Related papers: Numerical computation of convolutions in free prob…
This work proposes algorithms for computing additive and multiplicative free convolutions of two given measures. We consider measures with compact support whose free convolution results in a measure with a density function that exhibits a…
We introduce and study a new type of convolution of probability measures called the orthogonal convolution, which is related to the monotone convolution. Using this convolution, we derive alternating decompositions of the free additive…
In this thesis we study convolutions that arise from noncommutative probability theory. We prove several regularity results for free convolutions, and for measures in partially defined one-parameter free convolution semigroups. We discuss…
We extend to arbitrary measures results of Bao, Erd\"os, Schnelli, Moreillon, and Ji on the connectedness of the supports of additive convolutions of measures on \mathbb{R} and of free multiplicative convolutions of measures on…
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…
In this paper, we give subordination functions for free additive and free multiplicative deconvolutions in some domain of the complex half-plane, under the condition that the distributions admit moments, respectively, of second order for…
A comprehensive convergence and stability analysis of some probabilistic numerical methods designed to solve Cauchy-type inverse problems is performed in this study. Such inverse problems aim at solving an elliptic partial differential…
We develop analytic tools for studying the free multiplicative convolution of any measure on the real line and any measure on the nonnegative real line. More precisely, we construct the subordination functions and the $S$-transform of an…
It is a classical result in complex analysis that the class of functions that arise as the Cauchy transform of probability measures may be characterized entirely in terms of their analytic and asymptotic properties. Such transforms are a…
In a large class of statistical inverse problems it is necessary to suppose that the transformation that is inverted is known. Although, in many applications, it is unrealistic to make this assumption, the problem is often insoluble without…
We study the hydrodynamic limits of three kinds of one-dimensional stochastic log-gases known as Dyson's Brownian motion model, its chiral version, and the Bru-Wishart process studied in dynamical random matrix theory. We define the…
This paper contributes to the study of the free additive convolution of probability measures. It shows that under some conditions, if measures $\mu_i$ and $\nu_i, i=1,2$, are close to each other in terms of the L\'{e}vy metric and if the…
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…
We develop an efficient and convergent numerical method for solving the inverse problem of determining the potential of nonlinear hyperbolic equations from lateral Cauchy data. In our numerical method we construct a sequence of linear…
The Gauss-Minkowski correspondence in $\mathbb{R}^2$ states the existence of a homeomorphism between the probability measures $\mu$ on $[0,2\pi]$ such that $\int_0^{2\pi} e^{ix}d\mu(x)=0$ and the compact convex sets (CCS) of the plane with…
Numerical relativity (NR) enables the study of physics in strong and dynamical gravitational fields and provides predictions for the gravitational-wave signals produced by merging black holes. Despite the impressive accuracy of modern…
The Stieltjes (or sometimes called the Cauchy) transform is a fundamental object associated with probability measures, corresponding to the generating function of the moments. In certain applications such as free probability it is essential…
Different types of convolution operations involving large Vandermonde matrices are considered. The convolutions parallel those of large Gaussian matrices and additive and multiplicative free convolution. First additive and multiplicative…
Let $\mu$ and $\nu$ be probability measures on $\mathbb{R}$ with compact support, and let $\mu \boxplus \nu$ denote their additive free convolution. We show that for $z \in \mathbb{R}$ greater than the sum of essential suprema of $\mu$ and…
We develop the complex-analytic viewpoint on the tree convolutions studied by the second author and Weihua Liu in "An operad of non-commutative independences defined by trees" (Dissertationes Mathematicae, 2020, doi:10.4064/dm797-6-2020),…