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Stochastic partial differential equations (SPDEs) model the evolution in time of spatially extended systems subject to a random driving. Recent years have witnessed tremendous progress in the theory of so-called singular SPDEs. These equations feature a singular, distribution-valued driving term, a typical example being spacetime white noise, which makes them ill-posed as such. In many cases, it is however possible to make sense of these equations by applying a so-called renormalisation procedure, initially introduced in quantum field theory. This book gives a largely self-contained exposition of the subject of regular and singular SPDEs in the particular case of the Allen-Cahn equation, which models phase separation. Properties of the equation are discussed successively in one, two and three spatial dimensions, allowing to introduce new difficulties of the theory in an incremental way. In addition to existence and uniqueness of solutions, aspects of long-time dynamics such as invariant measures and metastability are discussed. A large part of the last chapter, about the three-dimensional case, is dedicated to the theory of regularity structures, which has been developed by Martin Hairer and co-authors in the last years, and allows to describe a large class of singular SPDEs. --

Stochastic partial differential equations. --- Stability. --- Structural stability --- Mathematical models.

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Effective Dynamics of Stochastic Partial Differential Equations focuses on stochastic partial differential equations with slow and fast time scales, or large and small spatial scales. The authors have developed basic techniques, such as averaging, slow manifolds, and homogenization, to extract effective dynamics from these stochastic partial differential equations. The authors' experience both as researchers and teachers enable them to convert current research on extracting effective dynamics of stochastic partial differential equations into concise and comprehensive chapters. The b

Differential equations. --- Mathematics. --- Stochastic partial differential equations. --- Stochastic partial differential equations --- Mathematics --- Physical Sciences & Mathematics --- Mathematical Statistics --- Banach spaces, Stochastic differential equations in --- Hilbert spaces, Stochastic differential equations in --- SPDE (Differential equations) --- Stochastic differential equations in Banach spaces --- Stochastic differential equations in Hilbert spaces --- Differential equations, Partial

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The aim of this book is to give a systematic and self-contained presentation of basic results on stochastic evolution equations in infinite dimensional, typically Hilbert and Banach, spaces. These are a generalization of stochastic differential equations as introduced by Itô and Gikham that occur, for instance, when describing random phenomena that crop up in science and engineering, as well as in the study of differential equations. The book is divided into three parts. In the first the authors give a self-contained exposition of the basic properties of probability measure on separable Banach and Hilbert spaces, as required later; they assume a reasonable background in probability theory and finite dimensional stochastic processes. The second part is devoted to the existence and uniqueness of solutions of a general stochastic evolution equation, and the third concerns the qualitative properties of those solutions. Appendices gather together background results from analysis that are otherwise hard to find under one roof. The book ends with a comprehensive bibliography that will contribute to the book's value for all working in stochastic differential equations.

Stochastic partial differential equations. --- Banach spaces, Stochastic differential equations in --- Hilbert spaces, Stochastic differential equations in --- SPDE (Differential equations) --- Stochastic differential equations in Banach spaces --- Stochastic differential equations in Hilbert spaces --- Differential equations, Partial

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Rigorous error estimates for amplitude equations are well known for deterministic PDEs, and there is a large body of literature over the past two decades. However, there seems to be a lack of literature for stochastic equations, although the theory is being successfully used in the applied community, such as for convective instabilities, without reliable error estimates at hand. This book is the first step in closing this gap. The author provides details about the reduction of dynamics to more simpler equations via amplitude or modulation equations, which relies on the natural separation of ti

Stochastic partial differential equations. --- Banach spaces, Stochastic differential equations in --- Hilbert spaces, Stochastic differential equations in --- SPDE (Differential equations) --- Stochastic differential equations in Banach spaces --- Stochastic differential equations in Hilbert spaces --- Differential equations, Partial

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This book is devoted to stochastic Navier–Stokes equations and more generally to stochasticity in fluid mechanics. The two opening chapters describe basic material about the existence and uniqueness of solutions: first in the case of additive noise treated pathwise and then in the case of state-dependent noise. The main mathematical techniques of these two chapters are known and given in detail for using the book as a reference for advanced courses. By contrast, the third and fourth chapters describe new material that has been developed in very recent years or in works now in preparation. The new material deals with transport-type noise, its origin, and its consequences on dissipation and well-posedness properties. Finally, the last chapter is devoted to the physical intuition behind the stochastic modeling presented in the book, giving great attention to the question of the origin of noise in connection with small-scale turbulence, its mathematical form, and its consequences on large-scale properties of a fluid.

Differential equations. --- Stochastic analysis. --- Functional analysis. --- Differential Equations. --- Stochastic Analysis. --- Functional Analysis. --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Analysis, Stochastic --- Mathematical analysis --- Stochastic processes --- 517.91 Differential equations --- Differential equations --- Fluid mechanics. --- Stochastic partial differential equations.