Positive and Positive-Definite Generalized Functions

Gel'fand, I. M.; Vilenkin, N. Ya.

doi:10.1016/b978-1-4832-2974-4.50007-x

Cited by 172 publications

(291 citation statements)

References 0 publications

Supporting

Mentioning

290

Contrasting

Unclassified

Order By: Relevance

“…This is very similar to the classical scalar white noise. In fact, as explained in [19], the above series converges in the sense of distributions to a generalized Gaussian random process called white noise. For this reason we will refer to the ε → 0 limit as the white-noise limit.…”

Section: Theorem 14mentioning

confidence: 99%

The Gauss-Bonnet-Chern theorem: A probabilistic perspective

Nicolaescu

Savale

2016

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. We prove that the Euler form of a metric connection on a real oriented vector bundle E over a compact oriented manifold M can be identified, as a current, with the expectation of the random current defined by the zero-locus of a certain random section of the bundle. We also explain how to reconstruct probabilistically the metric and the connection on E from the statistics of random sections of E.

show abstract

Section: Theorem 14mentioning

confidence: 99%

The Gauss-Bonnet-Chern theorem: A probabilistic perspective

Nicolaescu

Savale

2016

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…for a certain non-negative tempered measure S which is called the spectral measure of the generalized random field ( [16], page 264). If the spectral measure is absolutely continuous with respect to the Lebesgue measure on R 2 , its Lebesgue density s(x), x ∈ R 2 is called the spectral density.…”

Section: De Wijs Processmentioning

confidence: 99%

“…Technically, the de Wijs process is a generalized Gaussian random field ( [16], Chapter III), whose index set is a certain class of contrasts, that is, non-atomic signed Borel measures on the Euclidean plane with zero total mass. This process corresponds to the logarithmic variogram model and is a generalization of the Brownian motion in two dimensions.…”

Section: Mondalmentioning

confidence: 99%

Applying Dynkin’s isomorphism: An alternative approach to understand the Markov property of the de Wijs process

Mondal¹

2015

Bernoulli

View full text Add to dashboard Cite

Dynkin's (Bull. Amer. Math. Soc. 3 (1980) 975-999) seminal work associates a multidimensional transient symmetric Markov process with a multidimensional Gaussian random field. This association, known as Dynkin's isomorphism, has profoundly influenced the studies of Markov properties of generalized Gaussian random fields. Extending Dykin's isomorphism, we study here a particular generalized Gaussian Markov random field, namely, the de Wijs process that originated in Georges Matheron's pioneering work on mining geostatistics and, following McCullagh (Ann. Statist. 30 (2002) 1225-1310, is now receiving renewed attention in spatial statistics. This extension of Dynkin's theory associates the de Wijs process with the (recurrent) Brownian motion on the two dimensional plane, grants us further insight into Matheron's kriging formula for the de Wijs process and highlights previously unexplored relationships of the central Markov models in spatial statistics with Markov processes on the plane.

show abstract

“…where E* is the conjugate space of E. Consider a function C{ξ) on E defined by the formula (1) C(£) = e-lisil /a, ξ e E.…”

Section: Eaha E*mentioning

confidence: 99%