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Stochastic Partial Differential Equations - An Introduction (Paperback, 1st ed. 2021)
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Stochastic Partial Differential Equations - An Introduction (Paperback, 1st ed. 2021)
Series: SpringerBriefs in Mathematics
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This book gives a concise introduction to the classical theory of
stochastic partial differential equations (SPDEs). It begins by
describing the classes of equations which are studied later in the
book, together with a list of motivating examples of SPDEs which
are used in physics, population dynamics, neurophysiology, finance
and signal processing. The central part of the book studies SPDEs
as infinite-dimensional SDEs, based on the variational approach to
PDEs. This extends both the classical Ito formulation and the
martingale problem approach due to Stroock and Varadhan. The final
chapter considers the solution of a space-time white noise-driven
SPDE as a real-valued function of time and (one-dimensional) space.
The results of J. Walsh's St Flour notes on the existence,
uniqueness and Hoelder regularity of the solution are presented. In
addition, conditions are given under which the solution remains
nonnegative, and the Malliavin calculus is applied. Lastly,
reflected SPDEs and their connection with super Brownian motion are
considered. At a time when new sophisticated branches of the
subject are being developed, this book will be a welcome reference
on classical SPDEs for newcomers to the theory.
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