Dispersion rates under finite mode Kolmogorov flows
This paper extends results of the authors and the reviewer [Ann. Probab. 30, 2046-2080 (2002; Zbl 1017.60073), ibid. 28, 1852-1869 (2000) and Electron. Commun. Probab. 4, 91-101 (1999; Zbl 0938.60048)] on the expansion of sets under semimartingale flows, to the case of finite-mode approximations to Kolmogorov flows. Given a random compact starting set \(X\) with almost-sure positive volume, the result states that the maximum displacement of any point in \(X\) from the origin at time \(t\) grows linearly in \(t\), uniformly in all directions. That is, if \(\varphi(t,s,x)\) is the random mapping from time \(s\) to time \(t\), there are positive constants \(c\) and \(C\) such that \[ c\leq \inf_{u\in S^1} \liminf_{t\to\infty} t^{-1} \sup_{x\in X} \langle \varphi(t,0,x),u\rangle \leq \limsup_{t\to\infty} t^{-1} \sup_{x\in X} \|\varphi(t,0,x),u\|\leq C \] almost surely.
scientific article; zbMATH DE number 1906180