Abstract
It is shown that the geometrically correct investigation of regularity of nonlinear differential flows on manifolds and related parabolic equations requires the introduction of a new type of variations with respect to the initial data. These variations are defined via a certain generalization of a covariant Riemannian derivative to the case of diffeomorphisms. The appearance of curvature in the structure of high-order variational equations is discussed and a family of a priori nonlinear estimates of regularity of any order is obtained. By using the relationship between the differential equations on manifolds and semigroups, we study C ∞-regular properties of solutions of the parabolic Cauchy problems with coefficients increasing at infinity. The obtained conditions of regularity generalize the classical coercivity and dissipation conditions to the case of a manifold and correlate (in a unified way) the behavior of diffusion and drift coefficients with the geometric properties of the manifold without traditional separation of curvature.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
K. Itô, “Stochastic differential equations in a differentiable manifold,” Nagoya Math. J., 1, 35–47 (1950).
K. Itô, “The Brownian motion and tensor fields on Riemannian manifold,” in: Proc. Int. Cong. Math., Stockholm, 536–539 (1962).
N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland Publ., Dordrecht (1981).
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.
Y. I. Belopolskaja and Yu. L. Daletskii, Stochastic Equations and Differential Geometry, Kluwer, Berlin (1996).
J.-M. Bismut, “Large deviations and the Malliavin calculus,” Progress in Math., Birkhäuser, Basel, 45 (1984).
P. Malliavin, Stochastic Analysis, Springer, Berlin (1997).
L. C. G. Rogers and D. Williams, Diffusions, Markov Processes and Martingales, Wiley, New York (1987).
Yu. Gliklikh, Global Analysis in Mathematical Physics. Geometric and Stochastic Methods, Kluwer, Berlin (1996).
L. Schwartz, Semimartingales and their Stochastic Calculus on Manifolds, Press de l’Universite de Montreal (1984).
M. Emery, Stochastic Calculus in Manifolds, Springer, Berlin (1989).
B. K. Driver, “A Cameron-Martin type quasi-invariance theorem for Brownian motion on a compact Riemannian manifold,” J. Funct. Anal., 110, 272–376 (1992).
D. W. Stroock, “An introduction to the analysis of paths on Riemannian manifold,” AMS: Math. Surv. Monogr., 74 (2000).
D. Applebaum, “Towards a quantum theory of classical diffusions on Riemannian manifolds,” Quantum Probab. Appl., Ed. L. Accardi, Springer, Berlin, 6 (1990).
A. Val. Antoniouk and A. Vict. Antoniouk, “Nonlinear calculus of variations for differential flows on manifolds: geometrically correct generalization of covariant and stochastic variations,” Ukr. Math. Bull., 1, No. 4, 449–484 (2004).
P. A. Meyer, “Geometrie differentielle stochastique, Seminaire de Probabilites XVI,” Lect. Notes Math., 921 (1982).
A. Val. Antoniouk, “Nonlinear symmetries of variational calculus and regularity properties of differential flows on noncompact manifolds,” Symmetry in Nonlinear Math. Phys.: Proc. of the Fifth Internat. Conf. (Kiev, 2003), Kiev, 1228–1235 (2003).
E. Pardoux, “Stochastic partial differential equations and filtering of diffusion processes,” Stochastics, 3, 127–167 (1979).
N. V. Krylov and B. L. Rozovskii, “On the evolutionary stochastic equations,” Contemp. Problems Math., 14, VINITI, Moscow (1979), pp. 71–146.
A. Besse, Manifolds All of Whose Geodesics Are Closed, Springer, Berlin (1978).
A. Val. Antoniouk, “Upper bounds for general second-order operators acting on metric functions,” Ukr. Math. Bull., 1, No. 4, 449–484 (2004).
N. S. Sinukov, Geodesic Mappings of Riemannian Spaces, Nauka, Moscow (1979).
A. Val. Antoniouk, “Regularity of nonlinear flows on manifolds: nonlinear estimates for new-type variations or Why generalizations of classical covariant derivatives are required?,” Nonlinear Boundary Problems, 15, 3–20 (2005).
A. Val. Antoniouk and A. Vik. Antoniouk, Nonlinear Effects in the Regularity Problems for Infinite-Dimensional Evolutions of the Classical Gibbs Models, Institute of Mathematics, Ukrainian National Academy of Sciences, Kiev, 187 p. (2005).
A. Val. Antoniouk and A. Vict. Antoniouk, “Nonlinear estimates approach to the regularity properties of diffusion semigroups,” Nonlinear Analysis and Applications: To V. Lakshmikantham on His 80th Birthday, Eds R. P. Agarwal and D. O’Regan, 1, Kluwer, Berlin, 165–226 (2003).
H. Kunita, Stochastic Flows and Stochastic Differential Equations, Cambridge Univ. Press (1990).
A. Val. Antoniouk and A. Vict. Antoniouk, “How the unbounded drift shapes the Dirichlet semigroups behavior of non-Gaussian Gibbs measures,” J. Funct. Anal., 135, 488–518 (1996).
Author information
Authors and Affiliations
Additional information
Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 8, pp. 1011–1034, August, 2006.
Rights and permissions
About this article
Cite this article
Antonyuk, A.V., Antonyuk, A.V. Regularity of nonlinear flows on noncompact Riemannian manifolds: Differential geometry versus stochastic geometry or what kind of variations is natural?. Ukr Math J 58, 1145–1170 (2006). https://doi.org/10.1007/s11253-006-0126-1
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/s11253-006-0126-1