Large systems of diffusions interacting through their ranks

We study the limiting behavior of the empirical measure of a system of diffusions interacting through their ranks when the number of diffusions tends to infinity. We prove that under certain assumptions the limiting dynamics is given by a McKean–Vlasov evolution equation. Moreover, we show that the evolution of the cumulative distribution function under the limiting dynamics is governed by the generalized porous medium equation with convection. The implications of the results for rank-based models of capital distributions in financial markets are also explained.     

Comparison of interacting diffusions and an application to their ergodic theory

Summary A general comparison argument for expectations of certain multitime functionals of infinite systems of linearly interacting diffusions differing in the diffusion coefficient is derived. As an application we prove clustering occurs in the case when the symmetrized interaction kernel is recurrent, and the components take values in an interval bounded on one side. The technique also gives an alternative proof of clustering in the case of compact intervals.     

Multilevel large deviations and interacting diffusions

Summary Let ( ^N ) be a sequence of random variables with values in a topological space which satisfy the large deviation principle. For eachM and eachN, let ^{M, N} denote the empirical measure associated withM independent copies of ^N . As a main result, we show that ( ^{M, N} ) also satisfies the large deviation principle asM,N. We derive several representations of the associated rate function. These results are then applied to empirical measure processes ^{M, N} (t) =M ^–1 _i=1 ^N _i ^N (t) 0tT, where ( ₁ ^N ,..., _M ^N (t)) is a system of weakly interacting diffusions with noise intensity 1/N. This is a continuation of our previous work on the McKean-Vlasov limit and related hierarchical models ([4], [5]).Research partially supported by a Natural Science and Engineering Research Council of Canada operating grant     

Finite and infinite systems of interacting diffusions

Summary We study the problem of relating the long time behavior of finite and infinite systems of locally interacting components. We consider in detail a class of lincarly interacting diffusionsx(t)={x _i (t),i ∈ ℤ ^d } in the regime where there is a one-parameter family of nontrivial invariant measures. For these systems there are naturally defined corresponding finite systems, , with . Our main result gives a comparison between the laws ofx(t _N ) andx ^N (t _N ) for timest _N →∞ asN→∞. The comparison involves certain mixtures of the invariant measures for the infinite system. Partly supported by the U.S. Army Research Office through the Mathematical Sciences Institute of Cornell University, by the National Science Foundation, and by the National Security Agency Research supported in part by the DFG Partly supported by S.R.63540155 of Japan Ministry of Education     

Scaling limits of interacting diffusions in domains

We survey recent effort in establishing the hydrodynamic limits and the fluctuation limits for a class of interacting diffusions in domains. These systems are introduced to model the transport of positive and negative charges in solar cells. They are general microscopic models that can be used to describe macroscopic phenomena with coupled boundary conditions, such as the popula- tion dynamics of two segregated species under competition. Proving these two types of limits represents establishing the functional law of large numbers and the functional central limit theorem, respectively, for the empirical measures of the spatial positions of the particles. We show that the hydrodynamic limit is a pair of deterministic measures whose densities solve a coupled nonlinear heat equations, while the fluctuation limit can be described by a Gaussian Markov process that solves a stochastic partial differential equation.     

A variational approach to nonlinear and interacting diffusions

Abstract

The article presents a novel variational calculus to analyze the stability and the propagation of chaos properties of nonlinear and interacting diffusions. This differential methodology combines gradient flow estimates with backward stochastic interpolations, Lyapunov linearization techniques as well as spectral theory. This framework applies to a large class of stochastic models including nonhomogeneous diffusions, as well as stochastic processes evolving on differentiable manifolds, such as constraint-type embedded manifolds on Euclidian spaces and manifolds equipped with some Riemannian metric. We derive uniform as well as almost sure exponential contraction inequalities at the level of the nonlinear diffusion flow, yielding what seems to be the first result of this type for this class of models. Uniform propagation of chaos properties w.r.t. the time parameter is also provided. Illustrations are provided in the context of a class of gradient flow diffusions arising in fluid mechanics and granular media literature. The extended versions of these nonlinear Langevin-type diffusions on Riemannian manifolds are also discussed.     

Existence of equilibrium for infinite system of interacting diffusions

We develop and implement new probabilistic strategy for proving basic results about long-time behavior for interacting diffusion processes on unbounded lattice. The concept of the solution used is rather weak as we construct the process as a solution to suitable infinite-dimensional martingale problem. However, the techniques allow us to consider cases where the generator of the particle is degenerate elliptic operator. As a model example, we present the situation where the operator arises from Heisenberg group. In the last section, we provide further examples that can be handled using our methods.     

