We derive a universal, exact asymptotic form of the splitting probability for symmetric continuous jump processes, which quantifies the probability $ π_{0,\underline{x}}(x_0)$ that the process crosses $x$ before 0 starting from a given position $x_0\in[0,x]$ in the regime $x_0\ll x$. This analysis provides in particular a fully explicit determination of the transmission probability ($x_0=0$), in striking contrast with the trivial prediction $ π_{0,\underline{x}}(0)=0$ obtained by taking the continuous limit of the process, which reveals the importance of the microscopic properties of the dynamics. These results are illustrated with paradigmatic models of jump processes with applications to light scattering in heterogeneous media in realistic 3$d$ slab geometries. In this context, our explicit predictions of the transmission probability, which can be directly measured experimentally, provide a quantitative characterization of the effective random process describing light scattering in the medium.
Conditional splitting probabilities for hidden-state inference in drift-diffusive processes
Imperial College London（帝国理工学院）；Laboratoire de Physique de l’Ecole Normale Supérieure (LPENS)（巴黎高等师范学院物理实验室）；Max Planck Insitute for Dynamics and Self-Organization（马克斯·普朗克动力学与自组织研究所）
Modern view of activated rate processes: unidirectional fluxes at equilibrium, correlation functions, and splitting probabilities
University of Texas at Austin（德克萨斯大学奥斯汀分校）；Oden Institute for Computational Engineering and Sciences, University of Texas at Austin（德克萨斯大学奥斯汀分校奥登计算工程与科学学院）；Department of Chemistry, University of Texas at Austin（德克萨斯大学奥斯汀分校化学系）
