Dynamical Equations for Poisson Galton--Watson Trees and Component Densities of Sparse Inhomogeneous Random Graphs

We study Poisson Galton–Watson trees on a standard Borel type space when the offspring kernel is multiplied by a scalar parameter. On finite trees, we identify the Radon–Nikodym derivative between two parameter values and show that it remains measurable after projection to the total progeny measure. Under a uniform bound on the offspring intensities, differentiation yields exact differential and integral equations for the projected laws without irreducibility, reversibility, or a positive eigenfunction. With an additional positive eigenfunction bounded above and away from zero, we relate these equations to an infinite spinal tree, uniform pruning, the Doob transform, and the Aldous–Pitman ascension process. For a uniformly bounded offspring kernel, we also prove uniform exponential integrability of the total progeny throughout the spectrally subcritical regime. As an application, under the graphical-kernel assumptions of Bollobas, Janson and Riordan, the number $K_n$ of connected components satisfies $K_n\mathrm{/}n\to {\mathbb{E}}_{\pi }[1\mathrm{/}{T}_u]$ in probability and in $L^1$, where $T_u$ is the total progeny of the associated branching process and $1\mathrm{/}\mathrm{\infty }=0$. If $q_u(x)$ is its extinction probability from type $x$, re-rooting and extinction duality give the explicit limit $${\int }_S{q}_u(x)\pi (dx)-\frac{u}{2}{\int }_{S\times S}\kappa (x,y)q_u(x)q_u(y)\pi (dx)\pi (dy)\mathrm{.}$$ This extends the finite-type and compact-continuous formulas to the full BJR graphical-kernel setting, allowing separable noncompact type spaces and kernels that may be unbounded or reducible.