Conditions for global existence of solutions of ordinary differential, stochastic differential, and parabolic equations. (English) Zbl 1074.58017

Summary: First, we prove a necessary and sufficient condition for global in time existence of all solutions of an ordinary differential equation (ODE). It is a condition of one-sided estimate type that is formulated in terms of so-called proper functions on the extended phase space. A generalization of this idea to stochastic differential equations (SDE) and parabolic equations (PE) allows us to prove similar necessary and sufficient conditions for global in time existence of solutions of special sorts: \(L^1\)-complete solutions of SDE (this means that they belong to a certain functional space of \(L^1\) type) and the so-called complete Feller evolution families giving solutions of PE. The general case of equations on noncompact smooth manifolds is under consideration.


58J65 Diffusion processes and stochastic analysis on manifolds
35K15 Initial value problems for second-order parabolic equations
35R60 PDEs with randomness, stochastic partial differential equations
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
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