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.
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Üstünel, A.S., Zakai, M. Sufficient conditions for the invertibility of adapted perturbations of identity on the Wiener space. Probab. Theory Relat. Fields 139, 207–234 (2007). https://doi.org/10.1007/s00440-006-0044-z
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DOI: https://doi.org/10.1007/s00440-006-0044-z
Keywords
- Malliavin calculus
- H-C1 maps
- Multiplicity
- Change of variables formula
- Wiener space
- Wiener measure
- Adapted
- Perturbation of identity
- Carleman’s inequality
- Logarithmic sobolev inequality
Mathematics Subject Classifications (2000)
- 60H07
- 60H05
- 60H25
- 60G15
- 60G30
- 60G35
- 46G12
- 47H05
- 47H1
- 35J60
- 35B65
- 35A30
- 46N10
- 49Q20
- 58E12
- 26A16
- 28C20