Probabilistic aspects of critical growth-fragmentation equations

Abstract

The self-similar growth-fragmentation 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 self-similar case, using a probabilistic approach based on Lévy processes and positive self-similar 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 non-homogeneous case has some surprising features. In particular, using the fact that certain self-similar Markov processes can enter $(0,\infty )$ continuously from either $0$ or $\infty$, we exhibit unexpected spontaneous generation of mass in the solutions.

28 pages. v2 adds an expository section 6 and fixes some errors

References in corpus (2)

• Recurrent extensions of self-similar Markov processes and Cramér's condition II
• Time Asymptotics for a Critical Case in Fragmentation and Growth-Fragmentation Equations

Cited by in corpus (12)

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• Spectral gap for the growth-fragmentation equation via Harris's Theorem
• Asynchronous exponential growth of the growth-fragmentation equation with unbounded fragmentation rate
• Time Asymptotics for a Critical Case in Fragmentation and Growth-Fragmentation Equations
• Self-similar solutions of fragmentation equations revisited
• Probability tilting of compensated fragmentations
• A growth-fragmentation model related to Ornstein-Uhlenbeck type processes
• Strong laws of large numbers for a growth-fragmentation process with bounded cell sizes
• Well posedness and stochastic derivation of a diffusion-growth-fragmentation equation in a chemostat
• Growth-fragmentation processes and bifurcators
• A short remark on a Growth Fragmentation equation