Nonlinear calculus of variations for differential flows on manifolds: geometrically correct introduction of covariant and stochastic variations

Abstract

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 noncompact 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.