2. A related branching model with migration

There exists a population living in two colonies with labels $\mathbb{1,\,2}$ . Initially, each colony has n particles, spatially distributed in $\mathbb{R}$ . Write $k\sim t$ to denote the kth living particle at time t in each colony. For any $s\geq t$ , let $X_{k\sim t}^{n,\mathbb{i}}(s)$ denote the corresponding particle’s location at time s in colony $\mathbb{i}$ with $$\mathbb{i}\in\{\mathbb{1,\,2}\}$$ if it is alive up to time s. The motions of the particles during their lifetimes are modelled by independent Brownian motions. For any $\mathbb{i}\in\{\mathbb{1,2}\}$ , $k\sim t$ , accompanying the corresponding particle with a standard Brownian motion $\{B_{k\sim t}^{\mathbb{i}}(s)\,:\, s\geq 0\}$ , we have

\begin{equation*}X_{k\sim t}^{n,\mathbb{i}}(s) \,{\buildrel d \over=}\,X_{k\sim t}^{n,\mathbb{i}}(t)+B_{k\sim t}^{\mathbb{i}}(s)-B_{k\sim t}^{\mathbb{i}}(t) \quad \text{for all $ s\geq t$.}\end{equation*}

For $s<t$ , $X_{k\sim t}^{n,\mathbb{i}}(s)$ represents the corresponding particle’s location at time s if it is alive at s; otherwise the same notation represents its ancestor’s location at s.

We start by introducing the branching particle systems. In colony $\unicode{x1D7D9}$ there exist independent branching and emigration, and there are also independent branching and immigration in colony $\unicode{x1D7DA}$ . The branching mechanisms in the two colonies are also independent. However, the emigration and immigration are interactive. The particles in colony $\unicode{x1D7D9}$ can move to colony $\unicode{x1D7DA}$ , but not reciprocally. During branching/emigration/immigration events, all the particles move according to independent Brownian motions.

• (Measure-valued process in colony $\mathbb{i}$ with $$\mathbb{i}=\{\mathbb{1,\,2}\}$$ .) Let $\mu_t^{n,\mathbb{i}}$ be the empirical distribution of particles living in colony $\mathbb{i}$ , that is, for any $f\in C_{b}^{2}(\mathbb{R})$ , we have

  \begin{align*}\bigl\langle \mu_t^{n,\mathbb{i}},f\bigr\rangle =\frac{1}{n}\sum_{k\sim t}f\bigl(X_{k\sim t}^{n,\mathbb{i}}(t)\bigr),\end{align*}
  where the sum $k\sim t$ includes all those particles alive at t in each colony.

• (The behaviour of particles living in colony $\unicode{x1D7D9}$ .) For a particle k alive at time t in colony $\unicode{x1D7D9}$ , we consider (conditionally independent) random times $\tau_{k\sim t}^{\unicode{x1D7D9}}$ (corresponding to a reproduction event) and $\rho_{k\sim t}$ (corresponding to a migration event) such that

  \begin{align*}\mathbb{P}\bigl(\tau^{\unicode{x1D7D9}}_{k\sim t}>t+h\mid \mathcal{F}_t\bigr)={\textrm{e}}^{-\lambda_{n,\unicode{x1D7D9}}h}\end{align*}
  and
  \begin{align*}\mathbb{P}(\rho_{k\sim t}>t+h\mid \mathcal{F}_t)& = \mathbb{E}\biggl[\exp\biggl\{-\int_t^{t+h}\eta\bigl(X_{k\sim t}^{n,\unicode{x1D7D9}}(s), \mu_s^{n,\unicode{x1D7D9}},\mu_s^{n,\unicode{x1D7DA}}\bigr)\, {\textrm{d}} s\biggr\}\Bigm|\mathcal{F}_t\biggr] \\& \approx {\textrm{e}}^{-\eta\left(X_{k\sim t}^{n,\unicode{x1D7D9}}(t), \mu_t^{n,\unicode{x1D7D9}},\mu_t^{n,\unicode{x1D7DA}}\right)h}\end{align*}
  when h is sufficiently small, where $\lambda_{n,\unicode{x1D7D9}}$ is the branching rate of those particles living in colony $\unicode{x1D7D9}$ with ${\lambda_{n,\unicode{x1D7D9}}}/{n}\rightarrow\lambda_{\unicode{x1D7D9}}$ as $n\rightarrow\infty$ , and $\eta(\cdot, \cdot,\cdot)\in C_b^+(\mathbb{R}\times M_F(\mathbb{R})^2)$ is the migration intensity. If $\tau^{\unicode{x1D7D9}}_{k\sim t}<\rho_{k\sim t}$ , then at time $\tau^{\unicode{x1D7D9}}_{k\sim t}$ , particle k dies and gives birth to a random number $\xi^{n, \unicode{x1D7D9}}$ of offspring at the position of colony $\unicode{x1D7D9}$ where particle k died, with $\mathbb{E}[\xi^{n, \unicode{x1D7D9}}]=1+{\beta_{n,\unicode{x1D7D9}}}/{n}$ , $\operatorname{Var}\![\xi^{n, \unicode{x1D7D9}}]=\sigma^2_{n,\unicode{x1D7D9}}$ satisfying $\beta_{n,\unicode{x1D7D9}}\rightarrow\beta_{\unicode{x1D7D9}}$ and $\sigma_{n,\unicode{x1D7D9}}\rightarrow\sigma_{\unicode{x1D7D9}}$ as $n\rightarrow\infty$ . If $\rho_{k\sim t}<\tau^{\unicode{x1D7D9}}_{k\sim t}$ , then at time $\rho_{k\sim t}$ , particle k migrates to colony $\unicode{x1D7DA}$ and settles down at a random position according to a probability measure $\chi$ .

• (The behaviour of particles living in colony $\unicode{x1D7DA}$ .) For a particle k alive at time t in colony $\unicode{x1D7DA}$ , we consider random time $\tau_{k\sim t}^{\unicode{x1D7DA}}$ (corresponding to a reproduction event) such that

  \begin{align*}\mathbb{P}\bigl(\tau^{\unicode{x1D7DA}}_{k\sim t}>t+h\mid \mathcal{F}_t\bigr)={\textrm{e}}^{-\lambda_{n,\unicode{x1D7DA}}h},\end{align*}
  where $\lambda_{n,\unicode{x1D7DA}}$ is the branching rate of those particles living in colony $\unicode{x1D7DA}$ with ${\lambda_{n,\unicode{x1D7DA}}}/{n}\rightarrow\lambda_{\unicode{x1D7DA}}$ as $n\rightarrow\infty$ . Then, at time $\tau^{\unicode{x1D7DA}}_{k\sim t}$ , particle k dies and gives birth to a random number $\xi^{n, \unicode{x1D7DA}}$ of offspring at the position of colony $\unicode{x1D7DA}$ where particle k died, with $\mathbb{E}[\xi^{n, \unicode{x1D7DA}}]=1+{\beta_{n,\unicode{x1D7DA}}}/{n}$ , $\operatorname{Var}[\xi^{n, \unicode{x1D7DA}}]=\sigma^2_{n,\unicode{x1D7DA}}$ satisfying $\beta_{n,\unicode{x1D7DA}}\rightarrow\beta_{\unicode{x1D7DA}}$ and $\sigma_{n,\unicode{x1D7DA}}\rightarrow\sigma_{\unicode{x1D7DA}}$ as $n\rightarrow\infty$ .

3. Existence of a solution to the martingale problem

In this section we study the convergence of a sequence of measure-valued processes arising as the empirical measures of the proposed branching particle systems in Section 2, and show that the limit is a weak solution to the MP (1.3, 1.4). We denote $\sum_{i = 1}^0 f_i = 0$ for any $f_i$ . For any $t>0$ , let $h>0$ be sufficiently small. It follows from the construction of the branching particle systems that

\begin{align*}\bigl\langle \mu_{t+h}^{n,\unicode{x1D7D9}},f\bigr\rangle& = \frac{1}{n}\sum_{k\sim t}\bigl(\,f\bigl(X_{k\sim t}^{n,\unicode{x1D7D9}}(t+h)\bigr) +\Delta_{k\sim t}^{n,\unicode{x1D7D9},h}(\,f)-D_{k\sim t}^{n,\unicode{x1D7D9},h}(\,f)\bigr)+\frac{1}{n}\sum_{k\sim t}\Upsilon_{k\sim t}^{n,\unicode{x1D7D9},h}(\,f),\end{align*}

with

(3.1) \begin{align} \Delta_{k\sim t}^{n, \unicode{x1D7D9},{h}}(\,f) & =\Biggl[\sum_{i=1}^{\xi^{n, \unicode{x1D7D9}}_{k\sim t}}f\bigl(X_{k\sim t}^{n,\unicode{x1D7D9},i}(t+h)\bigr) - f\bigl(X_{k\sim t}^{n,\unicode{x1D7D9}}(t+h)\bigr)\Biggr]1{\hskip -2.5 pt}\hbox{I}_{\left\{\tau_{k\sim t}^{\unicode{x1D7D9}}< t+h<\rho_{k\sim t}\right\}},\end{align}

(3.2) \begin{align} D_{k\sim t}^{n, \unicode{x1D7D9},{h}}(\,f) & = f\bigl(X_{k\sim t}^{n,\unicode{x1D7D9}}(t+h)\bigr)1{\hskip -2.5 pt}\hbox{I}_{\left\{\rho_{k\sim t}< t+h<\tau_{k\sim t}^{\unicode{x1D7D9}}\right\}},\qquad\qquad\qquad\qquad\qquad\end{align}

where $\xi^{n, \unicode{x1D7D9}}_{k\sim t}$ , $k=1,2,\ldots$ are independent and identically distributed (i.i.d.) copies of $\xi^{n, \unicode{x1D7D9}}$ , $X_{k\sim t}^{n,\unicode{x1D7D9},i}(t+h)$ , $i=1,\ldots,\xi^{n, \unicode{x1D7D9}}_{k\sim t}$ are defined as i.i.d. copies of $X_{k\sim t}^{n,\unicode{x1D7D9}}(t+h)$ , $1{\hskip -2.5 pt}\hbox{I}_{\{\cdot\}}$ is the indicator function, and $\Upsilon_{k\sim t}^{n,\unicode{x1D7D9},h}(\,f)$ is the correction term due to particle k and its descendants having more than one branching or migration event in the time interval $(t,t+h]$ . Since f is bounded, one can check that the correction term satisfies

\begin{align*}\mathbb{E} \bigl[|\Upsilon_{k\sim t}^{n,\unicode{x1D7D9},h}(\,f)|\mid \mathcal{F}_{t}\bigr]\leq {\textrm{O}}(h^2),\end{align*}

where ${\textrm{O}}(h^2)$ is an infinitesimal of h with order no less than 2. In fact it is hard to obtain the explicit expression for the correction term. However, the order indicates that it is very tiny and can make itself disappear in the limit form. The conditional independence of $\rho_{k\sim t}$ and $\tau_{k\sim t}^{\unicode{x1D7D9}}$ implies that

\begin{align*}\mathbb{E}\bigl[1{\hskip -2.5 pt}\hbox{I}_{\left\{\tau_{k\sim t}^{\unicode{x1D7D9}}< t+h<\rho_{k\sim t}\right\}}\mid \mathcal{F}_t\bigr]& =\mathbb{P}\bigl(\tau_{k\sim t}^{\unicode{x1D7D9}}< t+h\mid \mathcal{F}_t\bigr)\mathbb{P}\bigl({t+h<\rho_{k\sim t}\mid \mathcal{F}_t }\bigr) \\& \approx \bigl(1-{\textrm{e}}^{-\lambda_{n,\unicode{x1D7D9}}h}\bigr){\textrm{e}}^{-\eta(X_{k\sim t}^{n,\unicode{x1D7D9}}(t), \mu_t^{n,\unicode{x1D7D9}},\mu_t^{n,\unicode{x1D7DA}})h}\\&\approx\lambda_{n,\unicode{x1D7D9}}h\end{align*}

and

\begin{align*}\mathbb{E}\bigl[1{\hskip -2.5 pt}\hbox{I}_{\left\{\rho_{k\sim t}< t+h<\tau_{k\sim t}^{\unicode{x1D7D9}}\right\}}\mid \mathcal{F}_t\bigr]& \approx \bigl(1-{\textrm{e}}^{-\eta(X_{k\sim t}^{n,\unicode{x1D7D9}}(t), \mu_t^{n,\unicode{x1D7D9}},\mu_t^{n,\unicode{x1D7DA}})h}\bigr){\textrm{e}}^{-\lambda_{n,\unicode{x1D7D9}}h} \\& \approx \eta\bigl(X_{k\sim t}^{n,\unicode{x1D7D9}}(t), \mu_t^{n,\unicode{x1D7D9}},\mu_t^{n,\unicode{x1D7DA}}\bigr)h.\end{align*}

The corresponding conditional probability can be replaced by the approximate value when h is sufficiently small. Applying Itô’s formula, we have

(3.3) \begin{align} f\bigl(X_{k\sim t}^{n,\unicode{x1D7D9}}(t+h)\bigr)& = f\bigl(X_{k\sim t}^{n,\unicode{x1D7D9}}(t)\bigr)+\int_{t}^{t+h}f^{\prime}\bigl(X_{k\sim t}^{n,\unicode{x1D7D9}}(s)\bigr){\textrm{d}} B^{\unicode{x1D7D9}}_{k\sim t}(s) +\frac12\int_{t}^{t+h}f^{\prime\prime}\bigl(X_{k\sim t}^{n,\unicode{x1D7D9}}(s)\bigr)\, {\textrm{d}} s.\end{align}

Consequently

\begin{align*}\bigl\langle \mu_{t+h}^{n,\unicode{x1D7D9}},f\bigr\rangle& = \bigl\langle \mu_{t}^{n,\unicode{x1D7D9}},f\bigr\rangle+\frac{1}{n}\sum_{k\sim t}\int_{t}^{t+h}f^{\prime}\bigl(X_{k\sim t}^{n,\unicode{x1D7D9}}(s)\bigr){\textrm{d}} B^{\unicode{x1D7D9}}_{k\sim t}(s) \\& \quad +\frac{1}{n}\sum_{k\sim t}\frac12\int_{t}^{t+h}f^{\prime\prime}\bigl(X_{k\sim t}^{n,\unicode{x1D7D9}}(s)\bigr)\, {\textrm{d}} s \\& \quad +\frac{1}{n}\sum_{k\sim t}\bigl(\Delta_{k\sim t}^{n,\unicode{x1D7D9},{h}}(\,f)-D_{k\sim t}^{n,\unicode{x1D7D9},{h}}(\,f)+\Upsilon_{k\sim t}^{n,\unicode{x1D7D9},h}(\,f)\bigr).\end{align*}

Given any $t>0$ , let us discretize the time interval [0, t] by the step size h. Set $j=\lfloor t/h\rfloor$ . Then

\begin{align*}\bigl\langle \mu_{t}^{n,\unicode{x1D7D9}},f\bigr\rangle & = \bigl\langle \mu_{0}^{n,\unicode{x1D7D9}},f\bigr\rangle +\sum_{\ell=0}^{j}\bigl(\bigl\langle \mu_{(\ell+1)h\wedge t}^{n,\unicode{x1D7D9}},f\bigr\rangle -\bigl\langle \mu_{\ell h}^{n,\unicode{x1D7D9}},f\bigr\rangle \bigr) \\& = \bigl\langle \mu_{0}^{n,\unicode{x1D7D9}},f\bigr\rangle +\sum_{\ell=0}^{j}\frac{1}{n}\sum_{k\sim \ell h}\int_{\ell h}^{ (\ell+1)h\wedge t}f^{\prime}\bigl(X_{k\sim \ell h}^{n,\unicode{x1D7D9}}(s)\bigr){\textrm{d}} B_{k\sim \ell h}^{\unicode{x1D7D9}}(s) \\& \quad +\sum_{\ell=0}^{j}\frac{1}{n}\sum_{k\sim \ell h}\frac12\int_{\ell h}^{(\ell+1)h\wedge t}f^{\prime\prime}\bigl(X_{k\sim \ell h}^{n,\unicode{x1D7D9}}(s)\bigr)\, {\textrm{d}} s \\& \quad +\sum_{\ell=0}^{j}\frac{1}{n}\sum_{k\sim \ell h}\bigl(\Delta_{k\sim\ell h}^{n, \unicode{x1D7D9},{h\wedge( t-\ell h)}}(\,f)-D_{k\sim\ell h}^{n, \unicode{x1D7D9},{h\wedge( t-\ell h)}}(\,f)+\Upsilon_{k\sim \ell h}^{n,\unicode{x1D7D9},{h\wedge( t-\ell h)}}(\,f)\bigr).\end{align*}

