Generalized Ornstein-Uhlenbeck Process Research Articles

Convergence of the fluctuations for interacting diffusions with jumps associated with boltzmann equations

We consider a nonlinear partial differential equation of the Boltzmann type, in which the nonlinearity appears in a bounded jump operator.We associate with this equation a probability measure P on the Skorohod path space, solution of a nonlinear martingale problem and with time-marginals (Pt ) solutions of the pde. Then some interacting stochastic particle systems are given, whose empirical measures converge to P In this paper we are interested in the convergence of the fluctuation processes We prove by martingale techniques and Sobolev embeddings that the processes converge in law to a generalized Ornstein-Uhlenbeck process in the space where W is a weighted Sobolev space completely described

Stochastic Burgers and KPZ Equations from Particle Systems

We consider two strictly related models: a solid on solid interface growth model and the weakly asymmetric exclusion process, both on the one dimensional lattice. It has been proven that, in the diffusive scaling limit, the density field of the weakly asymmetric exclusion process evolves according to the Burgers equation and the fluctuation field converges to a generalized Ornstein-Uhlenbeck process. We analyze instead the density fluctuations beyond the hydrodynamical scale and prove that their limiting distribution solves the (non linear) Burgers equation with a random noise on the density current. For the solid on solid model, we prove that the fluctuation field of the interface profile, if suitably rescaled, converges to the Kardar–Parisi–Zhang equation. This provides a microscopic justification of the so called kinetic roughening, i.e. the non Gaussian fluctuations in some non-equilibrium processes. Our main tool is the Cole-Hopf transformation and its microscopic version. We also develop a mathematical theory for the macroscopic equations.

Microscopic derivation of non-Markovian thermalization of a Brownian particle

In this paper, the first microscopic approach to the Brownian motion is developed in the case where the mass density of the suspending bath is of the same order of magnitude as that of the Brownian (B) particle. Starting from an extended Boltzmann equation, which describes correctly the interaction with the fluid, we derive systematicaly via the multiple time-scale analysis a reduced equation controlling the thermalization of the B particle, i.e. the relaxation towards the Maxwell distribution in velocity space. In contradistinction to the Fokker-Planck equation, the derived new evolution equation is non-local both in time and in velocity space, owing to correlated recollision events between the fluid and particle B. In the long-time limit, it describes a non-markovian generalized Ornstein-Uhlenbeck process. However, in spite of this complex dynamical behaviour, the Stokes-Einstein law relating the friction and diffusion coefficients is shown to remain valid. A microscopic expression for the friction coefficient is derived, which acquires the form of the Stokes law in the limit where the mean-free in the gas is small compared to the radius of particle B.

Asymptotic fluctuations and critical dimension for a two-level branching system

The high-density asymptotic behaviour of a two-level branching system in Ktd is studied. In the finitevariance case, a fluctuation limit process is obtained which is characterized as a generalized OrnsteinUhlenbeck process. In the case of critical branching at the two levels the long-time behaviour of the fluctuation limit process is shown to have critical dimension 2a, where CY is the index of the symmetric stable process representing the underlying particle motion. The same critical dimension has been obtained recently for the related (but qualitatively different) two-level superprocess. The fluctuation analysis uses different and simpler tools than the superprocess analysis.

Equilibrium fluctuations of Gradient reversible particle systems

The Boltzmann-Gibbs principle is known to be crucial in the study of the fluctuations of interacting particle systems. A new method is proposed in this paper which confirms this principle for models with gradient reversible dynamics in equilibrium. The method is simpler and can be applied to more general models than the conventional one which is developed by Brox et al. To illustrate the idea in more detail, we study the weakly asymmetric simple exclusion process of which the jump rates are slowly varying. As a consequence of the Boltzmann-Gibbs principle, the limit of the density fluctuation fields is identified as a generalized Ornstein-Uhlenbeck process. Finally, the extension to models with long range interactions is briefly discussed.

Interface fluctuations in the two-dimensional weakly asymmetric simple exclusion process

We consider the two-dimensional weakly asymmetric simple exclusion process, where the asymmetry is along the X-axis. The generator for such a process can be written as ε —2 L 0+ε —1 L α, ε>0, where L 0 and L α are the generators for the nearest neighbor symmetric simple exclusion and totally asymmetric simple exclusion, respectively. We prove propagation of chaos and convergence to Burgers equation with viscosity in the limit as ε goes to zero. The density fluctuation field converges to a generalized Ornstein–Uhlenbeck process. The covariance kernel for a class of travelling wave solutions is consistent with a phase boundary which fluctuates according to a linear stochastic partial differential equation.

