We are interested in the increment stationarity property of $L^2$-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 Ito , 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.
"Increment stationarity of $L^2$-indexed stochastic processes: spectral representation and characterization." Electron. Commun. Probab. 21 1 - 15, 2016. https://doi.org/10.1214/16-ECP4727