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16
Statistical Stability in Time Reversal
 SIAM J. APPL. MATH.
, 2004
"... When a signal is emitted from a source, recorded by an array of transducers, timereversed, and reemitted into the medium, it will refocus approximately on the source location. We analyze the refocusing resolution in a high frequency remotesensing regime and show that, because of multiple scatteri ..."
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Cited by 45 (21 self)
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When a signal is emitted from a source, recorded by an array of transducers, timereversed, and reemitted into the medium, it will refocus approximately on the source location. We analyze the refocusing resolution in a high frequency remotesensing regime and show that, because of multiple scattering in an inhomogeneous or random medium, it can improve beyond the diffraction limit. We also show that the backpropagated signal from a spatially localized narrowband source is selfaveraging, or statistically stable, and relate this to the selfaveraging properties of functionals of the Wigner distribution in phase space. Time reversal from spatially distributed sources is selfaveraging only for broadband signals. The array of transducers operates in a remotesensing regime, so we analyze time reversal with the parabolic or paraxial wave equation.
SelfAveraging in Time Reversal for the Parabolic Wave Equation
 Stochastics and Dynamics
, 2002
"... We analyze the selfaveraging properties of timereversed solutions of the paraxial wave equation with random coefficients, which we take to be Markovian in the direction of propagation. ..."
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Cited by 36 (16 self)
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We analyze the selfaveraging properties of timereversed solutions of the paraxial wave equation with random coefficients, which we take to be Markovian in the direction of propagation.
S.,Limit theorems for additive functionals of a Markov Chain, version 1
, 2008
"... Abstract. 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 αtails with respect to π, α ∈ (0, 2). We find sufficient conditions on the probability transition to prove convergence in law of N 1/α ∑ N n Ψ(Xn) to an αstabl ..."
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Cited by 19 (7 self)
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Abstract. 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 αtails with respect to π, α ∈ (0, 2). We find sufficient conditions on the probability transition to prove convergence in law of N 1/α ∑ N n Ψ(Xn) to an αstable law. A “martingale approximation ” approach and “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 −1/α ∫ Nt 0 V (Xs)ds to a stable process. The result is applied to show that an appropriately scaled limit of solutions of a linear Boltzman equation is a solution of the fractional diffusion equation. 1.
Selfaveraged scaling limits for random parabolic waves
 Archives of Rational Mechanics and Analysis
"... Abstract. We consider several types of scaling limits for the WignerMoyal equation of the parabolic waves in random media, the limiting cases of which include the radiative transfer limit, the diffusion limit and the whitenoise limit. We show under fairly general assumptions on the random refracti ..."
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Cited by 18 (11 self)
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Abstract. We consider several types of scaling limits for the WignerMoyal equation of the parabolic waves in random media, the limiting cases of which include the radiative transfer limit, the diffusion limit and the whitenoise limit. We show under fairly general assumptions on the random refractive index field that sufficient amount of medium diversity (thus excluding the whitenoise limit) leads to statistical stability or selfaveraging in the sense that the limiting law is deterministic and is governed by various transport equations depending on the specific scaling involved. The celebrated Schrödinger equation i � ∂Ψ
Selfaveraging from lateral diversity in the ItôSchrödinger equation
, 2006
"... We consider the random Schrödinger equation as it arises in the paraxial regime for wave propagation in random media. In the white noise limit it becomes the ItôSchrödinger stochastic partial differential equation (SPDE) which we analyze here in the high frequency regime. We also consider the large ..."
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Cited by 14 (9 self)
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We consider the random Schrödinger equation as it arises in the paraxial regime for wave propagation in random media. In the white noise limit it becomes the ItôSchrödinger stochastic partial differential equation (SPDE) which we analyze here in the high frequency regime. We also consider the large lateral diversity limit where the typical width of the propagating beam is large compared to the correlation length of the random medium. We use the Wigner transform of the wave field and show that it becomes deterministic in the large diversity limit when integrated against test functions. This is the selfaveraging property of the Wigner transform. It follows easily when the support of the test functions is of the order of the beam width. We also show with a more detailed analysis that the limit is deterministic when the support of the test functions
Asymptotics Problems for LaserMatter Modeling; Quantum and Classical Models
"... This paper is devoted to the asymptotic analysis of both quantum and classical models which describe the evolution of electrons subject to the potential of an atomic crystal perturbed by the highly oscillating potential of external electromagnetic waves. We derive either Einstein rate equations or ..."
