2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/131053Let $(W,H,μ)$ be the classical Wiener space. Assume that $U=I_W+u$ is an adapted perturbation of identity, i.e., $u:W\to H$ 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 $u\in \DD_{p,1}(H)$ is adapted and if $\exp({1/2}\|\nabla u\|_2^2-δu)\in L^q(μ)$, where $p^{-1}+q^{-1}=1$, then $I_W+u$ is almost surely invertible. As a consequence, if, there exists an integer $k\geq 1$ such that $\|\nabla^k u\|_{H^{\otimes(k+1)}}\in L^\infty(μ)$, then $I_W+u$ is again almost surely invertible.ProbabilityFunctional AnalysisStatistics Theory60H07, 60H05, 60H25, 60G15, 60G30, 60G35, 46G12, 47H05, 47H1, 35J60text