Let $M_{i}^{n,f}$ , $i=1,2,3$ be martingales with

(3.4) \begin{align}M_{1}^{n,f}(t)& = \sum_{\ell=0}^{j}\frac{1}{n}\sum_{k\sim \ell h}\int_{\ell h}^{(\ell+1)h\wedge t}f^{\prime}\bigl(X_{k\sim \ell h}^{n,\unicode{x1D7D9}}(s)\bigr){\textrm{d}} B^{\unicode{x1D7D9}}_{k\sim \ell h}(s),\end{align}

(3.5) \begin{align} M_{2}^{n,f}(t) & = \sum_{\ell=0}^{j}\frac{1}{n}\sum_{k\sim \ell h}\bigl\{\Delta_{k\sim\ell h}^{n, \unicode{x1D7D9},{h\wedge( t-\ell h)}}(\,f) - \mathbb{E}\bigl[\Delta_{k\sim\ell h}^{n, \unicode{x1D7D9},{h\wedge( t-\ell h)}}(\,f)\mid \mathcal{F}_{\ell h}\bigr]\bigr\},\end{align}

(3.6) \begin{align} M_{3}^{n,f}(t) & = \sum_{\ell=0}^{j}\frac{1}{n}\sum_{k\sim \ell h}\bigl\{D_{k\sim\ell h}^{n, \unicode{x1D7D9},{h\wedge( t-\ell h)}}(\,f) - \mathbb{E}\bigl[D_{k\sim\ell h}^{n, \unicode{x1D7D9},{h\wedge( t-\ell h)}}(\,f)\mid \mathcal{F}_{\ell h}\bigr]\bigr\}.\end{align}

Moreover, $A^{n, f}$ is defined as

\begin{align*}A^{n,f}(t)& = \sum_{\ell=0}^{j}\frac{1}{n}\sum_{k\sim \ell h}\frac12\int_{\ell h}^{(\ell+1)h\wedge t}f^{\prime\prime}\bigl(X_{k\sim \ell h}^{n,\unicode{x1D7D9}}(s)\bigr)\, {\textrm{d}} s+\sum_{\ell=0}^{j}\frac{1}{n}\sum_{k\sim \ell h}\Upsilon_{k\sim \ell h}^{n,\unicode{x1D7D9},{h\wedge( t-\ell h)}}(\,f) \\& \quad +\sum_{\ell=0}^{j}\frac{1}{n}\sum_{k\sim \ell h}\mathbb{E}\bigl[\Delta_{k\sim\ell h}^{n, \unicode{x1D7D9},{h\wedge( t-\ell h)}}(\,f)-D_{k\sim\ell h}^{n, \unicode{x1D7D9},{h\wedge( t-\ell h)}}(\,f)\mid \mathcal{F}_{\ell h}\bigr].\end{align*}

It follows from (3.3) that

\begin{align*}& \quad \mathbb{E}\bigl[f\bigl(X_{k\sim \ell h}^{n,\unicode{x1D7D9}}((\ell+1)h\wedge t)\bigr)\mid \mathcal{F}_{\ell h}\bigr]= f\bigl(X_{k\sim \ell h}^{n,\unicode{x1D7D9}}(\ell h)\bigr)+ {\textrm{O}}(h\wedge(t-\ell h)),\end{align*}

where ${\textrm{O}}(h\wedge(t-\ell h))\leq \frac{1}{2}\sup_{x \in \mathbb{R}} |f^{\prime\prime}(x)|h\leq Kh$ holds for any given h and n since $f \in C_b^2(\mathbb{R})$ . Combining with (3.1) and (3.2), we obtain the conditional expectations

\begin{align*}& \quad \mathbb{E}\big[\Delta_{k\sim\ell h}^{n, \unicode{x1D7D9},h\wedge(t-\ell h)}(\,f)\mid \mathcal{F}_{\ell h}\big]=\frac{{\beta}_{n,\unicode{x1D7D9}}\lambda_{n,\unicode{x1D7D9}}}{n}f\bigl(X_{k\sim \ell h}^{n,\unicode{x1D7D9}}(\ell h)\bigr)(h\wedge(t-\ell h))+ {\textrm{O}}((h\wedge(t-\ell h))^2)\end{align*}

and

\begin{align*}& \mathbb{E}\big[D_{k\sim\ell h}^{n, \unicode{x1D7D9},h\wedge(t-\ell h)}(\,f)\mid \mathcal{F}_{\ell h}\big] \\& \qquad\qquad= f\bigl(X_{k\sim \ell h}^{n,\unicode{x1D7D9}}(\ell h)\bigr)\eta\bigl(X_{k\sim \ell h}^{n,\unicode{x1D7D9}}(\ell h),{\mu}_{\ell h}^{n,\unicode{x1D7D9}},{\mu}_{\ell h}^{n,\unicode{x1D7DA}}\bigr)(h\wedge(t-\ell h))+{\textrm{O}}((h\wedge(t-\ell h))^2),\end{align*}

where ${\textrm{O}}((h\wedge(t-\ell h))^2)\leq Kh^2$ holds for any given h and n. Consequently

(3.7) \begin{align}A^{n,f}(t)& = \frac12\int_{0}^{t}\bigl\langle \mu_s^{n,\unicode{x1D7D9}},f^{\prime\prime}\bigr\rangle \, {\textrm{d}} s+\sum_{\ell=0}^{j}\frac{{\beta}_{n,\unicode{x1D7D9}}\lambda_{n,\unicode{x1D7D9}}}{n}\bigl\langle \mu_{\ell h}^{n,\unicode{x1D7D9}},f\bigr\rangle (h\wedge(t-\ell h)) \notag\\& \quad -\sum_{\ell=0}^{j}\bigl\langle \mu_{\ell h}^{n,\unicode{x1D7D9}},\eta\bigl(\cdot,\mu_{\ell h}^{n,\unicode{x1D7D9}},\mu_{\ell h}^{n,\unicode{x1D7DA}}\bigr)f\bigr\rangle (h\wedge(t-\ell h))\notag \\& \quad +\sum_{\ell=0}^{j}\bigl\langle \mu_{\ell h}^{n,\unicode{x1D7D9}},1\bigr\rangle \,{\textrm{O}}((h\wedge(t-\ell h))^2)\notag \\& \quad +\sum_{\ell=0}^{j}\frac{1}{n}\sum_{k\sim \ell h}\Upsilon_{k\sim \ell h}^{n,\unicode{x1D7D9},{h\wedge( t-\ell h)}}(\,f),\end{align}

with the last term satisfying

(3.8) \begin{align}\sum_{\ell=0}^{j}\frac{1}{n}\sum_{k\sim \ell h}\mathbb{E}\bigl[ |\Upsilon_{k\sim \ell h}^{n,\unicode{x1D7D9},{h\wedge( t-\ell h)}}(\,f)|\mid \mathcal{F}_{\ell h}\bigr]\leq \sum_{\ell=0}^{j}\bigl\langle \mu_{\ell h}^{n,\unicode{x1D7D9}},1\bigr\rangle \,{\textrm{O}}((h\wedge(t-\ell h))^2).\end{align}

One can see that

(3.9) \begin{align}\bigl\langle \mu_t^{n,\unicode{x1D7D9}}, f\bigr\rangle = \bigl\langle \mu_{0}^{n,\unicode{x1D7D9}},f\bigr\rangle +M_{1}^{n,f}(t)+M_{2}^{n,f}(t)-M_{3}^{n,f}(t)+A^{n,f}(t).\end{align}

Carrying out steps similar to those above in colony $\unicode{x1D7DA}$ , for any $g\in C_b^2 (\mathbb{R})$ , we have

(3.10) \begin{equation}\bigl\langle \mu_{t}^{n,\unicode{x1D7DA}},g\bigr\rangle =\bigl\langle \mu_{0}^{n,\unicode{x1D7DA}},g\bigr\rangle+\hat{M}_{1}^{n,g}(t)+\hat{M}_{2}^{n,g}(t)+\hat{M}_{3}^{n,g}(t)+\hat{A}^{n,g}(t),\end{equation}

where

(3.11) \begin{align}\hat{M}_{1}^{n,g}(t)=\sum_{\ell=0}^{\lfloor t/h\rfloor}\frac{1}{n}\sum_{k\sim \ell h}\int_{\ell h}^{(\ell+1)h\wedge t}g^{\prime}\bigl(X_{k\sim \ell h}^{n,\unicode{x1D7DA}}(s)\bigr){\textrm{d}} B^{\unicode{x1D7DA}}_{k\sim \ell h}(s)\end{align}

and

(3.12) \begin{align} \hat{M}_{2}^{n,g}(t) = \sum_{\ell=0}^{\lfloor t/h\rfloor}\frac{1}{n}\sum_{k\sim \ell h}\bigl\{\Delta_{k\sim \ell h}^{n, \unicode{x1D7DA},h\wedge(t-\ell h)}(g)-\mathbb{E}\bigl[\Delta_{k\sim\ell h}^{n, \unicode{x1D7DA},{h\wedge(t-\ell h)}}(g)\mid \mathcal{F}_{\ell h}\bigr]\bigr\}\end{align}

are martingales with

\begin{align*}\Delta_{k\sim \ell h}^{n, \unicode{x1D7DA},h\wedge(t-\ell h)}(g) =\Biggl[\sum_{i=1}^{\xi^{n, \unicode{x1D7DA}}_{k\sim \ell h}}g\bigl(X_{k\sim \ell h}^{n,\unicode{x1D7DA},i}((\ell+1)h\wedge t)\bigr) - g\bigl(X_{k\sim \ell h}^{n,\unicode{x1D7DA}}((\ell+1)h\wedge t)\bigr)\Biggr]1{\hskip -2.5 pt}\hbox{I}_{\{\tau_{k\sim \ell h}^{\unicode{x1D7DA}}< (\ell+1)h\wedge t\}}\end{align*}

and $X_{k\sim \ell h}^{n,\unicode{x1D7DA},i}((\ell+1)h\wedge t)$ having the same distribution as $X_{k\sim \ell h}^{n,\unicode{x1D7DA}}((\ell+1)h\wedge t)$ . Moreover,

(3.13) \begin{equation}\hat{M}_{3}^{n,g}(t)= \sum_{\ell=0}^{\lfloor t/h\rfloor}\frac{1}{n}\sum_{k\sim \ell h}\bigl\{D_{k\sim \ell h}^{n, \unicode{x1D7DA},{h\wedge(t-\ell h)}}(g)-\mathbb{E}\bigl[D_{k\sim\ell h}^{n, \unicode{x1D7DA},{h\wedge(t-\ell h)}}(g)\mid \mathcal{F}_{\ell h}\bigr]\bigr\}\end{equation}

is also a martingale with

\begin{align*}D_{k\sim\ell h}^{n, \unicode{x1D7DA},{h\wedge(t-\ell h)}}(g) = \langle\chi,g\rangle1{\hskip -2.5 pt}\hbox{I}_{\{ \rho_{k\sim\ell h}\leq(\ell+1)h\wedge t\}}.\end{align*}

We emphasize that the sum ‘ $k\sim\ell h$ ’ in (3.13) involves those particles living in colony $\unicode{x1D7D9}$ at time $\ell h$ . In addition,

\begin{align*}\hat{A}^{n,g}(t)& = \sum_{\ell=0}^{\lfloor t/h\rfloor}\frac{1}{n}\sum_{k\sim \ell h}\frac12\int_{\ell h}^{(\ell+1)h\wedge t}g^{\prime\prime}\bigl(X_{k\sim \ell h}^{n,\unicode{x1D7DA}}(s)\bigr)\, {\textrm{d}} s \\ & \quad + \sum_{\ell=0}^{\lfloor t/h\rfloor}\frac{1}{n}\sum_{k\sim \ell h}\mathbb{E}\bigl[\Delta_{k\sim \ell h}^{n, \unicode{x1D7DA},{h\wedge(t-\ell h)}}(g)\mid\mathcal{F}_{\ell h}\bigr]+ \sum_{\ell=0}^{\lfloor t/h\rfloor}\frac{1}{n}\sum_{k\sim \ell h}\mathbb{E}\bigl[D_{k\sim \ell h}^{n, \unicode{x1D7DA},h\wedge(t-\ell h)}(g)\mid\mathcal{F}_{\ell h}\bigr] \\& \quad +\sum_{\ell=0}^{\lfloor t/h\rfloor}\frac{1}{n}\sum_{k\sim \ell h}\Upsilon_{k\sim \ell h}^{n,\unicode{x1D7DA},{h\wedge( t-\ell h)}}(g)+\sum_{\ell=0}^{\lfloor t/h\rfloor}\frac{1}{n}\sum_{k\sim \ell h}\Xi_{k\sim \ell h}^{n,\unicode{x1D7DA},{h\wedge( t-\ell h)}}(g),\end{align*}

where the second and third terms are due to birth events in colony $\unicode{x1D7DA}$ and migration from colony $\unicode{x1D7D9}$ , respectively, the fourth term is the correction term due to those particles living in colony $\unicode{x1D7DA}$ and their descendants having more than one branching, and the last term is the correction term due to those particles living in colony $\unicode{x1D7D9}$ that have not only migrated to colony $\unicode{x1D7DA}$ but also given birth to offspring after settlements. Therefore the sum ‘ $k\sim\ell h$ ’ in the second or fourth term involves those particles living in colony $\unicode{x1D7DA}$ at $\ell h$ , and the sum ‘ $k\sim\ell h$ ’ in the third or last term involves those particles living in colony $\unicode{x1D7D9}$ at $\ell h$ . Considering the possibility of more than one branching or migration event in the time interval $(\ell h,(\ell+1)h\wedge t]$ and the boundedness of g, one can see that

\begin{align*}& \quad \mathbb{E}\bigl[|\Upsilon_{k\sim \ell h}^{n,\unicode{x1D7DA},h\wedge (t-\ell h)}(g)|\mid \mathcal{F}_{\ell h}\bigr]+\mathbb{E}\bigl[|\Xi_{k\sim \ell h}^{n,\unicode{x1D7DA},h\wedge (t-\ell h)}(g)|\mid \mathcal{F}_{\ell h}\bigr]\leq {\textrm{O}}((h\wedge(t-\ell h))^2). \end{align*}

Consequently

(3.14) \begin{align}\hat{A}^{n,g}(t)& = \frac12\int_{0}^{t}\bigl\langle \mu_s^{n,\unicode{x1D7DA}},g^{\prime\prime}\bigr\rangle \, {\textrm{d}} s+\sum_{\ell=0}^{\lfloor t/h\rfloor}\frac{{\beta}_{n,{\unicode{x1D7DA}}}\lambda_{n,{\unicode{x1D7DA}}}}{n}\bigl\langle \mu_{\ell h}^{n,\unicode{x1D7DA}},g\bigr\rangle (h\wedge(t-\ell h)) \notag \\& \quad +\sum_{\ell=0}^{\lfloor t/h\rfloor}\langle\chi, g\rangle\bigl\langle \mu_{\ell h}^{n,\unicode{x1D7D9}}, \eta\bigl(\cdot,\mu_{\ell h}^{n,\unicode{x1D7D9}},\mu_{\ell h}^{n,\unicode{x1D7DA}}\bigr)\bigr\rangle (h\wedge(t-\ell h)) \notag \\& \quad +\sum_{\ell=0}^{\lfloor t/h\rfloor}\bigl\langle \mu_{\ell h}^{n,\unicode{x1D7D9}}+ \mu_{\ell h}^{n,\unicode{x1D7DA}},1\bigr\rangle \,{\textrm{O}}((h\wedge(t-\ell h))^2) \notag \\& \quad +\sum_{\ell=0}^{\lfloor t/h\rfloor}\frac{1}{n}\sum_{k\sim \ell h}\Upsilon_{k\sim \ell h}^{n,\unicode{x1D7DA},{h\wedge( t-\ell h)}}(g)+\sum_{\ell=0}^{\lfloor t/h\rfloor}\frac{1}{n}\sum_{k\sim \ell h}\Xi_{k\sim \ell h}^{n,\unicode{x1D7DA},{h\wedge( t-\ell h)}}(g),\end{align}

with the terms in the last line satisfying

\begin{align*}& \quad \sum_{\ell=0}^{\lfloor t/h\rfloor}\frac{1}{n}\sum_{k\sim \ell h}\mathbb{E}\bigl[|\Upsilon_{k\sim \ell h}^{n,\unicode{x1D7DA},{h\wedge( t-\ell h)}}(g)|\mid\mathcal{F}_{\ell h}\bigr]+\sum_{\ell=0}^{\lfloor t/h\rfloor}\frac{1}{n}\sum_{k\sim \ell h}\mathbb{E}\bigl[|\Xi_{k\sim \ell h}^{n,\unicode{x1D7DA},{h\wedge( t-\ell h)}}(g)|\mid\mathcal{F}_{\ell h}\bigr] \\& \quad \quad\leq\sum_{\ell=0}^{\lfloor t/h\rfloor}\bigl\langle \mu_{\ell h}^{n,\unicode{x1D7D9}}+ \mu_{\ell h}^{n,\unicode{x1D7DA}},1\bigr\rangle \,{\textrm{O}}((h\wedge(t-\ell h))^2).\end{align*}

The following propositions give the quadratic variation and covariation processes for those martingales ${M}_{i}^{n,f}(t)$ and $\hat{M}_{i}^{n,g}(t)$ with $i=1,2,3$ .

