## Relations between $$L^{1}$$-completeness and continuity at infinity of stochastic flows and some applications.(English)Zbl 1190.58027

Let $$\xi(s)$$ be a stochastic flow of an SDE with smooth coefficients on a finite-dimensional manifold $$M$$ and $$\xi_{t,m}(s)$$ denote the orbit of flow with initial condition $$\xi_{t,m}(t)=m\in M$$. A stochastic flow is said to be complete if all its orbits are well-defined for all $$s>0$$.
The author shows that, for every complete flow, there exists a proper function $$\psi$$ on $$M$$ with the property that, for every orbit $$\xi_{t,m}(s)$$, the inequality $$E[\psi(\xi_{t,m}(s))]<\infty$$ for every $$s>t$$ holds. He then establishes relations between the notion of $$L^1$$-completeness (introduced by the author in previous work) and continuity in probability at infinity.

### MSC:

 58J65 Diffusion processes and stochastic analysis on manifolds 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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### References:

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