arXiv subjects

Wojciech Bozejko

Publications and source records attributed to Wojciech Bozejko.

2 recordsLinked to original sources

arXiv2024

Deformations and q-convolutions. Old and new results

This paper is the survey of some of our results related to $q$-deformations of the Fock spaces and related to $q$-convolutions for probability measures on the real line $\mathbb{R}$. The main idea is done by the combinatorics of moments of the measures and related $q$-cumulants of different types. The main and interesting $q$-convolutions are related to classical continuous (discrete) $q$-Hermite polynomial. Among them are classical ($q=1$) convolutions, the case $q=0$, gives the free and Boolean relations, and the new class of $q$-analogue of classical convolutions done by Carnovole, Koornwinder, Biane, Anshelovich, and Kula. The paper contains many questions and problems related to the positivity of that class of $q$-convolutions. The main result is the construction of Brownian motion related to $q$-Discrete Hermite polynomial of type I.

math-ph↗

arXiv2013

Generalized Gaussian processes and relations with random matrices and positive definite functions on permutation groups

The main purpose of this paper of the paper is an explicite construction of generalized Gaussian process with function $t_b(V)=b^{H(V)}$, where $H(V)=n-h(V)$, $h(V)$ is the number of singletons in a pair-partition $V \in \st{P}_2(2n)$. This gives another proof of Theorem of A. Buchholtz \cite{Buch} that $t_b$ is positive definite function on the set of all pair-partitions. Some new combinatorial formulas are also presented. Connections with free additive convolutions probability measure on $\mathbb{R}$ are also done. Also new positive definite functions on permutations are presented and also it is proved that the function $H$ is norm (on the group $S(\infty)=\bigcup S(n)$.

math.PR↗