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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Daffer, P.Z., Taylor, R.L.: Laws of large numbers forD[0, 1]. Ann. Probab.7, 85–95 (1979)
Hasegawa, Y.: Lévy's functional analysis in terms of an infinite dimensional Brownian motion. I. Osaka J. Math.19, 405–428 (1982)
Hasegawa, Y.: Lévy's functional analysis in terms of an infinite dimensional Brownian motion. II. Osaka J. Math.19, 549–570 (1982)
Hasegawa, Y.: Lévy's functional analysis in terms of an infinite dimensional Brownian motion. III. Nagoya Math. J.90, 155–173 (1983)
Lévy, P.: Problémes concrets d'analyse fonctionnelle. Paris: Gauthier-Villars 1951
Author information
Authors and Affiliations
Rights and permissions
About this article
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
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01794428