Density-scaled reaction jump and chemical Langevin generators

Fix finitely many stoichiometric vectors ${\nu }_1,\dots ,{\nu }_R\in {\mathbb{Z}}^d$, an open set $U\subseteq {\mathbb{R}}^d$, nonnegative propensity functions $a_j:U\to [0,\mathrm{\infty })$, and a volume parameter $V\ge 1$. At a density state $x\in U$ for which every positive-rate destination $x+{\nu }_j\mathrm{/}V$ lies in $U$, the density-scaled jump model has jump ${\nu }_j\mathrm{/}V$ at rate $V{a}_j(x)$. For a twice differentiable test function $\varphi$ defined at those destinations, its generator is ${\mathcal{L}}_V\varphi (x)=V{\sum }_{j:a_j(x)>0}{a}_j(x)(\varphi (x+{\nu }_j\mathrm{/}V)-\varphi (x))\mathrm{.}$

Put $b(x)={\sum }_j{\nu }_j{a}_j(x)$ and $C(x)={\sum }_j{a}_j(x){\nu }_j{\nu }_j^{\mathsf{T}}$. The associated chemical Langevin diffusion generator is ${\mathcal{G}}_V\varphi (x)=b(x)\cdot \mathrm{\nabla }\varphi (x)+\frac{1}{2V}{\sum }_{j=1}^R{a}_j(x){\nu }_j^{\mathsf{T}}{D}^2\varphi (x){\nu }_j\mathrm{.}$ A Markov diffusion with this generator is called the chemical Langevin model here. The definition does not assert existence, uniqueness, positivity, or approximation accuracy for either process. In an SDE representation the drift is $b$ and the covariance matrix per unit time is $C\mathrm{/}V$.