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 = IW + u is an adapted perturbation of identity, i.e., u : W → H is adapted to the canonical filtration ofW . We give some sufficient analytic conditions on u which imply the invertibility of the map U . In particular it is shown that if u ∈ IDp,1(H) is adapted and if exp( 2‖∇u‖2 − δu) ∈ Lq(μ), where p−1 + q−1 = 1, then IW + u is almost surely invertible. With the help of this result it is shown that if ∇u ∈ L∞(μ,H ⊗ H), then the Girsanov exponential of u times the Wiener measure satisfies the logarithmic Sobolev inequality and this implies the invertibility of U = IW + u. As a consequence, if, there exists an integer k ≥ 1 such that ‖∇u‖H⊗(k+1) ∈ L∞(μ), then IW +u is again almost surely invertible under the almost sure continuity hypothesis of t → ∇u̇t for i ≤ k − 1.