The physical relevance of fluctuations in a probabilistic language demands the illustration of basic mathematical tools, including the central limit theorem and the theory of large deviations. A short summary about random matrix theory precedes the model of generalized random walks, which includes Levy flights and walks as representations of anomalous diffusive processes. Einstein's approach to the role of fluctuations in thermodynamic processes is detailed for both an isolated and a thermalized thermodynamic system. An introduction to stochastic thermodynamics and to generalized fluctuation theorems is finally discussed.

An interface is rough if the mean square fluctuations of its position diverge at large times and system sizes. This may occur when the interface is driven out of equilibrium in the presence of some noise and the way roughness diverges defines suitable critical exponents. We introduce and discuss extensively two important universality classes: the Edwards–Wilkinson and the Kardar–Parisi–Zhang. The latter has been the subject of renewed interest since it was possible to determine analytically the whole spectrum of fluctuations and it was found an experimental system satisfying such predictions with great accuracy. The last part of the chapter is devoted to nonlocal models, specifically the celebrated Diffusion Limited Aggregation.

The phenomenological theory proposed by Einstein for interpreting the phenomenon of Brownian motion is described in detail. The alternative approaches due to Langevin and Fokker–Planck are also illustrated. The theory of Markov chains is also reported as a basic mathematical approach to stochastic processes in discrete space and time; various of its applications, for example, the Monte Carlo method, are also illustrated. The theory of stochastic equations, as a representation of stochastic processes in continuous space–time, is discussed and used for obtaining a generalized, rigorous formulation of the Langevin and Fokker–Planck equations for generalized fluctuating observables. The Arrhenius formula as an example of the first exit-time problem is also derived.

This chapter examines the related objectives of defining spatial clusters and delineating spatial boundaries in discontinuous data. The former often proceeds by grouping together adjacent locations when they have the most similar characteristics; the latter proceeds by estimating boundaries between locations that are most different. For this, there are several methods available that suggest ’boundary elements’ as possible components of a final division or complete boundary, depending on the kind of data (e.g. binary versus qualitative versus continuous quantitative) and the arrangement of the measured locations (e.g. regular lattice versus irregular spatial network). Once boundaries have been established, statistics are available to evaluate them, including boundary overlap measures. Clusters and boundaries represent two aspects of the same phenomenon, with the same challenge of formalizing similarity and difference in continuous spatial data.

This final chapter is a short introduction to pattern-forming systems, which highlights a few concepts and models rather than pretending to give a general overview (which is impossible in 40 pages). We focus on stationary bifurcations, distinguishing between scenarios where the critical wavevector vanishes and where it is a finite value, because they have different nonlinear behaviors. A few pages are devoted to describe some different experimental setups: thermal convection (a fluid heated from below, showing the rising of convection cells); unstable growth process (under particle deposition, with the formation of mounds); and a rotating mixture of granular systems (with their phase separation).