Mean field limits of particle-based stochastic reaction-drift-diffusion models

We consider particle-based stochastic reaction-drift-diffusion (PBSRDD) models where particles move via diffusion and drift induced by one- and two-body potential interactions. We formulate the dynamics of the particles as measure-valued stochastic processes (MVSPs), which describe the stochastic evolution of the concentration fields of each chemical species as a sum of $\delta-$functions encoding the position and type of each particle. Our goal is to formulate an appropriate MVSP model for such systems, and then rigorously investigate the large population limit of the MVSP dynamics, deriving partial-integral differential equations (PIDEs) that represent the limiting mean-field dynamics.

Particle-based stochastic reaction-diffusion (PBSRD) models have a long history of use in modeling the diffusion of, and reactions between, individual molecules. PBSRDD models are more macroscopic descriptions than millisecond-timescale quantum mechanical or molecular dynamics models of a few molecules [42], but more microscopic descriptions than deterministic 3D reaction-diffusion PDEs for the average concentration field of each chemical species. One of the most popular PBSRDD models for studying biological processes is the volume reactivity (VR) model of Doi [47, 9, 10]. In the Doi model, the positions of individual molecules are typically represented as points undergoing Brownian motion. Bimolecular reactions between two substrate molecules occur with a probability per unit time based on their current positions [9, 10]. Unimolecular reactions are typically assumed to represent internal processes, and as such are modeled as occurring with exponentially distributed times based on a specified reaction-rate constant.

In our prior work [28], we investigated the large population limit of PBSRD models (i.e. in the absence of drift). Allowing drift makes the models more relevant for applications, but complicates the model formulation and the analysis needed to prove the mean field limit. Many models of biological systems involve drift induced by background potential fields and/or by potential interactions. In modeling cellular processes, the former has been used to model how volume exclusion by DNA fibers impedes protein diffusion in the nucleus [29, 32], and to model membrane-bound protein motion induced by actin contraction in T cell synapses [43]. Two-body potential interaction fields have been used to more accurately account for attractive or repulsive interactions between molecules [23], including to model volume exclusion due to the physical size of molecules [4, 43]. More generally, such interactions can arise in diverse classes of agent-based models including models for the spread of infections or innovations within populations [20], or for interactions between cells. In addition, as our numerical studies in Section 7 demonstrate, the inclusion of potential interactions can have non-trivial effects on the behavior of the system. We therefore now investigate the large population limit PBSRDD models where particles move via diffusion and drift term induced by one-body and two-body potentials.

For simplicity, we will work in free space as in [28, 27], and as such, we will study the large population mean field limit via an increasing scaling parameter, $\gamma$, which can physically represent Avogadro’s number. The “large system size” limit $\gamma\rightarrow\infty$ is where one typically obtains more macroscopic coarse-grained partial-integral differential equation (PIDE), PDE, SPIDE, or SPDE models for biological systems that model diffusion and reaction via the evolution of continuous concentration fields [18, 28, 2, 36, 20, 38, 40]. To determine the limit for PBSRDD models we adopt the classical Stroock-Varadhan Martingale approach [12, 46] that previously allowed us to rigorously identify and prove the large population limit of the MVSP representation for PBSRDD models [28]. We note that this method has been successful in many instances to study large population dynamics and general interacting particle systems, see [14, 15, 6, 7, 8, 25, 37, 44]. We identify a new macroscopic system of partial integro-differential equations (PIDEs) whose solution corresponds to the large population limit of the MVSP, and we rigorously prove the convergence (in a weak sense) of the MVSP to this solution.

We also note here that the bottom-up approach that we take in this paper allows us to derive the new macroscopic PIDEs corresponding to the true population limit of the underlying spatial PBSRDD model. This is to be contrasted with the well-known macroscopic reaction-drift-diffusion PDE models of chemical reactions at a cellular scale, which are often derived by modifying standard ODE models for non-spatial reaction systems via the addition of drift-diffusion operators to give a (phenomenological) spatial model. Note, in the absence of potential interactions (i.e., particles move only via diffusion), in [27] we proved that such models can be seen as limits of the rigorous mean field limit we derived in [28] when bimolecular reaction kernels are short-range and averaging.

