Brownian motions on infinite dimensional quadric hypersurfaces

Summary A potential theory on an infinite dimensional quadric hypersurfaceS is developed following Lévy's limiting procedure. For a given real sequence {_n}_n=1 a quadratic fromh(x) on an infinite dimensional real sequence spaceE is defined by and a quadric hypersurfaceS is defined byS:={xE;h(x)=c}, and the Laplacian onS is introduced by the limiting procedure. Instead of a direct use of, the Brownian motion(t)=(₁(t)),₂(t),...), the diffusion process ((t),P^x) onS with the generator is constructed by solving a system of stochastic differential equations according to. The law of large numbers forX_n(t:=(_n,_n(t)) is proved, and ergodic properties are discussed.