We define a dynamical simple symmetric random walk in one dimension, and show that there almost surely exist exceptional times at which the walk tends to infinity. This is in contrast to the usual dynamical simple symmetric random walk in one dimension, for which such exceptional times are known not to exist. In fact we show that the set of exceptional times has Hausdorff dimension 1/2 almost surely, and give bounds on the rate at which the walk diverges at such times. We also show noise sensitivity of the event that our random walk is positive after n steps. In fact this event is maximally noise sensitive, in the sense that it is quantitatively noise sensitive for any sequence $$\varepsilon _n$$ ε n such that $$n\varepsilon _n\rightarrow \infty $$ n ε n → ∞ . This is again in contrast to the usual random walk, for which the corresponding event is known to be noise stable.

中文翻译：

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一维简单对称随机游走的噪声敏感性和特殊瞬态时间

我们在一维上定义了一个动态的简单对称随机游走，并表明几乎肯定存在游走趋于无穷大的特殊时间。这与通常的一维动态简单对称随机游走形成对比，已知这种特殊时间不存在。事实上，我们证明异常时间集几乎肯定具有 Hausdorff 维数 1/2，并给出了在这些时间游走发散率的界限。我们还展示了随机游走在 n 步后为正的事件的噪声敏感性。事实上，这个事件是最大的噪声敏感的，因为它对任何序列 $$\varepsilon _n$$ ε n 定量噪声敏感，使得 $$n\varepsilon _n\rightarrow \infty $$ n ε n → ∞ 。这再次与通常的随机游走形成对比，