Papers by Carlo Boldrighini
«Italia Nostra», a. LIV, vol. 435 (2008), pp. 24-25., 2008
Probability Theory and Related Fields, 1995
We consider a mechanical system in the plane, consisting of a vertical rod of length ℓ, with its ... more We consider a mechanical system in the plane, consisting of a vertical rod of length ℓ, with its center moving on the horizontal axis, subject to elastic collisions with the particles of a free gas, and to a constant forcef. Assuming a suitable initial measure we show that the evolution of the system as seen from the rod is described by an exponentially ergodic irreducible Harris chain, implying convergence to a stationary invariant measure ast→∞. We deduce that in the proper scaling the motion of the rod is described as a drift plus a diffusion. We prove in conclusion that the diffusion is nondegenerate and that the drift is nonzero iff≠0 and has the same sign off.
Probability Theory and Related Fields, 1997
We consider a mechanical model in the plane, consisting of a vertical rod, subject to a constant ... more We consider a mechanical model in the plane, consisting of a vertical rod, subject to a constant horizontal force f and to elastic collisions with the particles of a free gas which is “horizontally” in equilibrium at some inverse temperature β. In a previous paper we proved that, in the appropriate space and time scaling, the motion of the rod is described as a drift term plus a diffusion term. In this paper we prove that the drift d(f) and the diffusivity σ 2 (f) are continuous functions of f, and moreover that the Einstein relation holds, i.e., lim f → 0 d(f)f = β2 σ 2 (0) .
Journal of Statistical Physics, 1988
We prove the central limit theorem for the density fluctuation field of a one-dimensional mechani... more We prove the central limit theorem for the density fluctuation field of a one-dimensional mechanical system (hard rods with equal masses and lengths and elastic collisions) in the hydrodynamic limit on the Euler time scale. The limiting process is deterministic and is governed by the linearized Euler equations of the model.
Meccanica, 1977
In questo lavoro si considera un classico modello unidimensionale della fluidodinamica, consisten... more In questo lavoro si considera un classico modello unidimensionale della fluidodinamica, consistente in un'equazione alle derivate parziali parabolica quasilineare (equazione di Burgers). Si considera il problema in un intervallo con condizioni al contorno stazionarie. L'intento è quello di indagare se il modello presenta transizioni nel comportamento asintotico al variare della viscosità. Facendo uso della trasformazione di Hopf-Cole si trova che la soluzione stazionaria (che viene calcolata esplicitamente) è globalmente attrattiva per ogni valore della viscosità, sia nel caso omogeneo che in presenza di una forza. We study a quasilinear partial differential equation which is a classical unidimensional model of fluid motion (Burgers equation). We consider the problem in a finite interval with stationary boundary conditions. The aim is to see whether the model shows transitions in the asymptotic behaviour as viscosity varies. We show that there is always a unique stationary solution, which is explicitely exhibited, and, by using the Hopf-Cole transformation, that it is globally attractive for any value of the viscosity, both in the homogeneous and inhomogeneous case.
Acta Applicandae Mathematicae, 1991
An open problem in the theory of Fourier series is whether there are functions f ∈ L 1 such that ... more An open problem in the theory of Fourier series is whether there are functions f ∈ L 1 such that the partial sums S n(f, x) diverge faster than log log n, almost everywhere in x. For a class of particularly ‘bad’ functions Kahane proved that the rate of divergence is faster than o(log log n). We give here a probabilistic interpretation of the Kahane result, which shows that the record values of the sums S n(f, x) should behave essentially as the record values of a sequence of independent identically distributed random variables, for which we deduce the divergence rate log log n. Numerical computation is in good agreement with the prediction. One can argue that the Kahane examples are in some sense ‘optimal’, and conclude that, under this assumption, ...(log log n) is the highest possible rate for divergence almost everywhere of the Fourier partial sums for L 1 functions.
Communications in Mathematical Physics, 1980
We associate to the plane incompressible Euler equation with periodic conditions the correspondin... more We associate to the plane incompressible Euler equation with periodic conditions the corresponding Hopf equation, as an equation for measures on the space of solenoidal distributions. We define equilibrium states as the solutions of the stationary Hopf equation. We find a class of equilibrium states which corresponds to a class of infinitely divisible distributions, and investigate the properties of gaussian and poissonian states. Equilibrium dynamics for a class of poissonian states is constructed by means of the Onsager vortex equations.
Probability Theory and Related Fields, 1989
We study a one-dimensional semi-infinite system of identical particles, driven by a constant forc... more We study a one-dimensional semi-infinite system of identical particles, driven by a constant force acting only on the first particle. Particles interact through elastic collisions. At time zero all particles are at rest, and the interparticle distances are i.i.d. r.v.'s, the support of the distribution being in (d, ∞), d>0. We show that if d is large enough the dynamics has a strong cluster property, and prove, for large times, convergence to a limiting distribution for the system as seen from the first particle, as well as existence of drift velocity and invariance principle for the motion of the first particle.
