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

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

2006-05-16

arxiv.org/abs/math/0605433v3

Mathematics

Probability

Scientific paper

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 $\exp({1/2}\|\nabla u\|_2^2-\delta u)\in L^q(\mu)$, where $p^{-1}+q^{-1}=1$, then $I_W+u$ is almost surely invertible. As a consequence, if, there exists an integer $k\geq 1$ such that $\|\nabla^k u\|_{H^{\otimes(k+1)}}\in L^\infty(\mu)$, then $I_W+u$ is again almost surely invertible.

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