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Brownian motions on infinite dimensional quadric hypersurfaces
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  • Published: September 1989

Brownian motions on infinite dimensional quadric hypersurfaces

In the memory of my friend Ichiro Enomoto

  • Yoshihei Hasegawa1 

Probability Theory and Related Fields volume 80, pages 347–364 (1989)Cite this article

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Summary

A potential theory on an infinite dimensional quadric hypersurfaceS is developed following Lévy's limiting procedure. For a given real sequence {λ n } ∞ n=1 a quadratic fromh(x) on an infinite dimensional real sequence spaceE is defined by\(h(x): = \mathop {\lim }\limits_{N \to \infty } \frac{1}{N}\sum\limits_{n = 1}^N {\lambda _n x_n^2 ,x = (x_1 ,x_2 ,...) \in E} \) and a quadric hypersurfaceS is defined byS:={x∈E;h(x)=c}, and the Laplacian\(\bar \Delta _\infty \) onS is introduced by the limiting procedure. Instead of a direct use of\(\bar \Delta _\infty \), the Brownian motionξ(t)=(ξ 1(t)),ξ 2(t),...), the diffusion process (ξ(t),P x) onS with the generator\({{\bar \Delta _\infty } \mathord{\left/ {\vphantom {{\bar \Delta _\infty } 2}} \right. \kern-\nulldelimiterspace} 2}\) is constructed by solving a system of stochastic differential equations according to\(\bar \Delta _\infty \). The law of large numbers forX n (t:=(λ n ,ξ n (t)) is proved, and ergodic properties are discussed.

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References

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Authors and Affiliations

  1. Department of Mathematics, Nagoya Institute of Technology, Gokiso, Showa-ku, 466, Nagoya, Japan

    Yoshihei Hasegawa

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  1. Yoshihei Hasegawa
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Cite this article

Hasegawa, Y. Brownian motions on infinite dimensional quadric hypersurfaces. Probab. Th. Rel. Fields 80, 347–364 (1989). https://doi.org/10.1007/BF01794428

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  • Received: 05 November 1986

  • Revised: 04 July 1988

  • Issue Date: September 1989

  • DOI: https://doi.org/10.1007/BF01794428

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Keywords

  • Differential Equation
  • Stochastic Process
  • Brownian Motion
  • Probability Theory
  • Diffusion Process
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