Conservative fixed-pivot redistribution on a finite positive grid

Let $x_0<x_1<\cdots <x_M=R$ be a finite grid of positive pivot masses with integer $M\ge 1$, and let $\mathrm{\Delta }={\max }_{0\le i<M}(x_{i+1}-x_i)$. For $z\in [x_i,x_{i+1}]$, define the fixed-pivot probability measure

$P_{\mathrm{\Delta }}(z)=\frac{x_{i+1}-z}{x_{i+1}-x_i}{\delta }_{x_i}+\frac{z-x_i}{x_{i+1}-x_i}{\delta }_{x_{i+1}}\mathrm{.}$

At a grid node this rule returns the single Dirac mass at that node, including at both endpoints. For $z>R$, the rule returns an overflow record of mass $z$ and no last-bin particle. For $z<x_0$ it returns an invalid-grid status. The domain convention is therefore part of the method: every retained aggregation product and every daughter size must be at least $x_0$, and a product above $R$ is handled by the overflow rule.

For error analysis, the same barycentric formula is also used on a semi-infinite grid $x_0<x_1<\cdots$ with $x_i\to \mathrm{\infty }$ and ${\sup }_i(x_{i+1}-x_i)\le \mathrm{\Delta }$. On that grid it is defined for every $z\ge x_0$ and never overflows; this is an auxiliary unbounded-pivot dynamics, distinct from the finite-cutoff method above.