2026/07/18 by Bohan Hu, Wen Sun
#math.PR
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 Kn of connected components satisfies Kn/n → \mathbb Eπ[1/Tu] in probability and in L1, where Tu is the total progeny of the associated branching process and 1/∞=0. If qu(x) is its extinction probability from type x, re-rooting and extinction duality give the explicit limit ∫S qu(x) π(dx) - u\over 2∫S× Sκ(x,y)qu(x)qu(y) π(dx)π(dy). 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.