Fundamentals and Tools

Abstract

In our study of noise effects in spatially extended systems, we will make use of spatiotemporal Langevin equations. These are stochastic partial differential equations, usually nonlinear, that govern the dynamical evolution of the relevant variables of the system. Owing to the existence of noise, these variables become stochastic field processes themselves, which require for their study suitable statistical approaches, most times corresponding to generalizations of well-known techniques used in the study of zero-dimensional stochastic processes [VKa81, HL84, Ris84, Gard89, ST98]. This chapter presents a brief practical introduction to stochastic partial differential equations (Section 2.1), and to the set of techniques, both theoretical (Section 2.2) and numerical (Section 2.3), that are usually employed to deal with them.