Flows associated to adapted vector fields on the Wiener space

Abstract

The existence and uniqueness of a flow associated to an adapted vector field ξ on the Wiener space with d τ ξ α = a β α dω β (τ ) + b α dτ are proved by a modified Picard's iteration method, mainly under the conditions of exponential integrability concerning b α as well as the first-order Malliavin gradients of a β α and b α . A Newton-Leibnitz type inequality for a kind of Malliavin differentiable functionals is also proved, which is the key point to prove the above result, and has an independent interest.

Horizontal Diffusion in C 1 Path Space

“…The problem is similar to finding flows associated to derivative processes as studied in [6,7,8,9,13,12] and [11]. However it is transversal in the sense that in these papers diffusions with the same starting point are deformed along a drift which vanishes at time 0.…”