Smoothness of Itô maps and diffusion processes on path spaces (I)

Abstract

Let p ∈ [ 1 , 2 ) and α, ε > 0 be such that α ∈ ( p − 1 , 1 − ε ) . Let V, W be two Euclidean spaces. Let Ω p ( V ) be the space of continuous paths taking values in V and with finite p-variation. Let k ∈ N and f : W → Hom ( V , W ) be a Lip ( k + α + ε ) map in the sense of E.M. Stein [Stein E.M., Singular integrals and differentiability properties of functions, Princeton Mathematical Series, vol. 30, Princeton University Press, Princeton, NJ, 1970]. In this paper we prove that the Itô map, defined by I ( x ) = y , is a local C k , ε 1 + ε map (in the sense of Fréchet) between Ω p ( V ) and Ω p ( W ) , where y is the solution to the differential equation d y t = f ( y t ) d x t , y 0 = a . This result strengthens the continuity results and Lipschitz continuity results in [Lyons T., Differential equations driven by rough signals. I. An extension of an inequality of L.C. Young, Math. Res. Lett. 1 (4) (1994) 451–464; Lyons T., Qian Z., System Control and Rough Paths, Oxford Mathematical Monographs, Clarendon Press, Oxford, 2002] particularly to the non-integer case. It allows us to construct the fractional like Brownian motion and infinite dimensional Brownian motions on the space of paths with finite p-variation. As a corollary in the particular case where p = 1 , we obtain that the development from the space of finite 1-variation paths on R d to the space of finite 1-variation paths on a d-dimensional compact Riemannian manifold is a smooth bijection.