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

Vincent Bansaye (1)

1 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique

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 Poincaré's compactification technics for 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.

Mots clés: coming down from infinity; non expansivity; dynamical system; Approximation of stochastic processes; martingales; stochastic calculus; scaling limits

Domaines: Probabilités [math.PR]

Vincent Bansaye. Approximation of stochastic processes by non-expansive flows and coming down from infinity. 2016. ⟨hal-01232373v2⟩