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

Abstract

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 $${u\in {\rm ID}_{p,1}(H)}$$ 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 $${\nabla u\in L^\infty(\mu,H\otimes H)}$$ , 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 $${t\to\nabla^i \dot{u}_t}$$ for i ≤ k − 1.