Ursula Porod, Steve Zelditch
Semi-classical Limit of Random Walks II
Preprint series: ESI preprints

MSC:

58F06 Geometric quantization (applications of representation theory), See also {22E45, 81S10}

60B15 Probability measures on groups, Fourier transforms, factorization

60J15 Random walks

22E30 Analysis on real and complex Lie groups, See also {33C80,

43A30 Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc.

Abstract: Let $(G, \mu)$ be a symmetric random walk on a compact Lie group
$G$. We will call $(G, \mu)$ a {\it Lagrangean} random walk if the step
distribution $\mu$, a probability measure on $G$, is also a Lagrangean distribution
on $G$ with respect to some Lagrangean submanifold $\Lambda \subset T^*G$.
In particular, we are interested in the cases where $\mu$ is a smooth
$\delta$-function $ \delta_C$ along a `positively curved hypersurface' $C$ of $G$
or where $\mu$ is a sum of $\delta$-functions $\sum_j \delta_{C_j}$ along a finite
union of regular conjugacy classes $C_j$ in $G$. The Markov (transition) operator
$T_{\mu}$ of the Lagrangean random walk is then a Fourier integral
operator and our purpose is to apply microlocal techniques to study the
convolution powers $\mu^{*k}$ of $\mu.$

In cases where all convolution powers
are `clean' (such as for $\delta$-functions on positively curved hypersurfaces),
classical FIO methods will be used to determine
\begin{itemize}
\item the Sobolev smoothing order of $T_{\mu}$ on $W^s(G)$,

\item the
minimal power $k = k_{\mu}$ for which $\mu^{*k} \in L^2$,

\item the asympotics of the Fourier transform $\hat{\mu}(\rho)$ of $\mu$
along rays $L = \N \rho$ of representations.
\end{itemize}

In general, convolutions of Lagrangean measures are not `clean' and
there can occur a large variety of possible
singular behaviour in the convolution powers $\mu^{*k}$. 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 $\delta$-functions along unions of
regular conjugacy classes.
Keywords: random walks on compact Lie groups, geometric quantization, Szegoe limit theorems