Khoshnevisan D. Analysis of Stochastic Partial Differential Equations

New York: American Mathematical Society, 2014. — 116 p.The general area of stochastic PDEs is interesting to mathematicians because it contains an enormous number of challenging open problems. There is also a great deal of interest in this topic because it has deep applications in disciplines that range from applied mathematics, statistical mechanics, and theoretical physics, to theoretical neuroscience, theory of complex chemical reactions [including polymer science], fluid dynamics, and mathematical finance.
The stochastic PDEs that are studied in this book are similar to the familiar PDE for heat in a thin rod, but with the additional restriction that the external forcing density is a two-parameter stochastic process, or what is more commonly the case, the forcing is a "random noise," also known as a "generalized random field." At several points in the lectures, there are examples that highlight the phenomenon that stochastic PDEs are not a subset of PDEs. In fact, the introduction of noise in some partial differential equations can bring about not a small perturbation, but truly fundamental changes to the system that the underlying PDE is attempting to describe.
The topics covered include a brief introduction to the stochastic heat equation, structure theory for the linear stochastic heat equation, and an in-depth look at intermittency properties of the solution to semilinear stochastic heat equations. Specific topics include stochastic integrals à la Norbert Wiener, an infinite-dimensional Itô-type stochastic integral, an example of a parabolic Anderson model, and intermittency fronts.
There are many possible approaches to stochastic PDEs. The selection of topics and techniques presented here are informed by the guiding example of the stochastic heat equation.
A co-publication of the AMS and CBMS.
Readership: Graduate students and research mathematicians interested in stochastic PDEs.PreludeWiener integrals
White noise
Stochastic convolutions
Brownian sheet
Fractional Brownian motionA linear heat equation
A non-random heat equation
The mild solution
Structure theory
Approximation by interacting Brownian particles
Two or more dimensions
Non-linear equationsWalsh–Dalang integrals
The Brownian filtration
The stochastic integral
Integrable random fieldsA non-linear heat equation
Stochastic convolutions
Existence and uniqueness of a mild solution
Mild implies weakIntermezzo: A parabolic Anderson model
Brownian local times
A moment boundIntermittency
Some motivation
Intermittency and the stochastic heat equation
Renewal theory
Proof of Theorem 7.8Intermittency fronts
The problem
Some proofsIntermittency islands
The existence and size of tall islands
A tail estimate
On the upper bound of Theorem 9.1
On the lower bound of Theorem 9.1Correlation length
An estimate for the length of intermittency islands
A coupling for independenceSome special integrals
A Burkholder–Davis–Gundy inequality
Regularity theory
Garsia’s theorem
Kolmogorov’s continuity theorem