Quasi-invariance of product measures under Lie group perturbations: Fisher information and \(L^ 2\)-differentiability. (English) Zbl 0779.60039

Let \(G\) be a connected finite-dimensional Lie group acting continuously on a locally compact second countable space \(X\). Suppose that a probability measure \(\mu\) on \(X\) is quasi-invariant under the induced action of \(G\) on the measures, i.e. \(g\mu\sim\mu\) for all \(g\in G\). The paper studies sequences \((g_ n)_{n\in\mathbb{N}}\) in \(G\) such that \(\prod^ \infty_ ng_ n\mu\sim\prod^ \infty_ n\mu\). Let \(d\) be any Riemannian metric on \(G\) and assume for this review that \(g\mapsto g\mu\) is \(1-1\). The deepest result says that the map \(g\mapsto\left({dg\mu\over d\mu}\right)^{1/2}\in L^ 2(\mu)\) is differentiable if (and only if) for all \((g_ n)\) converging to \(e\in G\) the conditions \(\sum_ nd(g_ n,e)^ 2<\infty\) and \(\prod_ ng_ n\mu\sim\prod_ n\mu\) are equivalent. The proof uses representation theory of Lie groups, smoothing by the Brownian semigroup on \(G\) and the Fisher information formulation for Kakutani’s dichotomy.


60G30 Continuity and singularity of induced measures
60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
58J65 Diffusion processes and stochastic analysis on manifolds
62A01 Foundations and philosophical topics in statistics
62B15 Theory of statistical experiments
22E47 Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.)
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