffusion process with a molecular diffusion and random Markovian-Gaussian drift for which the usual (spatial) Peclet number is infinite. We introduce a temporal Peclet number and we prove that, under the finiteness of the temporal Peclet number, the laws of diffusions under the diffusive rescaling converge weakly to the law of a Brownia...

Let V(t,x), be a time-space stationary d-dimensional Markovian and Gaussian random field given over a probability space . Consider a diffusion with a random drift given by the stochastic differential equation , x(0)=0, where w(·) is a standard d-dimensional Brownian motion defined over another probability space . The so-called Lagrangian process,...

We study transport of a passive tracer particle in a time dependent turbulent flow in the medium with positive molecular diffusivity. We show that there exists then a probability measure equivalent to the underlying physical probability, corresponding to the Eulerian velocity field, under which the particle Lagrangian velocity observations are stat...

We formulate a stochastic differential equation describing the Lagrangian environment process of a passive tracer in Ornstein-Uhlenbeck velocity fields. We subsequently prove a local existence and uniqueness result when the velocity field is regular. When the Ornstein-Uhlenbeck velocity field is only spatially Hölder continuous we construct and ide...

We study the transport of a passive tracer particle by a random d-dimensional, Gaussian, compressible velocity field. It is well known, since the work of Lumley, see [13], and Port and Stone, see [20], that the observations of the velocity field from the moving particle, the so-called Lagrangian velocity process, are statistically stationary when t...

Resumen Resumen We studythe passive scalar transport in a class of nonmixing Markovian flows with power-law spectra and correlation times. We establish anew diffusion regime under an optimal condition (convergent Kubo formula)on the spatial/temporal structure of this family of flows. Under<sup> </sup...

We prove the law of large numbers for random walks in random environments on the d-dimensional integer lattice Z d . The environment is described in terms of a stationary random field of transition probabilities on the lattice, possessing a certain drift property, modeled on the Kalikov condition. In contrast to the previously considered models, we...

We prove the law of large numbers for random walks in random environments on the d-dimensional integer lattice Z^d. The environment is described in terms of a stationary random field of transition probabilities on the lattice, possessing a certain drift property, modeled on the Kalikov condition. In contrast to the previously considered m...

We study a diffusion with a random, time dependent drift. We prove the invariance principle when the spectral measure of the drift satisfies a certain integrability condition. This result generalizes the results of [13, 7]. 1.

In this paper we present the functional central limit theorem for a class of Markov processes, whose L2-generator satisfies the so-called graded sector condition. We apply the result to obtain homogenization theorems for certain classes of diffusions with a random Gaussian drift. Additionally, we present a result concerning the regularity of the ef...

We consider the asymptotic behavior of the solutions ofscaled convection-diffusion equations ∂ t u ɛ (t, x) = κΔ x (t, x) + 1/ɛV(t/ɛ2,xɛ) ·∇ x u ɛ (t, x) with the initial condition u ɛ(0,x) = u 0(x) as the parameter ɛ↓ 0. Under the assumptions that κ > 0 and V(t, x), (t, x) ∈R d is a d-dimensional,stationary, zero mean, incompressible, Gaussia...

W show that the motion of a particle advected by a random Gaussian velocity field with long-range correlations converges to a fractional Brownian motion in the long time limit.

We study transport in random undirectional wave-like velocity fields with nonlinear dispersion relations. For this simple model, we have several interesting findings: (1) In the absence of molecular diffusion the entire family of fractional Brownian motions (FBMs), persistent or anti-persistent, can arise in the scaling limit. (2) The infrared cuto...

In this talk we discuss the motion of a Brownian particle among random, Poisson like scattered traps. We shall first describe the asymptotic of the probability of survival of the particle among the traps in long times both for the typical configuration of the environment (the so called quenched asymptotic) and for the averaged case (the annealed as...

We prove the Taylor-Kubo formula for a class of isotropic, non-mixing flows with long-range correlation. For the proof, we develop the method of high-order correctors expansion.

We establish diffusion and fractional Brownian motion approximations for motions in a Markovian Gaussian random field with a nonzero mean.

We prove turbulent diffusion theorems for Markovian velocity fields which either are mixing in time or have stationary vector potentials.

Passive scalar motion in a family of random Gaussian velocity fields with long-range correlations is shown to converge to persistent fractional Brownian motions in long times.

We prove the Taylor-Kubo formula for a class of isotropic, non-mixing flows with long-range correlation. For the proof, we develop the method of high order correctors expansion.

We prove an almost sure invariance principle for diffusion driven by velocities with unbounded stationary vector potentials. The result generalizes to multiple particles motion, driven by a common velocity field and independent molecular Brownian motions.

We prove that the solution of a system of random ordinary differential equations dX(t) dt = V(t; X(t)) with diffusive scaling, X " (t) = "X( t " 2 ), converges weakly to a Brownian Motion when " # 0. We assume that V(t; x), t 2 R, x 2 R d is a d-dimensional, random, incompressible, stationary Gaussian field which has mean zero and decorrelates in f...

In this paper we deal with the solutions of Itô stochastic differential equationfor a small parameter [epsilon]. We prove that for 0[less-than-or-equals, slant][alpha]<1 and V a divergence-free, Gaussian random field, sufficiently strongly mixing in t variable the family of processes {X[epsilon](t)}t[greater-or-equal, slanted]0, [epsilon]>0 converg...

We study the asymptotic behavior of Brownian motion in steady, unbounded incompressible random flows. We prove an invariance principle for almost all realizations of random flows. The key compactness result is obtained by Moser’s iterative scheme in PDE theory.

In this paper we prove several theorems concerning the motion of a particle in a random environment. The trajectory of a particle is the solution of the differential equation $dx(t)/dt=V(x(t))$, where $V(x) = v + \varepsilon^{1-\alpha} F(x), \; 0\leq\alpha\leq 1,$ $v$ is a constant vector, $F$ is a mean-zero fluctuation field and $\varepsilon^{1-\a...

We investigate asymptotic behavior of a Markov chain given by a difference stochastic equation X n+1 =S(X n )+ξ n . We prove asymptotic periodicity of the Markov chain under the following assumptions: (H1) S:V→V is a Borel measurable transformation defined on a cone V⊆ℝ d , bounded on bounded subsets of V, (H2) there is a norm |·| defined in ℝ d su...

A class of Markov operators having a sweeping property is studied. Such operators appear in ergodic theory and applied mathematics.

Transport of a passive agent by turbulent flow is often modelled by convection- diusion equation with random coecients. Passing from the microscopic to macroscopic description of the phenomenon one can obtain in the limit the eective law of transport. We review the known results on the subject using perturbative as well as non-perturbative argu- me...

We present here some basic facts and problems connected with the so- called minimal displacement problem i.e. the problem of evaluation of the quantity inf∥x-Tx∥, where x varies over a closed bounded subset of a Banach space and T is a Lipschitzian mapping. Strictly connected with this is also the problem of finding a Lipschitzian retraction of the...

In this paper we rigorously establish the existence of the mobility coefficient for a tagged particle in a simple exclusion process with adsorption/desorption in a presence of an external forced field interacting with the particle. The proof is obtained using a perturbative argument. In addition, we show that for a constant external field the mobil...

Typescript. Thesis (Ph.D.)--New York University, Graduate School of Arts and Science, 1994. Includes bibliographical references (leaves: 68-70)

A paraître dans la série Grundlehren der mathematischen Wissenschaften. La version attachée est une version préliminaire de cet ouvrage. Les références des chapitres ne sont pas intégrées à ces versions, à l'exception du chapitre 2 dont les références sont rassemblées dans un fichier spécifique. oui