Proposition 3.1. The variation processes of ${M}_{i}^{n,f}(t)$ with $i=1,2,3$ are as follows:

(3.15) \begin{align} \bigl\langle M_{1}^{n,f}\bigr\rangle _{t}& = \sum_{\ell=0}^{\lfloor t/h\rfloor}\dfrac{1}{n^2}\sum_{k\sim \ell h}\int_{\ell h}^{(\ell+1)h\wedge t}\bigl|f^{\prime}\bigl(X_{k\sim \ell h}^{n,\unicode{x1D7D9}}(s)\bigr)\bigr|^{2}d s,\end{align}

(3.16) \begin{align}\notag\bigl\langle M_{2}^{n,f}\bigr\rangle _{t}&= \sum_{\ell=0}^{\lfloor t/h\rfloor}\bigl\langle \mu_{\ell h}^{n,\unicode{x1D7D9}},f^2\bigr\rangle\biggl(\sigma_{n,\unicode{x1D7D9}}^2+\frac{{\beta}^2_{n,\unicode{x1D7D9}}}{n^2}\biggr)\dfrac{\lambda_{n,{\unicode{x1D7D9}}}}{n}(h\wedge(t-\ell h))\\& \quad+\frac{1}{n}\sum_{\ell=0}^{\lfloor t/h\rfloor}\bigl\langle\mu_{\ell h}^{n,\unicode{x1D7D9}},1\bigr\rangle \,O((h\wedge(t-\ell h))^2),\end{align}

(3.17) \begin{align}\bigl\langle M_{3}^{n,f}\bigr\rangle _{t}\notag&=\frac{1}{n}\sum_{\ell=0}^{\lfloor t/h\rfloor}\bigl\langle \mu_{\ell h}^{n,\unicode{x1D7D9}},f^2({\cdot})\eta\bigl(\cdot,\mu_{\ell h}^{n,\unicode{x1D7D9}},\mu_{\ell h}^{n,\unicode{x1D7DA}}\bigr)\bigr\rangle (h\wedge(t-\ell h))\\&\quad+\frac{1}{n}\sum_{\ell=0}^{\lfloor t/h\rfloor}\bigl\langle\mu_{\ell h}^{n,\unicode{x1D7D9}},1\bigr\rangle \,O((h\wedge(t-\ell h))^2).\end{align}

Proof. Notice that (3.15) is obtained directly by (3.4). Moreover, by (3.1) one can see that

\begin{align*}\mathbb{E}\bigl[\bigl(\Delta_{k\sim\ell h}^{n, \unicode{x1D7D9},h\wedge(t-\ell h)}(\,f)\bigr)^2\mid \mathcal{F}_{\ell h}\bigr]& = \biggl(\sigma_{n,\unicode{x1D7D9}}^2+\frac{{\beta}^2_{n,\unicode{x1D7D9}}}{n^2}\biggr)f^2\bigl(X_{k\sim \ell h}^{n,\unicode{x1D7D9}}(\ell h)\bigr){\lambda_{n,\unicode{x1D7D9}}}(h\wedge(t-\ell h)) \\& \quad +{\textrm{O}}((h\wedge(t-\ell h))^2),\end{align*}

where ${\textrm{O}}((h\wedge(t-\ell h))^2)\leq Kh^2$ holds for any given h and n. Applying Lemma 8.12 of [Reference Xiong21], the quadratic variations of $M_{2}^{n,f}$ are as follows:

\begin{align*}&\bigl\langle M_{2}^{n,f}\bigr\rangle _{t} = \sum_{\ell=0}^{\lfloor t/h\rfloor}\frac{1}{n^2}\mathbb{E}\Biggl\{\Biggl[\sum_{k\sim \ell h}\big(\Delta_{k\sim\ell h}^{n, \unicode{x1D7D9}, h\wedge(t-\ell h)}(\,f)-\mathbb{E}\bigl(\Delta_{k\sim\ell h}^{n, \unicode{x1D7D9}, h\wedge(t-\ell h)}(\,f)\mid \mathcal{F}_{\ell h}\bigr)\bigr)\Biggr]^2\Bigm|\mathcal{F}_{\ell h}\Biggr\} \\& \quad = \sum_{\ell=0}^{\lfloor t/h\rfloor}\frac{1}{n^2}\sum_{k\sim \ell h}\biggl[f^2\bigl(X_{k\sim \ell h}^{n,\unicode{x1D7D9}}(\ell h)\bigr) \biggl(\sigma_{n,\unicode{x1D7D9}}^2+\frac{{\beta}^2_{n,\unicode{x1D7D9}}}{n^2}\biggr)\lambda_{n,\unicode{x1D7D9}}(h\wedge(t-\ell h))+{\textrm{O}}((h\wedge(t-\ell h))^2)\biggr] \\& \quad = \sum_{\ell=0}^{\lfloor t/h\rfloor}\bigl\langle \mu_{\ell h}^{n,\unicode{x1D7D9}},f^2\bigr\rangle\biggl(\sigma_{n,\unicode{x1D7D9}}^2+\frac{{\beta}^2_{n,\unicode{x1D7D9}}}{n^2}\biggr)\frac{\lambda_{n,{\unicode{x1D7D9}}}}{n}(h\wedge(t-\ell h))+\frac{1}{n}\sum_{\ell=0}^{\lfloor t/h\rfloor}\bigl\langle \mu_{\ell h}^{n,\unicode{x1D7D9}},1\bigr\rangle \,{\textrm{O}}((h\wedge(t-\ell h))^2).\end{align*}

Further, by (3.2) we have

\begin{align*}\mathbb{E}\bigl[\bigl(D_{k\sim\ell h}^{n, \unicode{x1D7D9},h\wedge(t-\ell h)}(\,f)\bigr)^2\mid \mathcal{F}_{\ell h}\bigr]& = f^2\bigl(X_{k\sim \ell h}^{n,\unicode{x1D7D9}}(\ell h)\bigr)\eta\bigl(X_{k\sim \ell h}^{n,\unicode{x1D7D9}}(\ell h),{\mu}_{\ell h}^{n,\unicode{x1D7D9}},{\mu}_{\ell h}^{n,\unicode{x1D7DA}}\bigr)(h\wedge(t-\ell h)) \\[2pt]& \quad +{\textrm{O}}((h\wedge(t-\ell h))^2).\end{align*}

In the same way, the quadratic variation process of $M_3^{n,f}(t)$ is derived as follows:

\begin{align*} \bigl\langle M_{3}^{n,f}\bigr\rangle _{t} & =\sum_{\ell=0}^{\lfloor t/h\rfloor}\frac{1}{n^2}\mathbb{E}\Biggl\{\Biggl[\sum_{k\sim \ell h}\bigl(D_{k\sim\ell h}^{n, \unicode{x1D7D9}, h\wedge(t-\ell h)}(\,f)-\mathbb{E}\bigl(D_{k\sim\ell h}^{n, \unicode{x1D7D9},h\wedge(t-\ell h)}(\,f)\mid \mathcal{F}_{\ell h}\bigr)\bigr)\Biggr]^2\Bigm|\mathcal{F}_{\ell h}\Biggr\}\\[4pt]& =\frac{1}{n}\sum_{\ell=0}^{\lfloor t/h\rfloor}\bigl\langle \mu_{\ell h}^{n,\unicode{x1D7D9}},f^2({\cdot})\eta\bigl(\cdot,\mu_{\ell h}^{n,\unicode{x1D7D9}},\mu_{\ell h}^{n,\unicode{x1D7DA}}\bigr)\bigr\rangle (h\wedge(t-\ell h))+\frac{1}{n}\sum_{\ell=0}^{\lfloor t/h\rfloor}\bigl\langle \mu_{\ell h}^{n,\unicode{x1D7D9}},1\bigr\rangle \,{\textrm{O}}((h\wedge(t-\ell h))^2).\end{align*}

This completes the proof.

As above, the quadratic variation processes for $\hat{M}_{i}^{n,g}(t)$ with $i=1,2,3$ are stated in the proposition below without proof.

Proposition 3.2. The variation processes of $\hat{M}_{i}^{n,g}(t)$ with $i=1,2,3$ are as follows:

(3.18) \begin{align} \bigl\langle \hat{M}_{1}^{n,g}\bigr\rangle _{t}&=\sum_{\ell=0}^{\lfloor t/h\rfloor}\frac{1}{n^2}\sum_{k\sim \ell h}\int_{\ell h}^{(\ell+1)h\wedge t}\bigl|g^{\prime}\bigl(X_{k\sim \ell h}^{n,\unicode{x1D7DA}}(s)\bigr)\bigr|^{2}d s,\end{align}

(3.19) \begin{align}\notag\bigl\langle \hat{M}_{2}^{n,g}\bigr\rangle _{t}&=\sum_{\ell=0}^{\lfloor t/h\rfloor}\bigl\langle \mu_{\ell h}^{n,\unicode{x1D7DA}},g^2\bigr\rangle \biggl(\sigma_{n,\unicode{x1D7DA}}^2+\frac{{\beta}^2_{n,\unicode{x1D7DA}}}{n^2}\biggr)\frac{\lambda_{n,\unicode{x1D7DA}}}{n}(h\wedge(t-\ell h))\\&\quad+\frac{1}{n}\sum_{\ell=0}^{\lfloor t/h\rfloor}\bigl\langle\mu_{\ell h}^{n,\unicode{x1D7DA}},1\bigr\rangle \,O((h\wedge(t-\ell h))^2),\end{align}

(3.20) \begin{align}\notag\bigl\langle \hat{M}_{3}^{n,g}\bigr\rangle _{t}&=\frac{1}{n}\sum_{\ell=0}^{\lfloor t/h\rfloor}\langle\chi, g\rangle^2\bigl\langle \mu_{\ell h}^{n,\unicode{x1D7D9}}, \eta\bigl(\cdot,\mu_{\ell h}^{n,\unicode{x1D7D9}},\mu_{\ell h}^{n,\unicode{x1D7DA}}\bigr)\bigr\rangle (h\wedge(t-\ell h))\\& \quad+\frac{1}{n}\sum_{\ell=0}^{\lfloor t/h\rfloor}\bigl\langle\mu_{\ell h}^{n,\unicode{x1D7D9}},1\bigr\rangle \,O((h\wedge(t-\ell h))^2).\end{align}

From the construction of the model, one can check that the martingales in Proposition 3.1 and 3.2 are mutually uncorrelated except for $M_{3}^{n,f}$ and $\hat{M}_{3}^{n,g}$ , whose covariation process is demonstrated in the following proposition.

Proposition 3.3. The covariation process of ${M}_{3}^{n,f}$ and $\hat{M}_{3}^{n,g}$ is

\begin{align*}\bigl\langle{M}_{3}^{n,f},\hat{M}_{3}^{n,g}\bigr\rangle_t&=\sum_{\ell=0}^{\lfloor t/h\rfloor}\frac{1}{n}\langle \chi, g\rangle\big\langle\mu^{n,\unicode{x1D7D9}}_{\ell h},f\eta\bigl(\cdot,\mu_{\ell h}^{n,\unicode{x1D7D9}},\mu^{n,\unicode{x1D7DA}}_{\ell h}\bigr)\bigr\rangle (h\wedge(t-\ell h))\\&\quad+\sum_{\ell=0}^{\lfloor t/h\rfloor}\frac{1}{n}\bigl\langle\mu^{n,\unicode{x1D7D9}}_{\ell h},1\bigr\rangle^2 \,{\textrm{O}}((h\wedge(t-\ell h))^2).\end{align*}

Proof. For simplicity of notation, we set

\begin{align*}I_1(\ell,k)&=\frac{1}{n}\bigl\{D_{k\sim\ell h}^{n, \unicode{x1D7D9},h\wedge(t-\ell h)}(\,f) - \mathbb{E}\bigl[D_{k\sim\ell h}^{n, \unicode{x1D7D9}, h\wedge(t-\ell h)}(\,f)\mid \mathcal{F}_{\ell h}\bigr]\bigr\},\\[3pt] I_2(\ell,k)&=\frac{1}{n}\bigl\{D_{k\sim\ell h}^{n, \unicode{x1D7DA},{h\wedge(t-\ell h)}}(g) - \mathbb{E}\bigl[D_{k\sim\ell h}^{n, \unicode{x1D7DA}, h\wedge(t-\ell h)}(g)\mid \mathcal{F}_{\ell h}\bigr]\bigr\}. \end{align*}

It follows from (3.6) and (3.13) that

\begin{align*}{M}_{3}^{n,f}(t)=\sum_{\ell=0}^{\lfloor t/h\rfloor}\sum_{k\sim \ell h}I_1(\ell,k)\quad\text{and}\quad\hat{M}_{3}^{n,g}(t)=\sum_{\ell=0}^{\lfloor t/h\rfloor}\sum_{k\sim \ell h}I_2(\ell,k),\end{align*}

where $\sum_{k\sim\ell h}$ includes all alive particles in colony $\unicode{x1D7D9}$ at time $\ell h$ . Again applying Lemma 8.12 of [Reference Xiong21], one can check that

\begin{align*}\bigl\langle {M}_{3}^{n,f},\hat{M}_{3}^{n,g}\bigr\rangle _t& = \sum_{\ell=0}^{\lfloor t/h\rfloor}\mathbb{E}\Biggl[\sum_{k_1\sim\ell h,k_2\sim\ell h}I_1(\ell,k)I_2(\ell,k)\Bigm|\mathcal{F}_{\ell h}\Biggr].\end{align*}

For $k_1=k_2=k$ , the kth living particle in colony $\unicode{x1D7D9}$ emigrates to colony $\unicode{x1D7DA}$ , and we have

\begin{align*}& \mathbb{E}[I_1(\ell,k)I_2(\ell,k)\mid\mathcal{F}_{\ell h}] \\& \quad =\frac{\langle \chi, g\rangle }{n^2} \bigl\{\mathbb{E}\bigl[D_{k\sim\ell h}^{n, \unicode{x1D7D9},{h\wedge(t-\ell h)}}(\,f)\mid\mathcal{F}_{\ell h}\bigr]\bigl(1-\mathbb{E}\bigl[{\text{$1{\hskip -2.5 pt}\hbox{I}$}}_{\{ \rho_{k\sim\ell h}\leq(\ell+1)h\wedge t\}}\mid\mathcal{F}_{\ell h}\bigr]\bigr)\bigr\} \\& \quad =\frac{\langle \chi, g\rangle }{n^2} f\bigl(X_{k\sim \ell h}^{n,\unicode{x1D7D9}}(\ell h)\bigr)\eta\bigl(X_{k\sim \ell h}^{n,\unicode{x1D7D9}}(\ell h),\mu^{n,\unicode{x1D7D9}}_{\ell h},\mu^{n,\unicode{x1D7DA}}_{\ell h}\bigr){(h\wedge(t-\ell h))} \\& \quad \quad \times \bigl[1 - \eta\bigl(X_{k\sim \ell h}^{n,\unicode{x1D7D9}}(\ell h),\mu^{n,\unicode{x1D7D9}}_{\ell h},\mu^{n,\unicode{x1D7DA}}_{\ell h}\bigr)(h\wedge (t-\ell h))\bigr]+ \frac{{\textrm{O}} ((h\wedge (t-\ell h))^2)}{n^2} \\& \quad =\frac{\langle \chi, g\rangle }{n^2} f\bigl(X_{k\sim \ell h}^{n,\unicode{x1D7D9}}(\ell h)\bigr)\eta\bigl(X_{k\sim \ell h}^{n,\unicode{x1D7D9}}(\ell h),\mu^{n,\unicode{x1D7D9}}_{\ell h},\mu^{n,\unicode{x1D7DA}}_{\ell h}\bigr){(h\wedge(t-\ell h))}+\frac{{\textrm{O}} ((h\wedge (t-\ell h))^2)}{n^2}.\end{align*}

