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. Some non-expansivity property of the flow of the dynamical is required, which allows us to deal with non-Lipschitz vector fields. We use the stochastic calculus and we follow the martingale technic 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 some general properties on the coming down from infinity of one-dimensional SDE. In particular, we recover and complement known results on Λ-coalescent and birth and death processes. Moreover, using Poincaré's compactification for dynamical systems close to infinity, we develop this approach in two dimensions for competitive Lotka Volterra diffusions and classify the coming down from infinity. Finally, we provide uniform estimates for scaling limits of competitive birth and death processes.

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

Domaines: Probabilités [math.PR]

Vincent Bansaye. Approximation of stochastic processes by non-expansive flows and coming down from infinity. 2015. ⟨hal-01232373v1⟩