A probabilistic approach to spectral analysis of growthfragmentation equations 
 
Authors:  Jean Bertoin Alexander R Watson 
 
Institution:  1. Institute of Mathematics, University of Zurich, Switzerland;2. School of Mathematics, University of Manchester, UK 
 
Abstract:  The growthfragmentation 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 asymptotic 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: we use a Feynman–Kac formula to relate the solution of the growthfragmentation 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 Malthus exponent and the asymptotic profile in terms of a related Markov process, and give a spectral interpretation in terms of the growthfragmentation operator and its dual. 
 
Keywords:  35Q92 47D06 45K05 47G20 60G51 Growthfragmentation equation Spectral analysis Malthus exponent Feynman–Kac formula 
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