TY - JOUR
T1 - Stochastic integrals for spde's: A comparison
AU - Dalang, Robert C.
AU - Quer-Sardanyons, Lluís
PY - 2011/1/1
Y1 - 2011/1/1
N2 - 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.
AB - 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.
KW - Cylindrical Wiener process
KW - Hilbert-space-valued Wiener process
KW - Martingale measure
KW - Random field solution
KW - Spatially homogeneous Gaussian noise
KW - Stochastic heat equation
KW - Stochastic partial differential equation
KW - Stochastic wave equation
U2 - 10.1016/j.exmath.2010.09.005
DO - 10.1016/j.exmath.2010.09.005
M3 - Article
SN - 0723-0869
VL - 29
SP - 67
EP - 109
JO - Expositiones Mathematicae
JF - Expositiones Mathematicae
ER -