Stochastic flows and Bismut formulas for stochastic Hamiltonian systems
Stochastic flows and Bismut formulas for stochastic Hamiltonian systems
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DOI：
10.1016/j.spa.2010.05.015
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发表时间：
2010-09
影响因子：
1.4
通讯作者：
Xicheng Zhang
中科院分区：
文献类型：
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作者：
Xicheng Zhang
首先考虑不具整体Lipschitz条件的随机微分方程，给出了其严格保守的充分条件。特别地，得到了由具有非全局Lipschitz系数的随机微分方程定义的随机流的一个判别准则。我们还使用Zvonkin的变换，推导出一个随机流的C1-同态的非退化的SDES与Hölder连续漂移。接下来，我们证明了一个Bismut型公式的某些退化的SDES。最后，我们将所得结果应用于随机Hamilton系统，特别是如下随机非线性振子方程其中c 0 ∈R，Θ∈C∞（R）有界一阶导数，w是一维布朗白色噪声.
We first consider the stochastic differential equations (SDE) without global Lipschitz conditions, and give sufficient conditions for the SDEs to be strictly conservative. In particular, a criteria for stochastic flows of diffeomorphisms defined by SDEs with non-global Lipschitz coefficients is obtained. We also use Zvonkin’s transformation to derive a stochastic flow of C1-diffeomorphisms for non-degenerate SDEs with Hölder continuous drifts. Next, we prove a Bismut type formula for certain degenerate SDEs. Lastly, we apply our results to stochastic Hamiltonian systems, which in particular covers the following stochastic nonlinear oscillator equation where c0∈R,Θ∈C∞(R) has a bounded first order derivative, and ẇtis a one dimensional Brownian white noise.