Basics on Noise and SPDEs

This chapter is devoted to introducing the notion of random noise, to informally describing SPDEs as extensions of PDEs and, through various examples, to giving motivations for the study of SPDEs. We put the focus on Gaussian random fields with an emphasis on isonormal Gaussian processes and (Gaussian) white noise on $$\mathbb {R}^{k}$$ , and the connections between these notions. Because of its importance throughout the book, we give particular attention to (Gaussian) space-time white noise and describe its many facets, including several series representations. Using the classical approach in several familiar examples of PDEs, we anticipate the form of the random field solutions to SPDEs which will be developed in later chapters. We close this chapter with a selection of examples of SPDEs that appear in models coming from various disciplines.