Further, for the case $k_1\neq k_2$ we obtain that

\begin{align*}\mathbb{E}\big[D_{k_1\sim\ell h}^{n, \unicode{x1D7D9}, h\wedge(t-\ell h)}(\,f)D_{k_2\sim\ell h}^{n, \unicode{x1D7D9},h\wedge (t-\ell h)}(g) \mid \mathcal{F}_{\ell h}\big]=0{,} \end{align*}

since the probability that two particles emigrate from colony $\unicode{x1D7D9}$ to $\unicode{x1D7DA}$ simultaneously is 0, which implies

\begin{align*}& \mathbb{E}\big[I_1(\ell,k_1)I_2(\ell,k_2)\mid \mathcal{F}_{\ell h}\big] \\& \quad =-\frac{\langle \chi, g\rangle }{n^2} \mathbb{E}\big[D_{{k_1}\sim\ell h}^{n, \unicode{x1D7D9},{h\wedge(t-\ell h)}}(\,f)\mid\mathcal{F}_{\ell h}\big]\cdot \mathbb{E}\big[{\text{$1{\hskip -2.5 pt}\hbox{I}$}}_{\{ \rho_{k_2\sim\ell h}\leq(\ell+1)h\wedge t\}}\mid\mathcal{F}_{\ell h}\big] \\& \quad =\frac{{\textrm{O}}((h\wedge (t-\ell h))^2)}{n^2}.\end{align*}

Therefore we obtain

\begin{align*}& \bigl\langle {M}_{3}^{n,f},\hat{M}_{3}^{n,g}\bigr\rangle _t \\& \quad=\sum_{\ell=0}^{\lfloor t/h\rfloor}\sum_{k\sim\ell h}\frac{\langle \chi, g\rangle }{n^2}f\bigl(X_{k\sim \ell h}^{n,\unicode{x1D7D9}}(\ell h)\bigr)\eta\bigl(X_{k\sim \ell h}^{n,\unicode{x1D7D9}}(\ell h),\mu^{n,\unicode{x1D7D9}}_{\ell h},\mu^{n,\unicode{x1D7DA}}_{\ell h}\bigr)(h\wedge(t-\ell h))\\& \quad \quad+\sum_{\ell=0}^{\lfloor t/h\rfloor}\sum_{\substack{k_1\sim \ell h, k_2\sim \ell h}}\frac{1}{{n^2}} \,{\textrm{O}}((h\wedge (t-\ell h))^2)\\& \quad =\sum_{\ell=0}^{\lfloor t/h\rfloor}\frac{\langle \chi, g\rangle }{n}\bigl\langle \mu^{n,\unicode{x1D7D9}}_{\ell h},f\eta\bigl(\cdot,\mu^{n,\unicode{x1D7D9}}_{\ell h},\mu^{n,\unicode{x1D7DA}}_{\ell h}\bigr)\bigr\rangle (h\wedge(t-\ell h))\\& \quad \quad+\sum_{\ell=0}^{\lfloor t/h\rfloor}\frac{1}{n}\bigl\langle\mu^{n,\unicode{x1D7D9}}_{\ell h},1\bigr\rangle^2 \,{\textrm{O}}((h\wedge(t-\ell h))^2).\end{align*}

The result follows.

In the following we make some estimations (Lemmas 3.1–3.4) and then prove the tightness of the empirical measure for the branching particle systems.

Lemma 3.1. Assume that

\begin{align*} \sup_{n}\mathbb{E}\bigl[\bigl\langle\mu_0^{n, \unicode{x1D7D9}}, 1\bigr\rangle^{2p} + \bigl\langle\mu_0^{n, \unicode{x1D7DA}}, 1\bigr\rangle^{2p}\bigr] < \infty \quad for\ some\ p \ge 1. \end{align*}

For any $T>0$ , there exists a constant $K = K(p, T)$ such that

\begin{align*}\sup_n\mathbb{E}\biggl[\sup_{t\leq T}\bigl(\bigl\langle\mu_t^{n,\unicode{x1D7D9}},1\bigr\rangle^{2p} +\bigl\langle\mu_t^{n,\unicode{x1D7DA}},1\bigr\rangle^{2p}\bigr)\biggr]\,<\,K.\end{align*}

Proof. Replacing f and g with 1 in (3.9) and (3.10), by Doob’s inequality one can check that

(3.20) \begin{align}\mathbb{E}\biggl[\sup_{t\leq T}\bigl\langle\mu_{t}^{n,\unicode{x1D7D9}},1\bigr\rangle^{2p}\biggr]& \leq {K_1 + K_1}\sum_{i = 1}^3\mathbb{E}\bigl[|M_{i}^{n,1}(T)|^{2p}\bigr]+ {K_1}\mathbb{E}\biggl[\sup_{t\leq T}|A^{n,1}(t)|^{2p}\biggr] \notag \\& = {K_1 + K_1}\sum_{i = 1}^3\mathbb{E}\bigl[\bigl\langle M_{i}^{n,1}\bigr\rangle_T^{p}\bigr]+ {K_1}\mathbb{E}\biggl[\sup_{t\leq T} |A^{n,1}(t)|^{2p}\biggr]\end{align}

and similarly

\begin{align*}\mathbb{E}\biggl[\sup_{t\leq T}\bigl\langle\mu_{t}^{n,\unicode{x1D7DA}},1\bigr\rangle^{2p}\biggr]& \leq{K_1} + {K_1}\sum_{i = 1}^3\mathbb{E}\bigl[\bigl\langle\hat{M}_{i}^{n,1}\bigr\rangle_T^{p}\bigr]+ {K_1}\mathbb{E}\biggl[\sup_{t\leq T}|\hat{A}^{n,1}(t)|^{2p}\biggr],\end{align*}

where $K_1$ is a constant depending on p. By 3.7, we have

\begin{align*}& \mathbb{E}\biggl[\sup_{t\leq T}|A^{n,1}(t)|^{2p}\biggr]\\& \quad \le\mathbb{E}\Biggl[\sup_{t\leq T}\Biggl|\sum_{\ell=0}^{\lfloor t/h\rfloor}\frac{{\beta}_{n,\unicode{x1D7D9}}\lambda_{n,\unicode{x1D7D9}}}{n}\bigl\langle \mu_{\ell h}^{n,\unicode{x1D7D9}},1\bigr\rangle (h\wedge(t-\ell h))-\sum_{\ell=0}^{\lfloor t/h\rfloor}\bigl\langle \mu_{\ell h}^{n,\unicode{x1D7D9}},\eta\bigl(\cdot,\mu_{\ell h}^{n,\unicode{x1D7D9}},\mu_{\ell h}^{n,\unicode{x1D7DA}}\bigr) \bigr\rangle (h\wedge(t-\ell h)) \\& \quad \quad+\sum_{\ell=0}^{\lfloor t/h\rfloor}\bigl\langle \mu_{\ell h}^{n,\unicode{x1D7D9}},1\bigr\rangle \,{\textrm{O}}((h\wedge(t-\ell h))^2)+\sum_{\ell=0}^{\lfloor t/h\rfloor}\frac{1}{n}\sum_{k\sim \ell h}\Upsilon_{k\sim \ell h}^{n,\unicode{x1D7D9},{h\wedge( t-\ell h)}}(\,f)\Biggr|^{2p}\Biggr]\\& \quad \leq\mathbb{E}\Biggl[\sup_{t\leq T}\Biggl|\sum_{\ell=0}^{\lfloor t/h\rfloor}\frac{{\beta}_{n,\unicode{x1D7D9}}\lambda_{n,\unicode{x1D7D9}}}{n}\bigl\langle \mu_{\ell h}^{n,\unicode{x1D7D9}},1\bigr\rangle (h\wedge(t-\ell h))-\sum_{\ell=0}^{\lfloor t/h\rfloor}\bigl\langle \mu_{\ell h}^{n,\unicode{x1D7D9}},\eta\bigl(\cdot,\mu_{\ell h}^{n,\unicode{x1D7D9}},\mu_{\ell h}^{n,\unicode{x1D7DA}}\bigr) \bigr\rangle (h\wedge(t-\ell h)) \\& \quad \quad+\sum_{\ell=0}^{\lfloor t/h\rfloor}\bigl\langle \mu_{\ell h}^{n,\unicode{x1D7D9}},1\bigr\rangle \,{\textrm{O}}((h\wedge(t-\ell h))^2) +\sum_{\ell=0}^{\lfloor t/h\rfloor}\frac{1}{n}\sum_{k\sim \ell h}\mathbb{E}\bigl[\Upsilon_{k\sim \ell h}^{n,\unicode{x1D7D9},{h\wedge( t-\ell h)}}(\,f)\mid \mathcal{F}_{\ell h}\bigr]\\ & \quad \quad+ \sum_{\ell=0}^{{\lfloor t/h\rfloor}}\frac{1}{n}\sum_{k\sim \ell h}\bigl(\Upsilon_{k\sim \ell h}^{n,\unicode{x1D7D9},{h\wedge( t-\ell h)}}(\,f)-\mathbb{E}\bigl[\Upsilon_{k\sim \ell h}^{n,\unicode{x1D7D9},{h\wedge( t-\ell h)}}(\,f)\mid\mathcal{F}_{\ell h}\bigr]\bigr)\Biggr|^{2p}\Biggr],\end{align*}

where $\Upsilon_{k\sim \ell h}^{n,\unicode{x1D7D9},{h\wedge( t-\ell h)}}(\,f)$ is expressed as the sum of two parts. The upper bound of the first part is given by (3.8). The second part can be treated in the same way as other martingales. Recall that ${\lambda_{n,\unicode{x1D7D9}}}/{n}\rightarrow\lambda_{\unicode{x1D7D9}}$ and $\beta_{n,\unicode{x1D7D9}}\rightarrow\beta_{\unicode{x1D7D9}}$ as $n\rightarrow\infty$ ; $\eta$ is bounded and the step size h is sufficiently small. Moreover,

\begin{align*}\sum_{\ell = 0}^{\lfloor T/h\rfloor}\bigl\langle\mu_{\ell h}^{n,\unicode{x1D7D9}},1\bigr\rangle(h\wedge(t-\ell h))\end{align*}

is bounded by

\begin{align*}\int_0^T\sup_{t\leq s}\bigl\langle\mu_{t}^{n,\unicode{x1D7D9}},1\bigr\rangle \, {\textrm{d}} s.\end{align*}

By Hölder’s inequality, there is a constant $K_2$ depending on p and T such that

\begin{align*}\mathbb{E}\biggl[\sup_{t\leq T}|A^{n,1}(t)|^{2p}\biggr] \le K_2 \int_0^T \mathbb{E}\biggl[\sup_{t\leq s}\bigl\langle\mu_t^{n,\unicode{x1D7D9}},1\bigr\rangle^{2p}\biggr] \, {\textrm{d}} s.\end{align*}

The other terms of (3.20) can be treated in the same way. Then we have

\begin{align*}\mathbb{E}\biggl[\sup_{t\leq T}\bigl\langle\mu_{t}^{n,\unicode{x1D7D9}},1\bigr\rangle^{2p}\biggr]& \leq {K_1 + K_2} \int_0^T \mathbb{E}\biggl[\sup_{t\leq s}\bigl\langle\mu_t^{n,\unicode{x1D7D9}},1\bigr\rangle^p\biggr] \, {\textrm{d}} s + {K_2} \int_0^T \mathbb{E}\biggl[\sup_{t\leq s}\bigl\langle\mu_t^{n,\unicode{x1D7D9}},1\bigr\rangle^{2p}\biggr] \, {\textrm{d}} s\\& \leq {(K_1 + K_2T)} + {K_2} \int_0^T \mathbb{E}\biggl[\sup_{t\leq s}\bigl\langle\mu_t^{n,\unicode{x1D7D9}},1\bigr\rangle^{2p}\biggr] \, {\textrm{d}} s,\end{align*}

where the last inequality follows from $x \le x^2 + 1$ for any $x \ge 0.$ Let $K_3 = K_1 + K_2T$ , which depends on p and T. Similarly, by (3.14), (3.18)–(3.20), one can see that

\begin{align*}\mathbb{E}\biggl[\sup_{t\leq T}\bigl\langle\mu_{t}^{n,\unicode{x1D7DA}},1\bigr\rangle^{2p}\biggr]\leq {K_3} + {K_2} \int_0^T\mathbb{E}\biggl[\sup_{t\leq s}\bigl(\bigl\langle\mu_t^{n,\unicode{x1D7DA}},1\bigr\rangle^{2p} + \bigl\langle\mu_t^{n,\unicode{x1D7D9}},1\bigr\rangle^{2p}\bigr)\biggr] \, {\textrm{d}} s.\end{align*}

As a consequence, we obtain

\begin{align*}\mathbb{E}\biggl[\sup_{t\leq T}\bigl(\langle\mu_{t}^{n,\unicode{x1D7D9}},1\bigr\rangle^{2p} + \bigl\langle\mu_{t}^{n,\unicode{x1D7DA}},1\bigr\rangle^{2p}\bigr)\biggr] \le{2K_3} + {2K_2}\int_0^T \mathbb{E}\biggl[\sup_{t\leq s}\bigl(\bigl\langle\mu_t^{n,\unicode{x1D7D9}},1\bigr\rangle^{2p} + \bigl\langle\mu_t^{n,\unicode{x1D7DA}},1\bigr\rangle^{2p}\bigr)\biggr] \, {\textrm{d}} s.\end{align*}

The result follows from Gronwall’s inequality.

Lemma 3.2. Under the condition of Lemma 3.1, for any $0\leq s\leq t\leq T$ , $f,g\in {C_b^2 (\mathbb{R})}$ , and $i=1,2,3$ , we have

\begin{align*}\mathbb{E}\bigl[\bigl|\bigl\langle{M_i}^{n,f}\bigr\rangle_t-\bigl\langle{M_i}^{n,f}\bigr\rangle_s\bigr|^p + \bigl|\bigl\langle{\hat{M}_i}^{n,g}\bigr\rangle_t-\bigl\langle{\hat{M}_i}^{n,g}\bigr\rangle_s\bigr|^p\bigr]\leq K|t - s|^p.\end{align*}

Proof. We start with the case of $i=2$ . It follows from (3.16) and Hölder’s inequality that

\begin{align*}&\mathbb{E}\bigl[\bigl|\bigl\langle{M_2}^{n,f}\bigr\rangle_t-\bigl\langle{M_2}^{n,f}\bigr\rangle_s\bigr|^p\bigr]\\&\quad=\mathbb{E}\Biggl|\bigl\langle \mu_{s}^{n,\unicode{x1D7D9}},f^2\bigr\rangle \biggl(\sigma_{n,\unicode{x1D7D9}}^2+\frac{{\beta}^2_{n,\unicode{x1D7D9}}}{n^2}\biggr)\lambda_{n,{\unicode{x1D7D9}}}((\lfloor s/h\rfloor+1)h-s)\\&\quad\quad+\sum_{\ell=\lfloor s/h\rfloor+1}^{\lfloor t/h\rfloor}\bigl\langle \mu_{\ell h}^{n,\unicode{x1D7D9}},f^2\bigr\rangle{\biggl(\sigma_{n,\unicode{x1D7D9}}^2+\frac{{\beta}^2_{n,\unicode{x1D7D9}}}{n^2}\biggr)}\frac{\lambda_{n,{\unicode{x1D7D9}}}}{n}{(h\wedge(t-\ell h))}\\&\quad\quad+ \frac{1}{n}\sum_{\ell= \lfloor s/h\rfloor+1}^{\lfloor t/h\rfloor}\bigl\langle \mu_{\ell h}^{n,\unicode{x1D7D9}},1\bigr\rangle \,{\textrm{O}}((h\wedge(t-\ell h))^2)+\frac{1}{n}\bigl\langle \mu_{s}^{n,\unicode{x1D7D9}},1\bigr\rangle \,{\textrm{O}}(((\lfloor s/h\rfloor+1)h-s)^2)\Biggr|^p\\&\quad\le K \mathbb{E}\biggl[\int_s^t \sup_{s\leq u\leq r}\bigl\langle\mu_{u}^{n, \unicode{x1D7D9}}, 1\bigr\rangle {\textrm{d}} r\biggr]^p \\&\quad \le K|t - s|^p \mathbb{E}\biggl[\biggl(\sup_{s \le r \le t} \bigl\langle \mu_{r}^{n, \unicode{x1D7D9}}, 1\bigr\rangle\biggr)^p\biggr]\\&\quad\le K|t - s|^p \biggl[\mathbb{E}\biggl[\biggl(\sup_{s \le r \le t} \bigl\langle \mu_{r}^{n, \unicode{x1D7D9}}, 1\bigr\rangle\biggr)^{2p}\biggr]\biggr]^{1/2}\\&\quad= K|t - s|^p \biggl[\mathbb{E}\biggl[ \sup_{s \le r \le t} \bigl\langle \mu_{r}^{n, \unicode{x1D7D9}}, 1\bigr\rangle^{2p}\biggr]\biggr]^{1/2}\\&\quad\le K|t - s|^p,\end{align*}

where the last inequality follows from Lemma 3.1. As above, a similar estimation can be carried out for $\langle{{\hat{M}}_2}^{n,g}\rangle_t$ , which implies the result for $i=2$ . For $i=1,3$ we can derive the results analogously.

By the same approach as for Lemma 3.2, the following lemmas are presented without proof.

