Zoia A, Dumonteil E, Mazzolo A
CEA/Saclay, DEN/DANS/DM2S/SERMA/LTSD, F-91191 Gif-sur-Yvette, France.
Phys Rev E Stat Nonlin Soft Matter Phys. 2012 Jan;85(1 Pt 1):011132. doi: 10.1103/PhysRevE.85.011132. Epub 2012 Jan 19.
By building upon a Feynman-Kac formalism, we assess the distribution of the number of collisions in a given region for a broad class of discrete-time random walks in absorbing and nonabsorbing media. We derive the evolution equation for the generating function of the number of collisions, and we complete our analysis by examining the moments of the distribution and their relation to the walker equilibrium density. Some significant applications are discussed in detail: in particular, we revisit the gambler's ruin problem and generalize to random walks with absorption the arcsine law for the number of collisions on the half-line.
通过基于费曼 - 卡茨形式体系，我们评估了在吸收性和非吸收性介质中一类广泛的离散时间随机游走在给定区域内碰撞次数的分布。我们推导了碰撞次数生成函数的演化方程，并通过研究分布的矩及其与游走者平衡密度的关系来完成我们的分析。详细讨论了一些重要应用：特别是，我们重新审视了赌徒破产问题，并将半直线上碰撞次数的反正弦定律推广到具有吸收的随机游走。
