  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.

Statistical Mechanics