Some Non-Linear S.P.D.E's That Are Second Order In Time

Abstract

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.

Sample Path Properties of Anisotropic Gaussian Random Fields

“…In the following we list as examples the solutions to the stochastic heat equation and stochastic wave equation, and discuss possible ways to study their sample path properties using general methods for Gaussian random fields. We refer to Walsh (1986), Dalang (1999), Dalang and Frangos (1998), Dalang and Mueller (2003), Mueller and Tribe (2002), Dalang andSanz-Solé (2005, 2007), Nualart (2007a, 2007b) and the articles in this volume for more information.…”

“…The stochastic wave equation in one spatial dimension [i.e., N = 1] 22) driven by the white noise was considered by Walsh (1986) and many other authors [see Dalang and Frangos (1998) and Dalang and Mueller (2003) for a list of the references]. In spatial dimension two or higher, however, the stochastic wave equation driven by the white noise has no solution in the space of real valued measurable processes [see Walsh (1986)].…”

“…For the stochastic wave equation with spatial dimension three, we refer to Dalang and Mueller (2003), and Dalang and Sanz-Solé (2007) for information on the existence of a process solution and its sample path regularities. It seems that, in all the cases considered so far, the questions on fractal properties, existence and regularity of the local times of the solutions remain to be investigated.…”

Analytically weak solutions to linear SPDEs with unbounded time-dependent differential operators and an application

“…For complementary results on stochastic beam equations (with operators not being time-dependent and without algebraic constraint) we refer to [5], [6] and [1] and references therein.…”

Stochastic fractional partial differential equations driven by Poisson white noise

“…However, few publications treat stochastic partial differential equations involving fractional derivatives. Most of them investigate evolution type equations, driven by a fractional power of the Laplacian (see [3], [10]). These operators generate symmetric stable semigroups when the order of derivation is less than 2.…”