Quasi-invariance of product measures under Lie group perturbations: Fisher information and \(L^ 2\)-differentiability (Q1201180)

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.