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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
##### Keywords:
absolute continuity; random process
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##### References:
  Yu. L. Daletskii and S. V. Fomin, Measures and Differential Equations in Infinite-Dimensional Spaces [in Russian], Nauka, Moscow (1983).  Yu. L. Daletskii and B. D. Maryanin, Dokl. Akad. Nauk SSSR,285, No. 6, 1297-1300 (1985).  Yu. L. Daletskii, Proceedings of the 4th International Vilnius Conference on Probability Theory and Math. Statistic,4, Vilnius (1985), pp. 58-61.  A. V. Skorokhod, Integration in a Hilbert Space [in Russian], Nauka, Moscow (1975). · Zbl 0355.62084  D. Bell, Trans. Am. Math. Soc.,290, No. 2, 851-855 (1985).
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