Singularity of compound stationary measures
Behrang Forghani, Vadim Kaimanovich
TL;DR
This work demonstrates that for the modular group $G=\mathrm{PSL}(2,\mathbb Z) \cong \mathbb Z_2 * \mathbb Z_3$, the class of filling probability measures is not closed under convolution or convex combination. By leveraging the mediant tree and boundary encoding, the authors describe Minkowski and Denjoy measure classes on the boundary and solve the Radon–Nikodym problem to parametrize harmonic measures $\nu$ of random walks via $\nu=\varkappa^{\alpha,p}$ with $\alpha=\pi_{ba}$ and $p=\dfrac{\pi_a}{1+\pi_a}$. They construct explicit finitely supported measures $\mu_1,\mu_2$ whose harmonic measures are equivalent, yet their convolution or convex combination yields a harmonic measure singular to that class, thereby answering several closure questions in the negative. The results illuminate how maximal entropy and boundary harmonic measures interact under simple compound operations and reveal delicate boundary-behavior distinctions in hyperbolic-group random walks. Overall, the paper provides concrete counterexamples and a detailed framework connecting group-theoretic structures, boundary measures, and stochastic processes on hyperbolic groups.
Abstract
We show that the product or convex combination of two Markov operators with equivalent stationary measures need not have a stationary measure from the same measure class. More specifically, we exhibit examples of a hitherto undescribed phenomenon: maximal entropy random walks for which the resulting compound random walks no longer have maximal entropy. The underlying group in these examples is $PSL(2,\mathbb Z)\cong{{\mathbb Z}_2}*{{\mathbb Z}_3}$, and the associated harmonic measures belong to the canonical Minkowski and Denjoy measure classes on the boundary. These examples also demonstrate that a number of other natural families of random walks are not closed under convolutions or convex combinations of step distributions.