Sufficient conditions for the invertibility of adapted perturbations of identity on the Wiener space

Let (W, H, mu) be the classical Wiener space. Assume that U = I-W + u is an adapted perturbation of identity, i.e., u : W -> H is adapted to the canonical filtration of W. We give some sufficient analytic conditions on u which imply the invertibility of the map U. In particular it is shown that if u. IDp, 1( H) is adapted and if exp(1/2 parallel to del u parallel to(2)(2) - delta u) is an element of L-q(mu), where p(-1) + q(-1) = 1, then I-W + u is almost surely invertible. With the help of this result it is shown that if del u is an element of L-infinity(mu, H circle times H), then the Girsanov exponential of u times the Wiener measure satisfies the logarithmic Sobolev inequality and this implies the invertibility of U = I-W + u. As a consequence, if, there exists an integer k >= 1 such that parallel to del(k)u parallel to(H circle times(k+1)) is an element of L-infinity(mu), then I-W + u is again almost surely invertible under the almost sure continuity hypothesis of t -> del(i)u(t) for i <= k - 1.