Let $X$ be a complex analytic curve. In this paper we prove that the subanalytic sheaf of tempered holomorphic solutions of $𝒟$-modules on $X$ induces a fully faithful functor on a subcategory of germs of formal holonomic $𝒟$-modules. Further, given a germ $ℳ$ of holonomic $𝒟$-module, we obtain some results linking the subanalytic sheaf of tempered solutions of $ℳ$ and the classical formal and analytic invariants of $ℳ$.

We study supercritical branching Brownian motion on the real line starting at the origin and with constant drift $c$. At the point $xgt;0$, we add an absorbing barrier, i.e. individuals touching the barrier are instantly killed without producing offspring. It is known that there is a critical drift $c_0$, such that this process becomes extinct almost surely if and only if $c\ge c_0$. In this case, if $Z_x$ denotes the number of individuals absorbed at the barrier, we give an asymptotic for $P(Z_x=n)$ as $n$ goes to infinity. If $c=c_0$...