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

Let (W, H, μ) 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 ${uin {rm ID}_{p,1}(H)}$ is adapted and if ${exp(frac{1}{2}|nabla u|_2^2-delta u)in L^q(mu)}$ , 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 ${nabla uin L^infty(mu,Hotimes H)}$ , 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 ${|nabla^k u|_{H^{otimes(k+1)}}in L^infty(mu)}$ , then I _W  +  u is again almost surely invertible under the almost sure continuity hypothesis of ${ttonabla^i dot{u}_t}$ for i ≤  k ? 1.