For our model, consider $\Gamma_{j}$ to be the set of indices of particles of species $j$. In the absence of reactions, suppose that the $i$th particle is of species $j$ and located at position $Q_{t}^{i}$ at time $t$. The $i$th particle then moves according to

Here $v_{j}(x)$ denotes the one-body potential experienced by a particle of type $j$ when at position $x$, and $u_{j,j^{\prime}}(\|x-y\|)/\gamma$ the two-body potential between particles at locations $x$ and $y$ of types $j$ and $j^{\prime}$ respectively. $N^{\zeta}(t)$ denotes the total number of particles at time $t$, and $\zeta=(\frac{1}{\gamma},\eta)$ is a vector consisting of the mean field limit scaling parameter, $\gamma$, and a displacement range parameter, $\eta$. In the large population mean field limit, $\gamma$ plays the role of a system size (e.g., Avogadro’s number, or in bounded domains the product of Avogadro’s number and the domain volume)[1]. $\eta$ represents a regularization parameter we introduce to rigorously handle $\delta$-function placement densities for reaction products, which are commonly used in many PBSRDD models, see Section 5. Finally, $D^{j}$ represents the diffusion coefficient for a particle of species $j$, and $\{W_{t}^{i}\}_{i\in\mathbb{N}_{+}}$ is a countable collection of standard independent Brownian motions in $\mathbb{R}^{d}$.

From (1), we see each particle follows the gradient of a potential landscape, the so-called suitability landscape, and also experiences an additional force derived from the two-body potentials. The scaling $\frac{1}{\gamma}$ in front of the pair-wise potentials is the mean field scaling, which, at least formally, preserves the total strength of the interaction at order one so that we have a well-defined large population mean field limit in the absence of reactions, see [33].

From a mathematical point of view, adding drift due to potential interactions makes the rigorous formulation of the particle model more complicated, and gives rise to a number of new terms that need to be appropriately handled for a coarse-grained mean field limit to exist. Specifically, in moving from PBSRD (purely-diffusive) to PBSRDD models (drift due to potential interactions) reactive interaction functions, which determine the probability per time substrates react and produce products at specified positions, may require modification. Such modifications ensure detailed balance of reaction fluxes, i.e., the statistical mechanical property that the pointwise forward and backward reaction fluxes should balance at equilibrium [31, 13, 17]. When formulating PBSRDD reactive interaction terms to be consistent with detailed balance at equilibrium, the needed modifications result in a dependence on the full potential of the system (hence adding a dependence on the positions of non-reactant particles). We specifically formulate these modifications as an extra rejection/acceptance probability appearing within reactive interaction functions, following the approach proposed in [13], but note our results should be straightforward to adapt should one instead modify reaction rates or product placement mechanisms to ensure detailed balance. One of the contributions of this work is that we make all of these modifications precise in a general fashion via the MVSP formulation. We illustrate our results for a specific, but important, choice of rejection/acceptance probabilities based on the functional choices proposed by Fröhner and Noé [13]. The modified reactive interaction functions we employ then give rise to additional nonlinear concentration-dependent coefficients within the limiting reactive terms of the resulting mean field PIDEs, giving rise to a new deterministic, macroscopic reaction-drift-diffusion model.

