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**Chasing balls through martingale fields.**
*(English)*
Zbl 1017.60073

The solution of the equation \(\varphi_t(x) = x + \int_0^t M(ds,\varphi_s(x)) \in {\mathbb R}^d\) driven by a field of continuous martingales \(M(t,x)\) constitutes a flow of homeomorphisms in the initial point \(x \in {\mathbb R}^d\). The asymptotic behavior of the image \(\varphi_t(X)\) of compact connected subsets \(X \subset {\mathbb R}^d\) with more than one point, and of the path \(\varphi_t(x)\) of individual points \(x\), is studied for the dimension \(d\) larger than 1. Under some basic continuity and uniform ellipticity conditions it is shown that with probability one the diameter of the image \(\varphi_t(X)\) either grows linearly with \(t\) or shrinks to \(0\) when the time \(t\) tends to infinity. The existence of ballistic points \(x\), i.e. points such that \(\|\varphi_t(x) \|\) grows linearly with \(t\), travelling in every direction is proved via the stronger concept of ball chasing. Let \(\psi\) be a Lipschitz continuous process adapted to the filtration generated by \(M(t,x)\) and with Lipschitz constant bounded by the speed of growths of \(\varphi_t(X)\). Then the existence of an exceptional point \(x \in X\) such that the set of time points \(t\) such that \(\|\varphi_t(x) - \psi(t) \|\leq \varepsilon(t)\) is unbounded, has asymptotically positive density or contains every \(t \geq T\) for some \(T\) is proved. Here the function \(\varepsilon(t)\) is constant or proportional to the logarithm of \(t\). The proofs rely on a retraction principle formulated in the paper. The paper extends previous results of the authors together with M. Cranston [Electron. Commun. Probab. 91-101 (1999; Zbl 0938.60048) and Ann. Probab. 28, No. 4, 1852-1869 (2000)].

Reviewer: Bo Markussen (Berlin)

### MSC:

60H20 | Stochastic integral equations |

60G44 | Martingales with continuous parameter |

60G17 | Sample path properties |

60G60 | Random fields |

### Keywords:

stochastic flows; martingale field; dichotomy; ballistic points; exceptional points; supermartingale inequalities; submartingale inequalities; retraction principle### Citations:

Zbl 0938.60048
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\textit{M. Scheutzow} and \textit{D. Steinsaltz}, Ann. Probab. 30, No. 4, 2046--2080 (2002; Zbl 1017.60073)

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### References:

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