Communications in Mathematical Physics, 1981
Journal of Statistical Physics, 1989
We study the evolution of the completely asymmetric simple exclusion process in one dimension, wi... more We study the evolution of the completely asymmetric simple exclusion process in one dimension, with particles moving only to the right, for initial configurations corresponding to average densityρ − (ρ +) left (right) of the origin,ρ −⩽ρ +. The microscopic shock position is identified by introducing a “second-class” particle. Results indicate that the shock profile is stable, and that the distribution as seen from the shock positionN(t) tends, as time increases, to a limiting distribution, which is locally close to an equilibrium distribution far from the shock. Moreover \(N(t)_{\frown \,}^\smile V \cdot t\) , withV=1−ρ −−ρ +, as predicted, and the dispersion ofN(t), σ2(t), behaves linearly, for not too small values ofρ +−ρ −, i.e., \(\sigma ^2 (t)_{\frown \,}^\smile S \cdot t\) , whereS is equal, up to a scaling factor, to the valueS WA predicted in the weakly asymmetric case. Forρ +=ρ − we find agreement with the conjecture \(\sigma ^2 (t)_{\frown \,}^\smile \bar S \cdot t^{4/3} \) .
Communications in Mathematical Physics, 1979
A five-modes truncation of the Navier-Stokes equations for a two dimensional incompressible fluid... more A five-modes truncation of the Navier-Stokes equations for a two dimensional incompressible fluid on a torus is considered. A computer analysis shows that for a certain range of the Reynolds number the system exhibits a stochastic behaviour, approached through an involved sequence of bifurcations.
Communications in Mathematical Physics, 1980
We associate to the plane incompressible Euler equation with periodic conditions the correspondin... more We associate to the plane incompressible Euler equation with periodic conditions the corresponding Hopf equation, as an equation for measures on the space of solenoidal distributions. We define equilibrium states as the solutions of the stationary Hopf equation. We find a class of equilibrium states which corresponds to a class of infinitely divisible distributions, and investigate the properties of gaussian and poissonian states. Equilibrium dynamics for a class of poissonian states is constructed by means of the Onsager vortex equations.
Nuclear Physics B, 1970
The Energy Citations Database (ECD) provides access to historical and current research (1948 to t... more The Energy Citations Database (ECD) provides access to historical and current research (1948 to the present) from the Department of Energy (DOE) and predecessor agencies.
Stochastic Processes and Their Applications, 1987
We consider the (kinetic) continuum limit for a class of stochastic interacting particle systems.... more We consider the (kinetic) continuum limit for a class of stochastic interacting particle systems. We prove propagation of chaos and convergence to macroscopic equations of the "Reaction-Diffusion" type. interacting particle systems * continuum limit * propagation of chaos * reaction-diffusion equations Research partially supported by CNR grant n.
Stochastic Processes and Their Applications, 2009
ABSTRACT We propose a discrete-time random walk on , d=1,2,..., as a variant of recent models of ... more ABSTRACT We propose a discrete-time random walk on , d=1,2,..., as a variant of recent models of random walk on in a random environment which is i.i.d. in space-time. We allow space correlations of the environment and develop an analytic method to deal with them. We prove, under some general assumptions, that if the random term is small, a "quenched" (i.e., for a fixed "history" of the environment) Central Limit Theorem for the displacement of the random walk holds almost-surely. Proofs are based on L2 estimates. We consider for brevity only the case of odd dimension d, as even dimension requires somewhat different estimates.
Probability Theory and Related Fields, 1997
We consider a model of random walk on ℤν, ν≥2, in a dynamical random environment described by a f... more We consider a model of random walk on ℤν, ν≥2, in a dynamical random environment described by a field ξ={ξ t (x): (t,x)∈ℤν+1}. The random walk transition probabilities are taken as P(X t +1= y|X t = x,ξ t =η) =P 0( y−x)+ c(y−x;η(x)). We assume that the variables {ξ t (x):(t,x) ∈ℤν+1} are i.i.d., that both P 0(u) and c(u;s) are finite range in u, and that the random term c(u;·) is small and with zero average. We prove that the C.L.T. holds almost-surely, with the same parameters as for P 0, for all ν≥2. For ν≥3 there is a finite random (i.e., dependent on ξ) correction to the average of X t , and there is a corresponding random correction of order to the C.L.T.. For ν≥5 there is a finite random correction to the covariance matrix of X t and a corresponding correction of order to the C.L.T.. Proofs are based on some new L p estimates for a class of functionals of the field.
Moscow Mathematical Journal, 2008
We consider a discrete-time random walk on Z d , d = 1, 2, . . . in a random environment with Mar... more We consider a discrete-time random walk on Z d , d = 1, 2, . . . in a random environment with Markov evolution in time. We complete and extend to all dimension d ≥ 1 the results obtained in [2] on the time decay of the correlations of the "environment from the point of view of the random walk".
Stochastic Processes and Their Applications, 1992
Journal of Statistical Physics, 1983
We study the evolution of the states for one-dimensional infinite harmonic systems, interacting t... more We study the evolution of the states for one-dimensional infinite harmonic systems, interacting through a translation invariant force of rapid decrease. We prove that for a large class of initial states convergence to a Gaussian limiting state, as time goes to infinity, is equivalent to convergence of the covariance. The main assumption on the initial states is a kind of weak dependence between distant regions (mixing condition). We prove also convergence of the covariance under some general assumptions. We show furthermore that there are two countable families of intensive constants of the motion, which are “inherited” from the corresponding finite systems. The translation invariant limiting states are in one-to-one correspondence with the admissible values of these constants of the motion. Moreover, under some additional regularity assumption, such states are shown to be Gibbs states, obtained by a “Boltzmann-Gibbs” prescription.
We consider a discrete-time random walk on Z d , d = 1, 2, . . . in a random environment with Mar... more We consider a discrete-time random walk on Z d , d = 1, 2, . . . in a random environment with Markov evolution in time. We complete and extend to all dimension d ≥ 1 the results obtained in [2] on the time decay of the correlations of the "environment from the point of view of the random walk".
Uploads
Papers by Carlo Boldrighini