Abstract
Two apparently very different approachs of Ito’s Calculus have been recently introduced : one is the Hida’s analysis of the white noise and the corresponding Hida distributions theory. The otherone refeers to the reformulation in terms of Sobolev spaces of Malliavin’s Calculus. The goal of the present paper is to show the relation between these two approachs, recalling also shortly under a semplified form some aspects of a distribution theory on Gaussian spaces developed in the years 1974–1978 : only real vector spaces are ae considered, the e and n seminorms on multiple tensor products are suppressed ; distributions and cylindrical distributions are simultanously studied. Also some complementary results are given. The A. thanks P.A.Meyer in general and also for communication of his manuscript [P.A.ME + J.A.YA 86]. The paper is organized in the following way :
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I
Nuclear and Complete Gaussian vector spaces. Examples connected with Ito’s Calculus.
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II
Distribution theory on Gaussian vector spaces.
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III
Differential Calculus in Sobolev Spaces and Ito’s Calculus.
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IV
Kernels and Symbols of Operators on Gaussian Vector Spaces.
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V
References.
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References
L. Schwartz, Théorie des distributions à valeurs vectorielles. Annales Institut Fourier (1957).
L. Schwartz, Séminaire à l’Ecole Polytechnique sur les applications radonifiantes (1970).
C. Dellacherie et P.A. Meyer, Théorie des martingales. Hermann (1980).
P.A. Meyer et Yan, Manuscrit 1986 sur les distributions sur l’espace de Wiener.
A. Badrikian, Lecture Notes in Mathematics n° 139 (1970).
T. Hida, Stationary Stochastic Processes, Princeton University Press (1970).
T. Dwyer, PDE in Fisher - Fock Spaces, Bull. AMS vol. 77, n°5, sept. 1971.
I. Segal Illinois J. of Math t6 (1982) p. 500–523.
I. Segal, Paper in the volume edited in the honour of Krein (1977).
B. Gaveau et Ph. Trauber (1982), Journal of Functional Analysis.
P.A. Meyer Lect. Notes in Math. N°921 and Lect. Notes in Control (Bengalore conf.). Proof of WP’k(X) Dom in LP of (1 + )kl2in the particular case k - 1 and of the inclusion C.
P.A. Meyer proves * in general Result requested and received by P. Malliavin the 8 mai 1983, published in LNM, Séminaire de Probabilités n°18 (1984).
M. et P. Krée, prove* in general and the continuity of δ. Compte rendu presented the 6 mai 1983 by P. Malliavin.
J. Diebolt, Compte rendu presented the 6 mai 1983 byP. Malliavin introduced δ in Malliavin’s integration by parts.
S. Sugita, Kyoto Math. Journal (1985) also proves the continuity of δ and apply this to the integration by parts; manuscript mentionning [M.D. KR 83] and received by Kyoto Math. J. the 21. 11. 1983.
N. Ikeda and S. Watanabe ∈ Stochastic Analysis; ed. bt K. Ito, North Holland 1984. Paper at Katata Conference (1982) written after novembre 1983 annoncing the results of [S. SU. 85].
S. Watanabe - Lecture at Tata Institute (Bombay) on Malliavin’s Calculus.
S. Ustunel. To appear in JFA.
J. Hida and L. Streit (Kyoto Math. Journal 1977) initiate the calculus on white noise.B - some references concerning the differential calculus in sobolev Spaces before malliavin’s calculus
C. de Witt (Comm in Math. Phys. 1972) and P. Krée (E Oberwolfach Conference 1971 on Approximation Theory-Birkhauser) introduce cylindrical distributions.
P. Krée (Comptes Rendus 1974. Lecture Notes in Math. N°410 and 473. Definition of the distribution operators ∇, ∇é, S . Definition of basic properties and definition of Sobolev Spaces WP’k(X) for ⋀ < p <∞, k=± 1, Explicit formula for the inver-sion of the decomposition in Wiener chaos.
P. Krée (Mémoire SMF n°46) simplifies the definition of ∇ and δ.
M. Krée (comptes Rendus 1974 et Bulletin Soc. Math. France 1977) proves * for p = 2 (result annonced in P. KR 74); gives another defintion of Sobolev Spaces and trace properties.
B. Lascar (Communications in PDE 1976) gives other properties of Sobolev Spaces : continuity of δ for p = 2 , partitions of unity, theorem of Paley-Wiener type for θ-transform.
L. Schwartz (International Conference of Jerusalem in Functional Analysis 1977) presents these results and also give a simplified definition of the distribution relative derivative.
P. Krée and R. Raczka (Annales Institut A. Poincaré - Physique 1978) gave a kernel and Symbol theory in the cylindrical case.
P. Krée in Lecture Notes in Math. N°644 (Dublin Conference 1977) gives the non cylindral version of [P. KR + R. RA 78]..
Séminaire EDP en dimension infinie n°3 1976–1977 : extension of kernel’s and Symbol’s theory to the anticommutative case.
P. Krée in Lecture Notes in Math. N° 843 (Rio Conference 1978) gives forε =± the formalism of contraction of forms.
P. Paclet (two Comptes-Rendus 1978 présented by G. Choc.~uet and paper in [EDP ∞ 78] redefines any potential ∈ W2’1(X) outside a set of zero capacity.
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Krée, P. (1990). Distributions, Sobolev Spaces on Gaussian Vector Spaces and Ito’s Calculus. In: Albeverio, S., Streit, L., Blanchard, P. (eds) Stochastic Processes and their Applications. Mathematics and Its Applications (Soviet Series), vol 61. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-2117-7_13
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DOI: https://doi.org/10.1007/978-94-009-2117-7_13
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