  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.
