Regularity of \(\ell\) 2-valued Ornstein-Uhlenbeck processes. (English) Zbl 0645.60046

Schmuland, B.

C. R. Math. Acad. Sci., Soc. R. Can. 10, No. 2, 119-124 (1988).

Let \(X_ i(t)\), \(i=1\),..., \(t\geq 0\), be a family of independent one- dimensional and stationary Ornstein-Uhlenbeck processes with drift and diffusion coefficients \((-\lambda_ i)\) and \((\delta_ i)\), respectively. Two properties of this family are proved.
1) Let \(X(t)=(X_ i(t))\) be an \(\ell\) 2-valued stochastic process. Then the process X(t) is continuous in \(\ell\) 2 provided \(\sum \sigma\) \(4_ i/\lambda_ i<\infty\). 2) Let for every fixed \(t\geq 0\), \(X_ i(t)\) tend to zero with probability one. Then there exists \(x_ 0\geq 0\) such that for some \(t\geq 0\) the sequence \(X_ i(t)\) clusters at each point of the interval \([-x_ 0\), \(x_ 0]\) with probability one.

Reviewer: B.Goldys

MSC:

60G17	Sample path properties
60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
60G15	Gaussian processes
60J60	Diffusion processes

Keywords:

Hilbert space; continuity; Dirichlet form; Ornstein-Uhlenbeck processes