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.


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


[1] DOI: 10.1155/S1085337502206053 · Zbl 1018.58028
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