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Particle approximations of Lyapunov exponents connected to Schrödinger operators and Feynman–Kac semigroups

Pierre Del Moral, L. Miclo (2003)

ESAIM: Probability and Statistics

We present an interacting particle system methodology for the numerical solving of the Lyapunov exponent of Feynman–Kac semigroups and for estimating the principal eigenvalue of Schrödinger generators. The continuous or discrete time models studied in this work consists of N interacting particles evolving in an environment with soft obstacles related to a potential function V . These models are related to genetic algorithms and Moran type particle schemes. Their choice is not unique. We will examine...

Particle approximations of Lyapunov exponents connected to Schrödinger operators and Feynman–Kac semigroups

Pierre Del Moral, L. Miclo (2010)

ESAIM: Probability and Statistics

We present an interacting particle system methodology for the numerical solving of the Lyapunov exponent of Feynman–Kac semigroups and for estimating the principal eigenvalue of Schrödinger generators. The continuous or discrete time models studied in this work consists of N interacting particles evolving in an environment with soft obstacles related to a potential function V. These models are related to genetic algorithms and Moran type particle schemes. Their choice is not unique. We...

Past, Present and Future of Brain Stimulation

J. Modolo, R. Edwards, J. Campagnaud, B. Bhattacharya, A. Beuter (2010)

Mathematical Modelling of Natural Phenomena

Recent technological advances including brain imaging (higher resolution in space and time), miniaturization of integrated circuits (nanotechnologies), and acceleration of computation speed (Moore’s Law), combined with interpenetration between neuroscience, mathematics, and physics have led to the development of more biologically plausible computational models and novel therapeutic strategies. Today, mathematical models of irreversible medical conditions...

Perturbing the hexagonal circle packing: a percolation perspective

Itai Benjamini, Alexandre Stauffer (2013)

Annales de l'I.H.P. Probabilités et statistiques

We consider the hexagonal circle packing with radius 1 / 2 and perturb it by letting the circles move as independent Brownian motions for time t . It is shown that, for large enough t , if 𝛱 t is the point process given by the center of the circles at time t , then, as t → ∞ , the critical radius for circles centered at 𝛱 t to contain an infinite component converges to that of continuum percolation (which was shown – based on a Monte Carlo estimate – by Balister, Bollobás and Walters to be strictly bigger than...

Pointwise convergence of Boltzmann solutions for grazing collisions in a Maxwell gas via a probabilitistic interpretation

Hélène Guérin (2004)

ESAIM: Probability and Statistics

Using probabilistic tools, this work states a pointwise convergence of function solutions of the 2-dimensional Boltzmann equation to the function solution of the Landau equation for Maxwellian molecules when the collisions become grazing. To this aim, we use the results of Fournier (2000) on the Malliavin calculus for the Boltzmann equation. Moreover, using the particle system introduced by Guérin and Méléard (2003), some simulations of the solution of the Landau equation will be given. This result...

Pointwise convergence of Boltzmann solutions for grazing collisions in a Maxwell gas via a probabilitistic interpretation

Hélène Guérin (2010)

ESAIM: Probability and Statistics


Using probabilistic tools, this work states a pointwise convergence of function solutions of the 2-dimensional Boltzmann equation to the function solution of the Landau equation for Maxwellian molecules when the collisions become grazing. To this aim, we use the results of Fournier (2000) on the Malliavin calculus for the Boltzmann equation. Moreover, using the particle system introduced by Guérin and Méléard (2003), some simulations of the solution of the Landau equation will be given. This result...

Potential confinement property of the parabolic Anderson model

Gabriela Grüninger, Wolfgang König (2009)

Annales de l'I.H.P. Probabilités et statistiques

We consider the parabolic Anderson model, the Cauchy problem for the heat equation with random potential in ℤd. We use i.i.d. potentials ξ:ℤd→ℝ in the third universality class, namely the class of almost bounded potentials, in the classification of van der Hofstad, König and Mörters [Commun. Math. Phys.267 (2006) 307–353]. This class consists of potentials whose logarithmic moment generating function is regularly varying with parameter γ=1, but do not belong to the class of so-called double-exponentially...

Probability and quanta: why back to Nelson?

Piotr Garbaczewski (1998)

Banach Center Publications

We establish circumstances under which the dispersion of passive contaminants in a forced flow can be consistently interpreted as a Markovian diffusion process.

Process-level quenched large deviations for random walk in random environment

Firas Rassoul-Agha, Timo Seppäläinen (2011)

Annales de l'I.H.P. Probabilités et statistiques

We consider a bounded step size random walk in an ergodic random environment with some ellipticity, on an integer lattice of arbitrary dimension. We prove a level 3 large deviation principle, under almost every environment, with rate function related to a relative entropy.

Propagation of chaos for the 2D viscous vortex model

Nicolas Fournier, Maxime Hauray, Stéphane Mischler (2014)

Journal of the European Mathematical Society

We consider a stochastic system of N particles, usually called vortices in that setting, approximating the 2D Navier-Stokes equation written in vorticity. Assuming that the initial distribution of the position and circulation of the vortices has finite (partial) entropy and a finite moment of positive order, we show that the empirical measure of the particle system converges in law to the unique (under suitable a priori estimates) solution of the 2D Navier-Stokes equation. We actually prove a slightly...

Propriétés dispersives pour des équations cinétiques et applications à l’équation de Vlasov-Poisson

Delphine Salort (2008/2009)

Séminaire Équations aux dérivées partielles

On considère l’équation de Vlasov-Poisson en dimension 3. On montre des résultats d’existence et d’unicité de solutions faibles de l’équation de Vlasov-Poisson avec densité bornée pour des données initiales ayant strictement moins de six moments dans L x , ξ 1 . La preuve est basée sur une nouvelle approche qui consiste à établir des effets de moments a priori pour des équations de transport avec des termes de force peu réguliers.

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