论文标题
松鼠可以记住几乎没有：随机步行与离散时间续订过程引起的跳跃反转
Squirrels can remember little: A random walk with jump reversals induced by a discrete-time renewal process
论文作者
论文摘要
我们考虑一类离散时间随机步行，并在整数线上具有定向单位步骤。在离散时间续订过程中，在事件的瞬间时，这些步骤的方向逆转，并保持在稳定的时间。该模型代表了电报过程的离散时间半马克维亚概括。我们使用生成功能为传播器得出精确的公式。我们证明，对于在扩散限制中分布的几何分布等待时间，此步行将收敛于古典电报过程。我们考虑了预期位置的大型渐进性：对于有限的等待时间密度，沃克在靠近出发地点的平均定位位置，而sublinear power-power-power-power-power-power-power-power-power-power-power-power-power-power-power-power-power-powerha则是脂肪尾的等待时间密度（即无限平均值的密度）。我们通过考虑“衰老效应”作为非马克维亚语的标志来探索异常的扩散特征，在这种标志上，“老化更新过程”的离散时间开始起作用。通过得出此过程的相关分布，我们在等待时间分布时获得了差异的显式公式。在这种情况下，通常用于脂肪尾的等待时间PDFS $ t^2 $ - 全球超级延展性缩放率在较大的时间限制中出现。相比之下，如果台阶逆转之间的等待时间PDF是轻尾（有限的平均值和方差的“狭窄”），则步行将表现出正常的扩散，对于“宽'等待时间PDF（有限的平均值和无限差异）超级延伸较大的时间缩放。我们还考虑了随时间变化的版本，其中步行属于连续的时间点过程，例如时间段的泊松过程。这定义了一类新的有偏见的连续时间随机步行，表现出几种异常扩散方案。
We consider a class of discrete-time random walks with directed unit steps on the integer line. The direction of the steps is reversed at the time instants of events in a discrete-time renewal process and is maintained at uneventful time instants. This model represents a discrete-time semi-Markovian generalization of the telegraph process. We derive exact formulae for the propagator using generating functions. We prove that for geometrically distributed waiting times in the diffusive limit, this walk converges to the classical telegraph process. We consider the large-time asymptotics of the expected position: For waiting time densities with finite mean the walker remains in the average localized close to the departure site whereas escapes for fat-tailed waiting-time densities (i.e. densities with infinite mean) by a sublinear power-law. We explore anomalous diffusion features by accounting for the `aging effect' as a hallmark of non-Markovianity where the discrete-time version of the `aging renewal process' comes into play. By deriving pertinent distributions of this process we obtain explicit formulae for the variance when the waiting-times are Sibuya-distributed. In this case and generally for fat-tailed waiting time PDFs a $t^2$-ballistic superdiffusive scaling emerges in the large time limit. In contrast if the waiting time PDF between the step reversals is light-tailed (`narrow' with finite mean and variance) the walk exhibits normal diffusion and for `broad' waiting time PDFs (with finite mean and infinite variance) superdiffusive large time scaling. We also consider time-changed versions where the walk is subordinated to a continuous-time point process such as the time-fractional Poisson process. This defines a new class of biased continuous-time random walks exhibiting several regimes of anomalous diffusion.
