Abstract
We show that a special choice of the Cameron–Martin direction in the characterization of the Wiener measure via the formula of integration by parts leads to a set of natural representations for derivatives of nonlinear diffusion semigroups. In particular, we obtain a final solution of the non-Lipschitz singularities in the Malliavin calculus.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
Yu. L. Daletskii, “Infinite-dimensional elliptic operators and parabolic equations connected with them,” Russian Math. Surveys 22, No. 4, 1–53 (1967).
K. D. Elworthy, Stochastic Differential Equations on Manifolds Cambridge University Press, London (1982).
P. Malliavin, “Stochastic calculus of variations and hypoelliptic operators,” in: Proceedings of the Int. Symp. SDE (Kyoto, 1976) Wiley, New York (1978), pp. 195–263.
P. Malliavin, “C k-hypoellipticity with degeneracy,” in: Proceedings of the Int. Conf. “Stochastic Analysis” (Northwestern Univ., Evanston, 1978) Academic Press, New York (1978), pp. 199–214.
P. Malliavin, Stochastic Analysis Springer, Paris (1997).
D. Nualart, The Malliavin Calculus and Related Topics Springer, Berlin (1995).
J.-M. Bismut, “Martingales, the Malliavin calculus and hypoellipticity under general Hörmander conditions,” Z. Wahrscheinlichkeitstheor. vew. Geb. 56, 468–505 (1981).
J.-M. Bismut, Large Deviations and the Malliavin Calculus Birkhäuser, Basel (1984).
D. Ocone, “Malliavin's calculus and stochastic integral representations of functionals of diffusion processes,” Stochastics 12, 161–185 (1984).
D. Stroock, “The Malliavin calculus, a functional analytic approach,” J. Funct. Anal. 44, 212–257 (1981).
A. Val. Antonyuk and A. Vik. Antonyuk, Non-Lipschitz Singularities in the Malliavin Calculus: Increase in Smoothness for Infinite-Dimensional Semigroups Preprint No. 96.23, Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1996).
N. V. Krylov and B. L. Rozovskii, “Cauchy problem for linear stochastic partial differential equations,” Izv. Akad. Nauk USSR 6, 1329–1347 (1977).
N. V. Krylov and B. L. Rozovskii, “On evolution stochastic equations,” in: VINITI Series in Contemporary Problems in Mathematics Vol. 14, VINITI, Moscow (1979), pp. 71–146.
E. Pardoux, “Stochastic partial differential equations and filtering of diffusion processes,” Stochastics 3, 127–167 (1979).
J. L. Doob, Stochastic Processes Wiley, New York (1953).
K. D. Elworthy, “Stochastic flows on Riemannian manifolds,” in: M. A. Pinsky and V. Wihstutz (editors), Diffusion Processes and Related Problems in Analysis Vol. 2, Birkhäuser, Basel (1992), pp. 37–72.
G. DaPrato, K. D. Elworthy, and J. Zabczyk, “Strong Feller property for stochastic semilinear equations,” Stochastic Anal. Appl. 13, 35–45 (1995).
G. DaPrato, D. Nualart, and J. Zabczyk, Strong Feller Property for Infinite-Dimensional Stochastic Equations Preprint No. 33, Scuola Normale Superiore Preprints, Pisa (1994).
K. D. Elworthy and X.-M. Li, “Formulae for the derivatives of heat semigroups,” J. Funct. Anal. 125, No. 1, 252–286 (1994).
A. Val. Antonyuk and A. Vik. Antonyuk, Quasicontractive Nonlinear Calculus of Variations and Smoothness of Discontinuous Semigroups Generated by Non-Lipschitz Stochastic Differential Equations Preprint No. 96.22, Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1996).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Antonyuk, A.V., Antonyuk, A.V. Higher-Order Relations for Derivatives of Nonlinear Diffusion Semigroups. Ukrainian Mathematical Journal 53, 134–140 (2001). https://doi.org/10.1023/A:1010401203717
Issue Date:
DOI: https://doi.org/10.1023/A:1010401203717