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 \(y\).
Reviewer: Hans Crauel (Frankfurt am Main)
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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