Abstract: Let X(s, t)=e^-\alpha s-\beta t\leftX_0+\sigma \int_0^s \int_0^t e^\alpha a+\beta b d w(a, b)\right be an Ornstein-Uhlenbeck Process with two parameters （oup₂）. Let l: t=λs+T（s≥0） be a ray, and λ and c two nonnegative constants. Y=X（s,λs+c）,s≥0, is the process induced by X（s, t） on the ray l. Y is a Marker Process and its transition density is calculated. It is proved that Y is oup₁ if and only if λ=0, c＞0, and Y is a weakly stationary Process if and only if \lambda=0, \quad c=\frac\ln \left(\sigma^2+4 \alpha \beta E X_0^2\right)-2 \ln \sigma2 \beta.