To illustrate the main result of this paper, consider the reversible $A+B\rightleftarrows C$ reaction as an example. Denote the forward $A+B\rightarrow C$ reaction by $\mathcal{R}_{1}$ and the backward $C\rightarrow A+B$ reaction by $\mathcal{R}_{2}$. Let us represent by $A(t)$ the stochastic process for the number of species A molecules at time $t$, and label the position of $i$th molecule of species A at time $t$ by the stochastic process $\bm{Q}_{i}^{A(t)}\subset\mathcal{R}^{d}$. The quantity

corresponds to the stochastic, singular molar concentration field of species A at point $x$ at time $t$. We define $B^{\gamma}(x,t)$ and $C^{\gamma}(x,t)$ in a similar fashion, and let $\mathbf{S}^{\gamma}(x,t)=(A^{\gamma}(x,t),B^{\gamma}(x,t),C^{\gamma}(x,t))$. For each species we assume a spatially-constant diffusivity, $D^{\mathrm{A}},D^{\mathrm{B}}$ and $D^{\mathrm{C}}$, respectively.

For $\mathcal{R}_{1}$, let $K^{\gamma}_{1}(x,y)$ denote the probability per time that one A molecule at $x$ and one $\mathrm{B}$ molecule at $y$ react. The product $C$ molecule is then placed at $z$ following the probability density $m^{\eta}_{1}(z|x,y)$, conditioning on substrates at $x$ and $y$. Here the displacement range parameter $\eta$ represents a mollification parameter for singular $\delta$-function placement densities. We further incorporate a rejection/acceptance mechanism via an acceptance probability $\pi^{\gamma}_{1}(z|x,y,\bm{q})$, which indicates the probability that the above reaction is accepted, i.e. alllowed to occur, and generates a product at $z$ given the positions of the substrates at $x$ and $y$, and of the non-substrate and non-product particles molecules at $\bm{q}$. We define $K_{2}^{\gamma}(z),m^{\eta}_{2}(x,y|z)$ and $\pi^{\gamma}_{2}(x,y|z,\bm{q})$ analogously for the reverse reaction $\mathcal{R}_{2}$. Finally, we assume that the acceptance probabilities can be equivalently rewritten as a function of the substrate positions, product positions, and the pre-reaction concentration fields, $\pi^{\gamma}_{1}\bigl{(}z|x,y,\mathbf{S}^{\gamma}(x^{\prime},t)dx^{\prime}\bigr{)}$ and $\pi^{\gamma}_{2}\bigl{(}x,y|z,\mathbf{S}^{\gamma}(x^{\prime},t)dx^{\prime}\bigr{)}$. In Section 6 we illustrate that this assumption holds for the specific choices of rejection/acceptance probabilities proposed in [13].

Note that $K_{1}^{\gamma},m_{1}^{\eta}$ and $\pi_{1}^{\gamma}$ all depend on $\zeta=(\frac{1}{\gamma},\eta)$. In the large population limit that $\gamma\rightarrow\infty$ and $\eta\rightarrow 0$ jointly, denoted as $\zeta\rightarrow 0$, let $K_{1},m_{1}$ and $\pi_{1}$ denote their respective (possibly rescaled) limits. We will specify our assumptions on the mean field limits and $\zeta$ scalings of $K_{1}^{\gamma},m_{1}^{\eta}$, $\pi_{1}^{\gamma}$, $K_{2}^{\gamma},m_{2}^{\eta}$ and $\pi_{2}^{\gamma}$, as well as specific functional examples for them, in Sections 2, 4 and  6.

In this work, we derive the large population (thermodynamic) limit and prove, in a weak sense, that as $\zeta\rightarrow 0$,

with $\mathbf{S}(x,t)=(A(x,t),B(x,t),C(x,t))$ representing the limiting deterministic mean-field molar concentration fields. The latter satisfy the system of reaction-drift-diffusion PIDEs that