Scaling limits of interacting diffusions with arbitrary initial distributions

Summary It is remarked that for Brownian particles interacting with a smooth repulsive pair potential the nonlinear diffusion equation which S. Varadhan has derived under an entropy bound for initial densities is valied whatever initial distribution they start with.Research partially supported by Japan Society for the Promotion of Science     

Diffusive clustering in an infinite system of hierarchically interacting diffusions

Summary We study a countable system of interacting diffusions on the interval [0,1], indexed by a hierarchical group. A particular choice of the interaction guaranties, we are in the diffusive clustering regime. This means clusters of components with values either close to 0 or close to 1 grow on various different scales. However, single components oscillate infinitely often between values close to 0 and close to 1 in such a way that they spend fraction one of their time together and close to the boundary. The processes in the whole class considered and starting with a shift-ergodic initial law have the same qualitative properties (universality).     

Darcy's law as long-time limit of adiabatic porous media flow

It is conjectured that Darcy's law governs the motion of compressible porous media flow in large time. This has been justified for one-dimensional isentropic flows. In this work, we show the conjecture is true for one-dimensional adiabatic flows with generic small smooth initial data.     

Smoothness of the intensity measure density for interacting branching diffusions with immigrations

We consider the invariant measure for finite systems of interacting branching diffusions with immigrations. We use Malliavin calculus in order to show that the intensity measure of the invariant measure admits a density which is continuous, one times partially differentiable and bounded provided the immigration measure is absolute continuous.     

Tightness problem and stochastic evolution equation arising from fluctuation phenomena for interacting diffusions

The central limit (or fluctuation) phenomena are discussed in the interacting diffusion system. The tightness in the Kolmogorov-Prokhorov sense is proved for a sequence of distribution valued processes arising from finite particle systems. Further, the stochastic differential equation for the limit process is derived by constructing an infinite dimensional Brownian motion.     

Local limit theorems for transition densities of Markov chains converging to diffusions

We consider triangular arrays of Markov chains that converge weakly to a diffusion process. Local limit theorems for transition densities are proved. Received: 28 August 1998 / Revised version: 6 September 1999 / Published online: 14 June 2000     

Central limit theorem for linearly dependent fields of continuous elements

Summary The central limit theorem for stationary linearly dependent sequences is extended for elements in the space of continuous functions on a compact metric space. The proof is based on a new estimate for exponential-type moments of sums of independent random variables.     

Concentration for multidimensional diffusions and their boundary local times

We prove that probability laws of certain multidimensional semimartingales which includes time-inhomogenous diffusions, under suitable assumptions, satisfy quadratic transportation cost inequality under the uniform metric. From this we derive concentration properties of Lipschitz functions of process paths that depend on the entire history. In particular, we estimate concentration of boundary local time of reflected Brownian motions on a polyhedral domain. We work out explicit applications of consequences of measure concentration for the case of Brownian motion with rank-based drifts.     

Domain of attraction of quasi-stationary distribution for one-dimensional diffusions

We study quasi-stationarity for one-dimensional diffusions killed at 0, when 0 is a regular boundary and +∞ is an entrance boundary. We give a necessary and sufficient condition for the existence of exactly one quasistationary distribution, and we also show that this distribution attracts all initial distributions.     

A note on the quasi-ergodic distribution of one-dimensional diffusions

In this note, we study quasi-ergodicity for one-dimensional diffusions on $(0,\infty )$, where 0 is an exit boundary and +∞ is an entrance boundary. Our main aim is to improve some results obtained by He and Zhang (2016) [3]. In simple terms, the same main results of the above paper are obtained with more relaxed conditions.     

Existence,semiclassical limit and long-time behavior of weak solution to quantum drift-diffusion model

The existence, semiclassical limit and long-time behavior of weak solutions to the transient quantum drift-diffusion model are studied. Using semi-discretization in time and entropy estimate, we get the global existence and semiclassical limit of nonnegative weak solutions to one-dimensional isentropic model with nonnegative initial and homogeneous Neumann (or periodic) boundary conditions. Furthermore, by a logarithmic Sobolev inequality, we obtain an inequality of the periodic weak solution to this model (or its isothermal case) which shows that the solution exponentially approaches its mean value as time increases to infinity.