Blow-up of a Stable Stochastic Differential Equation
Blow-up of a Stable Stochastic Differential Equation
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DOI：
10.1007/s10884-015-9467-5
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发表时间：
2014-08
影响因子：
1.3
通讯作者：
Matti Leimbach;M. Scheutzow
中科院分区：
文献类型：
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作者：
Matti Leimbach;M. Scheutzow
我们研究了一个在有限时间内具有爆炸的二维常微分方程。如果把它看作是一个带有加性白色噪声的振荡器，那么它就是完全的--在这个意义上说，对于每一个初始条件，几乎肯定没有爆炸。此外，相关的马尔可夫过程甚至承认一个不变的概率测度。另一方面，正如我们将证明的，相应的局部随机流几乎肯定不是强完备的，即存在解在有限时间内爆炸的（随机）初始条件。
We examine a 2-dimensional ODE which exhibits explosion in finite time. Considered as an SDE with additive white noise, it is known to be complete—in the sense that for each initial condition there is almost surely no explosion. Furthermore, the associated Markov process even admits an invariant probability measure. On the other hand, as we will show, the corresponding local stochastic flow will almost surely not be strongly complete, i.e. there exist (random) initial conditions for which the solutions explode in finite time.