Probabilistic aspects of critical growthfragmentation equations
Abstract
The selfsimilar growthfragmentation equation describes the evolution of a medium in which particles grow and divide as time proceeds, with the growth and splitting of each particle depending only upon its size. The critical case of the equation, in which the growth and division rates balance one another, was considered by Doumic and Escobedo in the homogeneous case where the rates do not depend on the particle size. Here, we study the general selfsimilar case, using a probabilistic approach based on Lévy processes and positive selfsimilar Markov processes which also permits us to analyse quite general splitting rates. Whereas existence and uniqueness of the solution are rather easy to establish in the homogeneous case, the equation in the nonhomogeneous case has some surprising features. In particular, using the fact that certain selfsimilar Markov processes can enter $(0,\infty)$ continuously from either $0$ or $\infty$, we exhibit unexpected spontaneous generation of mass in the solutions.
 Publication:

arXiv eprints
 Pub Date:
 June 2015
 arXiv:
 arXiv:1506.09187
 Bibcode:
 2015arXiv150609187B
 Keywords:

 Mathematics  Probability;
 35Q92;
 45K05;
 60G18;
 60G51
 EPrint:
 28 pages. v2 adds an expository section 6 and fixes some errors