Tomasz Komorowski - Academia.edu (original) (raw)

Papers by Tomasz Komorowski

arXiv (Cornell University), Jul 18, 2023

We consider a chain consisting of n+1 pinned harmonic oscillators subjected on the right to a tim... more We consider a chain consisting of n+1 pinned harmonic oscillators subjected on the right to a time dependent periodic force F(t) while Langevin thermostats are attached at both endpoints of the chain. We show that for long times the system is described by a Gaussian measure whose covariance function is independent of the force, while the means are periodic. We compute explicitly the work and energy due to the periodic force for all n including n → ∞.

arXiv (Cornell University), Sep 29, 2022

We consider the limit of solutions of scaled linear kinetic equations with a reflectiontransmissi... more We consider the limit of solutions of scaled linear kinetic equations with a reflectiontransmission-absorption condition at the interface. Both the coefficient describing the probability of absorption and the scattering kernel degenerate. We prove that the long-time, large-space limit is the unique solution of a version of the fractional in space heat equation that corresponds to the Kolmogorov equation for a symmetric stable process, which is reflected, or transmitted while crossing the interface and is killed upon the first hitting of the interface. The results of the paper are related to the work in [19], where the case of a non-degenerate probability of absorption has been considered.

Annales Henri Poincaré

We consider an infinite chain of coupled harmonic oscillators with a Poisson thermostat at the or... more We consider an infinite chain of coupled harmonic oscillators with a Poisson thermostat at the origin. In the high frequency limit, we establish the reflection-transmission-scattering coefficients for the wave energy scattered off the thermostat. Unlike the case of the Langevin thermostat [5], in the macroscopic limit the Poissonian thermostat scattering generates a continuous cloud of waves of frequencies different from that of the incident wave.

arXiv (Cornell University), Apr 28, 2021

In this paper, we study the KPZ equation on the torus and derive Gaussian fluctuations in large t... more In this paper, we study the KPZ equation on the torus and derive Gaussian fluctuations in large time.

Studia Mathematica, 2022

We consider a massless tracer particle moving in a random, stationary, isotropic and divergence f... more We consider a massless tracer particle moving in a random, stationary, isotropic and divergence free velocity field. We identify a class of fields, for which the limit of the laws of appropriately scaled tracer trajectory processes is non-Gaussian but a Rosenblatt type process.

Consider a Markov chain {Xn}n≥0 with an ergodic probability measure π. Let Ψ be a function on the... more Consider a Markov chain {Xn}n≥0 with an ergodic probability measure π. Let Ψ be a function on the state space of the chain, with α-tails with respect to π, α ∈ (0,2). We find sufficient conditions on the probability transition to prove convergence in law of N ∑N n Ψ(Xn) to an α-stable law. A “martingale approximation” approach and a “coupling” approach give two different sets of conditions. We extend these results to continuous time Markov jump processes Xt, whose skeleton chain satisfies our assumptions. If waiting times between jumps have finite expectation, we prove convergence of N ∫ Nt

The Annals of Probability, 2020

We consider the limit of a linear kinetic equation, with reflection-transmissionabsorption at an ... more We consider the limit of a linear kinetic equation, with reflection-transmissionabsorption at an interface, with a degenerate scattering kernel. The equation arise from a microscopic chain of oscillators in contact with a heat bath. In the absence of the interface, the solutions exhibit a superdiffusive behavior in the long time limit. With the interface, the long time limit is the unique solution of a version of the fractional in space heat equation, with reflection-transmission-absorption at the interface. The limit problem corresponds to a certain stable process that is either absorbed, reflected, or transmitted upon crossing the interface.

Archive for Rational Mechanics and Analysis, 2020

SIAM Journal on Mathematical Analysis, 2018

We consider a semi-linear advection equation driven by a highly-oscillatory space-time Gaussian r... more We consider a semi-linear advection equation driven by a highly-oscillatory space-time Gaussian random field, with the randomness affecting both the drift and the nonlinearity. In the linear setting, classical results show that the characteristics converge in distribution to a homogenized Brownian motion, hence the point-wise law of the solution is close to a functional of the Brownian motion. Our main result is that the nonlinearity plays the role of a random diffeomorphism, and the point-wise limiting distribution is obtained by applying the diffeomorphism to the limit in the linear setting.

