# On a class of Time-fractional Continuous-state Branching Processes

## TL;DR

This paper introduces a new class of non-Markov population models that generalize classical branching processes, incorporating fractional and time-changed dynamics, with detailed analysis of specific cases like the fractional Yule process.

## Contribution

It develops a unified framework for non-Markovian branching processes using rescaled, time-changed Galton--Watson processes, extending classical models with fractional dynamics.

## Key findings

- Expressions for moments of the processes
- Branching inequality for process evolution
- Analysis of the fractional Yule process

## Abstract

We propose a class of non-Markov population models with continuous or discrete state space via a limiting procedure involving sequences of rescaled and randomly time-changed Galton--Watson processes. The class includes as specific cases the classical continuous-state branching processes and Markov branching processes. Several results such as the expressions of moments and the branching inequality governing the evolution of the process are presented and commented. The generalized Feller branching diffusion and the fractional Yule process are analyzed in detail as special cases of the general model.