Coupling Iterated Kolmogorov Diffusions

Abstract

The Kolmogorov-1934 diffusion is the two-dimensional diffusion generated by real Brownian motion and its time integral. In this paper we construct successful co-adapted couplings for iterated Kolmogorov diffusions defined by adding iterated time integrals as further components to the original Kolmogorov diffusion. A Laplace-transform argument shows it is not possible successfully to couple all iterated time integrals at once; however we give an explicit construction of a successful co-adapted coupling method for Brownian motion, its time integral, and its twice-iterated time integral; and a more implicit construction of a successful co-adapted coupling method which works for finite sets of iterated time integrals.