The multiparametric Brownian motion. (Russian) Zbl 0637.60062

Molchan, G. M.

Teor. Veroyatn. Mat. Stat., Kiev 36, 88-101 (1987).

As generalizations of multiparameter Brownian motions, a class of random fields on Riemannian spaces, denoted by \({\mathcal M}_{\gamma}\), is introduced. A random field x(t) belongs to \({\mathcal M}_{\gamma}\), if 1) x(t) is a Gaussian random field with mean zero and structure function \(E| x(t)-x(s)|\) \(2=\Delta (\theta (t,s))\), where \(\theta\) (t,s) is the Riemann metric, \(\Delta\) (\(\theta)\) is continuous at \(\theta =0\); and 2) the restriction of x(t) on a geodesic \(\gamma\) is a quasi-Markov process.
All forms of the structure functions of random fields with the \({\mathcal M}_{\gamma}\)-property on two-point homogeneous spaces are given. In some special cases, the spectral functions and the factorization representations in terms of white noises are given, too.

Reviewer: He Shengwu

MSC:

60G60	Random fields
60J65	Brownian motion
60G15	Gaussian processes
60J60	Diffusion processes

Keywords:

multiparameter Brownian motions; random fields on Riemannian spaces; Gaussian random field; factorization representations