White Noise

Hida, Takeyuki; Kuo, Hui-Hsiung; Potthoff, Jürgen; Streit, Ludwig

doi:10.1007/978-94-017-3680-0

Cited by 291 publications

(134 citation statements)

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“…Corollary 1 generalizes results as Corollary 4.22 in [5]. It seems impossible to extend the corollary to Kondratiev spaces with & b À1 since there is noconstant K such that m n3 Km3n3 for all m, n P N.…”

supporting

confidence: 55%

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Variational methods for PDEs aplied to stochastic partial differential equations

Våge¹

1998

MATH. SCAND.

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supporting

confidence: 55%

“…One can show that for every x P R d the sum in (8) converges to the usual de¢nition, h3Y 0 x i, in L 2 " (resp. the space of Hida test functions, s, see [5]…”

Section: Introductionmentioning

confidence: 99%

Variational methods for PDEs aplied to stochastic partial differential equations

Våge¹

1998

MATH. SCAND.

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“…White noise as a H s loc -valued random variable Gaussian white noise has been studied extensively in the literature, see e.g. [34], and there are different procedures to construct the corresponding probability measure. Let us mention the classical approaches by Hida [33] and the abstract Wiener space construction by Gross [30].…”

Section: Ergodicity By Time Translationsmentioning

confidence: 99%

Inverse Problem for the Wave Equation with a White Noise Source

Helin

Lassas

Oksanen

2014

Commun. Math. Phys.

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Abstract. We consider a smooth Riemannian metric tensor g on R n and study the stochastic wave equation for the LaplaceBeltrami operator ∂ 2 t u − ∆ g u = F . Here, F = F (t, x, ω) is a random source that has white noise distribution supported on the boundary of some smooth compact domain M ⊂ R n . We study the following formally posed inverse problem with only one measurement. Suppose that g is known only outside of a compact subset of M int and that a solution u(t, x, ω 0 ) is produced by a single realization of the source F (t, x, ω 0 ). We ask what information regarding g can be recovered by measuring u(t, x, ω 0 ) on R + × ∂M ? We prove that such measurement together with the realization of the source determine the scattering relation of the Riemannian manifold (M, g) with probability one. That is, for all geodesics passing through M , the travel times together with the entering and exit points and directions are determined. In particular, if (M, g) is a simple Riemannian manifold and g is conformally Euclidian in M , the measurement determines the metric g in M .

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“…The positive definite functions we consider will have a form similar to that in the expression on the left hand side in (2.1), but with a different quadratic form occurring in the exponent. For background references we recommend [10], [11], [14] and [25].…”

Section: Preliminariesmentioning

confidence: 99%

Stochastic Processes Induced by Singular Operators

Alpay¹,

Jørgensen²

2012

Numerical Functional Analysis and Optimization

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Abstract. In this paper we study a general family of multivariable Gaussian stochastic processes. Each process is prescribed by a fixed Borel measure σ on R n . The case when σ is assumed absolutely continuous with respect to Lebesgue measure was studied earlier in the literature, when n = 1. Our focus here is on showing how different equivalence classes (defined from relative absolute continuity for pairs of measures) translate into concrete spectral decompositions of the corresponding stochastic processes under study. The measures σ we consider are typically purely singular. Our proofs rely on the theory of (singular) unbounded operators in Hilbert space, and their spectral theory.

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White Noise

Cited by 291 publications

References 0 publications

Variational methods for PDEs aplied to stochastic partial differential equations

Variational methods for PDEs aplied to stochastic partial differential equations

Inverse Problem for the Wave Equation with a White Noise Source

Stochastic Processes Induced by Singular Operators

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