Flow of Homeomorphisms and Stochastic Transport Equations

Abstract

We consider Stratonovich stochastic differential equations with drift coefficient A ₀ satisfying only the condition of continuity

where r is a positive C ¹ function defined on a neighborhood ]0, c ₀] of 0 such that (Osgood condition), and s → r(s) is decreasing while s → sr(s ²) is increasing. We prove that the equation defines a flow of homeomorphisms if the diffusion coefficients A ₁,…, A _N are in . If , we prove limit theorems for Wong–Zakai approximation as well as for regularizing the drift A ₀. As an application, we solve a class of stochastic transport equations.

Keywords:

• Flow of homeomorphisms
• Non-Lipschitz condition
• Stochastic differential equation
• Stochastic transport equation
• Wong–Zakai approximation

Mathematics Subject Classification:

• Primary: 60J60
• Secondary: 60H10
• 34A12