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Blow-up of a Stable Stochastic Differential Equation

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Abstract

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.

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Acknowledgments

We would like to acknowledge the DFG Research training group 1845 Stochastic Analysis with applications in biology, finance and physics for its financial support.

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Authors and Affiliations

  1. Technische Universität Berlin, MA 7-5, Str. des 17. Juni 135, 10623, Berlin, Germany

    Matti Leimbach & Michael Scheutzow

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  1. Matti Leimbach
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  2. Michael Scheutzow
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Correspondence to Matti Leimbach.

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Leimbach, M., Scheutzow, M. Blow-up of a Stable Stochastic Differential Equation. J Dyn Diff Equat 29, 345–353 (2017). https://doi.org/10.1007/s10884-015-9467-5

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  • DOI: https://doi.org/10.1007/s10884-015-9467-5

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