Abstract: A stochastic process X=X_t, t∈R₊^N is called N-parameter Ornstein-Uhlenbeck process （OUP）, if X_t=e^-\alpha, t\leftX_0+\sigma \iint_0, t e^\alpha, a d w(a)\right Where X₀~N（0, 1）, W is the N-parameter Wiener process and X₀ is independent of W. X^N,d=（X_t¹,…, X_t^d）:t∈R₊^N is defind ad the N-parameter d dimensional OUP. We prove that when d＜2N, the range of X^N,d has a positive d-dimensional volume a.s. The fact is a foundation for further discussion of sample paths of X^N,d.