A probabilistic approach to spectral analysis of growth-fragmentation equations
Résumé
The growth-fragmentation equation describes a system of growing and dividing particles, and arises in models of cell division, protein polymerisation and even telecommunications protocols. Several important questions about the equation concern the asymptotic behaviour of solutions at large times: at what rate do they converge to zero or infinity, and what does the asymp-totic profile of the solutions look like? Does the rescaled solution converge to its asymptotic profile at an exponential speed? These questions have traditionally been studied using analytic techniques such as entropy methods or splitting of operators. In this work, we present a probabilistic approach to the study of this asymptotic behaviour. We use a Feynman–Kac formula to relate the solution of the growth-fragmentation equation to the semigroup of a Markov process, and characterise the rate of decay or growth in terms of this process. We then identify the spectral radius and the asymptotic profile in terms of a related Markov process, and give a spectral interpretation in terms of the growth-fragmentation operator and its dual. In special cases, we obtain exponential convergence.
Mots clés
growth-fragmentation equation
transport equations
cell division equation
one-parameter semigroups
spectral analysis
spectral radius
Feynman–Kac formula
piecewise-deterministic Markov processes
Lévy processes
one-
parameter semigroups
Malthus exponent
piecewise-
deterministic Markov processes
2010 Mathematics Subject Classification: 35Q92
47D06
45K05
47G20
60G51
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