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Absolute continuity of smooth measures. (English. Russian original) Zbl 0693.58035
Funct. Anal. Appl. 22, No. 2, 149-150 (1988); translation from Funkts. Anal. Prilozh. 22, No. 2, 77-78 (1988).
Let X be a smooth separable Banach manifold, \({\mathfrak a}\) be a smooth vector field on X having a global flow \(T_ t\) (t\(\in {\mathbb{R}})\). The logarithmic derivative \(\rho\) (\(\mu\),\({\mathfrak a},x)\) is defined by the relation \[ (d/dt)\int_{X}\phi (T_ tx)\mu (dx)|_{t=0}=- \int_{X}\phi (x)\rho (\mu,{\mathfrak a},x)_{\mu}(dx) \] for the class C(x) functions.
Let \({\mathfrak a}(t)\) be a vector field, depending on time, and \(T_{t,\tau}: x(\tau)\mapsto x(t)\) be the evolutionary family of X smooth transformations, defined by the differential equation \(dx/dt={\mathfrak a}(t,x)\). Let \(\mu_ t=\mu \circ T^{(-1)}_{t,0}\), v(t) be the vector field on X, \(T_{t,0}\)-compatible with a(t): \(v(t,T_{t,0}x)=T'_{t,0}{\mathfrak a}(t,x)\). If the logarithmic derivative \(\rho\) (\(\mu\),v(t),x) is continuous with respect to t, then the measures \(\mu\) and \(\mu_ t\) are equivalent and \[ \mu_ t(dx)/\mu (dx)=\exp \{- \int^{t}_{0}\phi (\mu,v(s),T^{-1}_{s_ 0}x)ds\}. \] Then let B be a separable Banach space, H be a Hilbert space and \(B^*\subset H\subset B\) be the equipped Hilbert space with 2-absolutely summing inclusion maps, \(\mu\) be a Borel measure on B, differentiable along the directions \(h\in B^*\) and \(\rho (\mu,h,\kappa)=<\lambda (x,h)>\) where \(<,>\) is the canonical pairing of B and \(B^*\). If f: \(B\to B\) is an invertible mapping such that \(f^{(-1)}: y\mapsto x=y+F(y)\), F: \(B\to B^*\) is a differentiable mapping, F and \(F^ 1\) are bounded, then \(\mu^{(f)}=\mu \circ f^{(-1})\) and \(\mu\) are equivalent and \[ \frac{\mu^{(f)}(dx)}{\mu (dx)}=\det (I+F'(y))\exp \{<\int^{1}_{0}\lambda (y+sF(y))ds,F(y)>\}. \]
Reviewer: Yu.L.Daletskij

MSC:
58J65 Diffusion processes and stochastic analysis on manifolds
28A15 Abstract differentiation theory, differentiation of set functions
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References:
[1] Yu. L. Daletskii and S. V. Fomin, Measures and Differential Equations in Infinite-Dimensional Spaces [in Russian], Nauka, Moscow (1983).
[2] Yu. L. Daletskii and B. D. Maryanin, Dokl. Akad. Nauk SSSR,285, No. 6, 1297-1300 (1985).
[3] Yu. L. Daletskii, Proceedings of the 4th International Vilnius Conference on Probability Theory and Math. Statistic,4, Vilnius (1985), pp. 58-61.
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[5] D. Bell, Trans. Am. Math. Soc.,290, No. 2, 851-855 (1985).
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