On a family of critical growth-fragmentation semigroups and refracted Lévy processes

Strong laws of large numbers for a growth-fragmentation process with bounded cell sizes

“…In past work on growth-fragmentation equations [10,15], a key element of the analysis was an auxiliary function defined in terms of return times of Y . In our context, we may define this as follows.…”

“…This homogeneous case was studied in [9], where it was observed that no such simple asymptotic profile exists. However, Cavalli [15] has shown that a simple change in the drift, making it piecewise constant, can yield convergence of averages to an explicit asymptotic profile. In this work, we use this approach to take the results a step further: not only do we obtain explicit expressions and show that the system averages exhibit this convergence, but we are also able to prove this behaviour for the stochastic system of cell masses via a strong law of large numbers.…”

“…which, thanks to (3), agrees with (5). One approach used in [9,15] is to consider a single tagged particle via a Feynman-Kac formula, and to analyse the Laplace transform of the hitting time of points of this particle. Other recent approaches make use of Harris-style theorems for nonconservative semigroups [5], and quasi-stationary methods [16].…”

The strong Malthusian behavior of growth-fragmentation processes

“…Actually (15), which is a weaker assumption than (14), not only ensures that the process in continuous time W t = e −λt Z t , h is a martingale, but also that the same holds for the process indexed by generations (W k : k ∈ N). Indeed, from the very definition of the function L x 0 ,x 0 , (15) states that the expected value under P x 0 of the nonnegative martingale e −λt ℓ(X t )E t , evaluated at the first return time H(x 0 ), equals 1, and therefore the stopped martingale e −λt∧H(x 0 ) ℓ(X t∧H(x 0 ) )E t∧H(x 0 ) , t ≥ 0 is uniformly integrable. Plainly, the first jump time of X, T 1 occurs before H(x 0 ), and the optional sampling theorem yields…”

“…A stochastic approach for establishing (2), which is based on the Feynman-Kac formula and circumvents spectral theory, has been developed by the authors in [11,8] and Cavalli in [15]. To carry out this programme, we introduce, under the assumption sup x>0 c(x)/x < ∞, the unique strong Markov process X on (0, ∞) with generator…”