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On the notion of \(L^{ 1}\)-completeness of a stochastic flow on a manifold. (English) Zbl 1018.58028

Fix a finite-dimensional manifold \(M\), \(T> 0\), \(M^T:= [0,T]\times M\), a generator \(A\) on \(M\), and \(A^T:= {\partial\over\partial t}+ A\). For \((t,x)\in M^T\), denote by \(\xi_{t,x}\) the stochastic flow on \(M\) (associated with \(A\)) such that \(\xi_{t,x}(t)= x\).
The flow \(\xi\) is said to be “\(L^1\)-complete on \([0,T]\)” if: it is complete on \([0,T]\); there exists a smooth proper \(\nu: M\to \mathbb{R}_+\) such that \(\mathbb{E}[\nu(\xi_{t,x}(T))]< \infty\) for all \((t,x)\in M^T\); for any \(K> 0\) there exists a compact \(C_{K,T}\subset M\) such that \(\mathbb{E}[\nu(\xi_{t,x}(T))]< K\Rightarrow x\in C_{K,T}\); \((t,x)\mapsto\mathbb{E}[\nu(\xi_{t,x}(T))]\) is smooth.
Then it is proved that \(\xi\) is \(L^1\)-complete on \([0,T]\) if and only if there exists a smooth proper \(u: M^T\to \mathbb{R}_+\) such that: \(A^Tu\) is constant and for all \((t,x)\in M^T\) the variables \(u(T\wedge\tau_n,\xi_{t,x}(T\wedge\tau_n))\) are uniformly integrable, where \(\tau_n\), \(n\in\mathbb{N}\), denotes the time at which \(s\mapsto u(s,\xi_{t,x}(s))\) exists on \([0,n]\).

MSC:

58J65 Diffusion processes and stochastic analysis on manifolds
58J35 Heat and other parabolic equation methods for PDEs on manifolds
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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