Nonequilibrium fluctuations in particle systems modelling reaction-diffusion equations

We consider a class of stochastic evolution models for particles diffusing on a lattice and interacting by creation-annihilation processes. The particle number at each site is unbounded. We prove that in the macroscopic (continuum) limit the particle density satisfies a reaction-diffusion PDE, and that microscopic fluctuations around the average are described by a generalized Ornstein-Uhlenbeck process, for which the covariance kernel is explicitely exhibited.

Fluctuations from the hydrodynamical limit for the symmetric simple exclusion in [formula omitted

We show that the fluctuation field of the simple exclusion process on Z d converges to a mean zero generalized Ornstein-Uhlenbeck process.

Gravity in one dimension: Diffusion in acceleration

The one-dimensional gravitational system consists ofN parallel sheets of constant mass density. The sheets move perpendicular to their surface solely under their mutual gravitational attraction. When a pair has an encounter, they simply pass through each other. In this paper I consider the motion of a single sheet in an equilibrium ensemble. Under the assumption that the times separating encounters are random, I show that the acceleration and velocity(A, V) of a labeled sheet form a Markovian pair. Further, I prove that, in the limit of largeN, (1)the(A, V) process is deterministic, (2) the(A, V) process obeys Vlasov dynamics, and (3) that scaled fluctuations in(A, V) comprise a diffusion which obeys a generalized Ornstein-Uhlenbeck process with time-dependent drift and diffusion tensors.

A mean field limit of the contact process with large range

A mean field limit of the contact process is obtained as the rangeM approaches ∞. Fluctuations about the deterministic limit are identified as a Generalized Ornstein Uhlenbeck process.

The demographic variation process of branching random fields

A branching random field with immigration is considered. The demographic variation process is a non-Markovian signed measure-valued process which measures the changes in the system due to branchings and deaths in the population. The asymptotic behavior of this process under various scalings is studied. It is shown that the fluctuation limits are generalized Gaussian processes which are Markovian if and only if the branching is critical, in which case they are non-stationary generalized Ornstein-Uhlenbeck processes.

High density limit theorems for nonlinear chemical reactions with diffusion

The solutionX of a nonlinear reaction-diffusion equation on then-dimensional unit cubeS is approximated by a space-time jump Markov processX v,N (law of large numbers (LLN)).X v,N is constructed on a gridS N onS ofN cells, wherev is proportional to the initial number of particles in each cell. The deviation ofX v,N fromX is computed by a central limit theorem (CLT). The assumptions on the parametersv, N are for the LLN: υ → ∞, asN → ∞, and for the CLT:\(\frac{N}{\upsilon } \to 0\), asN → ∞. The limitY =Y X in the CLT, which is a generalized Ornstein-Uhlenbeck process, is represented as the mild solution of a linear stochastic partial differential equation (SPDE) and its best possible state spaces are described. The problem of stationary solutions ofY X in dependence ofX is also investigated.

Limit theorems for supercritical branching random fields with immigration

A measure-valued process which carries genealogical information is defined for a supercritical branching random field with immigration. This process counts the particles present at a final time whose ancestors had specified locations at given times in the past. A law of large numbers and a fluctuation limit theorem are proved for this process under a space-time scaling. The fluctuation limit is a nonstationary generalized Ornstein-Uhlenbeck process. An example of interest in transport theory and polymer chemistry is given.

High Density Limit Theorems for Infinite Systems of Unscaled Branching Brownian Motions

The fluctuations of an infinite system of unscaled branching Brownian motions in $R^d$ are shown to converge weakly under a spatial central limit normalization when the initial density of particles tends to infinity. The limit is a generalized Gaussian $M$ which can be written as $M = M^I + M^{II}$, where $M^I$ is the fluctuation limit of a Poisson system of Brownian motions obtained by Martin-Lof, and $M^{II}$ arises from the spatial central limit normalization of the demographic variation process of the system. In the critical case $M^I$ and $M^{II}$ are independent and $M^{II}$ coincides with the generalized Ornstein-Uhlenbeck found by Dawson and by Holley and Stroock as the renormalization limit of an infinite system of critical branching Brownian motions when $d \geq 3$. Generalized Langevin equations for $M, M^I$ and $M^{II}$ are given.