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On the Upper and Lower Class For Gaussian Processes with Several Parameters

Published online by Cambridge University Press:  22 January 2016

Takayuki Kawada
Takayuki Kawada*
Affiliation:
Kobe College of Commerce
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1.In the study on Hölder-continuity of Brownian motion, A.N.Kol-mogorov introduced the concept of upper and lower classes and presented a criterion with the integral form to test whether some function belongs to upper or lower class; the so-called Kolmogorov’s test (I.Petrovesky [10]). P.Lévy considered the upper and lower class with regard to the uniform continuity of Brownian motion. We shall recall the definition of the upper and lower classes. We shall call <p(t) a function belonging to the upper class with regard to the uniform continuity of Brownian motion x(t) if there exists a positive number s{w) such that, for almost all w,

implies

(1.1)

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 35 , July 1969 , pp. 109 - 132
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1969

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