Lemma 3.3. Under the conditions of Lemma 3.1, for any $0\leq s\leq t\leq T$ , $f,g\in {C_b^2 (\mathbb{R})}$ , we have

\begin{align*}\mathbb{E}\bigl[\bigl|\bigl\langle{M_3}^{n,f},{\hat{M}_3}^{n,g}\bigr\rangle_t-\bigl\langle{M_3}^{n,f},{\hat{M}_3}^{n,g}\bigr\rangle_s\bigr|^p \bigr]\leq K|t - s|^p.\end{align*}

Lemma 3.4. Under the conditions of Lemma 3.1, for any $0\leq s\leq t\leq T$ , $f,g\in {C_b^2 (\mathbb{R})}$ , we have

\begin{align*}\mathbb{E}\bigl[|A^{n,f}(t)-A^{n,f}(s)|^{2p}+|\hat{A}^{n,g}(t)-\hat{A}^{n,g}(s)|^{2p}\bigr]\leq K|t - s|^{2p}.\end{align*}

Theorem 3.1. Assume that

\begin{align*} \sup_{n}\mathbb{E}\bigl[\bigl\langle\mu_0^{n, \unicode{x1D7D9}}, 1\bigr\rangle^{2p} + \bigl\langle\mu_0^{n, \unicode{x1D7DA}}, 1\bigr\rangle^{2p}\bigr] < \infty\quad for\ some\ p \ge 1. \end{align*}

The sequence

\begin{align*} \bigl\{\bigl(\mu_t^{n,\unicode{x1D7D9}},\mu_t^{n,\unicode{x1D7DA}}\bigr)_{t\in[0,T]}\,:\, n \ge 1\bigr\} \end{align*}

is tight in $D([0,T],M_F(\mathbb{R})^2)$ . Furthermore, the limit $\bigl(\mu_t^{\unicode{x1D7D9}},\mu_t^{\unicode{x1D7DA}}\bigr)_{t\geq 0}$ is a solution to the MP (1.3, 1.4) with $b_{\unicode{x1D7D9}}=\beta_{\unicode{x1D7D9}}\lambda_{\unicode{x1D7D9}}$ , $b_{\unicode{x1D7DA}}=\beta_{\unicode{x1D7DA}}\lambda_{\unicode{x1D7DA}}$ , $\gamma_{\unicode{x1D7D9}}=\sigma^2_{\unicode{x1D7D9}}\lambda_{\unicode{x1D7D9}}$ , and $\gamma_{\unicode{x1D7DA}}=\sigma^2_{\unicode{x1D7DA}}\lambda_{\unicode{x1D7DA}}.$

Proof. Suppose that $\{h_n\,:\, n \ge 1\}$ is a sequence in $(0, + \infty)$ satisfying $\lim_{n \rightarrow +\infty}h_n=0$ . For simplicity, we use $M_i^{n,f}$ and $\hat{M}_i^{n,g}$ to denote the martingales defined by (3.4)–(3.6) and (3.11)–(3.13) with respect to $h = h_n.$ In fact, by Jakubowski’s criterion (see e.g. Dawson [Reference Dawson2, Theorem 3.6.4]), the tightness of

\begin{align*} \bigl\{\bigl(\mu_t^{n,\unicode{x1D7D9}},\mu_t^{n,\unicode{x1D7DA}}\bigr)_{t\in[0,T]}\,:\, n \ge 1\bigr\} \end{align*}

in $D([0,T],M_F(\mathbb{R})^2)$ is obtained by the tightness of

\begin{align*} \bigl\{\bigl(\bigl\langle\mu_t^{n,\unicode{x1D7D9}},f\bigr\rangle,\bigl\langle\mu_t^{n,\unicode{x1D7DA}},g\bigr\rangle\bigr)_{t\in[0,T]}\,:\, n \ge 1\bigr\} \end{align*}

in $D([0,T],\mathbb{R}^2)$ for any $f,g\in {C_b^2 (\mathbb{R})}$ .

Denote

\begin{align*}M^{n,f}(t)=\sum_{i=1}^3 M^{n,f}_i(t)\quad\text{and}\quad \hat{M}^{n,g}(t)=\sum_{i=1}^3 \hat{M}^{n,g}_i(t).\end{align*}

Then

\begin{align*}\langle M^{n,f}\rangle_{t}=\sum_{i=1}^3\langle M^{n,f}_i\rangle_{t},\quad\langle \hat{M}^{n,g}\rangle_{t}=\sum_{i=1}^3\langle \hat{M}^{n,g}_i\rangle_{t}\quad\text{and}\quad\langle {M}^{n,f}, \hat{M}^{n,g}\rangle_{t}=\bigl\langle {M}_3^{n,f}, \hat{M}_3^{n,g}\bigr\rangle_{t}.\end{align*}

For any $ 0\leq s\leq t\leq T$ and $p \geq 1,$ by Hölder’s inequality and Lemmas 3.2 and 3.4, we have

\begin{align*} &\mathbb{E}\bigl[\bigl|\bigl\langle\mu_t^{n,\unicode{x1D7D9}}, f\bigr\rangle - \bigl\langle\mu_s^{n,\unicode{x1D7D9}}, f\bigr\rangle\bigr|^{2p}\bigr] \\ &\quad \leq K \mathbb{E}\bigl[|A^{n,f}(t)-A^{n,f}(s)|^{2p}\bigr] + K \mathbb{E}\bigl[|\langle{M}^{n,f}\rangle_t-\langle{M}^{n,f}\rangle_s|^p\bigr]\\&\quad \leq K \mathbb{E}\bigl[|A^{n,f}(t)-A^{n,f}(s)|^{2p}\bigr] + K \mathbb{E}\bigl[|\langle{M_2}^{n,f}\rangle_t-\langle{M_2}^{n,f}\rangle_s|^p\bigr]\\&\quad \quad + K \mathbb{E}\bigl[|\langle{M_1}^{n,f}\rangle_t-\langle{M_1}^{n,f}\rangle_s|^p\bigr] + K \mathbb{E}\bigl[|\langle{M_3}^{n,f}\rangle_t-\langle{M_3}^{n,f}\rangle_s|^p\bigr]\\&\quad \leq K|t - s|^p + K|t - s|^{2p}.\end{align*}

On the other hand, it follows that

\begin{align*}\mathbb{E}\bigl[|\langle\mu_t^{n,\unicode{x1D7DA}}, g\rangle - \langle\mu_s^{n,\unicode{x1D7DA}}, g\rangle|^{2p}\bigr]& \leq K|t - s|^p + K|t - s|^{2p}.\end{align*}

Combining with Lemma 3.3, the tightness of

\begin{gather*}\bigl(\bigl\langle\mu_t^{n,\unicode{x1D7D9}}, f\bigr\rangle\bigr)_{t\in[0,T]},\quad\bigl(\bigl\langle\mu_t^{n,\unicode{x1D7DA}}, g\bigr\rangle\bigr)_{t\in[0,T]},\quad(A^{n,f}(t))_{t\in[0,T]},\quad(\hat{A}^{n,g}(t))_{t\in[0,T]},\\(\langle M^{n, f}\rangle_t)_{t\in[0,T]},\quad(\langle \hat{M}^{n, g}\rangle_t)_{t\in[0,T]}\quad\text{and}\quad(\langle {M}^{n, f},\hat{M}^{n, g}\rangle_t)_{t\in[0,T]}\end{gather*}

follows from [Reference Jacod and Shiryaev9, Theorem VI.4.1], which implies that $\bigl(\mu_t^{n, \unicode{x1D7D9}}, \mu_t^{n, \unicode{x1D7DA}}\bigr)_{t\in[0,T]}$ is tight. Thus there is a subsequence $\bigl(\mu_t^{n_k, \unicode{x1D7D9}}, \mu_t^{n_k, \unicode{x1D7DA}}\bigr)_{t\in[0,T]}$ converging in law as $k \rightarrow \infty$ . Suppose that $\bigl(\mu_t^{\unicode{x1D7D9}}, \mu_t^{\unicode{x1D7DA}}\bigr)_{t\in[0,T]}$ is the weak limit. For any $f, g \in C_b^2(\mathbb{R})$ , we have

\begin{align*}&\bigl(\langle\mu^{n_k, \unicode{x1D7D9}}, f\rangle, A^{n_k, f}, \langle M^{n_k, f}\rangle, \langle\mu^{n_k, \unicode{x1D7DA}}, g\rangle, \hat{A}^{n_k, g}, \langle \hat{M}^{n_k, g}\rangle,\langle{M}^{n_k, f},\hat{M}^{n_k, g}\rangle\bigr)\\&\quad \longrightarrow\bigl(\langle\mu^{\unicode{x1D7D9}}, f\rangle, A^{f}, \langle M^{f}\rangle, \langle\mu^{\unicode{x1D7DA}}, g\rangle, \hat{A}^{g}, \langle \hat{M}^{g}\rangle, \langle{M}^{f},\hat{M}^{g}\rangle\bigr)\end{align*}

weakly as $k \rightarrow \infty$ , where

\begin{align*}A^{f}(t)& = \frac12\int_{0}^{t}\bigl\langle \mu_s^{\unicode{x1D7D9}},f^{\prime\prime}\bigr\rangle \, {\textrm{d}} s+b_{\unicode{x1D7D9}}\int_{0}^{t}\bigl\langle \mu_s^{\unicode{x1D7D9}},f\bigr\rangle \, {\textrm{d}} s-\int_{0}^{t}\bigl\langle \mu_s^{\unicode{x1D7D9}},\eta\bigl(\cdot,\mu_s^{\unicode{x1D7D9}},\mu_s^{\unicode{x1D7DA}}\bigr)f\bigr\rangle \, {\textrm{d}} s, \\\hat{A}^{g}(t)& = \frac12\int_{0}^{t}\bigl\langle \mu_s^{\unicode{x1D7DA}},g^{\prime\prime}\bigr\rangle \, {\textrm{d}} s+b_{\unicode{x1D7DA}}\int_{0}^{t}\bigl\langle \mu_s^{\unicode{x1D7DA}},g\bigr\rangle \, {\textrm{d}} s+\int_{0}^{t}\langle \chi,g\rangle \bigl\langle \mu_s^{\unicode{x1D7D9}},\eta\bigl(\cdot,\mu_s^{\unicode{x1D7D9}},\mu_s^{\unicode{x1D7DA}}\bigr)\bigr\rangle \, {\textrm{d}} s\end{align*}

with $b_{\unicode{x1D7D9}}=\beta_{\unicode{x1D7D9}}\lambda_{\unicode{x1D7D9}}$ , $b_{\unicode{x1D7DA}}=\beta_{\unicode{x1D7DA}}\lambda_{\unicode{x1D7DA}}$ . For any $f, g \in {C_b^2(\mathbb{R})}$ and $t\in[0,T]$ , we see that

\begin{align*}\langle{M}^{n_k, f},\hat{M}^{n_k, g}\rangle_t=\bigl\langle{M}^{n_k, f}_3,\hat{M}^{n_k, g}_3\bigr\rangle_t\rightarrow 0\quad \text{as $k\rightarrow\infty$}\end{align*}

by Proposition 3.3. Moreover, $\langle{M_i}^{n_k,f}\rangle_t \rightarrow 0$ and $\langle{\hat{M}_i}^{n_k,g}\rangle_t \rightarrow 0$ with $i=1,3$ as $k \rightarrow \infty$ by (3.15), (3.17), (3.18), and (3.20). We can pass to the limit to conclude that $M^f(t)$ and $\hat{M}^g(t)$ are martingales with quadratic variations

\begin{align*}\langle M^{f}\rangle_t=\gamma_{\unicode{x1D7D9}}\int_0^t\bigl\langle\mu_s^{\unicode{x1D7D9}},f^2\bigr\rangle \, {\textrm{d}} s\quad\text{and}\quad\langle{\hat{M}}^{g}\rangle_t=\gamma_{\unicode{x1D7DA}}\int_0^t\bigl\langle \mu_s^{\unicode{x1D7DA}},g^2\bigr\rangle \, {\textrm{d}} s,\end{align*}

where $\gamma_{\unicode{x1D7D9}}=\sigma^2_{\unicode{x1D7D9}}\lambda_{\unicode{x1D7D9}}$ and $\gamma_{\unicode{x1D7DA}}=\sigma^2_{\unicode{x1D7DA}}\lambda_{\unicode{x1D7DA}}$ . Letting $T\rightarrow\infty$ , it implies that $\bigl(\mu_t^{\unicode{x1D7D9}},\,\mu_t^{\unicode{x1D7DA}}\bigr)_{t\geq 0}$ is a solution to the MP (1.3, 1.4). The result follows.

4. Uniqueness of the solution to the martingale problem

In this section we first derive the SPDEs satisfied by the distribution function-valued processes of the mutually interacting superprocesses with migration, and then establish its equivalence with the MP (1.3, 1.4). Moreover, the pathwise uniqueness of the SPDEs is proved by an extended Yamada–Watanabe argument.

4.1. A related system of SPDEs

For any $y\in\mathbb{R}$ , we write

(4.1) \begin{equation}u_t^{\unicode{x1D7D9}}(y)=\mu_t^{\unicode{x1D7D9}}(({-}\infty,y]) \quad\text{and}\quad u_t^{\unicode{x1D7DA}}(y)=\mu_t^{\unicode{x1D7DA}}(({-}\infty,y])\end{equation}

as the distribution function-valued processes for the mutually interacting superprocesses with migration $\bigl(\mu_t^{\unicode{x1D7D9}},\mu_t^{\unicode{x1D7DA}}\bigr)_{t\geq0}$ . For any $x,y\in\mathbb{R}\cup\{\pm\infty\}$ , $\nu_1,\nu_2\in M_{F} (\mathbb{R})$ , and $\eta(\cdot,\cdot,\cdot)\in C_b^{+}(\mathbb{R}\times M_F(\mathbb{R})^2)$ , let

\begin{align*}\dot{\eta}(y,\nu_1,\nu_2)=\int_{-\infty}^{y}\eta(x,\nu_1,\nu_2)\nu_1({\textrm{d}} x).\end{align*}

Let $W^{\mathbb{i}}({\textrm{d}} s\,{\textrm{d}} a)$ be independent space–time white noise random measures on $\mathbb{R}_{+}\times\mathbb{R}$ with intensity ${\textrm{d}} s\,{\textrm{d}} a$ and $\mathbb{i}\in\{\mathbb{1,2}\}$ . We consider the following SPDEs: for any $t\ge 0$ and $y\in\mathbb{R}$ ,

(4.2) \begin{equation}\left\{\begin{aligned}u_t^{\unicode{x1D7D9}}(y)&=u_0^{\unicode{x1D7D9}}(y)+\sqrt{\gamma_{\unicode{x1D7D9}}}\int_0^t\int_{0}^{u_s^{\unicode{x1D7D9}}(y)}W^{\unicode{x1D7D9}}({\textrm{d}} s\, {\textrm{d}} a)+\int_0^t\biggl(\dfrac{\Delta}{2} u_s^{\unicode{x1D7D9}}(y)+b_{\unicode{x1D7D9}}u_s^{\unicode{x1D7D9}}(y)\biggr)\, {\textrm{d}} s \\&\quad-\int_0^t \dot{\eta}\bigl(y,\mu_s^{\unicode{x1D7D9}},\mu_s^{\unicode{x1D7DA}}\bigr)\, {\textrm{d}} s,\\u_t^{\unicode{x1D7DA}}(y)&=u_0^{\unicode{x1D7DA}}(y)+\sqrt{\gamma_{\unicode{x1D7DA}}}\int_0^t\int_{0}^{u_s^{\unicode{x1D7DA}}(y)}W^{\unicode{x1D7DA}}({\textrm{d}} s\, {\textrm{d}} a)+\int_0^t\biggl(\dfrac{\Delta}{2} u_s^{\unicode{x1D7DA}}(y)+b_{\unicode{x1D7DA}}u_s^{\unicode{x1D7DA}}(y)\biggr) \, {\textrm{d}} s\\&\quad +\dot{\chi}(y)\int_0^t\dot{\eta}\bigl(+\infty,\mu_s^{\unicode{x1D7D9}},\mu_s^{\unicode{x1D7DA}}\bigr)\, {\textrm{d}} s,\end{aligned}\right.\end{equation}

where $\dot{\chi}$ is the distribution function of $\chi$ , i.e. $\dot{\chi}(y)=\chi(({-}\infty,y])$ .

