Processus linéaires, processus généralisés

Fernique, Xavier

doi:10.5802/aif.249

Cited by 89 publications

(76 citation statements)

References 2 publications

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“…An example is given therein of a metric space A'for which Mt = M", yet Zis not a F-space. Fernique's example [10] of a separable Hubert space with weak topology is a <r-compact non-F-space. Varadarajan [39] gives an example of a countable non-F-space.…”

Section: Definitions and Preliminariesmentioning

confidence: 99%

Bounded continuous functions on a completely regular space

Sentilles¹

1972

Trans. Amer. Math. Soc.

119

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Abstract. Three locally convex topologies on C(X) are introduced and developed, and in particular shown to coincide with the strict topology on locally compact A1 and yield dual spaces consisting of tight, -r-additive and x where p. is a bounded regular Borel measure on X. This yields a very satisfactory relationship between the topology on X, the space C(X), a natural class of linear functionals on it, and those measures on Jfthat measure at least the sets determined by the topology on X in the usual way, the Borel sets.This kind of representation was subsequently extended to locally compact spaces : first to functionals on the space C0(X) of continuous functions vanishing at infinity, and then further, to the bounded continuous functions on X. The last result, due to R. C. Buck, demanded the use of a locally convex topology, the strict topology, rather than a norm topology. In both extensions the same satisfactory relationship between measure and topology was obtained.In this paper we begin the development of locally convex topologies for C(X) which extend this kind of representation to its last reasonable setting, completely regular Hausdorff spaces. This setting appears to be ultimate in the sense of Hewitt's example of a regular space upon which the only continuous functions are constants. Definitions and preliminaries.The actual work of integral representation of linear forms has been done by other authors, going back to Aleksandrov [1] and, following his work, by Varadarajan [39], and later Knowles [21] and more recently Kirk [20] and Moran ([24], [25]). Our work relies heavily on theirs and will not extend the representations they have obtained but will relate these works to earlier

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Section: Definitions and Preliminariesmentioning

confidence: 99%

Bounded continuous functions on a completely regular space

Sentilles¹

1972

Trans. Amer. Math. Soc.

119

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“…The theory of generalized stochastic processes are thoroughly studied in I. M. Gel'fand and N. Ja. Vilenkin [5] and in X. Fernique [3]. In this section we review those parts of the theory which are relevant to this paper.…”

Section: Introductionmentioning

confidence: 99%

Generalized Stochastic Integrals and Equations

Dawson¹

1970

Transactions of the American Mathematical Society

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“…An exhaustive introduction on the first topic can be found in Rabier and Rheinboldt [10], while the basic theory of generalised processes can be found in Dawson [1], Fernique [2], or Chapter 3 in Gel'fand and Vilenkin [3].…”

Section: Preliminaries On Dae and Generalised Processesmentioning

confidence: 99%

Linear stochastic differential-algebraic equations with constant coefficients

Alabert

Ferrante

2006

Electron. Commun. Probab.

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We consider linear stochastic differential-algebraic equations with constant coefficients and additive white noise. Due to the nature of this class of equations, the solution must be defined as a generalised process (in the sense of Dawson and Fernique). We provide sufficient conditions for the law of the variables of the solution process to be absolutely continuous with respect to Lebesgue measure.

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Processus linéaires, processus généralisés

Cited by 89 publications

References 2 publications

Bounded continuous functions on a completely regular space

Bounded continuous functions on a completely regular space

Generalized Stochastic Integrals and Equations

Linear stochastic differential-algebraic equations with constant coefficients

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