Some dichotomy results for central Gaussian convolution semigroups on compact groups

This is a survey article on Brownian motions on compact connected groups and the associated Gaussian convolution semigroups. The emphasize is on infinite dimensional groups such as the infinite dimensional torus and infinite products of special orthogonal groups. We discuss the existence of Brownian motions having nice properties such as marginales having a continuous density with respect to Haar measure. We relate the existence of these Brownian motions to the algebraic structure of the group. The results we describe reflect the conflicting effects of, on the one hand, the infinite dimensionality and, on the other hand, the compact nature of the underlying group.

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Analysis on compact Lie groups of large dimension and on connected compact groups

Saloff‐Coste

2010

Colloq. Math.

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Abstract. The study of Gaussian convolution semigroups is a subject at the crossroad between abstract and concrete problems in harmonic analysis. This article suggests selected open problems that are in large part motivated by joint work with Alexander Bendikov.This paper is dedicated to the memory of Andrzej Hulanicki. My first scientific encounter with Andrzej was through his students, which is fitting given the importance they had to him throughout his career. During my Ph.D., I came across the work of Pawe l G lowacki (he told me later that he was the referee of my first research paper, published by Studia Mathematica and based on my Ph.D. thesis). Later, I met Waldemar Hebisch, in Boston, in the late nineteen eighties. In 1991, Andrzej invited me (together with my wife, Cathy) to Wroc law for a month. We met for the first time at the Wroc law train station where he picked us up, carrying a mathematical book so that we could recognize him. This turned out to be a beginning of a very enjoyable personal and scientific relation with Andrzej, his many students and collaborators, and with the Mathematical Institute in Wroc law. As many others, I benefited from attending the conferences and schools organized by Andrzej's group in Tuczno and, later, in Zakopane. My first Ph.D. student, AndrzejŻuk, wrote a Master thesis under Andrzej's supervision in Wroc law before following Andrzej's suggestion to work with me in Toulouse (in fact,Żuk prepared his thesis under the joint supervision of Andrzej and myself, spending half his time in Toulouse and the other half in Wroc law, as required by his doctoral fellowship). Throughout his career, Andrzej, in addition to pursuing his own research in mathematics, put a tremendous energy into organizing and supporting mathematics in Poland and in Wroc law. He maintained many long term scientific international relations, including during periods when it was not necessarily an easy thing to do. He, relentlessly, 2010 Mathematics Subject Classification: 22A10, 43A70, 60B15.

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Some dichotomy results for central Gaussian convolution semigroups on compact groups

Cited by 3 publications

References 12 publications

On the convergence to equilibrium of Brownian motion on compact simple Lie groups

On the convergence to equilibrium of Brownian motion on compact simple Lie groups

Central Gaussian Convolution Semigroups on Compact Groups: A Survey

Analysis on compact Lie groups of large dimension and on connected compact groups

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