Definition 4.1. We say that the SPDEs (4.2) have a weak solution if there exists a $C_{b,m} (\mathbb{R})^2$ -valued process $\bigl(u_t^{\unicode{x1D7D9}},u_t^{\unicode{x1D7DA}}\bigr)_{t\geq0}$ on a stochastic basis such that for any $f,\,g\in C_0^2 (\mathbb{R})$ and $t\ge 0,$ the following holds:

\begin{align*}\left\{\begin{aligned}\bigl\langle u_t^{\unicode{x1D7D9}},f\bigr\rangle_1 & =\bigl\langle u_0^{\unicode{x1D7D9}},f\bigr\rangle_1 +\sqrt{\gamma_{\unicode{x1D7D9}}}\int_0^t\int_{0}^{\infty}\int_{\mathbb{R}}f(y){\text{$1{\hskip -2.5 pt}\hbox{I}$}}_{\{a\leq u_s^{\unicode{x1D7D9}}(y)\}}{\textrm{d}} yW^{\unicode{x1D7D9}}({\textrm{d}} s\, {\textrm{d}} a)\\& \quad +\int_0^t\biggl[ \biggl\langle \dfrac{\triangle}{2} u_s^{\unicode{x1D7D9}},f\biggr\rangle_1 +b_{\unicode{x1D7D9}}\bigl\langle u_s^{\unicode{x1D7D9}},f\bigr\rangle_1 - \bigl\langle\dot{\eta}\bigl(\cdot,\mu_{s}^{\unicode{x1D7D9}},\mu_{s}^{\unicode{x1D7DA}}\bigr),f\bigr\rangle_1\biggr]\, {\textrm{d}} s,\\\bigl\langle u_t^{\unicode{x1D7DA}},g\bigr\rangle_1 & =\bigl\langle u_0^{\unicode{x1D7DA}},g\bigr\rangle_1 +\sqrt{\gamma_{\unicode{x1D7DA}}}\int_0^t\int_{0}^{\infty}\int_{\mathbb{R}}g(y){\text{$1{\hskip -2.5 pt}\hbox{I}$}}_{\{a\leq u_s^{\unicode{x1D7DA}}(y)\}}{\textrm{d}} yW^{\unicode{x1D7DA}}({\textrm{d}} s\, {\textrm{d}} a)\\&\quad+\int_0^t\biggl[ \biggl\langle \frac{\triangle}{2}u_s^{\unicode{x1D7DA}}, g\biggr\rangle_1 +b_{\unicode{x1D7DA}}\bigl\langle u_s^{\unicode{x1D7DA}},g\bigr\rangle_1+\langle\dot\chi,g\rangle_1\dot{\eta}\bigl(+\infty,\mu_{s}^{\unicode{x1D7D9}},\mu_{s}^{\unicode{x1D7DA}}\bigr)\biggr]\, {\textrm{d}} s.\end{aligned}\right.\end{align*}

Proposition 4.1. Suppose that $\bigl(u_t^{\unicode{x1D7D9}},u_t^{\unicode{x1D7DA}}\bigr)_{t\geq0}$ is a weak solution to the system of SPDEs (4.2). Then the corresponding measure-valued process $\bigl(\mu_t^{\unicode{x1D7D9}},\mu_t^{\unicode{x1D7DA}}\bigr)_{t\geq 0}$ is a solution to the MP (1.3, 1.4).

Proof. For a non-decreasing continuous function h on $\mathbb{R}$ , the inverse function is defined as $h^{-1}(a)=\inf\{x\,:\, h(x)>a\}.$ Then, for any $f,g\in C_0^3 (\mathbb{R})$ , we have

\begin{align*}\bigl\langle\mu_t^{\unicode{x1D7D9}},f\bigr\rangle &=-\bigl\langle u_t^{\unicode{x1D7D9}},f^{\prime}\bigr\rangle_1\\&= -\bigl\langle u_0^{\unicode{x1D7D9}},f^{\prime}\bigr\rangle_1 -\sqrt{\gamma_{\unicode{x1D7D9}}}\int_0^t\int_{0}^{\infty}\int_{\mathbb{R}}f^{\prime}(y){\text{$1{\hskip -2.5 pt}\hbox{I}$}}_{\{a\leq u_s^{\unicode{x1D7D9}}(y)\}}{\textrm{d}} yW^{\unicode{x1D7D9}}({\textrm{d}} s\, {\textrm{d}} a)\\&\quad-\int_0^t\biggl( \biggl\langle \dfrac{\triangle}{2} u_s^{\unicode{x1D7D9}},f^{\prime}\biggr\rangle_1 +b_{\unicode{x1D7D9}}\bigl\langle u_s^{\unicode{x1D7D9}},f^{\prime}\bigr\rangle_1-\bigl\langle\dot{\eta}\bigl(\cdot,\mu_s^{\unicode{x1D7D9}},\mu_s^{\unicode{x1D7DA}}\bigr),f^{\prime}\bigr\rangle_1\biggr)\, {\textrm{d}} s\\&=\bigl\langle\mu_0^{\unicode{x1D7D9}},f\bigr\rangle+\sqrt{\gamma_{\unicode{x1D7D9}}}\int_0^t\int_{0}^{\infty}f\bigl(\bigl(u_s^{\unicode{x1D7D9}}\bigr)^{-1}(a)\bigr)W^{\unicode{x1D7D9}}({\textrm{d}} s\, {\textrm{d}} a)\\&\quad+\int_0^t\biggl( \biggl\langle \mu_s^{\unicode{x1D7D9}},\frac12f^{\prime\prime}\biggl\rangle+b_{\unicode{x1D7D9}}\bigl\langle \mu_s^{\unicode{x1D7D9}},f\bigr\rangle -\bigl\langle\mu_s^{\unicode{x1D7D9}},\eta\bigl(\cdot,\mu_s^{\unicode{x1D7D9}},\mu_s^{\unicode{x1D7DA}}\bigr)f\bigr\rangle\biggr)\, {\textrm{d}} s\end{align*}

and

\begin{align*}\bigl\langle\mu_t^{\unicode{x1D7DA}},g\bigr\rangle &=-\bigl\langle u_t^{\unicode{x1D7DA}},g^{\prime}\bigr\rangle_1\\[2pt]&= -\bigl\langle u_0^{\unicode{x1D7DA}},g^{\prime}\bigr\rangle_1 -\sqrt{\gamma_{\unicode{x1D7DA}}}\int_0^t\int_{0}^{\infty}\int_{\mathbb{R}}g^{\prime}(y){\text{$1{\hskip -2.5 pt}\hbox{I}$}}_{\{a\leq u_s^{\unicode{x1D7DA}}(y)\}}{\textrm{d}} yW^{\unicode{x1D7DA}}({\textrm{d}} s\, {\textrm{d}} a)\\[2pt]&\quad-\int_0^t\biggl( \biggl\langle \frac{\triangle}{2} u_s^{\unicode{x1D7DA}},g^{\prime}\biggr\rangle_1 +b_{\unicode{x1D7DA}}\bigl\langle u_s^{\unicode{x1D7DA}},g^{\prime}\bigr\rangle_1 + \langle\dot\chi,g^{\prime} \rangle_1\dot{\eta}\bigl(+\infty,\mu_s^{\unicode{x1D7D9}},\mu_s^{\unicode{x1D7DA}}\bigr)\biggr)\, {\textrm{d}} s\\[2pt]&=\bigl\langle \mu_0^{\unicode{x1D7DA}},g\bigr\rangle+\sqrt{\gamma_{\unicode{x1D7DA}}}\int_0^t\int_{0}^{\infty}g\bigl(\bigl(u_s^{\unicode{x1D7DA}}\bigr)^{-1}(a)\bigr)W^{\unicode{x1D7DA}}({\textrm{d}} s\, {\textrm{d}} a)\\[2pt]&\quad+\int_0^t\biggl( \biggl\langle \mu_s^{\unicode{x1D7DA}},\frac12g^{\prime\prime}\biggr\rangle+b_{\unicode{x1D7DA}}\bigl\langle \mu_s^{\unicode{x1D7DA}},g\bigr\rangle +\langle\chi,g\rangle\dot{\eta}\bigl(+\infty,\mu_s^{\unicode{x1D7D9}},\mu_s^{\unicode{x1D7DA}}\bigr)\biggr)\, {\textrm{d}} s.\end{align*}

Thus $M_t^f$ and $\hat{M}_t^g$ are martingales with quadratic variation processes

\begin{align*}\langle M^f\rangle_t &= \gamma_{\unicode{x1D7D9}}\int_0^t\int_0^{\infty}f^2\bigl(\bigl(u_s^{\unicode{x1D7D9}}\bigr)^{-1}(a)\bigr)\, {\textrm{d}} s\,{\textrm{d}} a\\[2pt]&= \gamma_{\unicode{x1D7D9}}\int_0^t\int_{\mathbb{R}}f^2(y){\textrm{d}} s\, {\textrm{d}} \bigl(u_s^{\unicode{x1D7D9}}(y)\bigr)\\[2pt]&=\gamma_{\unicode{x1D7D9}}\int_0^t\bigl\langle\mu_s^{\unicode{x1D7D9}},f^2\bigr\rangle \, {\textrm{d}} s\end{align*}

and

\begin{align*}\langle \hat{M}^g\rangle_t &=\gamma_{\unicode{x1D7DA}}\int_0^t\int_0^{\infty}g^2\bigl(\bigl(u_s^{\unicode{x1D7DA}}\bigr)^{-1}(a)\bigr)\, {\textrm{d}} s\,{\textrm{d}} a\\[2pt]&=\gamma_{\unicode{x1D7DA}}\int_0^t\int_{\mathbb{R}}g^2(y){\textrm{d}} s\, {\textrm{d}} \bigl(u_s^{\unicode{x1D7DA}}(y)\bigr)\\[2pt]&=\gamma_{\unicode{x1D7DA}}\int_0^t\bigl\langle\mu_s^{\unicode{x1D7DA}},g^2\bigr\rangle \, {\textrm{d}} s.\end{align*}

The independence of $W^{\unicode{x1D7D9}}$ and $W^{\unicode{x1D7DA}}$ leads to $\langle M^f, \hat{M}^g\rangle_t=0.$ Therefore $\bigl(\mu_t^{\unicode{x1D7D9}},\mu_t^{\unicode{x1D7DA}}\bigr)_{t\geq 0}$ is a solution to the MP (1.3, 1.4). This completes the proof.

Proposition 4.2. Suppose that $\bigl(\mu_t^{\unicode{x1D7D9}},\mu_t^{\unicode{x1D7DA}}\bigr)_{t\geq 0}$ is a solution to the MP (1.3, 1.4) and ${\eta}(\cdot,\nu_1,\nu_2)\in C_0^{1} (\mathbb{R})$ for any $\nu_1,\nu_2\in M_F (\mathbb{R})$ . Then the random field $\bigl(u_t^{\unicode{x1D7D9}},u_t^{\unicode{x1D7DA}}\bigr)_{t\geq0}$ defined by (4.1) is a weak solution to the SPDEs (4.2).

Proof. Let $f, g\in C_0^2 (\mathbb{R})$ and set

\begin{align*}\tilde{f}(y)=\int_y^{\infty}f(x){\textrm{d}} x,\quad \tilde{g}(y)=\int_y^{\infty}g(x){\textrm{d}} x.\end{align*}

Then we have

\begin{align*}\bigl\langle u_t^{\unicode{x1D7D9}}, f\bigr\rangle_1& = \bigl\langle\mu_t^{\unicode{x1D7D9}},\tilde{f}\bigr\rangle\\& =\bigl\langle\mu_0^{\unicode{x1D7D9}},\tilde{f}\bigr\rangle+\int_0^t\biggl\langle\mu_s^{\unicode{x1D7D9}},\frac12{\tilde{f}}^{\prime\prime}\biggr\rangle \, {\textrm{d}} s+b_{\unicode{x1D7D9}}\int_0^t\bigl\langle\mu_s^{\unicode{x1D7D9}},\tilde{f}\bigr\rangle \, {\textrm{d}} s\\&\quad -\int_0^t\bigl\langle\mu_s^{\unicode{x1D7D9}},{\eta}\bigl(\cdot,\mu_s^{\unicode{x1D7D9}},\mu_s^{\unicode{x1D7DA}}\bigr)\tilde{f}\bigr\rangle \, {\textrm{d}} s+M_t^{\tilde{f}}\\& = \bigl\langle u_0^{\unicode{x1D7D9}},{f}\bigr\rangle_1+\int_0^t\biggl\langle u_s^{\unicode{x1D7D9}},\frac12{{f}}^{\prime\prime} \biggr\rangle_1 {\textrm{d}} s+b_{\unicode{x1D7D9}}\int_0^t\bigl\langle u_s^{\unicode{x1D7D9}},{f}\bigr\rangle_1 {\textrm{d}} s\\&\quad +\int_0^t\bigl\langle u_s^{\unicode{x1D7D9}},\bigl({\eta}\bigl(\cdot,\mu_s^{\unicode{x1D7D9}},\mu_s^{\unicode{x1D7DA}}\bigr){\tilde{f}}\bigr)^{\prime}\bigr\rangle_1 {\textrm{d}} s+M_t^{\tilde{f}}.\end{align*}

Note that

\begin{align*}\bigl\langle u_s^{\unicode{x1D7D9}},\bigl({\eta}\bigl(\cdot,\mu_s^{\unicode{x1D7D9}},\mu_s^{\unicode{x1D7DA}}\bigr){\tilde{f}}\bigr)^{\prime}\bigr\rangle_1&=\bigl\langle u_s^{\unicode{x1D7D9}},{\eta}^{\prime}\bigl(\cdot,\mu_s^{\unicode{x1D7D9}},\mu_s^{\unicode{x1D7DA}}\bigr)\tilde{f}-{\eta}\bigl(\cdot,\mu_s^{\unicode{x1D7D9}},\mu_s^{\unicode{x1D7DA}}\bigr)f\bigr\rangle_1\\&=\bigl\langle u_s^{\unicode{x1D7D9}}{\eta}^{\prime}\bigl(\cdot,\mu_s^{\unicode{x1D7D9}},\mu_s^{\unicode{x1D7DA}}\bigr),\tilde{f}\bigr\rangle_1 - \bigl\langle u_s^{\unicode{x1D7D9}},{\eta}\bigl(\cdot,\mu_s^{\unicode{x1D7D9}},\mu_s^{\unicode{x1D7DA}}\bigr)f\bigr\rangle_1\\&=\biggl\langle \int_{-\infty}^{\cdot}u_s^{\unicode{x1D7D9}}(x){\eta}^{\prime}\bigl(x,\mu_s^{\unicode{x1D7D9}},\mu_s^{\unicode{x1D7DA}}\bigr){\textrm{d}} x,{f}\biggr\rangle_1-\bigl\langle u_s^{\unicode{x1D7D9}},{\eta}\bigl(\cdot,\mu_s^{\unicode{x1D7D9}},\mu_s^{\unicode{x1D7DA}}\bigr)f\bigr\rangle_1\\&=-\biggl\langle\int_{-\infty}^{\cdot}{\eta}\bigl(x,\mu_s^{\unicode{x1D7D9}},\mu_s^{\unicode{x1D7DA}}\bigr){\textrm{d}} u_s^{\unicode{x1D7D9}}(x),f\biggr\rangle_1\\&=-\bigl\langle\dot{\eta}\bigl(\cdot,\mu_s^{\unicode{x1D7D9}},\mu_s^{\unicode{x1D7DA}}\bigr),f\bigr\rangle_1.\end{align*}

Therefore we continue to have

(4.3) \begin{align} \bigl\langle u_t^{\unicode{x1D7D9}}, f\bigr\rangle_1& = \bigl\langle u_0^{\unicode{x1D7D9}},{f}\bigr\rangle_1+\frac12\int_0^t\bigl\langle u_s^{\unicode{x1D7D9}},{{f}}^{\prime\prime}\bigr\rangle_1 {\textrm{d}} s \notag \\& \quad + b_{\unicode{x1D7D9}}\int_0^t\bigl\langle u_s^{\unicode{x1D7D9}},{f}\bigr\rangle_1 {\textrm{d}} s-\int_0^t\bigl\langle\dot{\eta}\bigl(\cdot,\mu_s^{\unicode{x1D7DA}},\mu_s^{\unicode{x1D7D9}}\bigr),f\bigr\rangle_1 {\textrm{d}} s+M_t^{\tilde{f}}.\end{align}

Similarly, we will have

(4.4) \begin{align} \bigl\langle u_t^{\unicode{x1D7DA}}, g\bigr\rangle_1& = \bigl\langle u_0^{\unicode{x1D7DA}},{g}\bigr\rangle_1 + \frac12\int_0^t\bigl\langle u_s^{\unicode{x1D7DA}},{{g}}^{\prime\prime}\bigr\rangle_1 {\textrm{d}} s+b_{\unicode{x1D7DA}}\int_0^t\bigl\langle u_s^{\unicode{x1D7DA}},{g}\bigr\rangle_1 {\textrm{d}} s \notag \\& \quad +\langle\dot\chi,{g}\rangle_1\int_0^t \dot{\eta}\bigl(+\infty,\mu_s^{\unicode{x1D7D9}},\mu_s^{\unicode{x1D7DA}}\bigr)\, {\textrm{d}} s+\hat{M}_t^{\tilde{g}}.\end{align}

