Finite-channel reaction propensity systems and their generator

Fix positive integers $d$ and $R$. A reaction propensity system consists of a state set $E\subseteq {\mathbb{Z}}^d$ that is finite or countable, increments ${\nu }_j\in {\mathbb{Z}}^d$ for $1\le j\le R$, and propensities $a_j:E\to [0,\mathrm{\infty })$. Here “countable” means finite or countably infinite, as in Finite, countably infinite, countable, uncountable. The total propensity $\lambda (x):={\sum }_{j=1}^R{a}_j(x)$ is finite for every $x\in E$, and $a_j(x)>0$ implies $x+{\nu }_j\in E$. Here $\mathbb{N}$ and $\mathbb{Z}$ have the meanings in The natural numbers $\mathbb{N}$ (von Neumann) and The integers as equivalence classes of pairs of naturals, and addition in ${\mathbb{Z}}^d$ is coordinatewise.

A reaction event of channel $j$ at state $x$ records the marked transition $x\mapsto x+{\nu }_j$. The counting process $N_j(t)$ records the number of channel-$j$ events by time $t$. A zero increment ${\nu }_j=0$ is allowed: it records a reaction event without changing the state.

For a time-homogeneous system, the reaction generator operator on bounded functions $f:E\to \mathbb{R}$ is $(Qf)(x):={\sum }_{j:a_j(x)>0}{a}_j(x)(f(x+{\nu }_j)-f(x))\mathrm{.}$ When $E$ is finite, set $\mathrm{\Lambda }:=\max (\{0\}\cup \{\lambda (x):x\in E\})<\mathrm{\infty }$; thus $\mathrm{\Lambda }=0$ for $E=\mathrm{\varnothing }$ and $\mathrm{\Lambda }={\max }_{x\in E}\lambda (x)$ otherwise. If no active reaction has zero increment, the corresponding matrix has off-diagonal entries $q_{xy}={\sum }_{j:x+{\nu }_j=y}{a}_j(x)$ and diagonal $q_{xx}=-\lambda (x)$. With zero-increment events, the operator formula remains valid and the matrix diagonal is instead minus the sum of state-changing rates; the zero-increment channels remain part of the marked event process.

For a finite horizon $T$, a time-dependent propensity system replaces $a_j(x)$ by functions $a_j:[0,T]\times E\to [0,\mathrm{\infty })$ whose sections $t\mapsto a_j(t,x)$ are measurable for the Borel sigma-algebra $\mathcal{B}([0,T])$ of the usual topology on $[0,T]$ (The Borel sigma-algebra of a topological space). Require finite total rate $\lambda (t,x):={\sum }_j{a}_j(t,x)$. Its time-dependent generator operator is $(Q_{t}f)(x):={\sum }_{j:a_j(t,x)>0}{a}_j(t,x)(f(x+{\nu }_j)-f(x))\mathrm{.}$ This definition specifies a stochastic model. It does not assert that a physical chemical system follows that model.