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Lack of strong completeness for stochastic flows. (English) Zbl 1235.60066

An example of a stochastic differential equation with locally Lipschitz continuous coefficients with linear growth is constructed which does not generate a stochastic flow of homeomorphisms. The example is two-dimensional and of the form \(dx=\sigma(x,y)\,dW(t)\), \(dy=0\), where \(\sigma\) has increasingly strong oscillations for \(x\to\infty\) with frequencies depending on \(y\).

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
37H10 Generation, random and stochastic difference and differential equations
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
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