Monotone and Boolean Convolutions for Non-compactly Supported Probability Measures

Monotone, free, and boolean cumulants: a shuffle algebra approach

The theory of cumulants is revisited in the "Rota way", that is, by following a combinatorial Hopf algebra approach. Monotone, free, and boolean cumulants are considered as infinitesimal characters over a particular combinatorial Hopf algebra. The latter is neither commutative nor cocommutative, and has an underlying unshuffle bialgebra structure which gives rise to a shuffle product on its graded dual. The moment-cumulant relations are encoded in terms of shuffle and half-shuffle exponentials. It is then shown how to express concisely monotone, free, and boolean cumulants in terms of each oth…

Regularity of monotone transport maps between unbounded domains

The regularity of monotone transport maps plays an important role in several applications to PDE and geometry. Unfortunately, the classical statements on this subject are restricted to the case when the measures are compactly supported. In this note we show that, in several situations of interest, one can to ensure the regularity of monotone maps even if the measures may have unbounded supports.

随机单调删失下的对数Sobolev不等式

Log-Sobolev under random monotone censoring

本文证明布尔立方体的对数Sobolev不等式在随机单调删失下保持稳定，并推导出典型单调子集上的超收缩性、高斯集中和O(n log n)混合时间，解决了Ding和Mossel的猜想。