Ustunel, A. S. Stochastic Feynman-Kac formula. (English) Zbl 0541.60062 J. Anal. Math. 42, 155-165 (1983). 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 Keywords:stochastic Feynman-Kac formula; semimartingale with the strong Markov property; Schrödinger equation PDFBibTeX XMLCite \textit{A. S. Ustunel}, J. Anal. Math. 42, 155--165 (1983; Zbl 0541.60062) Full Text: DOI References: [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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.