论文标题
在无限间隔的随机普通微分方程上平均原理
Averaging Principle on Infinite Intervals for Stochastic Ordinary Differential Equations
论文作者
论文摘要
与在有限间隔上进行随机平均的现有作品相反，我们在整个真实轴上建立了一个平均原理，即所谓的第二个Bogolyubov定理，用于在希尔伯特空间中具有poisson稳定的希尔伯特空间中的半线性随机差分方程（特别是定期，Quasi-Periodic，几乎是周期性的，等于周期性的）。在某些适当的条件下，我们证明存在针对原始方程的独特复发解决方案，该方程具有与系数相同的复发属性，在平均方程的固定解的一个固定溶液的较小邻域中，并且该复发溶液会在时间尺度接近时，在整个真实轴上均匀地平均方程的固定解决方案。
In contrast to existing works on stochastic averaging on finite intervals, we establish an averaging principle on the whole real axis, i.e. the so-called second Bogolyubov theorem, for semilinear stochastic ordinary differential equations in Hilbert space with Poisson stable (in particular, periodic, quasi-periodic, almost periodic, almost automorphic etc) coefficients. Under some appropriate conditions we prove that there exists a unique recurrent solution to the original equation, which possesses the same recurrence property as the coefficients, in a small neighborhood of the stationary solution to the averaged equation, and this recurrent solution converges to the stationary solution of averaged equation uniformly on the whole real axis when the time scale approaches zero.
