Some results on time-varying fractional partial differential equations and birth-death processes

Some results on specific types of branching processes are presented. Firstly, linear pure birth and birth-death processes governed by partial differential equations with time-varying coefficients are analysed. Such processes are constructed by inserting the fractional time derivative into the p.d.e. governing the law of fractional Brownian motion. We consider also pure birth processes stopped at first-passage time of Brownian motion and present the related distributions and the governing equations. Some explicit results on the mean values and low-order probabilities are obtained in terms of generalised Mittag-Leffler functions.