138 research outputs found
Some non-linear s.p.d.e.'s that are second order in time
We extend Walsh's theory of martingale measures in order to deal with
hyperbolic stochastic partial differential equations that are second order in
time, such as the wave equation and the beam equation, and driven by spatially
homogeneous Gaussian noise. For such equations, the fundamental solution can be
a distribution in the sense of Schwartz, which appears as an integrand in the
reformulation of the s.p.d.e. as a stochastic integral equation. Our approach
provides an alternative to the Hilbert space integrals of Hilbert-Schmidt
operators. We give several examples, including the beam equation and the wave
equation, with nonlinear multiplicative noise terms
Multiple points of the Brownian sheet in critical dimensions
It is well known that an -parameter -dimensional Brownian sheet has no
-multiple points when , and does have such points when
. We complete the study of the existence of -multiple points by
showing that in the critical cases where , there are a.s. no
-multiple points.Comment: Published at http://dx.doi.org/10.1214/14-AOP912 in the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Stochastic integrals for spde's: a comparison
We present the Walsh theory of stochastic integrals with respect to
martingale measures, alongside of the Da Prato and Zabczyk theory of stochastic
integrals with respect to Hilbert-space-valued Wiener processes and some other
approaches to stochastic integration, and we explore the links between these
theories. 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
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