Self-Similarity

Abstract

Self-similarity is a major part of the mathematics. One can refer to [53] for a general reference. The self-similarity literature is quite confusing for beginners since the statement of very elementary facts may look very similar to deep theorems. On the one hand if you assume that you observe a self-similar phenomenon, then the self-similarity is an invariance property and you expect your phenomenon to be easier to study than general phenomena with no structure. On the other hand if you want to have a complete classification of self-similar fields then we can find in the literature a lot of counter-examples that prevent to draw even an heuristic picture of what is true for every self-similar fields. Following [138] a classical tool to simplify the study of the self-similarity in the stochastic case is to assume that the fields have stationary increments.