The paper is organized as follows. In Section 2, we go over the notation and definitions that are used throughout this work. We then present the stochastic equation for the PBSRDD model, which describes the evolution of the empirical measure (MVSP) of the chemical species in path space, in Section 3. In Section 4, we summarize the basic assumptions we make about the form of the reaction rate functions, product placement densities, and acceptance probabilities. In Section 5, we present our main result on the mean field limit, Theorem 5.1, which describes the evolution equation satisfied by the empirical measures for the molar concentration of each species in the large population limit for general reaction networks. As illustrative examples, we also present the mean field limits for specific chemical systems. In Section 6, we discuss a specific formulation of the acceptance probability in the generalized Fröhner-Noé model [13], and examine its large-population limit. We also examine a number of reversible reactions and give the corresponding acceptance probabilities. In Section 7, we present numerical studies illustrating the empirical convergence of numerical solutions to the PBSRDD model to the derived mean field PIDEs as $\zeta\rightarrow 0$. Our numerical examples also demonstrate that the potential interactions can have significant impacts on the statistical and dynamical behavior of the system. Finally, in Section 8, we give the proof of Theorem 5.1. Appendix 9 includes the verification that the Fröhner-Noé type of acceptance probabilities satisfy the relevant assumptions made in Section 4, and provides the proof of a mollification-type result used in Section 8.

We consider a collection of particles with $J$ different types, and for the rest of the paper, we will interchangeably use the terms particle or molecule and type or species. Let $\mathcal{S}=\{S_{1},\cdots,S_{J}\}$ denote the set of different possible types, with $p_{i}\in\mathcal{S}$ the value of the species of the $i$th particle. We also assume an underlying probability triple, $(\Omega,\mathcal{F},\mathbb{P})$, on which all random variables are defined.

In our model, molecules diffuse in space $\mathbb{R}^{d}$ subject to drift arising from potential interactions, and can undergo $L$ possible types of reactions, denoted as $\mathcal{R}_{1},\cdots,\mathcal{R}_{L}$. We use non-negative integer stoichiometric coefficients $\{\alpha_{\ell j}\}_{j=1}^{J}$ and $\{\beta_{\ell j}\}_{j=1}^{J}$ to describe the $\mathcal{R}_{\ell}$th reaction, $\ell\in\{1,\ldots,L\}$, as

and the multi-index vectors $\bm{\alpha}{}^{(\ell)}=$ $\bigl{(}\alpha_{\ell 1},\alpha_{\ell 2},\cdots,\alpha_{\ell J}\bigr{)}$ and $\bm{\beta}^{(\ell)}=\bigl{(}\beta_{\ell 1},\beta_{\ell 2},\cdots,\beta_{\ell J}\bigr{)}$ to collect the coefficients of the $\ell$th reaction. We denote the substrate and product orders of the reaction by $|\bm{\alpha}^{(\ell)}|\doteq\sum_{i=1}^{J}\alpha_{\ell i}\leq 2$ and $|\bm{\beta}^{(\ell)}|\doteq\sum_{j=1}^{J}\beta_{\ell j}\leq 2$. The implicit assumption that all reactions are at most second order is justified by the assumption that the probability that three substrates in a dilute system simultaneously have the proper configuration and energy levels to react is small. It is further justified since reactions of order three and above are often considered to be approximations of sequences of bimolecular reactions in biological models. For subsequent notational purposes, we label the reactions such that the first $\tilde{L}$ reactions correspond to those that have no products, i.e., annihilation reactions of the form

for $\ell\in\{1,\ldots,\tilde{L}\}$. We assume that the remaining $L-\tilde{L}$ reactions have one or more product particles.

