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- W135744175 abstract "In this notes we discuss Brownian motions on compact groups. For a compact group G, these are G-valued stochastic processes having independent stationary increments, continuous paths, and some additional natural properties such as symmetry and bi-invariance. See also Heyer’s treaty [26] for background and historical remarks. We outline recent results concerning two types of problems: (a) existence of Brownian motions having certain prescribed properties such as one-dimensional marginals having continuous densities; (b) relations between various properties of the one-dimensional marginals. We then illustrate these results by looking at some specific examples of compact, connected, locally connected groups that are not Lie groups. Our general goal is to develop a better understanding of Brownian motions on compact groups. The results presented below solve a number of questions that emerged in the sixties and seventies concerning these processes and their associated harmonic sheaves. For instance, which are the groups that carry bi-invariant Brelot harmonic sheaves? which are the groups that carry Brownian motions with one dimensional marginals having a continuous density? For Brownian motions with one dimensional marginals having a continuous density μt(·), what are the possible behaviors of μt(e) as t tends to zero? Among other natural questions is the problem of studying the regularity of the paths of such processes. Although this will not be discussed in the sequel, it should be clear that our results are important building blocks in this direction. We conjecture that any connected locally connected compact group G having a countable basis for its topology admits Brownian motion having Holder continuous paths with Holder exponent α for all α < 1/2. We will present results in these directions in a forthcoming paper. Our study naturally belongs to the general area known as Probability on Algebraic Structures. As an identified field of study, Probability on Algebraic Structures can be traced back to the late fifties and early sixties. Two references with pointers to the early papers on the" @default.
- W135744175 created "2016-06-24" @default.
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- W135744175 date "2000-01-01" @default.
- W135744175 modified "2023-09-24" @default.
- W135744175 title "Brownian motion on compact groups in infinite dimension" @default.
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