Kunita, H. Stochastic flows acting on Schwartz distributions. (English) Zbl 0818.60044 J. Theor. Probab. 7, No. 2, 247-278 (1994). Consider a Schwartz distribution \(T\) and the stochastic flow \(\varphi_{s,t}\) generated by a stochastic differential equation; the composition of \(T\) with \(\varphi\) is defined as a random distribution; the spatial regularity of this variable is studied in terms of Sobolev spaces. For the time regularity, after defining a stochastic integral, a generalized Itô’s formula is proved when \(T = T(t)\) is a semimartingale. These results are applied to the regularity of semigroups, and to the existence and spatial regularity of a local time for a one-dimensional flow. Moreover, the relation with the pull-back defined by Watanabe is discussed. Reviewer: J.Picard (Aubière) Cited in 1 ReviewCited in 30 Documents MSC: 60H07 Stochastic calculus of variations and the Malliavin calculus 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) Keywords:stochastic flow; Schwartz distributions; generalized Itô’s formula; local time × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Hörmander, L. (1967). Hypoelliptic second order differential equations,Acta Math. 119, 147–171. · Zbl 0156.10701 · doi:10.1007/BF02392081 [2] Ikeda, N., and Watanabe, S. (1989).Stochastic Differential Equations and Diffusion Processes, Second edition, North-Holland Kodansha. · Zbl 0684.60040 [3] Kunita, H. (1970). Stochastic integrals based on martingales taking values in Hilbert space,Nagoya Math. J. 38, 41–52. · Zbl 0234.60071 [4] Kunita, H. (1990).Stochastic Flows and Stochastic Differential Equations, Cambridge University Press. · Zbl 0743.60052 [5] Kusuoka, S., and Stroock, D. W. (1985). Applications of the Malliavin calculus, Part II,J. Fac. Sci. University Tokyo Sect. IA, Math. 32, 1–76. · Zbl 0568.60059 [6] Métivier, M. (1982).Semimartingales: A Course on Stochastic Processes, Walter de Gruyter, Berlin. [7] Meyer, P. A. (1976). Un cours sur les integrales stochastiques,Seminaire Proba. X.Lecture Notes in Math. 511, 246–400. [8] Protter, P. (1990).Stochastic Integration and Differential Equations, Springer Verlag. · Zbl 0694.60047 [9] Watanabe, S. (1984).Lectures on Stochastic Differential Equations and Malliavin Calculus, Tata Institute of Fundamental Research/Springer. 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.