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On a Family of Critical Growth-Fragmentation Semigroups and Refracted Lévy Processes

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Abstract

The growth-fragmentation equation models systems of particles that grow and split as time proceeds. An important question concerns the large time asymptotic of its solutions. Doumic and Escobedo (Kinet. Relat. Models, 9(2):251–297, [12]) observed that when growth is a linear function of the mass and fragmentations are homogeneous, the so-called Malthusian behaviour fails. In this work we further analyse the critical case by considering a piecewise linear growth, namely

$$c(x) = \textstyle\begin{cases} a_{{-}} x \quad x < 1 \\ a_{{+}} x \quad x \geq 1, \end{cases} $$

with \(0 < a_{{+}} < a_{{-}}\). We give necessary and sufficient conditions on the coefficients ensuring the Malthusian behaviour with exponential speed of convergence to an asymptotic profile, and also provide an explicit expression of the latter. Our approach relies crucially on properties of so-called refracted Lévy processes that arise naturally in this setting.

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Notes

  1. In the sequel we will refer to it also using the term ‘mass’.

  2. The existence of eigenelements has been proved also in the case of much more general generators. In particular, the fragmentation kernel does not need to be homogeneous.

  3. This means that its finite-dimensional distributions are given in the following way. Let \(0 \leq t_{1} < \cdots < t_{n} \leq t\), and \(F: \mathbb{R}^{n} \to \mathbb{R_{+}}\). Then,

    $$ \tilde{\mathbb{E}}_{x} \bigl[F(Y_{t_{1}}, \dots , Y_{t_{n}} )\bigr] = \mathbb{E}_{x}\bigl[ \mathcal{M}'_{t} F(X_{t_{1}}, \dots , X_{t_{n}} ) \bigr], \quad x > 0. $$

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  1. Institut für Mathematik, Universität Zürich, Zürich, Switzerland

    Benedetta Cavalli

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  1. Benedetta Cavalli
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Cavalli, B. On a Family of Critical Growth-Fragmentation Semigroups and Refracted Lévy Processes. Acta Appl Math 166, 161–186 (2020). https://doi.org/10.1007/s10440-019-00261-5

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