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Stochastic Feynman-Kac formula. (English) Zbl 0541.60062

The stochastic partial differential equation called stochastic Feynman- Kac formula is investigated. For any distribution \(g\in {\mathcal D}'(R^ d)\) the author proves that the equation has a unique solution g exp- \(\int^{t}_{0}V(\cdot +w_ s)ds\) defining a semimartingale with the strong Markov property with continuous trajectories in \({\mathcal D}'(R^ d)\), and the infinitesimal generator of the semigroup is explicitly evaluated. Similar methods are applied to the Schrödinger equation.
Reviewer: G.Toscani

MSC:

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60G44 Martingales with continuous parameter
35R60 PDEs with randomness, stochastic partial differential equations
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[1] Badrikian, A., Séminaire sur les Fonctions Aléatoires Linéaires et les Mesures Cylindriques, Lecture Notes in Math. (1970), Berlin-Heidelberg-New York: Springer-Verlag, Berlin-Heidelberg-New York · Zbl 0209.48402
[2] Dellacherie, C.; Meyer, P. A., Probabilités et Potentiel, Chapitres I.IV (1975), Paris: Hermann, Paris · Zbl 0323.60039
[3] Doss, H., Sur une resolution stochastique de l’équation de Schrödinger à coefficients analytiques, Comm. Math. Phys., 73, 247-264 (1980) · Zbl 0427.60099 · doi:10.1007/BF01197701
[4] Dellacherie, C.; Meyer, P. A., Probabilités et Potentiel, Chapitres V.VII (1980), Paris: Hermann, Paris · Zbl 0464.60001
[5] A. Grothendieck,Produits tensoriels topologiques et espaces nucléaires, Mem. Am. Math. Soc. N∮ 6 (1955). · Zbl 0064.35501
[6] M. Metivier and J. Pellaumail,Stochastic Integration, Academic Press, 1980. · Zbl 0463.60004
[7] Schaefer, H. H., Topological Vector Spaces, Graduate Texts in Math. (1970), Berlin-Heidelberg-New York: Springer-Verlag, Berlin-Heidelberg-New York · Zbl 0212.14001
[8] Schwartz, L., Théorie des Distributions (1973), Paris: Hermann, Paris
[9] Schwartz, L., Processus de Markov et désintégrations régulières, Ann. Inst. Fourier, Grenoble, 27, 211-277 (1977) · Zbl 0356.60016
[10] A. S. Ustunel,Calcul stochastique sur les espaces nucléaires et ses applications, Thèse de Doctorat d’Etat, Université de Paris VI, 1981.
[11] Ustunel, A. S., Formule de Feynman-Kac stochastique, C.R. Acad. Sci. Paris, 292, 595-597 (1981) · Zbl 0466.60048
[12] Ustunel, A. S., Stochastic integration on nuclear spaces and its applications, Ann. Inst. H. Poincaré, 18, 165-200 (1982) · Zbl 0506.60050
[13] Ustunel, A. S., A characterization of semimartingales on the nuclear spaces, Z. Wahrscheinlichkeitstheor. Verw. Geb., 60, 21-39 (1982) · Zbl 0466.60006 · doi:10.1007/BF01957095
[14] A. S. Ustunel,Some applications of stochastic integration in infinite dimension I, preprint, to appear in Stochastics.
[15] A. S. Ustunel,Some applications of stochastic integration in infinite dimension II, preprint, to appear in Stochastics.
[16] A. S. Ustunel,Applications of integration by parts formula for infinite dimensional semimartingale, preprint. · Zbl 0602.60049
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