An estimation problem in compact Lie groups (Q1105542)

Let G be a compact, connected, simply connected semisimple Lie group. The author considers the G-valued Brownian motion \((Y_ t)\) with initial values supported on the exponential image of a fundamental domain \(\Delta\) w.r.t. the affine Weyl group (acting on a Cartan subalgebra). Let \(t_ 0>0\) be fixed and \(Z=f(Y_{t_ 0})\) a measurement, where the function \(f: G\to {\mathbb{R}}^ k\) with \(k=Rank\) G is constant on conjugacy classes and essentially yields diffeomorphisms when restricted to a finite partition of \(\Delta\). Under these hypotheses an explicit expression for the conditional probability of the initial value for known Z is given in terms of the root lattice and the Weyl group.