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Cited by 4 (1 self)
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This paper is devoted to the asymptotic analysis of both quantum and classical models which describe the evolution of electrons subject to the potential of an atomic crystal perturbed by the highly oscillating potential of external electromagnetic waves. We derive either Einstein rate equations or diffusion equations with respect to the energy variable, depending on whether the initial model is quantum or classical. We point out the analogies and differences in the treatment of the two models, considering successively the cases of (quasi)periodic perturbations or random ones. We point out the different role of the relaxation effects according to the nature of the perturbation. Key words. Vlasov equation. Bloch equation. Energy diffusion equation. Einstein rate equations.
Stability of time reversed waves in changing media, Disc
 Cont. Dyn. Syst. A
"... We analyze the refocusing properties of time reversed waves that propagate in two different media during the forward and backward stages of a timereversal experiment. We consider two regimes of wave propagation modeled by the paraxial wave equation with a smooth random refraction coefficient and th ..."
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Cited by 4 (2 self)
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We analyze the refocusing properties of time reversed waves that propagate in two different media during the forward and backward stages of a timereversal experiment. We consider two regimes of wave propagation modeled by the paraxial wave equation with a smooth random refraction coefficient and the ItôSchrödinger equation, respectively. In both regimes, we rigorously characterize the refocused signal in the high frequency limit and show that it is statistically stable, that is, independent of the realizations of the two media. The analysis is based on a characterization of the high frequency limit of the Wigner transform of two fields propagating in different media. The refocusing quality of the backpropagated signal is determined by the cross correlation of the two media. When the two media decorrelate, two distinct defocusing effects are observed. The first one is a purely absorbing effect due to the loss of coherence at a fixed frequency. The second one is a phase modulation effect of the refocused signal at each frequency. This causes defocusing of the backpropagated signal in the time domain. 1
Large Time Dynamics of a Classical System Subject to a Fast Varying Force
 COMM. MATH. PHYS
, 2007
"... We investigate the asymptotic behavior of solutions to a kinetic equation describing the evolution of particles subject to the sum of a fixed, confining, Hamiltonian, and a small, timeoscillating, perturbation. The equation also involves an interaction operator which acts as a relaxation in the e ..."
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Cited by 4 (0 self)
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We investigate the asymptotic behavior of solutions to a kinetic equation describing the evolution of particles subject to the sum of a fixed, confining, Hamiltonian, and a small, timeoscillating, perturbation. The equation also involves an interaction operator which acts as a relaxation in the energy variable. This paper aims at providing a classical counterpart to the derivation of rate equations from the atomic Bloch equations. In the present classical setting, the homogenization procedure leads to a diffusion equation in the energy variable, rather than a rate equation, and the presence of the relaxation operator regularizes the limit process, leading to finite diffusion coefficients. The key assumption is that the timeoscillatory perturbation should have welldefined long time averages: our procedure includes general “ergodic” behaviors, amongst which periodic, or quasiperiodic potentials only are a particular case.
The quantum scattering limit for a regularized Wigner equation
 Methods and Applications of Analysis
, 2004
"... We consider a regularized Wigner equation with an oscillatory kernel, the regularization acts in the space variable to damp high frequencies. The oscillatory kernel is directly derived from the Schrödinger equation with an oscillatory potential. The problem therefore contains three scales, ε the osc ..."
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Cited by 4 (0 self)
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We consider a regularized Wigner equation with an oscillatory kernel, the regularization acts in the space variable to damp high frequencies. The oscillatory kernel is directly derived from the Schrödinger equation with an oscillatory potential. The problem therefore contains three scales, ε the oscillation length, θ the regularization parameter, δ the potential lattice. We prove that the homogenized limit (as ε vanishes) of this equation is a scattering equation with discrete jumps. As δ vanishes, the discrete scattering kernel boils down to a standard regular scattering kernel. As θ vanishes we recover the quantum scattering operator with collisions preserving energy sphere.