Summary
In this work we study the absolute continuity of the image of the Wiener measure under the transformations of the formT(ω)=ω+u(ω), the shiftu is a random variable with values in the Cameron-Martin spaceH and is monotone in the sense that (T(ω+h-T(ω),h) H ≥0 a.s. for allh inH.
References
N. Bouleau, F. Hirsch: Dirichlet Forms and Analysis on Wiener Space. De Gruyter Studies in Math., Vol. 14, Berlin-New York, 1991.
F. E. Browder: On the solvability of nonlinear functional equations. Duke Math. J.30, 557–566, 1963
R. Buckdahn, E. Pardoux: Monotonicity methods for white noise driven quasi-linear SPDEs. Progress in Probability, M. Pinsky (Editor), Vol. 22. Birkhauser, 1993
S. Kusuoka: The nonlinear transformation of Gaussian measures on Banach space and its absolute continuity, I.
G. J. Minty: Monotone (non-linear) operators in Hilbert space. Duke Math. J.29, 34–346, 1962
G. J. Minty: Two theorems on nonlinear functional equations in Hilbert space. Bull. Am. Math. Soc.69, 691–692, 1963
A. S. Üstünel: Intégrabilité exponentielle de fonctionnelles de Wiener. CRAS, Paris, Série I, Vol.315, pp. 279–282, 1992
A. S. Üstünel: Some exponential moment inequalities for the Wiener functionals. Jour. Func. Analysis.136, 154–170 (1996)
A. S. Üstünel: An Introduction to Analysis on Wiener Space. Lecture Notes in Mathematics, Vol. 1610. Springer, 1995
A. S. Üstünel, M. Zakai: Transformations of Wiener measure under anticipative flows. Probab. Theory Relat. Fields93, 91–136, 1992
A. S. Üstünel, M. Zakai: Applications of the degree theorem to absolute continuity on Wiener space. Probab. Theory Relat. Fields,95, 509–520, 1993
A. S. Üstünel, M. Zakai: Transformation of the Wiener measure under non-invertible shifts. Probab. Theory Relat. Fields99, 485–500, 1994
A. S. Üstünel, M. Zakai: Absolute continuity of monotone shifts on Wiener space. Proceedings of the 5-th Silivri Workshop on Stochastic Analysis. Progress in Probability, Vol. 38, pp. 269–281. Birkhäuser, 1995
M. M. Vainberg: Variational Method and Method of Monotone Operators in the Theory of Nonlinear Equations. John Wiley & Sons, New York-Toronto, 1973
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Süleyman Üstünel, A., Zakai, M. Measures induced on Wiener space by monotone shifts. Probab. Th. Rel. Fields 105, 545–563 (1996). https://doi.org/10.1007/BF01191913
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DOI: https://doi.org/10.1007/BF01191913
Mathematics Subject Classification (1991)
- 60G30
- 60H07
- 60H15
- 35G20