On the scaling limits of Galton Watson processes in varying environment

Renormalized sequences of Galton Watson processes converge to Continuous State Branching Processes (CSBP), characterized by a L\'evy triplet of two numbers and a measure. This paper investigates the case of Galton Watson processes in varying environment and provides an explicit sufficient condition for finite-dimensional convergence in terms of convergence of a characteristic triplet of measures. We recover then classical results on the convergence of Galton Watson processes and we can add exceptional environments provoking positive or negative jumps at fixed times. We also apply this result to derive new results on the Feller diffusion in varying environment and branching processes in random environment. Our approach relies on the backward differential equation satisfied by the Laplace exponent and provides results about explosion, absorption and extinction. Thus, this paper exhibits a general class of CSBP in varying environment which is characterized by a triplet of measures. This provides a first step towards characterizing time-inhomogeneous, continuous-time and continuous state space processes which satisfy the branching property.