Stochastic integrals for spde's: A comparison

Robert C. Dalang, Lluís Quer-Sardanyons

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72 Citations (Scopus)

Abstract

We present the Walsh theory of stochastic integrals with respect to martingale measures, and various extensions of this theory, alongside of the Da Prato and Zabczyk theory of stochastic integrals with respect to Hilbert-space-valued Wiener processes, and we explore the links between these theories. Somewhat surprisingly, the end results of both theories turn out to be essentially equivalent. We then show how each theory can be used to study stochastic partial differential equations, with an emphasis on the stochastic heat and wave equations driven by spatially homogeneous Gaussian noise that is white in time. We compare the solutions produced by the different theories. © 2010 Elsevier GmbH.
Original languageEnglish
Pages (from-to)67-109
JournalExpositiones Mathematicae
Volume29
DOIs
Publication statusPublished - 1 Jan 2011

Keywords

  • Cylindrical Wiener process
  • Hilbert-space-valued Wiener process
  • Martingale measure
  • Random field solution
  • Spatially homogeneous Gaussian noise
  • Stochastic heat equation
  • Stochastic partial differential equation
  • Stochastic wave equation

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