Boolean convolution of probability measures on the unit circle

Additive processes on the unit circle and Loewner chains

This paper defines the notion of generators for a class of decreasing radial Loewner chains which are only continuous with respect to time. For this purpose, "Loewner's integral equation" which generalizes Loewner's differential equation is defined and analyzed. The definition of generators is motivated by the Lévy-Khintchine representation for additive processes on the unit circle. Actually, we can and do introduce a homeomorphism between the above class of Loewner chains and the set of the distributions of increments of additive processes equipped with suitable topologies. On the other hand,…

Spectra of Bernoulli convolutions as multipliers in $L^p$ on the circle

It is shown that the closure of the set of Fourier coefficients of the Bernoulli convolution $μ_θ$ parameterized by a Pisot number $θ$, is countable. Combined with results of Salem and Sarnak, this proves that for every fixed $θ>1$ the spectrum of the convolution operator $f\mapsto μ_θ*f$ in $L^p(S^1)$ (where $S^1$ is the circle group) is countable and is the same for all $p\in(1,\infty)$, namely, $\bar{\{\hat{μ_θ}(n) : n\in\mathbb{Z}\}}$. Our result answers the question raised by P. Sarnak in \cite{Sar}. We also consider the sets $\bar{\{\hat{μ_θ}(rn) : n\in\mathbb{Z}\}}$ for $r>0$ which corr…

Multiplicative monotone convolutions

Recently, Bercovici has introduced multiplicative convolutions based on Muraki's monotone independence and shown that these convolution of probability measures correspond to the composition of some function of their Cauchy transforms. We provide a new proof of this fact based on the combinatorics of moments. We also give a new characterisation of the probability measures that can be embedded into continuous monotone convolution semigroups of probability measures on the unit circle and briefly discuss a relation to Galton-Watson processes.

Free Brownian motion and free convolution semigroups: multiplicative case

We consider a pair of probability measures $μ,ν$ on the unit circle such that $Σ_λ(η_ν(z))=z/η_μ(z)$. We prove that the same type of equation holds for any $t\geq 0$ when we replace $ν$ by $ν\boxtimesλ_t$ and $μ$ by $\mathbb{M}_t(μ)$, where $λ_t$ is the free multiplicative analogue of the normal distribution on the unit circle of $\mathbb{C}$ and $\mathbb{M}_t$ is the map defined by Arizmendi and Hasebe. These equations are a multiplicative analogue of equations studied by Belinschi and Nica. In order to achieve this result, we study infinite divisibility of the measures associated with subord…

通过扰动反向热过程证明塔拉甘的卷积猜想

Talagrand's convolution conjecture up to loglog via perturbed reverse heat

本文证明在布尔超立方体上的热半群下，非负函数的尾界优于马尔可夫不等式，解决了塔拉甘卷积猜想，方法包括反向热过程、耦合构造和时间平滑反集中估计。

On the free convolution with a free multiplicative analogue of the normal distribution

We obtain a formula for the density of the free convolution of an arbitrary probability measure on the unit circle of $\mathbb{C}$ with the free multiplicative analogues of the normal distribution on the unit circle. This description relies on a characterization of the image of the unit disc under the subordination function, which also allows us to prove some regularity properties of the measures obtained in this way. As an application, we give a new proof for Biane's classic result on the densities of the free multiplicative analogue of the normal distributions. We obtain analogue results for…

欧氏空间与球面上具有对称性的乘法卷积

Multiplicative convolution with symmetries in Euclidean space and on the sphere

本文研究欧氏空间和球面上带坐标轴反射对称性的乘法卷积运算，刻画了固定一个测度时运算关于另一测度的单射性，并推广了 Molchanov 和 Nagel 关于凸体的相关定理。

关于自由卷积的支撑集

On the support of free convolutions

该研究将自由卷积支撑集连通性的已有结果推广至任意测度，无需绝对连续性或有界支撑假设，还拓展了自由加法卷积分量数结果，并给出单位圆上自由乘法卷积对应结果的推导方法。