We are interested in the long-time behavior of infinite-dimensional branching processes and in the applications of these limiting results. Such stochastic processes are particle systems where particles move independently according to a Markov process. When a branching event occurs, a particle is replaced by a random number of new particles. Times of branching events, numbers of offspring and offspring locations may depend on the ancestor position.

We focus on models with non-local branching: the offspring's location at birth is chosen through a Markov kernel. Examples are cell division or evolution model. More precisely, we will study

1. growth-fragmentation processes: size of cells grow exponentially and cells divide into two new cells at random times depending on their sizes.
2. models for the evolution of phenotype: individuals possess a trait that they pass on to their descendants up to a small variation.

Non-local branching involves non-self-adjoint operators making their study much more difficult than
classical branching diffusions. We propose to develop new approaches to prove law of large numbers
type results, leveraging ideas from functional analysis, partial differential equation study and
probability theory and then derive from those results new statistical estimators.

The project is structured around 3 challenges:

1. The first challenge is based on the partial differential equations (PDE) theory. Our aim is to develop new tools, close to Krein-Rutman theorem or entropy methods, to study the long time behavior of linear but non-local evolutionary PDE.
2. The second challenge is based on the probability theory. Its aim is to establish new law of large number results and central limit theorems for those particle systems.
3. Finally, the last challenge is focused on the applications. Our aim is to compare our theoretical results with concrete applications. In particular, we will develop new statistical tools to estimate the various parameters of our models.