Mass-action tracking and convergence through nodes

We give the tracking and low-population steps separately. The same pathwise inventory and variation estimates hold after conditioning on the initial census; expectations below average that census as well as reaction clocks. Fix a cutoff $0<\delta\le1$ and, on each edge, run a companion signed queue from the same empty state and the same exports with service function

All divided-difference slopes lie between $a_0=\kappa_-\delta$ and $a_1=\kappa_+$. The companion is driven by adapted coefficients; they are not asserted independent of either process. Since each physical reaction uses one of the $NL+O(1)$ packets, the sum of absolute population jumps is at most $2L+O(N^{-1})$. Hence the variations of $a_\pm$ are bounded independently of $N$ at fixed $\delta$.

For $m^*=\mu_NZ^*/N$, compensated reaction counting yields

Let $y$ solve the same adapted finite-variation equation without $M$, and let $y_J$ omit both $M$ and $de^N$. Scalar monotonicity and $\|e^N\|_\infty\le c_0/N$ give

For the first inequality, write the difference equation with a measurable divided-difference coefficient $a(t)\in[a_0,a_1]$. Its solution at $t$ is $\mu_N\int_0^t\exp[-\mu_N\int_s^t a(u)du]\,de^N(s)$. This is a pathwise Stieltjes identity, not an anticipative stochastic integral. Integration by parts bounds its absolute value by $2\mu_N\|e^N\|_\infty$. The second inequality follows because the drift points toward $[-J_*/a_0,J_*/a_0]$.

The instantaneous root $f(t)=[J(t)]_+/a_+(t)-[-J(t)]_+/a_-(t)$ satisfies

Contraction between jumps of $f$ and summation of its jumps imply

For $\xi=m^*-y$, production jumps cancel. The square-jump identity and monotonicity, with $V=\mathbb E\xi^2$ and $\alpha=\mu_N/N$, give

Young's inequality absorbs the square-root term into $\mu_Na_0V$ and gives $\sup_tV\le C_\delta(\alpha+\alpha^2)$. Thus

This controls directional flux because a signed queue exposes only one orientation, and

It remains to remove $\delta$ without assuming the desired population closeness. At time $t$ let $K=\{r:x_r<\delta\}$, and let $I_K,O_K$ be normalized queued charge directed into and out of $K$. Explicitly, with $z_e=Z_e/N$ and $e=(r,q)$,

Internal edges contribute zero to the cut balance. Summing (17) over $K$ gives

An incoming queue has origin outside $K$, so its origin population is at least $\delta$. Its expected service count therefore bounds

Consequently

The target directional current whose physical origin lies in $K$ is therefore bounded, using (16) and Cauchy–Schwarz, by

Couple physical and companion queues with identical exports. For a normalized queue $z=Z/N$, write the physical baseline rates as

In direction $s\in\{+,-\}$, use common baseline services at rate $\min\{b_s(z),b_s(z^*)\}$ and assign the remaining baseline rate only to the relevant queue. The companion also has extra rates

All coefficients use the physical population immediately before the event. At a physical service, choose the carrier and any retained service mark from the original conditional transition kernel. The split rates therefore preserve the complete physical marginal, including its dependence on the tag. The companion uses that same adapted population and does not supply a second population process. Every unmatched baseline service contracts their absolute signed queue difference by $1/N$; an extra companion service can enlarge it by at most $1/N$. Starting from equality, the expected number of baseline mismatches is at most the expected number of extra services. The latter normalized count is at most $R_{\delta,N}+D_{\delta,N}$, since it is supported on low-population origins. Comparing the two directional flux vectors thus costs at most twice this count. Adding the direct companion error gives

Take $N\to\infty$ at fixed $\delta$, then $\delta\downarrow0$, to prove flux convergence. Finally the physical population process has drift $B(\Phi^{N,+}-\Phi^{N,-})$ and an $O(N^{-1})$ quadratic-variation budget, because it has at most $NL+O(1)$ jumps of size $1/N$. The martingale maximal inequality, initial calibration and integrated flux convergence prove (19) tends to zero.

Each open set $\{t:w_r(t)>0\}$ is a countable union of intervals. No finiteness of the nodal set is inferred from piecewise $C^1$ regularity. On compact subintervals of these positive-weight components, standard integrated-hazard first-jump construction is unique until explosion or a nodal boundary; the increasing compact localization defines the minimal process on their union. Killing at such boundaries gives the minimal forward solution. The nonnegative vector $w$ solves its forward balance equation, so successive first-jump iteration, or positive Volterra iteration, bounds each partial-transition sum by $w$; with initial $\nu$ the bound is $Cw$. If a holding path remains at $r$ while $w_r$ tends to zero, write incoming and outgoing positive currents as $I_r,O_r$. Then $\dot w_r=I_r-O_r$ and $\lambda^B_{\rm out}(r)=O_r/w_r\ge-\dot w_r/w_r$. Integrating shows that the holding survival to that zero is zero. The dominated killed law also gives

Thus neither explosion nor nodal killing loses mass. For initial $w(0)$, normalization and domination imply equality with $w(t)$. For general $\nu$, normalization gives the asserted dominated process. The first-jump construction determines its law uniquely. This is the finite-graph existence argument underlying the standard Bell process [8, 3]; it is not a selection of that process among all event laws.

Let $N_\varepsilon$ count crossings of the deterministic weights through a regular level $\varepsilon$. One-dimensional coarea gives $\int_0^1N_\varepsilon d\varepsilon\le\sum_r\Var(w_r)$. There is a sequence $\varepsilon_k\downarrow0$ such that $\varepsilon_kN_{\varepsilon_k}\to0$: otherwise $N_\varepsilon$ would have a nonintegrable $c/\varepsilon$ lower bound near zero. The probability that the Bell path visits a sector while its weight is at most $\varepsilon$ is bounded by initial small-weight mass, jump influx into those sectors, and deterministic downcrossings at which the path is already in that sector. By Lemma 7.2 and (16), a valid bound is

Jump influx uses the integrated current bound at a small destination; each downcrossing contributes at most $C\varepsilon$.

Initialize the target at the same vertex as the tag. Until their first disagreement, split every physical tagged transition $r\to q$ at rate $\lambda^N_{qr}$ into a shared event of rate $\min\{\lambda^N_{qr},\lambda^B_{qr}\}$ and a physical-only residual. Add a target-only residual of rate $[\lambda^B_{qr}-\lambda^N_{qr}]_+$. A shared or physical-only event uses the original physical conditional transition kernel, including its queue decrement and retained marks. All other physical transitions retain their original rates and kernels. After disagreement, continue both marginals with their prescribed intensities. This construction preserves the complete physical marginal and gives the target its deterministic Markov rates, even though the physical queues depend on the tag's past.

For the estimate, stop at the first disagreement, at the first visit by the target to a sector of weight at most $\varepsilon$, or at the adapted first exit

Before this stopping time, $w_r>\varepsilon$ and $x_r^N>\varepsilon/2$ at the shared occupied origin, so

Markov's inequality bounds $\Prb(\tau_x\le T)$ by $2\epsilon_{x,N}/\varepsilon$. A union and compensator bound gives

Take $N\to\infty$ at each fixed $\varepsilon_k$, then $k\to\infty$. The queue proof's cutoff $\delta$ was already removed; the two localizations are not interchanged. This proves the whole-path claim.