Let $D^{i}$ label the diffusion coefficient for the $i$th particle, taking values in $\{D_{1},\ldots,D_{J}\}$, where $D_{j}$ is the diffusion coefficient for species $S_{j},j=1,\cdots,J$. We assume drift is imparted to particles via one- and two-body potentials. Let $v_{j}(x)$ denote a background potential that imparts drift to a particle of species $j$ located at $x$. Similarly, we let $u^{\gamma}_{j,j^{\prime}}(\left\lVert x-y\right\rVert):=u_{j,j^{\prime}}(\left\lVert x-y\right\rVert)/\gamma$ represent a two-body potential experienced between a particle of species $j$ at $x$ and a particle of species $j^{\prime}$ at $y$. Note, the inverse $\gamma$ scaling here is the standard mean field scaling to keep the total strength of the interaction fixed as more particles are added in the mean field limit [33]. We denote by $Q_{t}^{i}\in\mathbb{R}^{d}$ the position of the $i$th particle, $i\in\mathbb{N}_{+}$, at time $t$. In the absence of reactions, the dynamics for $Q_{t}^{i}$ are governed by (1). A particle’s state can be represented as a vector in $\hat{P}=\mathbb{R}^{d}\times\mathcal{S}$, the combined space encoding particle position and type. This state vector is subsequently denoted by $\hat{Q}_{t}^{i}\stackrel{{\scriptstyle\text{ def }}}{{=}}\bigl{(}Q_{t}^{i},p_{i}\bigr{)}$.

We now formulate our representation for the (number) concentration, equivalently number density, fields of each species. Let $E$ be a complete metric space and $M(E)$ the collection of measures on $E$. Let $\mathcal{M}(E)$ be the subset of $M(E)$ consisting of all finite, non-negative point measures of the form

For $f:E\mapsto\mathbb{R}$ and $\mu\in M(E)$, define

We will frequently have $E=\mathbb{R}^{d}$, in which case we omit the subscript $E$ and simply write $\langle f,\mu\rangle$. For each $t\geq 0$, we define the concentration (i.e., number density) of particles in the system at time $t$ by the distribution

where borrowing notation from [3], $N(t)=\langle 1,\nu_{t}\rangle_{\hat{P}}$ represents the stochastic process for the total number of particles at time $t$. To investigate the behavior of different types of particles, we denote the marginal distribution on the $j$th type, i.e., the concentration field for species $j$, by

a distribution on $\mathbb{R}^{d}$. $N_{j}(t)=\langle 1,\nu_{t}^{j}\rangle$ will similarly label the total number of particles of type $S_{j}$ at time $t$. Note that in the remainder, in any rigorous calculation $\nu_{t}$ and $\nu_{t}^{j}$ will be measures and treated as such. We will, however, abuse notation and also refer to them as concentration fields, i.e., number densities. Strictly speaking, the latter should refer to the densities associated with such measures, but we ignore this distinction in subsequent discussions. For $\nu$ any fixed particle distribution of the form (3), we will also use an alternative representation in terms of the marginal distributions $\nu^{j}\in\mathcal{M}(\mathbb{R}^{d})$ for particles of type $j$,

In considering the mean field large population limit, we will take a simultaneous limit in which the population scaling parameter $\gamma\to\infty$, and the (convenience) displacement range parameter $\eta\to 0$ (see Section 4 for the definition of the latter). As mentioned in the introduction, this dual limit is encoded via the vector limit parameter

In studying this limit we will work with rescaled measures for each species, denoted by

When $\gamma$ corresponds to Avogadro’s number, $\mu_{t}^{\zeta,j}$ physically corresponds to the measure for which the associated density would represent the molar concentration field for species $j$ at time $t$ (but we will again abuse notation and also refer to $\mu_{t}^{\zeta,j}$ as the molar concentration field). We similarly let

and define the vector of the molar concentrations for each species by

In the remainder, we will often write $N^{\zeta}(t)=\langle 1,\gamma\mu_{t}^{\zeta}\rangle_{\hat{P}}$ and $N^{\zeta}_{j}(t)=\langle 1,\gamma\mu_{t}^{\zeta,j}\rangle$ to make explicit that $N$ and $N_{j}$ depend on $\zeta$.

