论文标题
通过非均匀的Wiener-HOPF积分方程的连续时间随机步行的平均第一通道时间的精确计算
Exact calculation of the mean first-passage time of continuous-time random walks by nonhomogeneous Wiener-Hopf integral equations
论文作者
论文摘要
我们研究了在连续空间中的不对称连续时间随机步行的平均第一学期时间（MFPT），其特征是有限平均值的等待时间，并且具有有限平均值和有限差异的跳跃大小。在渐近极限中，此良好控制的过程受对流扩散方程的控制，而当对流速度位于边界方向时，MFPT结果是有限的。我们得出了一个非均匀的Wiener-HopF积分方程，该方程允许通过避免渐近限制来确切计算MFPT，并且它仅取决于跳跃大小的整体分布以及仅在等待时间的均值分布，因此它可以容纳一般的非马克维亚随机步行。通过对跳跃大小的不对称分布家族的案例研究，该家族的跳跃大小指数指数朝着边界呈指数，并在相反的方向上进行任意，我们表明MFPT确实独立于与边界相反方向上的跳跃尺寸分布。此外，我们还表明存在一个长度尺度，这仅取决于沿边界方向跳跃分布的特征，因此，对于边界附近的起点，MFPT取决于跳跃大小的特定整体分布，而与普遍性相反，这与远离边界的起点相反。
We study the mean first-passage time (MFPT) for asymmetric continuous-time random walks in continuous-space characterised by waiting-times with finite mean and by jump-sizes with both finite mean and finite variance. In the asymptotic limit, this well-controlled process is governed by an advection-diffusion equation and the MFPT results to be finite when the advecting velocity is in the direction of the boundary. We derive a nonhomogeneous Wiener-Hopf integral equation that allows for the exact calculation of the MFPT by avoiding asymptotic limits and it emerges to depend on the whole distribution of the jump-sizes and on the mean-value only of the waiting-times, thus it holds for general non-Markovian random walks. Through the case study of a quite general family of asymmetric distributions of the jump-sizes that is exponential towards the boundary and arbitrary in the opposite direction, we show that the MFPT is indeed independent of the jump-sizes distribution in the opposite direction to the boundary. Moreover, we show also that there exists a length-scale, which depends only on the features of the distribution of jumps in the direction of the boundary, such that for starting points near the boundary the MFPT depends on the specific whole distribution of jump-sizes, in opposition to the universality emerging for starting points far-away from the boundary.
