Stochastic Partial Differential Equations, Space-Time White Noise and Random Fields

This open access book provides a comprehensive introduction to the theory of stochastic partial differential equations (SPDEs). The focus is on SPDEs driven by Gaussian space-time white noise. The book covers both linear and nonlinear SPDEs, with Lipschitz and locally Lipschitz coefficients and multiplicative noise.  It provides a modern presentation of the theory of stochastic integration with respect to space-time white noise and unifies many results in the literature. The book discusses fundamental topics such as existence and uniqueness of random field solutions, along with their space-time sample path regularity properties. The book also presents a selection of additional topics such as weak solutions in law to SPDEs, space-time Markov properties, asymptotic bounds on moments, comparison theorems, a study of polarity of points for SPDEs with additive noise, and a study of SPDEs with rough initial conditions that includes the parabolic and hyperbolic Anderson models and their intermittency properties. In the context of the stochastic heat equation, the book discusses additional important topics including invariant and limit measures, reversible measures and their relationship to bridge measures, irreducibility properties, and large interval asymptotics. The appendices gather results from analysis and stochastic processes that are used throughout the core of the book, including key elements from the general theory of stochastic processes, a detailed presentation of Kolmogorov’s anisotropic continuity criterion, numerous integrability properties of the fundamental solutions and Green's functions associated to the heat and wave partial differential operators, explicit calculations of some space-time convolution series and some useful Gronwall-type lemmas. The book aims to be a reference for established researchers in the field of SPDEs, as well as for those who are interested in entering the field and becoming familiar with its techniques.  In particular, graduate and postgraduate students with a background in stochastic analysis will find here a comprehensive and self-contained source of information which provides essential expertise in the subject.

Robert C. Dalang is Professor at the Ecole Polytechnique Fédérale de Lausanne (EPFL), Switzerland, specializing in stochastic analysis and stochastic control. His research concerns fundamental properties of linear and nonlinear systems of stochastic partial differential equations, including sample path regularity, geometric properties of sample paths, probabilistic potential theory using Malliavin calculus, and long-time asymptotic properties. His publication record includes articles in the Annals of Probability, Probability Theory and Related Fields, Transactions of the AMS, Acta Mathematica, two Memoirs of the AMS and a monograph on sequential stochastic optimization. In 2019, he was elected Fellow of the Institute of Mathematical Statistics (USA).

Marta Sanz-Solé is Professor Emeritus at the University of Barcelona specializing in stochastic analysis, in particular in stochastic partial differential equations (SPDEs). Her research papers address a range of topics spanning from analysis on the Wiener space and Malliavin calculus to large deviations, support theorems, sample path properties and numerical approximations. Her publication record includes articles in the Annals of Probability, Probability Theory and Related Fields, Journal of Functional Analysis, Annals of the Institut Henri Poincaré among others, two Memoirs of the AMS and a monograph on Malliavin calculus.   Sanz-Solé is member of the Institute of Catalan Studies and of the Royal Academy of Sciences and Arts of Barcelona, Academia Europaea and the European Academy of Sciences. In 1998 she was awarded the Narcís Monturiol Medal of Scientific and Technological Excellence by the Catalan Government. In 2011, she was elected Fellow of the Institute of Mathematical Statistics (USA), and in 2017 she was the recipient of the Medal of the Royal Spanish Mathematical Society.