  In this article, we consider three important classes of $n$-variate fields
indexed by the set of $N$ dimensional integers, namely stationary, stationary
increment and self-similar fields. These classes are connected through
bijective transformations. In addition, we introduce generalized AR$(1)$ type
equations, whose unique stationary solutions are obtained via these
transformations. Lastly, we apply the transformations in order to construct
stationary fractional Ornstein-Uhlenbeck fields of the first and second kind.