In addition to having notations for representing particle concentration fields, we will also use state vectors to store the positions of particles of a given type. Define the particle index maps $\{\sigma_{j}(k)\}_{k=1}^{N_{j}(t)}$, which encode a fixed ordering for the positions of particles of species $j$, $Q^{\sigma_{j}(1)}\preceq\cdots\preceq Q^{\sigma_{j}(N_{j}(t))}$, arising from an (assumed) fixed underlying ordering on $\mathbb{R}^{d}$. Following the notation established in [3] (see Section 6.3 therein), we let $\mathbb{N}^{*}=\mathbb{N}\setminus\{0\}$ and let $H=(H^{1},\cdots,H^{k},\cdots):\mathcal{M}(\mathbb{R}^{d})\mapsto\left(\mathbb{R}^{d}\right)^{\mathbb{N}^{*}}$ such that

$H(\nu^{j}_{t})$ represents the position state vector for type $j$ particles. We analogously let $H^{i}\bigl{(}\nu_{t}^{j}\bigr{)}\in\mathbb{R}^{d}$ label the $i$th entry of the vector $H\bigl{(}\nu_{t}^{j}\bigr{)}$. Note that the zero entries after the $Q_{t}^{\sigma_{j}(N_{j}(t))}$ term merely serve as placeholders. As commented in [3], this function $H$ allows us to address a notational issue. In particular, choosing a particle of a type $j$ uniformly among all particles in $\nu^{j}_{t}\in\mathcal{M}(\mathbb{R}^{d})$ amounts to choosing an index uniformly in the set $\{1,\cdots,\langle 1,\nu_{t}^{j}\rangle\}$, and then choosing the individual particle from the arbitrary fixed ordering.

As particles of the same type are assumed indistinguishable, there is no ambiguity in the value of $H\bigl{(}\nu_{t}^{j}\bigr{)}$ in the case that two particles of type $j$ have the same position. Using this notation we will often write the marginal distribution of species $j$ as

With the preceding definitions, and analogously to [28], we introduce a system of notation to encode substrate and particle positions and configurations that are needed to later specify reaction processes.

To formulate the process-level model, it is necessary to specify more concretely the reaction process between individual particles. We make assumptions on $K_{\ell},m_{\ell}^{\eta}$ and $\pi_{\ell}^{\gamma}$ that are analogous to what we assumed in the introduction for the $A+B\rightleftarrows C$ reaction.

For reaction $\mathcal{R}_{\ell}$, denote by $K_{\ell}^{\gamma}(\bm{x})$ the rate (i.e., probability per time) that substrate particles with positions $\bm{x}\in\mathbb{X}^{(\ell)}$ react. As described in the next section, we assume this rate function has a specific scaling dependence on $\gamma$. Let $m_{\ell}^{\eta}(\bm{y}|\bm{x})$ be the placement density when the substrates at $\bm{x}\in\mathbb{X}^{(\ell)}$ react and generate products at $\bm{y}\in\mathbb{Y}^{(\ell)}$. We assume that for each $\bm{x}$ and fixed $\eta>0,m_{\ell}^{\eta}(\cdot|\bm{x})$ is bounded. We let $\pi_{\ell}^{\gamma}(\bm{y}|\bm{x},\bm{q})$ denote the probability that a candidate reaction between the substrates located at $\bm{x}\in\mathbb{X}^{(\ell)}$, given the non-substrate and non-product particles at $\bm{q}$, is accepted and produces products at $\bm{y}\in\mathbb{Y}^{(\ell)}$. We assume that this acceptance probability can equivalently be written in terms of $\bm{x}$, $\bm{y}$, and the molar concentration of each species (rather than the specific positions, $\bm{q},$ of each non-substrate and non-product particle). Therefore, we will usually write the acceptance probability as $\pi_{\ell}^{\gamma}\bigl{(}\bm{y}|\bm{x},\bm{\mu}_{t}^{\zeta}(dx^{\prime})\bigr{)}$. Note that we assume the acceptance probability may have an explicit dependence on $\gamma$. See Section 6 for explicit examples that demonstrate this dependence.

