Approximation of stochastic processes by non-expansive flows and coming down from infinity

TL;DR

This paper develops a method to approximate finite-dimensional stochastic processes using non-expansive dynamical flows, providing uniform trajectorial estimates and analyzing the phenomena of coming down from infinity in various stochastic models.

Contribution

It introduces a novel approach leveraging non-expansivity to approximate stochastic processes and extends the analysis of coming down from infinity to multi-dimensional systems.

Findings

Uniform trajectorial estimates for stochastic processes.

Classification of coming down from infinity in Lotka-Volterra diffusions.

Extension of results to two-dimensional competitive stochastic models.

Abstract

We approximate stochastic processes in finite dimension by dynamical systems. We provide trajectorial estimates which are uniform with respect to the initial condition for a well chosen distance. This relies on some non-expansivity property of the flow, which allows to deal with non-Lipschitz vector fields. We use the stochastic calculus and follow the martingale technics initiated in Berestycki and al [5] to control the fluctuations. Our main applications deal with the short time behavior of stochastic processes starting from large initial values. We state general properties on the coming down from infinity of one-dimensional SDEs, with a focus on stochastically monotone processes. In particular, we recover and complement known results on Lambda-coalescent and birth and death processes. Moreover, using Poincar{\'e}'s compactification technicsfor dynamical systems close to infinity, we develop this approach in two dimensions for competitive stochastic models. We classify the coming down from infinity of Lotka-Volterra diffusions and provide uniform estimates for the scaling limits of competitive birth and death processes.