Multifractal formalism for self-similar bridges

Abstract

We derive the thermodynamics of self-similar paths (or bridges) joining the two points and of the plane. These paths may be constituted with both macroscopic and microscopic fragments, each deserving its specific statistics, while remaining continuous. Such discontinuous paths are also studied with some information related to the statistics of their jumps. If the bridges under study are bound to be non-decreasing -paths, this study coincides with the one of multifractal measures on the unit interval. Relaxing this condition leads to an extension of the multifractal formalism whose main lines are derived here.