BEYOND WENTZELL-FREIDLIN: SEMI-DETERMINISTIC APPROXIMATIONS FOR DIFFUSIONS WITH SMALL NOISE AND A REPULSIVE CRITICAL BOUNDARY POINT

Florin Avram (1) , Jacky Cresson (1)

1 LMAP - Laboratoire de Mathématiques et de leurs Applications [Pau]

We extend below a limit theorem [3] for diffusion models used in population theory. A diffusion with small noise is defined as the solution of a stochastic differential equation (SDE) driven by standard Brownian motion B t (·) (defined on a probability space and progressively measurable with respect to an increasing filtration)        dX ε t = µ(X ε t)dt + √ εσ(X ε t)dB t , t ≥ 0, X ε 0 = x 0 = ε, X ε t ∈ I := (0, r) (1) where 0 < r ≤ +∞, ε > 0, µ : I → R, σ : I → R >0 and µ, σ satisfy conditions ensuring that (1) has a strong unique solution (for example, µ is locally Lifshitz and σ satisfies the Yamada-Watanabe conditions [18, (2.13), Ch.5.2.C]). § When ε → 0, (1) is a small perturbation of the dynamical system/ordinary differential equation (ODE): dx t dt = µ(x t), t ≥ 0, (2) which will also be supposed to admit a unique continuous solution x t , t ∈ R + subject to any x 0 ∈ (0, r), and the flow of which will be denoted by φ t (x). A basic result in the field is the "fluid limit", which states that when (1) admits a strong unique solution, the effect of noise is negligible as ε → 0, on any fixed time interval [0, T ]: § For reviews discussing the existence of strong and weak solutions, see for example [9, 17, 12].

Mots clés: dynamical systems; small noise; linearization; semi-deterministic fluid ap- proximation AMS classification: AMS 60J60 §1 Introduction

Domaines: Systèmes dynamiques [math.DS]

Florin Avram, Jacky Cresson. BEYOND WENTZELL-FREIDLIN: SEMI-DETERMINISTIC APPROXIMATIONS FOR DIFFUSIONS WITH SMALL NOISE AND A REPULSIVE CRITICAL BOUNDARY POINT. Publicaciones del Seminario Matemático García Galdeano, In press. ⟨hal-02390307⟩