Linear SPDEs Driven by Space-Time White Noise

This chapter initiates the study of SPDEs in an elementary setting. We consider a space-time white noise as a random forcing, and we mostly restrict the spatial dimension to $$k=1$$ . We start by introducing two notions of solution: random field solutions and weak solutions. Although in this book, we mostly emphasize the former notion, the latter is also widely present in the theory of PDEs and of SPDEs. We will then consider SPDEs with a linear differential operator driven by additive noise. We study two fundamental examples, namely the stochastic heat and wave equations in several different settings (on the real line, on finite intervals, etc.) and we prove sharp regularity properties of their sample paths.