Archive for Rational Mechanics and Analysis, 2017

We consider the Schrödinger equation with a time-independent weakly random potential of a strengt... more We consider the Schrödinger equation with a time-independent weakly random potential of a strength ε ≪ 1, with Gaussian statistics. We prove that when the initial condition varies on a scale much larger than the correlation length of the potential, the compensated wave function converges to a deterministic limit on the time scale t ∼ ε −2. This is shown under the sharp assumption that the correlation function R(x) of the random potential decays slower than 1/|x| 2 , which ensures that the effective potential is finite. When R(x) decays slower than 1/|x| 2 we establish an anomalous diffusive behavior for the averaged wave function on a time scale shorter than ε −2 , as long as the initial condition is "sufficiently macroscopic". We also consider the kinetic regime when the initial condition varies on the same scale as the random potential and obtain the limit of the averaged wave function for potentials with the correlation functions decaying faster than 1/|x| 2. We use random potentials of the Schonberg class which allows us to bypass the oscillatory phase estimates.

Nonlinearity, 2016

We consider a one dimensional infinite acoustic chain of harmonic oscillators whose dynamics is p... more We consider a one dimensional infinite acoustic chain of harmonic oscillators whose dynamics is perturbed by a random exchange of velocities, such that the energy and momentum of the chain are conserved. Consequently, the evolution of the system has only three conserved quantities: volume, momentum and energy. We show the existence of two space-time scales on which the energy of the system evolves. On the hyperbolic scale (tǫ −1 , xǫ −1) the limits of the conserved quantities satisfy a Euler system of equations, while the thermal part of the energy macroscopic profile remains stationary. Thermal energy starts evolving at a longer time scale, corresponding to the superdiffusive scaling (tǫ −3/2 , xǫ −1) and follows a fractional heat equation. We also prove the diffusive scaling limit of the Riemann invariants-the so called normal modes, corresponding to the linear hyperbolic propagation.

Probability Theory and Related Fields, 2001

We consider the asymptotic behavior of the solutions of scaled convectiondiffusion equations ∂ t ... more We consider the asymptotic behavior of the solutions of scaled convectiondiffusion equations ∂ t u ε (t, x) = κ x u ε (t, x) + 1/εV(t/ε 2 , x/ε) • ∇ x u ε (t, x) with the initial condition u ε (0, x) = u 0 (x) as the parameter ε ↓ 0. Under the assumptions that κ > 0 and V(t, x), (t, x) ∈ R d is a d-dimensional, stationary, zero mean, incompressible, Gaussian random field, Markovian and mixing in t we show that the laws of u ε (t, •), t ≥ 0 in an appropriate functional space converge weakly, as ε ↓ 0, to a δ-type measure concentrated on a solution of a certain constant coefficient heat equation.

Communications in Mathematical Physics, 2015

We consider a one dimensional infinite chain of harmonic oscillators whose dynamics is perturbed ... more We consider a one dimensional infinite chain of harmonic oscillators whose dynamics is perturbed by a stochastic term conserving energy and momentum. We prove that in the unpinned case the macroscopic evolution of the energy converges to the solution of the fractional diffusion equation ∂ t u = −| | 3/4 u. For a pinned system we prove that its energy evolves diffusively, generalizing some results of Basile and Olla (J.

The Annals of Applied Probability, 1999

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

Kinetic & Related Models, 2010

2.1.3. Random flows with spatial-temporal dependence. Let us now turn to a more complex situation... more 2.1.3. Random flows with spatial-temporal dependence. Let us now turn to a more complex situation when the random flow in (2.1) depends both on time and space: ∂φ ∂t + εV (t, x) • ∇φ = 0, φ(0, x) = φ 0 (x), (2.12)

Stochastic Processes and their Applications, 2005

We consider a tracer particle performing a continuous time nearest neighbor random walk on Z d in... more We consider a tracer particle performing a continuous time nearest neighbor random walk on Z d in dimension d ≥ 3 with random jump rates. The kind of a walk considered here models the motion of an electrically charged particle under a constant external electric field. We prove the existence of the mobility coefficient, and that it equals to the diffusivity coefficient of the particle.

Probability Theory and Related Fields, 2001

We study a diffusion with a random, time dependent drift. We prove the invariance principle when ... more 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]. ‫ޒ‬ d V(t, x; ω) • ∇ x ϕ(x) dx = 0 P − a.s., with P the underlying probability measure. In order to guarantee the existence of the solution of (1.1) we will also assume that V(t, x; ω) is (P −a.s.) locally Lipschitz in x. We are interested in proving an invariance principle for x(t), i.e. the convergence in distribution of the process εx(ε −2 t) to a Brownian motion with a certain co-variance matrix, sometimes referred to as the effective diffusivity, D ≥ 2I. This problem has been widely studied under various conditions on the random flow. Typically the flow is assumed to be the divergence of a stationary random anti-symmetric matrix valued field H(t, x; ω) = {H p,q (t, x; ω)}-the so-called stream matrix: V(t, x; ω) = ∇ x • H(t, x; ω) .

