Absolute continuity of smooth measures
type: Item
instance of: scholarly article
title: Absolute continuity of smooth measures
zbMATH Open document ID: 0693.58035
DOI: 10.1007/BF01077613
author: G. A. Sokhadze
author: Yuri L. Daletskii
publication date: 1988-01-01T00:00:00Z
review text: 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 \[ \frac{\mu^{(f)}(dx)}{\mu (dx)}=\det (I+F'(y))\exp \{<\int^{1}_{0}\lambda (y+sF(y))ds,F(y)>\}. \]
Mathematics Subject Classification ID: 58J65
Mathematics Subject Classification ID: 28A15
zbMATH DE Number: 4135086
zbMATH Keywords: absolute continuity
zbMATH Keywords: random process
MaRDI profile type: Publication
cites work: Q4127232
published in: Functional Analysis and Its Applications
description: scientific article; zbMATH DE number 4135086