We study the stochastic stability in the zero-noise limit from a quantitative point of view.We consider smooth expanding maps of the circle perturbed by additive noise. We show that in this case the zero-noise limit has a quadratic speed of convergence, as suggested by numerical experiments and heuristics published by Lin, in 2005 (see [25]). This is obtained by providing an explicit formula for the first and second term in the Taylor's expansion of the response of the stationary measure to the small noise perturbation. These terms depend on important features of the dynamics and of the noise which is perturbing it, as its average and variance.We also consider the zero-noise limit from a quantitative point of view for piecewise expanding maps showing estimates for the speed of convergence in this case.

中文翻译：

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零噪声极限下不变测度的二次响应和收敛速度

我们从定量的角度研究了零噪声极限下的随机稳定性。我们考虑了受加性噪声扰动的圆的平滑扩展图。我们表明，在这种情况下，零噪声极限具有二次收敛速度，正如 Lin 在 2005 年发表的数值实验和启发式方法所表明的那样（参见 [25]）。这是通过为平稳测量对小噪声扰动的响应的泰勒展开式中的第一项和第二项提供明确的公式而获得的。这些项取决于动力学和扰动它的噪声的重要特征，作为它的平均值和方差。这个案例。