To describe a reaction $\mathcal{R}_{\ell}$ with no products, i.e., $1\leq\ell\leq\tilde{L}$, we associate with it a Poisson point measure $dN_{\ell}(s,i,\theta)$ on $\mathbb{R}_{+}\times\mathbb{I}^{(\ell)}\times\mathbb{R}_{+}$. Here $\bm{i}\in\mathbb{I}^{(\ell)}$ gives the sampled substrate configuration, with $i_{r}^{(j)}$ labeling the $r$th sampled index of species $j$. The corresponding intensity measure of $dN_{\ell}$ is given by $d\bar{N}_{\ell}(s,i,\theta)=ds\bigl{(}\bigwedge_{j=1}^{J}\bigl{(}\bigwedge_{r=1}^{\alpha_{\ell j}}\bigl{(}\sum_{k\geq 0}\delta_{k}(i_{r}^{(j)})\bigr{)}\bigr{)}\bigr{)}d\theta$. Analogously, for each reaction $\mathcal{R}_{\ell}$ with products, i.e., $\tilde{L}+1\leq\ell\leq L$, we associate with it a Poisson point measure $dN_{\ell}(s,\bm{i},\bm{y},\theta_{1},\theta_{2})$ on $\mathbb{R}_{+}\times\mathbb{I}^{(\ell)}\times\mathbb{Y}^{(\ell)}\times\mathbb{R}_{+}\times\mathbb{R}_{+}$. Here $\bm{i}\in\mathbb{I}^{(\ell)}$ gives the sampled substrate configuration, with $i_{r}^{(j)}$ labeling the $r$th sampled index of species $j$. $\bm{y}\in\mathbb{Y}^{(\ell)}$ gives the sampled product configuration, with $y_{r}^{(j)}$ labeling the sampled position for the $r$th newly created particle of species $j$. The corresponding intensity measure is given by $d\bar{N}_{\ell}(s,\bm{i},\bm{y},\theta_{1},\theta_{2})=$ $ds\bigl{(}\bigwedge_{j=1}^{J}\bigl{(}\bigwedge_{r=1}^{\alpha_{\ell j}}\bigl{(}\sum_{k\geq 0}\delta_{k}(i_{r}^{(j)})\bigr{)}\bigr{)}\bigr{)}d\bm{y}d\theta_{1}d\theta_{2}$

The existence of the Poisson point measure follows as the intensity measure is $\sigma$-finite (see Theorem $1.8.1$ in [24] or Corollary $9.7$ in [35]). Let $d\tilde{N}_{\ell}(s,\bm{i},\bm{y},\theta_{1},\theta_{2})=dN_{\ell}(s,\bm{i},\bm{y},\theta_{1},\theta_{2})-d\bar{N}_{\ell}(s,\bm{i},\bm{y},\theta_{1},\theta_{2})$ be the compensated Poisson measure, for $\tilde{L}+1\leq\ell\leq L$. For any measurable set $A\in\mathbb{I}^{(\ell)}\times\mathbb{Y}^{(\ell)}\times\mathbb{R}_{+}\times\mathbb{R}_{+}$ such that $\bar{N}_{\ell}(\cdot,A)<\infty$, which is true if for example A is bounded, $N_{\ell}(\cdot,A)$ is a Poisson process and $\tilde{N}_{\ell}(\cdot,A)$ is a martingale (see Proposition $9.18$ in [35]). Similarly, we can define $d\tilde{N}_{\ell}(s,i,\theta)=dN_{\ell}(s,i,\theta)-d\bar{N}_{\ell}(s,i,\theta)$, for $1\leq\ell\leq\tilde{L}$. In this case, given any measurable set $A\in\mathbb{I}^{(\ell)}\times\mathbb{R}_{+}$ such that $\bar{N}_{\ell}(\cdot,A)<\infty$, we then have that $N_{\ell}(\cdot,A)$ is a Poisson process and $\tilde{N}_{\ell}(\cdot,A)$ is a martingale.