Self-similarity and Lamperti transformation for random fields

We define multi-self-similar random fields, that is, random fields that are self-similar component-wise. We characterize them, relate them to stationary random fields using a Lampertitype transformation and study these stationary fields. We also extend the notions of local stationarity and local stationarity reducibility to random. fields. Our work is motivated by applications arising from climatological and environmental sciences. We illustrate these new concepts with the fractional Brownian sheet and the Levy fractional Brownian random field.