Semi-classical limit of random walks II

Let (G, μ) be a symmetric random walk on a compact Lie group G. We will call (G, μ) a Lagrangean random walk if the step distribution μ, a probability measure on G, is also a Lagrangean distribution on G with respect to some Lagrangean submanifold Λ ⊂ T* G. In particular, we are interested in the cases where μ is a smooth δ-function δ_C along a 'positively curved hypersurface' C of G or where μ is a sum of δ-functions Σ_j δ_Cj along a finite union of regular conjugacy classes C_j in G. The Markov (transition) operator Tμ of the Lagrangean random walk is then a Fourier integral operator and our purpose is to apply microlocal techniques to study the convolution powers μ^*k of μ. In cases where all convolution powers are 'clean' (such as for δ-functions on positively curved hypersurfaces), classical FIO methods will be used to determine • the Sobolev smoothing order of Tμ on W^S(G), • the minimal power k = k_μ for which μ^*k ∈ L², • the asympotics of the Fourier transform μ̂(ρ) of μ along rays L = ℕρ of representations. In general, convolutions of Lagrangean measures are not 'clean' and there can occur a large variety of possible singular behaviour in the convolution powers μ^*k. Classical FIO methods are then no longer sufficient to analyze the asymptotic properties of Lagrangean random walks. However, it is sometimes possible to restore the simple 'clean convolution' behaviour by restricting the random walk to a fixed 'ray of representations'. In such cases, classical Toeplitz methods can be used to determine restricted versions of the above features along the ray. We will illustrate with the case of sums of δ-functions along unions of regular conjugacy classes.