## 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)].

### MSC:

 60H20 Stochastic integral equations 60G44 Martingales with continuous parameter 60G17 Sample path properties 60G60 Random fields

Zbl 0938.60048
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### References:

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