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arXiv · 2503.09770

Singularity of compound stationary measures

Behrang Forghani·Vadim Kaimanovich
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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.

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Behrang Forghani, Vadim Kaimanovich. 2025-03-12. Singularity of compound stationary measures. https://arxiv.org/abs/2503.09770

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