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 \(y\).


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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