Compensated fragmentation processes and limits of dilated fragmentations

A new class of fragmentation-type random processes is introduced, in which, roughly speaking, the accumulation of small dislocations which would instantaneously shatter the mass into dust, is compensated by an adequate dilation of the components. An important feature of these compensated fragmentations is that the dislocation measure $\nu$ which governs their evolutions has only to fulfill the integral condition $\int_{\mathit{p}}$ (1-$\mathit{p}_{1}$)$^{2}\nu$(d$\mathbf{p}$ < $\infty$, where $\mathbf{p}$ = ($\mathit{p}_{1}$,…) denotes a generic mass-partition. This is weaker than the necessary and sufficient condition $\int_{\mathit{p}}$ (1-$\mathit{p}_{1}$)$^{2}\nu$(d$\mathbf{p}$ < $\infty$ for $\nu$ to be the dislocation measure of a homogeneous fragmentation. Our main results show that such compensated fragmentations naturally arise as limits of homogeneous dilated fragmentations, and bear close connections to spectrally negative Lévy processes