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Brownian motion parametrized with metric space of constant curvature

Published online by Cambridge University Press:  22 January 2016

Shigeo Takenaka
Affiliation:
Department of Mathematics, Faculty of Science, Nagoya University
Izumi Kubo
Affiliation:
Department of Mathematics, Faculty of Science, Nagoya University
Hajime Urakawa
Affiliation:
Department of Mathematics, College of General Education, Tôhoku University
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Extract

P. Lévy introduced a generalized notion of Brownian motion in his monograph “Processus stochastiques et mouvement brownien” by taking the time parameter space to be a general metric space. Let (M, d) be a metric space and let O be a fixed point of M called the origin. Following his definition, a Brownian motion parametrized with the metric space (M, d) is a Gaussian system ℬ = {B(m); mM} such that the difference B(m) − B(m′) is a random variable with mean zero and variance d(m, m′), and that B(O) = 0.

Type
Research Article
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1981

References

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