Length of Galton–Watson trees and blow-up of semilinear systems

Abstract

By lower estimates of the functionals 𝔼[e S t K t N t ], where S t and N t denote the total length up to time t and the number of individuals at time t in a Galton-Watson tree, we obtain sufficient criteria for the blow-up of semilinear equations and systems of the type ∂w t /∂t = A w t + V w t β. Roughly speaking, the growth of the tree length has to win against the ‘mobility’ of the motion belonging to the generator A, since, in the probabilistic representation of the equations, the latter results in small K(t) as t → ∞. In the single-type situation, this gives a re-interpretation of classical results of Nagasawa and Sirao(1969); in the multitype scenario, part of the results obtained through analytic methods in Escobedo and Herrero (1991) and (1995) are re-proved and extended from the case A = Δ to the case of α-Laplacians.