## 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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