Multi-parameter infinite-dimensional \((r,\delta)\)-OU process (Q701606)

In terms of a stochastic integral relative to a space-time white noise, the author defines a multi-parameter Ornstein-Uhlenbeck process \(\{X_t(\cdot): t\in R^n_\delta\}\) taking values in the Wiener space \(W=C([0,1],R)\), where \((\delta_1, \dots, \delta_n) \in R^n\) and \(R^n_\delta = \{(s_1, \dots, s_n) \in R^n: s_i > \delta_i\}\). Let \(\mu_t\) denote the distribution of \(X_t(\cdot)\) on \(W\). A necessary and sufficient condition for \(\mu_t\) and \(\mu_{t^\prime}\) to be absolutely continuous is given and the limit in distribution of \(X_t(\cdot)\) as \(|t|\to \infty\) is discussed. Several extensions of the results are also given.