Nonlinear Differential Equations and Applications NoDEA, 2013

We study the principal Dirichlet eigenvalue of the operator LA = Δ α/2 + Ab(x) • ∇, on a bounded ... more We study the principal Dirichlet eigenvalue of the operator LA = Δ α/2 + Ab(x) • ∇, on a bounded C 1,1 regular domain D. Here α ∈ (1, 2), Δ α/2 is the fractional Laplacian, A ∈ R, and b is a bounded d-dimensional divergence-free vector field in the Sobolev space W 1,2d/(d+α) (D). We prove that the eigenvalue remains bounded, as A → +∞, if and only if b has non-trivial first integrals in the domain of the quadratic form of Δ α/2 for the Dirichlet condition.

Journal of Statistical Physics, 2012

We consider the long time, large scale behavior of the Wigner transform W (t, x, k) of the wave f... more We consider the long time, large scale behavior of the Wigner transform W (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 et al. in Phys. Rev. Lett. 96 (2006) to describe a system of interacting linear oscillators with a weak noise that conserves locally the kinetic energy and the momentum. The kinetic limit for the Wigner transform has been shown in Basile et al. in Arch. Rat. Mech. 195(1):171-203 (2009). In the present paper we prove that in the unpinned case there exists γ 0 > 0 such that for any γ ∈ (0, γ 0 ] the weak limit of W (t/ 3/2γ , x/ γ , k), as 1, satisfies a one dimensional fractional heat equation ∂ t W (t, x) = −ĉ(−∂ 2 x) 3/4 W (t, x) withĉ > 0. In the pinned case an analogous result can be claimed for W (t/ 2γ , x/ γ , k) but the limit satisfies then the usual heat equation.

Discrete & Continuous Dynamical Systems - B, 2012

We consider energy fluctuations for solutions of the Schrödinger equation with an Ornstein-Uhlenb... more We consider energy fluctuations for solutions of the Schrödinger equation with an Ornstein-Uhlenbeck random potential when the initial data is spatially localized. The limit of the fluctuations of the Wigner transform satisfies a kinetic equation with random initial data. This result generalizes that of [13] where the random potential was assumed to be white noise in time.

arXiv (Cornell University), Jul 18, 2023

We consider a chain consisting of n+1 pinned harmonic oscillators subjected on the right to a tim... more We consider a chain consisting of n+1 pinned harmonic oscillators subjected on the right to a time dependent periodic force F(t) while Langevin thermostats are attached at both endpoints of the chain. We show that for long times the system is described by a Gaussian measure whose covariance function is independent of the force, while the means are periodic. We compute explicitly the work and energy due to the periodic force for all n including n → ∞.

arXiv (Cornell University), Sep 29, 2022

We consider the limit of solutions of scaled linear kinetic equations with a reflectiontransmissi... more We consider the limit of solutions of scaled linear kinetic equations with a reflectiontransmission-absorption condition at the interface. Both the coefficient describing the probability of absorption and the scattering kernel degenerate. We prove that the long-time, large-space limit is the unique solution of a version of the fractional in space heat equation that corresponds to the Kolmogorov equation for a symmetric stable process, which is reflected, or transmitted while crossing the interface and is killed upon the first hitting of the interface. The results of the paper are related to the work in [19], where the case of a non-degenerate probability of absorption has been considered.

Annales Henri Poincaré

We consider an infinite chain of coupled harmonic oscillators with a Poisson thermostat at the or... more We consider an infinite chain of coupled harmonic oscillators with a Poisson thermostat at the origin. In the high frequency limit, we establish the reflection-transmission-scattering coefficients for the wave energy scattered off the thermostat. Unlike the case of the Langevin thermostat [5], in the macroscopic limit the Poissonian thermostat scattering generates a continuous cloud of waves of frequencies different from that of the incident wave.

arXiv (Cornell University), Apr 28, 2021

In this paper, we study the KPZ equation on the torus and derive Gaussian fluctuations in large t... more In this paper, we study the KPZ equation on the torus and derive Gaussian fluctuations in large time.

Studia Mathematica, 2022

We consider a massless tracer particle moving in a random, stationary, isotropic and divergence f... more We consider a massless tracer particle moving in a random, stationary, isotropic and divergence free velocity field. We identify a class of fields, for which the limit of the laws of appropriately scaled tracer trajectory processes is non-Gaussian but a Rosenblatt type process.

