The small noise limit of order-based diffusion processes

Benjamin Jourdain,Julien Reygner

doi:10.1214/ejp.v19-2906

Abstract

We introduce order-based diffusion processes as the solutions to multidimensional stochastic differential equations, with drift coefficient depending only on the ordering of the coordinates of the process and diffusion matrix proportional to the identity. These processes describe the evolution of a system of Brownian particles moving on the real line with piecewise constant drifts, and are the natural generalization of the rank-based diffusion processes introduced in stochastic portfolio theory or in the probabilistic interpretation of nonlinear evolution equations. Owing to the discontinuity of the drift coefficient, the corresponding ordinary differential equations are ill-posed. Therefore, the small noise limit of order-based diffusion processes is not covered by the classical Freidlin-Wentzell theory. The description of this limit is the purpose of this article. We first give a complete analysis of the two-particle case. Despite its apparent simplicity, the small noise limit of such a system already exhibits various behaviours. In particular, depending on the drift coefficient, the particles can either stick into a cluster, the velocity of which is determined by elementary computations, or drift away from each other at constant velocity, in a random ordering. The persistence of randomness in the zero noise limit is of the very same nature as in the pioneering works by Veretennikov (Mat. Zametki, 1983) and Bafico and Baldi (Stochastics, 1981) concerning the so-called Peano phenomenon. In the case of rank-based processes, we use a simple convexity argument to prove that the small noise limit is described by the sticky particle dynamics introduced by Brenier and Grenier (SIAM J. Numer. Anal., 1998), where particles travel at constant velocity between collisions, at which they stick together. In the general case of order-based processes, we give a sufficient condition on the drift for all the particles to aggregate into a single cluster, and compute the velocity of this cluster. Our argument consists in turning the study of the small noise limit into the study of the long time behaviour of a suitably rescaled process, and then exhibiting a Lyapunov functional for this rescaled process.

Highlights

For particular examples of such ordinary differential equations (ODEs), it was proved in [37] and [2] that the small noise limit of the law of Xǫ concentrates on the set of extremal solutions {x+, x−} and the weights associated with each such solution was explicitely computed
The small noise limit ξ turns out to coincide with the sticky particle dynamics introduced by Brenier and Grenier [6], which describes the evolution of a system of particles with unit mass, travelling at constant velocity between collisions, and such that, at each collision, the colliding particles stick together and form a cluster, the velocity of which is determined by the global conservation of momentum
In Subsection 4.1, we provide an extension of the stability condition of Lemma 3.3 for the rankbased case, which ensures that the particles aggregate into a single cluster in the small noise limit

Summary

Introduction

Xnǫ (t))t≥0 to (1) shall generically be called order-based diffusion process, as it describes the evolution of a system of n particles moving on the real line with piecewise constant drift depending on their ordering. By a simple convexity argument, we prove that the limit of Y ǫ when ǫ vanishes is the deterministic process ξ with the same drift b, normally reflected at the boundary of Dn. The small noise limit ξ turns out to coincide with the sticky particle dynamics introduced by Brenier and Grenier [6], which describes the evolution of a system of particles with unit mass, travelling at constant velocity between collisions, and such that, at each collision, the colliding particles stick together and form a cluster, the velocity of which is determined by the global conservation of momentum. The deterministic process (t)t≥0 shall be denoted by t

The two-particle case

The rank-based case

The order-based case

Conclusion