Splitting Probabilities of Jump Processes

We derive a universal, exact asymptotic form of the splitting probability for symmetric continuous jump processes, which quantifies the probability \( \pi_{0,\underline{x}}(x_0)\) 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 \( \pi_{0,\underline{x}}(0)=0\) 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.