Let $S^{\prime} (\mathbb{R})$ be the space of Schwarz distributions and define the $S^{\prime} (\mathbb{R})$ -valued processes $N_t$ and $\hat{N}_t$ by $N_t(\,f)=M_t^{\tilde{f}}$ and $\hat{N}_t(g)=\hat{M}_t^{\tilde{g}}$ for any $f,g\in C_0^{\infty} (\mathbb{R})$ . Then $N_t$ and $\hat{N}_t$ are $S^{\prime} (\mathbb{R})$ -valued continuous square-integrable martingales with

\begin{align*} \langle N(\,f)\rangle_t &= \langle M^{\tilde{f}}\rangle_t\\ &=\gamma_{\unicode{x1D7D9}}\int_0^t\int_{\mathbb{R}}\tilde{f}^2(y)\mu_s^{\unicode{x1D7D9}}({\textrm{d}} y)\, {\textrm{d}} s\\&= \int_0^t\int_{0}^{\infty}\bigl(\sqrt{\gamma_{\unicode{x1D7D9}}}\bigr)^2\tilde{f}^2\bigl(\bigl(u_s^{\unicode{x1D7D9}}\bigr)^{-1}(a)\bigr){\textrm{d}} a\, {\textrm{d}} s\\&= \int_0^t\int_{0}^{\infty}\biggl(\sqrt{\gamma_{\unicode{x1D7D9}}}\int_{\mathbb{R}}1{\hskip -2.5 pt}\hbox{I}_{\{a\leq u_s^{\unicode{x1D7D9}}(y)\}}f(y){\textrm{d}} y\biggr)^2{\textrm{d}} a\, {\textrm{d}} s\end{align*}

and

\begin{align*} \langle \hat{N}(g)\rangle_t&= \langle \hat{M}^{\tilde{g}}\rangle_t\\ &=\gamma_{\unicode{x1D7DA}}\int_0^t\int_{\mathbb{R}}\tilde{g}^2(y)\mu_s^{\unicode{x1D7DA}}({\textrm{d}} y)\, {\textrm{d}} s\\&= \int_0^t\int_{0}^{\infty}\bigl(\sqrt{\gamma_{\unicode{x1D7DA}}}\bigr)^2\tilde{g}^2\bigl(\bigl(u_s^{\unicode{x1D7DA}}\bigr)^{-1}(a)\bigr){\textrm{d}} a\, {\textrm{d}} s\\&= \int_0^t\int_{0}^{\infty}\biggl(\sqrt{\gamma_{\unicode{x1D7DA}}}\int_{\mathbb{R}}1{\hskip -2.5 pt}\hbox{I}_{\{a\leq u_s^{\unicode{x1D7DA}}(y)\}}g(y){\textrm{d}} y\biggr)^2{\textrm{d}} a\, {\textrm{d}} s.\end{align*}

Moreover, one can see that $\langle{N}(\,f),\hat{N}(g)\rangle_t=\langle{M}^{\tilde{f}},\hat{M}^{\tilde{g}}\rangle_t=0.$ By Theorem III-7 and Corollary III-8 of [Reference Karoui and Méléard10], on some extension of the probability space, one can define two independent Gaussian white noises $W^{\mathbb{i}}({\textrm{d}} s\, {\textrm{d}} a)$ , $\mathbb{i}=\mathbb{1,2}$ on $\mathbb{R}_{+}\times\mathbb{R}$ with intensity ${\textrm{d}} s\,{\textrm{d}} a$ such that

\begin{align*}N_t(\,f)=\int_0^t\int_0^{\infty}\int_{\mathbb{R}}\sqrt{\gamma_{\unicode{x1D7D9}}}1{\hskip -2.5 pt}\hbox{I}_{\{a\leq u_s^{\unicode{x1D7D9}}(x)\}}f(x){\textrm{d}} xW^{\unicode{x1D7D9}}({\textrm{d}} s\,{\textrm{d}} a)\end{align*}

and

\begin{align*}\hat{N}_t(g)=\int_0^t\int_0^{\infty}\int_{\mathbb{R}}\sqrt{\gamma_{\unicode{x1D7DA}}}1{\hskip -2.5 pt}\hbox{I}_{\{a\leq u_s^{\unicode{x1D7DA}}(x)\}}g(x){\textrm{d}} xW^{\unicode{x1D7DA}}({\textrm{d}} s\,{\textrm{d}} a).\end{align*}

Plugging back into (4.3) and (4.4), one can see that $\bigl(u^{\unicode{x1D7D9}}_t,u^{\unicode{x1D7DA}}_t\bigr)_{t\geq0}$ is a solution to (4.2).

4.2. Uniqueness for SPDEs

This subsection is devoted to proving the pathwise uniqueness of the solution to the system of SPDEs (4.2). By Propositions 4.1 and 4.2, the uniqueness of the solution to the MP (1.3, 1.4) is then a direct consequence. We apply the approach of an extended Yamada–Watanabe argument to smooth functions. This is an adaptation to the argument of Proposition 3.1 of [Reference Mytnik and Xiong19].

Before going deep into the uniqueness theorem, we introduce some notation. Let $\Phi\in C_0^{\infty} (\mathbb{R})^{+}$ such that supp $(\Phi)\subseteq({-}1,1)$ and the total integral is 1. Define $\Phi_m(x)=m\Phi(mx)$ . Notice that $\Phi_m\rightarrow\delta_0$ weakly in the sense that for all $f\in C_b (\mathbb{R})$ ,

\begin{align*}\lim_{m \rightarrow \infty}\int_{\mathbb{R}}\Phi_m(x)f(x){\textrm{d}} x = \int_{\mathbb{R}}\delta_0(x)f(x){\textrm{d}} x = f(0).\end{align*}

Let $\{a_k\}$ be a decreasing sequence defined recursively by $a_0=1$ and $\int_{a_k}^{a_{k-1}}z^{-1}{\textrm{d}} z=k$ for $k\geq 1.$ Let $\psi_k$ be non-negative functions in $C_0^{\infty} (\mathbb{R})$ such that supp $(\psi_k)\subseteq(a_k,a_{k-1})$ and $\int_{a_k}^{a_{k-1}}\psi_k(z){\textrm{d}} z=1$ and $\psi_k(z)\leq 2(kz)^{-1}$ for all $z\in\mathbb{R}.$ Let

\begin{align*}\phi_k(z)=\int_0^{|z|}{\textrm{d}} y\int_0^{y}\psi_k(x){\textrm{d}} x \quad\text{for all $ z\in\mathbb{R}$.}\end{align*}

Then $\phi_k(z)\uparrow |z|$ , $|\phi^{\prime}_k(z)|\leq 1$ and $|z|\phi^{\prime\prime}_k (z)\leq 2k^{-1}.$ Let

\begin{align*}J(x)=\int_{\mathbb{R}}\,{\textrm{e}}^{-|x|}\varrho(x-y){\textrm{d}} y,\end{align*}

where $\varrho$ is the mollifier given by $\varrho(x)=C\exp\{-1/(1-x^2)\}1{\hskip -2.5 pt}\hbox{I}_{\{|x|<1\}},$ and C is a constant such that $\int_{\mathbb{R}}\varrho(x){\textrm{d}} x=1$ . Then, for any $m\in{\mathbb{Z}_{+}}$ , there are positive constants $c_m$ and $C_m$ such that

(4.5) \begin{equation}c_m\,{\textrm{e}}^{-|x|}\leq |J^{(m)}(x)|\leq C_m\,{\textrm{e}}^{-|x|}\quad\text{for all $ x\in\mathbb{R}$.}\end{equation}

Suppose that $\bigl(u_t^{\unicode{x1D7D9}},u_t^{\unicode{x1D7DA}}\bigr)_{t\geq 0}$ and $\bigl({\tilde{u}}_t^{\unicode{x1D7D9}},{\tilde{u}}_t^{\unicode{x1D7DA}}\bigr)_{t\geq 0}$ are two weak solutions to the system of SPDEs (4.2) with the same initial values; $\bigl(\mu_t^{\unicode{x1D7D9}}, \mu_t^{\unicode{x1D7DA}}\bigr)_{t \ge 0}$ and $\bigl({\tilde{\mu}}_t^{\unicode{x1D7D9}},{\tilde{\mu}}_t^{\unicode{x1D7DA}}\bigr)_{t \ge 0}$ stand for their corresponding measure-valued processes, namely $u^{\mathbb{i}}_t(y)=\mu^{\mathbb{i}}_t({-}\infty, y]$ and $\tilde{u}^{\mathbb{i}}_t(y)=\tilde{\mu}^{\mathbb{i}}_t({-}\infty, y]$ for $\mathbb{i}=\mathbb{1,2}$ . Let

\begin{align*} v_t^{\mathbb{i}}(y) =u_t^{\mathbb{i}}(y)-{\tilde{u}}_t^{\mathbb{i}}(y)\quad\text{and}\quad\bar{G}_s^{\mathbb{i}}(a,y)=1{\hskip -2.5 pt}\hbox{I}_{\{a\leq u_s^{\mathbb{i}}(y)\}} - 1{\hskip -2.5 pt}\hbox{I}_{\{a\leq {\tilde{u}}_s^{\mathbb{i}}(y)\}}.\end{align*}

Moreover, we denote

(4.6) \begin{align}&I_1^{m,k,\mathbb{i}}=\frac{1}{2}\mathbb{E}\biggl[\int_0^t\int_{\mathbb{R}}\phi^{\prime}_k\bigl(\bigl\langle v_s^{\mathbb{i}},\Phi_m(x-\cdot)\bigr\rangle _1\bigr)\bigl\langle v_s^{\mathbb{i}},\Delta_{y}\Phi_m(x-\cdot)\bigr\rangle _1 J(x){\textrm{d}} x\, {\textrm{d}} s\biggr], \notag \\&I_2^{m,k,\mathbb{i}}=\mathbb{E}\biggl[\int_0^t\int_{\mathbb{R}}\phi^{\prime}_k\bigl(\bigl\langle v_s^{\mathbb{i}},\Phi_m(x-\cdot)\bigr\rangle _1\bigr)\bigl\langle v_s^{\mathbb{i}},\Phi_m(x-\cdot)\bigr\rangle _1 J(x){\textrm{d}} x\, {\textrm{d}} s\biggr],\end{align}

\begin{align}&{I}_3^{m,k,\mathbb{i}}=\mathbb{E}\biggl[\int_0^t\int_{\mathbb{R}}\int_{\mathbb{R}^{+}}\phi^{\prime\prime}_k\bigl(\bigl\langle v_s^{\mathbb{i}},\Phi_m(x-\cdot)\bigr\rangle _1\bigr) \biggl|\int_{\mathbb{R}}\bar{G}_s^{\mathbb{i}}(a,y)\Phi_m(x-y){\textrm{d}} y\biggr|^2 {\textrm{d}} aJ(x){\textrm{d}} x\, {\textrm{d}} s\biggr].\notag \end{align}

Proposition 4.3. For $\mathbb{i}=\mathbb{1,2}$ we have

\begin{align*}\mathbb{E}\biggl[\int_{\mathbb{R}}\phi_k\bigl(\bigl\langle v_t^{\mathbb{i}},\Phi_m(x-\cdot)\bigr\rangle _1\bigr)J(x){\textrm{d}} x\biggr]=I_{1}^{m,k,\mathbb{i}}+b_\mathbb{i}I_{2}^{m,k,\mathbb{i}}+\frac{\gamma_\mathbb{i}}{2}I_3^{m,k,\mathbb{i}}+I_{4}^{m,k,\mathbb{i}},\end{align*}

where $I_{1}^{m,k,\mathbb{i}}$ , $I_{2}^{m,k,\mathbb{i}}$ , and $I_3^{m,k,\mathbb{i}}$ are given by (4.6),

(4.7) \begin{align}{{I}}_4^{m,k,\unicode{x1D7D9}}=-\mathbb{E}\biggl[\int_0^t\int_{\mathbb{R}}\int_{\mathbb{R}}\phi^{\prime}_k\bigl(\bigl\langle v_s^{\unicode{x1D7D9}},\Phi_m(x-\cdot)\bigr\rangle _1\bigr)\Phi_m(x-y) \overline{\xi}_s(y) {\textrm{d}} yJ(x){\textrm{d}} x\, {\textrm{d}} s\biggr]\end{align}

and

(4.8) \begin{align}{{I}}_4^{m,k,\unicode{x1D7DA}}=\mathbb{E}\biggl[\int_0^t\int_{\mathbb{R}}\phi^{\prime}_k\bigl(\bigl\langle v_s^{\unicode{x1D7DA}},\Phi_m(x-\cdot)\bigr\rangle _1\bigr)\bigl\langle \dot{\chi},\Phi_m(x-\cdot)\bigr\rangle _1\overline{\xi}_s(\infty) J(x) {\textrm{d}} x\, {\textrm{d}} s\biggr],\end{align}

with $\overline{\xi}_s({\cdot}) = \dot{\eta}\bigl(\cdot,\mu_s^{\unicode{x1D7D9}},\mu_s^{\unicode{x1D7DA}}\bigr)-\dot{\eta}\bigl(\cdot,\tilde{\mu}_s^{\unicode{x1D7D9}},\tilde{\mu}_s^{\unicode{x1D7DA}}\bigr).$

Proof. It follows from (4.2) that

\begin{align*}v_t^{\unicode{x1D7D9}}(y)& = \sqrt{\gamma_{\unicode{x1D7D9}}}\int_0^t\int_0^{\infty}\bar{G}_s^{\unicode{x1D7D9}}(a,y)W^{\unicode{x1D7D9}}({\textrm{d}} s\,{\textrm{d}} a)+\int_0^t\biggl(\frac{\Delta_{y}}{2}v_s^{\unicode{x1D7D9}}(y)+b_{\unicode{x1D7D9}}v_s^{\unicode{x1D7D9}}(y)-\overline{\xi}_s(y) \biggr)\, {\textrm{d}} s\end{align*}

and

\begin{align*}v_t^{\unicode{x1D7DA}}(y)& = \sqrt{\gamma_{\unicode{x1D7DA}}}\int_0^t\int_0^{\infty}\bar{G}_s^{\unicode{x1D7DA}}(a,y)W^{\unicode{x1D7DA}}({\textrm{d}} s\,{\textrm{d}} a)+\int_0^t\biggl(\frac{\Delta_{y}}{2}v_s^{\unicode{x1D7DA}}(y)+b_{\unicode{x1D7DA}}v_s^{\unicode{x1D7DA}}(y)+\dot{\chi}(y)\overline{\xi}_s(\infty)\biggr)\, {\textrm{d}} s.\end{align*}

Consequently we have

(4.9) \begin{align}\bigl\langle v_t^{\unicode{x1D7D9}},\Phi_m(x-\cdot)\bigr\rangle _1& = \sqrt{\gamma_{\unicode{x1D7D9}}}\int_0^t\int_{\mathbb{R}^{+}}\int_{\mathbb{R}}\bar{G}_s^{\unicode{x1D7D9}}(a,y)\Phi_m(x-y){\textrm{d}} yW^{\unicode{x1D7D9}}({\textrm{d}} s\,{\textrm{d}} a) \notag \\& \quad + b_{\unicode{x1D7D9}}\int_0^t\bigl\langle v_s^{\unicode{x1D7D9}},\Phi_m(x-\cdot)\bigr\rangle _1 {\textrm{d}} s -\int_0^t\int_{\mathbb{R}}\Phi_m(x-y)\overline{\xi}_s(y){\textrm{d}} y\,{\textrm{d}} s \notag \\& \quad + \frac{1}{2}\int_0^t\bigl\langle v_s^{\unicode{x1D7D9}},\Delta_{y}\Phi_m(x-\cdot)\bigr\rangle _1 {\textrm{d}} s\end{align}

and

(4.10) \begin{align}\bigl\langle v_t^{\unicode{x1D7DA}},\Phi_m(x-\cdot)\bigr\rangle _1& = \sqrt{\gamma_{\unicode{x1D7DA}}}\int_0^t\int_{\mathbb{R}^{+}}\int_{\mathbb{R}}\bar{G}_s^{\unicode{x1D7DA}}(a,y)\Phi_m(x-y){\textrm{d}} yW^{\unicode{x1D7DA}}({\textrm{d}} s\,{\textrm{d}} a) \notag \\& \quad +\frac{1}{2}\int_0^t \bigl\langle v_s^{\unicode{x1D7DA}},\Delta_{y}\Phi_m(x-\cdot)\bigr\rangle _1 {\textrm{d}} s+b_{\unicode{x1D7DA}}\int_0^t\bigl\langle v_s^{\unicode{x1D7DA}},\Phi_m(x-\cdot)\bigr\rangle _1 {\textrm{d}} s \notag \\& \quad +\int_0^t\langle \dot{\chi},\Phi_m(x-\cdot)\rangle _1\overline{\xi}_s(\infty) \, {\textrm{d}} s.\end{align}

