Relations between smooth measures and their logarithmic gradients and derivatives. (Q1432114)

In this paper, \(E\) is a Hausdorff locally convex space, \(E'\) its topological dual, \( \mathcal M (E)\) the space of all countably additive measures on the \(\sigma\)-algebra generated by cylindrical sets in \(E\), \( \nu \in \mathcal M (E)\), and \(C\) is a vector space of bounded continuous functions on \(E\), having bounded continuous derivatives in all directions \( h \in E \). The measure \(\nu\) is called \(C\)-differentiable along \( h \in E \) if there is a \(\beta^{\nu}(h,.) \in \mathcal L^{1}(\nu) \) such that \(\int \phi (.) \beta^{\nu}(h,.) d \nu = \int \phi '(.) (h) d \nu \) for every \( \phi \in C \). \( H_{\nu}\) denotes the vector space of all \( h \in E \) along which \(\nu\) is \(C\)-differtiable and \(H\) is a vector subspace of \(H_{\nu}\); \(\beta^{\nu}(h,.) \) is called the logarithmic gradient of \(\nu\) along \( H\). Furthermore, \(H\) is assumed to be a Hilbert space, continuously and densely embedded in \(E\), which is also assumed to be a Hilbert space; \( i_{H}: H \to E \) is the embedding mapping and \( i_{H}^{*} \) its conjugate. The logarithmic gradient of a measure \(\nu\) with respect to \(H\) is the mapping \(b_{H}^{\nu} : E \to E \) such that for all \( z \in E' \) and \(\nu\)-almost all \( x \in E \), \(\beta^{\nu}(i_{H}^{*}(z),x)= z (b_{H}^{\nu}(x))\). The authors discuss the logarithmic derivatives and logarithmic gradients of smooth measures. They obtain necessary and sufficient conditions for certain Gaussian measures to be uniquely determined by their logarithmic derivatives. Moreover, some results about the existence of logarithmic gradients of smooth measures are obtained. Some other related results about these measures are also established.