  We are interested in the increment stationarity property for $L^2$-indexed
stochastic processes, which is a fairly general concern since many random
fields can be interpreted as the restriction of a more generally defined
$L^2$-indexed process. We first give a spectral representation theorem in the
sense of \citet{Ito54}, and see potential applications on random fields, in
particular on the $L^2$-indexed extension of the fractional Brownian motion.
Then we prove that this latter process is characterized by its increment
stationarity and self-similarity properties, as in the one-dimensional case.
