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Conformal invariance of white noise

Published online by Cambridge University Press:  22 January 2016

Takeyuki Hida ,
Ke-Seung Lee  and
Sheu-San Lee
Takeyuki Hida
Affiliation:
Department of Mathematics, Faculty of Science, Nagoya University, Chikusa-ku, Nagoya 464, Japan
Ke-Seung Lee
Affiliation:
College of Leberal arts and Sciences, Department of Mathematics, Korea University, Korea
Sheu-San Lee
Affiliation:
Shenyang Chemical Engineering Institute, Shenyang, China
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The remarkable link between the structure of the white noise and that of the infinite dimensional rotation group has been exemplified by various approaches in probability theory and harmonic analysis. Such a link naturally becomes more intricate as the dimension of the time-parameter space of the white noise increases. One of the powerful method to illustrate this situation is to observe the structure of certain subgroups of the infinite dimensional rotation group that come from the diffeomorphisms of the time-parameter space, that is the time change. Indeed, those subgroups would shed light on the probabilistic meanings hidden behind the usual formal observations. Moreover, the subgroups often describe the way of dependency for Gaussian random fields formed from the white noise as the time-parameter runs over the basic parameter space.

The main purpose of this note is to introduce finite dimensional subgroups of the infinite dimensional rotation group that have important probabilistic meanings and to discuss their roles in probability theory. In particular, we shall see that the conformal invariance of white noise can be described in terms of the conformal group which is a finite dimensional Lie subgroup of the infinite dimensional rotation group.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 98 , June 1985 , pp. 87 - 98
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1985

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