## The Annals of Probability

### Continuity of $l^2$-Valued Ornstein-Uhlenbeck Processes

#### Abstract

A stationary $l^2$-valued Ornstein-Uhlenbeck process is considered which is given formally by $dX_t = -AX_t dt + \sqrt 2a dB_t$, where $A$ is a positive self-adjoint operator on $l^2, B_t$ is a cylindrical Brownian motion on $l^2$ and $a$ is a positive diagonal operator on $l^2$. A simple criterion is given for the almost-sure continuity of $X_t$ in $l^2$ which is shown to be quite sharp. Furthermore, in certain special cases, we obtain simple necessary and sufficient conditions for the almost-sure continuity of $X_t$ in $l^2$.

#### Article information

Source
Ann. Probab., Volume 18, Number 1 (1990), 68-84.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176990938

Digital Object Identifier
doi:10.1214/aop/1176990938

Mathematical Reviews number (MathSciNet)
MR1043937

Zentralblatt MATH identifier
0699.60052

JSTOR
Iscoe, I.; Marcus, M. B.; McDonald, D.; Talagrand, M.; Zinn, J. Continuity of $l^2$-Valued Ornstein-Uhlenbeck Processes. Ann. Probab. 18 (1990), no. 1, 68--84. doi:10.1214/aop/1176990938. https://projecteuclid.org/euclid.aop/1176990938