Brownian motions on infinite dimensional quadric hypersurfaces
A potential theory on an infinite dimensional quadric hypersurface S is developed following Lévy's limiting procedure. For a given real sequence \(\{\lambda _ n\}^{\infty}_{n=1}\) a quadratic form h(x) on an infinite-dimensional real sequence space \({\mathbb{E}}\) is defined by \[ h(x):=\lim _{N\to \infty}N^{-1}\sum ^{N}_{n=1}\lambda _ nx^ 2_ n,\quad x=(x_ 1,x_ 2,...)\in {\mathbb{E}}, \] and a quadratic hypersurface S is defind by \(S:=\{x\in {\mathbb{E}}\); \(h(x)=c\}\), and the Laplacian \({\bar \Delta}{}_{\infty}\) on S is introduced by the limiting procedure. Instead of direct use of \({\bar \Delta}{}_{\infty}\), the Brownian motion \(\xi (t)=(\xi _ 1(t),\xi _ 2(t),...)\), the diffusion process (\(\xi\) (t),P x) on S with the generator \({\bar \Delta}{}_{\infty}/2\), is constructed by solving a system of stochastic differential equations according to \({\bar \Delta}{}_{\infty}\). The law of large numbers for \(X_ n(t):=(\lambda _ n,\xi _ n(t))\) is proved, and ergodic properties are discussed.
potential theory on an infinite dimensional quadric hypersurface
scientific article; zbMATH DE number 4040992