Stochastic Partial Differential Equations, Space-Time White Noise and Random Fields
- Open Access
- 2026
- Open Access
- Buch
- Verfasst von
-
Robert C. Dalang
Robert C. Dalang
- Institut de Mathématiques, École Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
-
Marta Sanz-Solé
Marta Sanz-Solé
- Facultat de Matemàtiques i Informàtica, Universitat de Barcelona, Barcelona, Spain
- Buchreihe
- Springer Monographs in Mathematics
- Verlag
- Springer Nature Switzerland
Über dieses Buch
Über dieses Buch
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.
Inhaltsverzeichnis
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Chapter 1. Basics on Noise and SPDEs
- Open Access
PDF-Version jetzt herunterladenAbstractThis 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. -
Chapter 2. Stochastic Integrals with Respect to Space-Time White Noise
- Open Access
PDF-Version jetzt herunterladenAbstractIn this chapter, we develop a theory of stochastic integration that is suitable for integrating random functions of space and time with respect to space-time white noise \(W=(W(A),\, A\in \mathcal B^f_{\mathbb {R}\times D})\). -
Chapter 3. Linear SPDEs Driven by Space-Time White Noise
- Open Access
PDF-Version jetzt herunterladenAbstractThis 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. -
Chapter 4. Non-linear SPDEs Driven by Space-Time White Noise
- Open Access
PDF-Version jetzt herunterladenAbstractThis chapter is devoted to the study of non-linear SPDEs defined by a linear partial differential operator \(\mathcal {L}\) on \(\mathbb {R}_+\times \mathbb {R}^{k}\) and driven by space-time white noise. -
Chapter 5. Asymptotic Behaviors in the Stochastic Heat Equation
- Open Access
PDF-Version jetzt herunterladenAbstractWe devote this chapter to the study of long-time behavior of solutions to SPDEs, building on ideas that originate in deterministic dynamical systems and in the theory of Markov processes. -
Chapter 6. Selected Results on SPDEs Driven by Space-Time White Noise
- Open Access
PDF-Version jetzt herunterladen
- Titel
- Stochastic Partial Differential Equations, Space-Time White Noise and Random Fields
- Verfasst von
-
Robert C. Dalang
Marta Sanz-Solé
- Copyright-Jahr
- 2026
- Verlag
- Springer Nature Switzerland
- Electronic ISBN
- 978-3-032-01650-8
- Print ISBN
- 978-3-032-01649-2
- DOI
- https://doi.org/10.1007/978-3-032-01650-8
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