Strong p-completeness of stochastic differential equations and the existence of smooth flows on noncompact manifolds

Abstract

We introduce strong p-completeness and use them for studying the continuous dependence of solutions of SDE's on non-compact manifolds. We obtain conditions for the existence of global smooth solution flow, and prove their diffeomorphism properties. The criterion is in terms of the growth of the solutions of the linearized equations (i.e. the derivative flows). It is also given in terms of the driving vector fields and their derivatives. For the Euclidean spaces there are three principle classes of examples: (1) the driving vector fields grow at most $|x|^{1-2\alpha}$ and the derivatives of the driving vector are allowed to be the order $|x|^\alpha$; (2) the coefficients of the SDE grows linearly while its derivative allowed to grow logarithmically, (3) The coefficients are bounded while the derivatives are allowed to grow linearly. We also given test on non-explosions.