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

de F. Marques, Mauro S.; San Martin, Luiz

Ann. Probab. 20, No. 3, 1420-1435 (1992).

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.

Reviewer: H.von Weizsäcker (Kaiserslautern)

MSC:

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.)

Keywords:

Kakutani dichotomy; Riemannian metric; Brownian motion on groups; quasi- invariance