Sufficient Conditions for the Invertibility of Adapted Perturbations of Identity on the Wiener Space

Abstract

Let $(W,H,\mu )$ be the classical Wiener space. Assume that $U=I_W+u$ is an adapted perturbation of identity, i.e., $u:W\to H$ 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 $\exp (1/2\Vert \nabla u{\Vert }_2^2-\delta u)\in L^q(\mu )$, where $p^{-1}+q^{-1}=1$, then $I_W+u$ is almost surely invertible. As a consequence, if, there exists an integer $k\ge 1$ such that $\Vert {\nabla }^{k}u{\Vert }_{H^{\otimes (k+1)}}\in L^{\infty }(\mu )$, then $I_W+u$ is again almost surely invertible.