Ukrainian Mathematical Bulletin

Volume 1, Issue 4, 2004 pp. 453-488.

Nonlinear Calculus of Variations for Differential Flows on Manifolds: Geometrically Correct Introduction of Covariant and Stochastic Variations

Authors: Alexander Val. Antoniouk and Alexandra Vict. Antoniouk
Author institution: Department of Nonlinear Analysis, Institute of Mathematics NAS of Ukraine, Tereschenkivska 3, 01601 MSP Kiev-4, Ukraine

Summary: We consider flows, generated by nonlinear differential equations on manifold that could also contain random terms and correspond to the second order parabolic equations. We demonstrate that the rigorous statement of the regularity problems for differential flows on non-compact manifolds requires the geometrically rigorous revision of definition of the high order variation with respect to the initial data and parameters. The main attention is devoted to the study of influence of the geometry and nonlinearities of coefficients on the regularity properties. To reach this aim we use the nonlinear symmetries of high order differential calculus and study a set of corresponding nonlinear estimates on variations. The arising conditions on regularity generalize the Krylov-Rosovskii-Pardoux conditions from linear space to the manifold setting. They also lead to the smooth and smoothing properties of associated Feller semigroups.

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