Consider a Markov chain {Xn}n≥0 with an ergodic probability measure π. Let Ψ be a function on the... more Consider a Markov chain {Xn}n≥0 with an ergodic probability measure π. Let Ψ be a function on the state space of the chain, with α-tails with respect to π, α ∈ (0,2). We find sufficient conditions on the probability transition to prove convergence in law of N ∑N n Ψ(Xn) to an α-stable law. A “martingale approximation” approach and a “coupling” approach give two different sets of conditions. We extend these results to continuous time Markov jump processes Xt, whose skeleton chain satisfies our assumptions. If waiting times between jumps have finite expectation, we prove convergence of N ∫ Nt

The Annals of Probability, 2020

We consider the limit of a linear kinetic equation, with reflection-transmissionabsorption at an ... more We consider the limit of a linear kinetic equation, with reflection-transmissionabsorption at an interface, with a degenerate scattering kernel. The equation arise from a microscopic chain of oscillators in contact with a heat bath. In the absence of the interface, the solutions exhibit a superdiffusive behavior in the long time limit. With the interface, the long time limit is the unique solution of a version of the fractional in space heat equation, with reflection-transmission-absorption at the interface. The limit problem corresponds to a certain stable process that is either absorbed, reflected, or transmitted upon crossing the interface.

Archive for Rational Mechanics and Analysis, 2020

SIAM Journal on Mathematical Analysis, 2018

We consider a semi-linear advection equation driven by a highly-oscillatory space-time Gaussian r... more We consider a semi-linear advection equation driven by a highly-oscillatory space-time Gaussian random field, with the randomness affecting both the drift and the nonlinearity. In the linear setting, classical results show that the characteristics converge in distribution to a homogenized Brownian motion, hence the point-wise law of the solution is close to a functional of the Brownian motion. Our main result is that the nonlinearity plays the role of a random diffeomorphism, and the point-wise limiting distribution is obtained by applying the diffeomorphism to the limit in the linear setting.

Archive for Rational Mechanics and Analysis, 2017

We consider the Schrödinger equation with a time-independent weakly random potential of a strengt... more We consider the Schrödinger equation with a time-independent weakly random potential of a strength ε ≪ 1, with Gaussian statistics. We prove that when the initial condition varies on a scale much larger than the correlation length of the potential, the compensated wave function converges to a deterministic limit on the time scale t ∼ ε −2. This is shown under the sharp assumption that the correlation function R(x) of the random potential decays slower than 1/|x| 2 , which ensures that the effective potential is finite. When R(x) decays slower than 1/|x| 2 we establish an anomalous diffusive behavior for the averaged wave function on a time scale shorter than ε −2 , as long as the initial condition is "sufficiently macroscopic". We also consider the kinetic regime when the initial condition varies on the same scale as the random potential and obtain the limit of the averaged wave function for potentials with the correlation functions decaying faster than 1/|x| 2. We use random potentials of the Schonberg class which allows us to bypass the oscillatory phase estimates.

Nonlinearity, 2016

We consider a one dimensional infinite acoustic chain of harmonic oscillators whose dynamics is p... more We consider a one dimensional infinite acoustic chain of harmonic oscillators whose dynamics is perturbed by a random exchange of velocities, such that the energy and momentum of the chain are conserved. Consequently, the evolution of the system has only three conserved quantities: volume, momentum and energy. We show the existence of two space-time scales on which the energy of the system evolves. On the hyperbolic scale (tǫ −1 , xǫ −1) the limits of the conserved quantities satisfy a Euler system of equations, while the thermal part of the energy macroscopic profile remains stationary. Thermal energy starts evolving at a longer time scale, corresponding to the superdiffusive scaling (tǫ −3/2 , xǫ −1) and follows a fractional heat equation. We also prove the diffusive scaling limit of the Riemann invariants-the so called normal modes, corresponding to the linear hyperbolic propagation.

Probability Theory and Related Fields, 2001

We consider the asymptotic behavior of the solutions of scaled convectiondiffusion equations ∂ t ... more We consider the asymptotic behavior of the solutions of scaled convectiondiffusion equations ∂ t u ε (t, x) = κ x u ε (t, x) + 1/εV(t/ε 2 , x/ε) • ∇ x u ε (t, x) with the initial condition u ε (0, x) = u 0 (x) as the parameter ε ↓ 0. Under the assumptions that κ > 0 and V(t, x), (t, x) ∈ R d is a d-dimensional, stationary, zero mean, incompressible, Gaussian random field, Markovian and mixing in t we show that the laws of u ε (t, •), t ≥ 0 in an appropriate functional space converge weakly, as ε ↓ 0, to a δ-type measure concentrated on a solution of a certain constant coefficient heat equation.

