×

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)
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] DOI: 10.1155/S1085337502206053 · Zbl 1018.58028
[2] DOI: 10.1155/S016117120430503X · Zbl 1074.58017
[3] Schwartz L, Ann. Institut Fourier Univ. Grenoble 27 pp 211– (1977) · Zbl 0356.60016
[4] Gliklikh YuE, Abstract Appl. Anal. 2006 pp 1– (2006)
[5] Elworthy, KD, Jan, YLe and Li, X-M.On the Geometry of Diffusion Operators and Stochastic Flows, Springer-Verlag, Lect. Notes Math. 1720, 1999
[6] Belopolskaya YI, Stochastic Processes and Differential Geometry (1989)
[7] Elworthy KD, Stochastic Differential Equations on Manifolds, Lecture Notes of London Mathematical Society 70 (1982)
[8] Gliklikh YuE, Global Analysis in Mathematical Physics. Geometric and Stochastic Methods (1997)
[9] Kunita H, Stochastic Flows and Stochastic Differential Equations (1990)
[10] Clark JMC, Geometric Methods in System Theory pp 131– (1973)
[11] Gikhman II, Introduction to the Theory of Stochastic Processes (1977)
[12] Azencott R, Bull. Soc. Math. France 102 pp 193– (1974)
[13] Ikeda N, Stochastic Differential Equations and Diffusion Processes (1981)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.