Functional inequalities provide information on the structure of a probability measure and on the relaxation of associated stochastic dynamics to equilibrium. In this talk, we will describe a multiscale analysis for decomposing high-dimensional measures into simpler structures and derive from it functional inequalities. The strategy is based on the renormalization group method used in statistical physics to study the distribution of interacting particle systems. We will explain how this decomposition of measures can be interpreted in terms of optimal transport. Finally, we will review other related developments among which numerical methods for sampling high-dimensional probabilities and generating images.