Communications in Mathematical Physics, 2015

We consider a one dimensional infinite chain of harmonic oscillators whose dynamics is perturbed ... more We consider a one dimensional infinite chain of harmonic oscillators whose dynamics is perturbed by a stochastic term conserving energy and momentum. We prove that in the unpinned case the macroscopic evolution of the energy converges to the solution of the fractional diffusion equation ∂ t u = −| | 3/4 u. For a pinned system we prove that its energy evolves diffusively, generalizing some results of Basile and Olla (J.

The Annals of Applied Probability, 1999

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

Kinetic & Related Models, 2010

2.1.3. Random flows with spatial-temporal dependence. Let us now turn to a more complex situation... more 2.1.3. Random flows with spatial-temporal dependence. Let us now turn to a more complex situation when the random flow in (2.1) depends both on time and space: ∂φ ∂t + εV (t, x) • ∇φ = 0, φ(0, x) = φ 0 (x), (2.12)

Stochastic Processes and their Applications, 2005

We consider a tracer particle performing a continuous time nearest neighbor random walk on Z d in... more We consider a tracer particle performing a continuous time nearest neighbor random walk on Z d in dimension d ≥ 3 with random jump rates. The kind of a walk considered here models the motion of an electrically charged particle under a constant external electric field. We prove the existence of the mobility coefficient, and that it equals to the diffusivity coefficient of the particle.

Probability Theory and Related Fields, 2001

We study a diffusion with a random, time dependent drift. We prove the invariance principle when ... more 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]. ‫ޒ‬ d V(t, x; ω) • ∇ x ϕ(x) dx = 0 P − a.s., with P the underlying probability measure. In order to guarantee the existence of the solution of (1.1) we will also assume that V(t, x; ω) is (P −a.s.) locally Lipschitz in x. We are interested in proving an invariance principle for x(t), i.e. the convergence in distribution of the process εx(ε −2 t) to a Brownian motion with a certain co-variance matrix, sometimes referred to as the effective diffusivity, D ≥ 2I. This problem has been widely studied under various conditions on the random flow. Typically the flow is assumed to be the divergence of a stationary random anti-symmetric matrix valued field H(t, x; ω) = {H p,q (t, x; ω)}-the so-called stream matrix: V(t, x; ω) = ∇ x • H(t, x; ω) .

Nonlinear Differential Equations and Applications NoDEA, 2013

We study the principal Dirichlet eigenvalue of the operator LA = Δ α/2 + Ab(x) • ∇, on a bounded ... more We study the principal Dirichlet eigenvalue of the operator LA = Δ α/2 + Ab(x) • ∇, on a bounded C 1,1 regular domain D. Here α ∈ (1, 2), Δ α/2 is the fractional Laplacian, A ∈ R, and b is a bounded d-dimensional divergence-free vector field in the Sobolev space W 1,2d/(d+α) (D). We prove that the eigenvalue remains bounded, as A → +∞, if and only if b has non-trivial first integrals in the domain of the quadratic form of Δ α/2 for the Dirichlet condition.

Journal of Statistical Physics, 2012

We consider the long time, large scale behavior of the Wigner transform W (t, x, k) of the wave f... more We consider the long time, large scale behavior of the Wigner transform W (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 et al. in Phys. Rev. Lett. 96 (2006) to describe a system of interacting linear oscillators with a weak noise that conserves locally the kinetic energy and the momentum. The kinetic limit for the Wigner transform has been shown in Basile et al. in Arch. Rat. Mech. 195(1):171-203 (2009). In the present paper we prove that in the unpinned case there exists γ 0 > 0 such that for any γ ∈ (0, γ 0 ] the weak limit of W (t/ 3/2γ , x/ γ , k), as 1, satisfies a one dimensional fractional heat equation ∂ t W (t, x) = −ĉ(−∂ 2 x) 3/4 W (t, x) withĉ > 0. In the pinned case an analogous result can be claimed for W (t/ 2γ , x/ γ , k) but the limit satisfies then the usual heat equation.

Discrete & Continuous Dynamical Systems - B, 2012

We consider energy fluctuations for solutions of the Schrödinger equation with an Ornstein-Uhlenb... more We consider energy fluctuations for solutions of the Schrödinger equation with an Ornstein-Uhlenbeck random potential when the initial data is spatially localized. The limit of the fluctuations of the Wigner transform satisfies a kinetic equation with random initial data. This result generalizes that of [13] where the random potential was assumed to be white noise in time.