"A Random Walk with a Skip-Free Component and the Lagrange Inversion Formula" by Viskov presents connections among Lagrange inversion and measures of random Lévy processes. The freely available original is in Cyrillic and I have no access to the SIAM translation, so I can't summarize the content directly, but here is the SIAM abstract:
The paper shows that for a random walk with a skip-free component, distributions of certain first passage times and hitting points are infinitely divisible. The proofs are elementary and based on an algebraic approach to the classical Lagrange formula. This approach permits us to write explicitly the respective Lévy measures.
And, in "Summary: Lagrange inversion and random forests", Gábor Pete notes
A nice interesting paper is by Viskov [Vis00]. He gives an algebraic proof of the Lagrange inversion formulae, with the representation theory of the Heisenberg–Weyl algebra, as the underlying idea. He deduces a new, exponential version of the inversion formula, which allows him to prove that if $h(z)$ is a basic Lagrangian distribution, i.e. the total progeny of a single GW-tree with offspring p.g.f. $g(0)\neq 0$, then it is infinitely divisible, with a possible positive mass at infinity. In fact, for $G(z) = h(z)/z$, $G^\lambda(z)$ is the generating function of a compound Poisson process $Y_{\lambda}$, $\lambda > 0$
$$P(Y_\lambda = m) = \lambda \frac{\partial_x^m}{m!} \; \frac{g^{m+\lambda}}{m+\lambda}.$$
Compositional inversion is intimately related to the combinatorics of the associahedra and noncrossing partitions; iterated Lie derivatives and flow functions; characterization of the Kac–Schwarz diff ops related to the action of Virasoro–Heisenberg ops; the inviscid Burgers–Hopf and KdV partial differential evolution equations and associated integrable hierarchies, or infinite set of conservation laws / differential identities w.r.t. parameters; and free probability with its association with random matrices and quantum field theories. This is a rather large constellation of ideas usually dealt with somewhat piecemeal, so it's rather optimistic to hope for a general survey article/book encompassing all of the stars (but hope springs eternal) and I would happily settle for a user-friendly update focusing mainly on the circle of ideas found in Viskov (in English foremost, but I could muddle through French).
(Qualifier: long before Heisenberg and Weyl, normal ordering of diff ops was explored by Scherk, Graves, Sylvester, Cayley, Pincherle (ladder ops also) and most likely others.)
