With the Douady–Earle extension of vector fields, see Douady and Earle (Acta Math 157(1–2):23–48, 1986) and Earle (Proc Am Soc 102(1):145–149, 1988), we construct the extension inside and outside the unit disk for the stochastic differential equation (SDE) defining the canonic Brownian motion above the homeomorphisms of the circle. The inside extension had been explicited in Airault et al. (J Funct Anal 259(12):3037–3079, 2010). We prove that the same extended SDE is valid inside and outside the unit disk. For this extended SDE, we discuss the behaviour of the solutions, in particular we prove that a solution with initial point inside (or outside) the unit disk, almost surely never reaches the unit circle in finite time. We obtain stochastic log-Lipschitz flows of homeomorphisms defined inside the unit disk as in Airault et al. (J Math Pures Appl (9) 83(8):955–1018, 2004), and also outside the unit disk.
Analysis and Mathematical Physics – Springer Journals
Published: Nov 8, 2011