Unbounded-grid fixed-pivot coagulation-fragmentation dynamics

Adopt all kernel and daughter-measure hypotheses of Continuous-size coagulation-fragmentation weak model: in particular, $K$ is symmetric, nonnegative, bounded and Lipschitz, $\beta$ is nonnegative, bounded and Lipschitz, and $B_x$ is a nonnegative Borel kernel with $B_x(E)\le D$, support in $[a,x]$, ${\int }_{E}z{B}_x(dz)=x$, and the stated bounded-Lipschitz dependence and convention below $2a$. Let $G=\{x_i:i\ge 0\}$ be a strictly increasing grid with $x_0=a$, $x_i\to \mathrm{\infty }$, and $x_{i+1}-x_i\le \mathrm{\Delta }\le 1$. Use the no-cutoff pivot rule $P_{\mathrm{\Delta }}$ from Conservative fixed-pivot redistribution on a finite positive grid on every cell of this semi-infinite grid. For every signed measure $\mu \in {\mathcal{M}}_w(E)$, write $P_{\mathrm{\Delta }}\mu =\int P_{\mathrm{\Delta }}(x)\mu (dx)$ for the linear projection of a measure on $E=[a,\mathrm{\infty })$ onto $G$.

For a signed grid measure $\nu \in {\mathcal{M}}_w(E)$, define $F_{\mathrm{\Delta }}(\nu )=\frac{1}{2}\iint K(x,y)(P_{\mathrm{\Delta }}(x+y)-{\delta }_x-{\delta }_y)\nu (dx)\nu (dy)+\int \beta (x)(\int P_{\mathrm{\Delta }}(z)B_x(dz)-{\delta }_x)\nu (dx)\mathrm{.}$ All products are projected, with no overflow; positive states are the population dynamics. The finite-cutoff algorithm instead sends products above its last pivot to the reported reservoir. By the barycentric identities, each projection has total number one, reproduces its first moment, and adds at most ${\mathrm{\Delta }}^2\mathrm{/}4$ to the second moment of each projected size.