Applying Itô’s formula to (4.9) and (4.10), we can easily get

\begin{align*}& \phi_k\bigl(\bigl\langle v_t^{\unicode{x1D7D9}},\Phi_m(x-\cdot)\bigr\rangle _1\bigr) \\& \quad =\sqrt{\gamma_{\unicode{x1D7D9}}}\int_0^t\int_{\mathbb{R}^{+}}\phi^{\prime}_k\bigl(\bigl\langle v_s^{\unicode{x1D7D9}},\Phi_m(x-\cdot)\bigr\rangle _1\bigr)\int_{\mathbb{R}}\bar{G}_s^{\unicode{x1D7D9}}(a,y)\Phi_m(x-y){\textrm{d}} yW^{\unicode{x1D7D9}}({\textrm{d}} s\,{\textrm{d}} a)\\& \quad \quad +\int_0^t\phi^{\prime}_k\bigl(\bigl\langle v_s^{\unicode{x1D7D9}},\Phi_m(x-\cdot)\bigr\rangle _1\bigr)\biggl[\frac12\bigl\langle v_s^{\unicode{x1D7D9}}, \Delta_{y}\Phi_m(x-\cdot)\bigr\rangle _1 +b_{\unicode{x1D7D9}} \bigl\langle v_s^{\unicode{x1D7D9}},\Phi_m(x-\cdot)\bigr\rangle _1\biggr] \, {\textrm{d}} s\\& \quad \quad +\frac{\gamma_{\unicode{x1D7D9}}}{2}\int_0^t\int_{\mathbb{R}^{+}}\phi^{\prime\prime}_k\bigl(\bigl\langle v_s^{\unicode{x1D7D9}},\Phi_m(x-\cdot)\bigr\rangle _1\bigr)\biggl|\int_{\mathbb{R}}\bar{G}_s^{\unicode{x1D7D9}}(a,y)\Phi_m(x-y){\textrm{d}} y\biggr|^2{\textrm{d}} a\, {\textrm{d}} s\\& \quad \quad -\int_0^t\int_{\mathbb{R}}\phi^{\prime}_k\bigl(\bigl\langle v_s^{\unicode{x1D7D9}},\Phi_m(x-\cdot)\bigr\rangle _1\bigr)\Phi_m(x-y) \overline{\xi}_s(y) {\textrm{d}} y\,{\textrm{d}} s\end{align*}

and

\begin{align*}& \phi_k\bigl(\bigl\langle v_t^{\unicode{x1D7DA}},\Phi_m(x-\cdot)\bigr\rangle _1\bigr) \\& \quad =\sqrt{\gamma_{\unicode{x1D7DA}}}\int_0^t\int_{\mathbb{R}^{+}}\phi^{\prime}_k\bigl(\bigl\langle v_s^{\unicode{x1D7DA}},\Phi_m(x-\cdot)\bigr\rangle _1\bigr)\int_{\mathbb{R}}\bar{G}_s^{\unicode{x1D7DA}}(a,y)\Phi_m(x-y){\textrm{d}} yW^{\unicode{x1D7DA}}({\textrm{d}} s\,{\textrm{d}} a)\\& \quad \quad+\int_0^t\phi^{\prime}_k\bigl(\bigl\langle v_s^{\unicode{x1D7DA}},\Phi_m(x-\cdot)\bigr\rangle _1\bigr)\biggl[\frac12\bigl\langle v_s^{\unicode{x1D7DA}}, \Delta_{y}\Phi_m(x-\cdot)\bigr\rangle _1 +b_{\unicode{x1D7DA}}\bigl\langle v_s^{\unicode{x1D7DA}},\Phi_m(x-\cdot)\bigr\rangle _1\biggr] \, {\textrm{d}} s\\& \quad \quad +\frac{\gamma_{\unicode{x1D7DA}}}{2}\int_0^t\int_{\mathbb{R}^{+}}\phi^{\prime\prime}_k\bigl(\bigl\langle v_s^{\unicode{x1D7DA}},\Phi_m(x-\cdot)\bigr\rangle _1\bigr)\biggl|\int_{\mathbb{R}}\bar{G}_s^{\unicode{x1D7DA}}(a,y)\Phi_m(x-y){\textrm{d}} y\biggr|^2{\textrm{d}} a\, {\textrm{d}} s\\& \quad \quad +\int_0^t\phi^{\prime}_k\bigl(\bigl\langle v_s^{\unicode{x1D7DA}},\Phi_m(x-\cdot)\bigr\rangle _1\bigr)\langle \dot{\chi},\Phi_m(x-\cdot)\rangle_1\overline{\xi}_s(\infty) \, {\textrm{d}} s.\end{align*}

Taking the expectations of $\bigl\langle \phi_k\bigl(\bigl\langle v_t^{\mathbb{i}},\Phi_m(x-\cdot)\bigr\rangle _1\bigr),\,J(x)\bigr\rangle _1$ with $\mathbb{i}=\mathbb{1,2}$ , we obtain the desired results.

Lemma 4.1. For $\mathbb{i}=\mathbb{1,2}$ we have

\begin{align*}2I_1^{m,k,\mathbb{i}}\leq \mathbb{E}\biggl[\int_0^t\int_{\mathbb{R}}\bigl\langle |v_s^{\mathbb{i}}|,\Phi_m(x-\cdot)\bigr\rangle _1 |J^{\prime\prime}(x)|{\textrm{d}} x\, {\textrm{d}} s\biggr].\end{align*}

Proof. Note that

\begin{align*}2I_1^{m,k,\mathbb{i}}& = \mathbb{E}\biggl[\int_0^t\int_{\mathbb{R}}\phi^{\prime}_k\bigl(\bigl\langle v_s^{\mathbb{i}},\Phi_m(x-\cdot)\bigr\rangle _1\bigr)\bigl\langle v_s^{\mathbb{i}},\Delta_{y}\Phi_m(x-\cdot)\bigr\rangle _1 J(x){\textrm{d}} x\, {\textrm{d}} s\biggr].\end{align*}

Since $\Delta_{y}\Phi_m(x-y)=\Delta_{x}\Phi_m(x-y)$ , we have

\begin{align*}2I_1^{m,k,\mathbb{i}}& = \mathbb{E}\biggl[\int_0^t\int_{\mathbb{R}}\phi^{\prime}_k\bigl(\bigl\langle v_s^{\mathbb{i}},\Phi_m(x-\cdot)\bigr\rangle _1\bigr)\Delta_{x}\bigl\langle v_s^{\mathbb{i}},\Phi_m(x-\cdot)\bigr\rangle _1 J(x){\textrm{d}} x\, {\textrm{d}} s\biggr] \\& = -\mathbb{E}\biggl[\int_0^t\int_{\mathbb{R}}\biggl(\frac{\partial}{\partial x}\bigl\langle v_s^{\mathbb{i}},\Phi_m(x-\cdot)\bigr\rangle _1\biggr)^2\phi^{\prime\prime}_k\bigl(\bigl\langle v_s^{\mathbb{i}},\Phi_m(x-\cdot)\bigr\rangle _1\bigr) J(x){\textrm{d}} x\, {\textrm{d}} s\biggr] \\& \quad -\mathbb{E}\biggl[\int_0^t\int_{\mathbb{R}}\frac{\partial}{\partial x}\bigl\langle v_s^{\mathbb{i}},\Phi_m(x-\cdot)\bigr\rangle _1 \phi^{\prime}_k\bigl(\bigl\langle v_s^{\mathbb{i}},\Phi_m(x-\cdot)\bigr\rangle _1\bigr) J^{\prime}(x){\textrm{d}} x\, {\textrm{d}} s\biggr] \\& \leq-\mathbb{E}\biggl[\int_0^t\int_{\mathbb{R}}\frac{\partial}{\partial x}\bigl\langle v_s^{\mathbb{i}},\Phi_m(x-\cdot)\bigr\rangle _1 \phi^{\prime}_k\bigl(\bigl\langle v_s^{\mathbb{i}},\Phi_m(x-\cdot)\bigr\rangle _1\bigr) J^{\prime}(x){\textrm{d}} x\, {\textrm{d}} s\biggr] \\& = -\mathbb{E}\biggl[\int_0^t\int_{\mathbb{R}}\frac{\partial}{\partial x}\bigl( \phi_k\bigl(\bigl\langle v_s^{\mathbb{i}},\Phi_m(x-\cdot)\bigr\rangle _1\bigr)\bigr)J^{\prime}(x){\textrm{d}} x\, {\textrm{d}} s\biggr] \\& = \mathbb{E}\biggl[\int_0^t\int_{\mathbb{R}}\phi_k\bigl(\bigl\langle v_s^{\mathbb{i}},\Phi_m(x-\cdot)\bigr\rangle _1\bigr)J^{\prime\prime}(x){\textrm{d}} x\, {\textrm{d}} s\biggr],\end{align*}

where the inequality follows from the fact that $\phi^{\prime\prime}_k(z)=\psi_k(|z|)\geq0$ . Use $\phi_k(z)\leq |z|$ to get

\begin{align*}\phi_k\bigl(\bigl\langle v_s^{\mathbb{i}},\Phi_m(x-\cdot)\bigr\rangle _1\bigr)\leq \bigl|\bigl\langle v_s^{\mathbb{i}},\Phi_m(x-\cdot)\bigr\rangle _1\bigr|\leq \bigl\langle |v_s^{\mathbb{i}}|,\Phi_m(x-\cdot)\bigr\rangle _1.\end{align*}

This implies the result.

Lemma 4.2. For $\mathbb{i}=\mathbb{1,2}$ we have

\begin{align*}I_3^{m,k,\mathbb{i}} \le 4\mathbb{E}\biggl[\int_0^t\int_{\mathbb{R}}\phi^{\prime\prime}_k \bigl(\bigl\langle v_s^{\mathbb{i}},\Phi_m(x-\cdot)\bigr\rangle _1\bigr)\bigl\langle|v_s^{\mathbb{i}}|,\Phi_m(x-\cdot)\bigr\rangle _1J(x){\textrm{d}} x\, {\textrm{d}} s\biggr].\end{align*}

Proof. It is easy to see that

\begin{align*}I_3^{m,k,\mathbb{i}}& \leq\mathbb{E}\biggl[\int_0^t\int_{\mathbb{R}}\int_{\mathbb{R}^{+}}\phi^{\prime\prime}_k \bigl(\bigl\langle v_s^{\mathbb{i}},\Phi_m(x-\cdot)\bigr\rangle _1\bigr)\int_{\mathbb{R}}\bigl(\bar{G}_s^{\mathbb{i}}(a,y)\bigr)^2\Phi_m(x-y){\textrm{d}} y\, {\textrm{d}} aJ(x){\textrm{d}} x\, {\textrm{d}} s\biggr] \\& \leq 4\mathbb{E}\biggl[\int_0^t\int_{\mathbb{R}}\phi^{\prime\prime}_k \bigl(\bigl\langle v_s^{\mathbb{i}},\Phi_m(x-\cdot)\bigr\rangle _1\bigr)\int_{\mathbb{R}}|v_s^{\mathbb{i}}(y)|\Phi_m(x-y){\textrm{d}} yJ(x){\textrm{d}} x\, {\textrm{d}} s\biggr].\end{align*}

The result follows.

Theorem 4.1. Assume that there exists a constant K such that

\begin{align*}|{\xi}(x,\nu_1,\nu_2)-{\xi}(x,\tilde{\nu}_1,\tilde{\nu}_2)|\leq K[\rho(\nu_1,\tilde{\nu}_1)+\rho(\nu_2,\tilde{\nu}_2)]\end{align*}

for any $x\in\mathbb{R}\cup\{\pm\infty\}$ and $\nu_i,\tilde{\nu}_i \in M_{F} (\mathbb{R})$ with $i = 1, 2$ . Then pathwise uniqueness holds for the SPDEs (4.2), namely, if (4.2) has two weak solutions defined on the same stochastic basis with the same initial values, then the solutions coincide almost surely.

Proof. Suppose $\bigl(u_t^{\unicode{x1D7D9}},u_t^{\unicode{x1D7DA}}\bigr)_{t\geq0}$ and $\bigl({\tilde{u}}_t^{\unicode{x1D7D9}},{\tilde{u}}_t^{\unicode{x1D7DA}}\bigr)_{t\geq0}$ are two weak solutions on the same stochastic basis with the same initial values. It is sufficient to show that $\bigl(u_t^{\unicode{x1D7D9}},u_t^{\unicode{x1D7DA}}\bigr) = \bigl({\tilde{u}}_t^{\unicode{x1D7D9}},{\tilde{u}}_t^{\unicode{x1D7DA}}\bigr)$ for all $t\geq0$ almost surely. Recall that $v_t^{\mathbb{i}}(y) =u_t^{\mathbb{i}}(y)-{\tilde{u}}_t^{\mathbb{i}}(y)$ . We subsequently estimate the values of $I_{\ell}^{m,k,\mathbb{i}}$ with $\ell=1,2,3,4$ and $\mathbb{i} = \unicode{x1D7D9},\unicode{x1D7DA}.$ Since

\begin{align*}\lim_{m\rightarrow\infty}\bigl\langle v_s^{\mathbb{i}},\Phi_m(x-\cdot)\bigr\rangle _1 =v_s^{\mathbb{i}}(x)\quad\text{and}\quad \lim_{m\rightarrow\infty}\bigl\langle |v_s^{\mathbb{i}}|,\Phi_m(x-\cdot)\bigr\rangle _1=|v_s^\mathbb{i}(x)|\end{align*}

for Lebesgue-a.e. x and any $s\geq 0$ almost surely, by Lemma 4.1 and the dominated convergence theorem, we have

\begin{align*}\limsup_{k,m\rightarrow\infty}2I_1^{m,k,\mathbb{i}}\leq \mathbb{E}\biggl[\int_0^t\int_{\mathbb{R}}|v_s^{\mathbb{i}}(x)|\cdot |J^{\prime\prime}(x)|{\textrm{d}} x\, {\textrm{d}} s\biggr].\end{align*}

By (4.5), there exists a constant K such that

(4.11) \begin{equation}\limsup_{k,m\rightarrow\infty}2I_1^{m,k,\mathbb{i}}\leq K\mathbb{E}\biggl[\int_0^t\int_{\mathbb{R}}|v_s^{\mathbb{i}}(x)|\cdot |J(x)|{\textrm{d}} x\, {\textrm{d}} s\biggr].\end{equation}

Using $|\phi^{\prime}_k(z)|\leq 1$ and the dominated convergence theorem, we can easily get

(4.12) \begin{equation}\limsup_{k,m\rightarrow\infty}\bigl|I_2^{m,k,\mathbb{i}}\bigr|\leq \mathbb{E}\biggl[\int_0^t\int_{\mathbb{R}}|v_s^{\mathbb{i}}(x)|\cdot |J(x)|{\textrm{d}} x\, {\textrm{d}} s\biggr].\end{equation}

Recall that $\phi^{\prime\prime}_k(z)|z|\leq 2k^{-1}$ . By Lemma 4.2 one can prove that

(4.13) \begin{equation}\limsup_{m\rightarrow\infty}I_3^{m,k,\mathbb{i}}\leq K\mathbb{E}\biggl[\int_0^t\int_{\mathbb{R}}\phi^{\prime\prime}_k\bigl(v_s^{\mathbb{i}}(x)\bigr)|v_s^{\mathbb{i}}(x)|J(x){\textrm{d}} x\, {\textrm{d}} s\biggr]={\textrm{O}}(k^{-1}).\end{equation}

Recall that $\chi$ is a finite measure on $\mathbb{R}$ and $|\phi^{\prime}_k(z)|\leq 1$ . By (4.7), (4.8) we have

(4.14) \begin{align}\limsup_{k,m\rightarrow\infty}|I_4^{m,k,\mathbb{i}}| & \leq \mathbb{E}\biggl[\int_0^t\int_{\mathbb{R}}\big[|\overline{\xi}_s(x)| + |\overline{\xi}_s(\infty)|\big]\cdot |J(x)|{\textrm{d}} x\, {\textrm{d}} s\biggr] \notag \\&\leq K\int_0^t\mathbb{E}\big[\rho\bigl({\mu}_s^{\unicode{x1D7D9}},\tilde{\mu}_s^{\unicode{x1D7D9}}\bigr)+\rho\bigl({\mu}_s^{\unicode{x1D7DA}},\tilde{\mu}_s^{\unicode{x1D7DA}}\bigr)\big]\, {\textrm{d}} s \notag \\&\leq K\mathbb{E}\biggl[\int_0^t\int_{\mathbb{R}}\bigl(|v_s^{\unicode{x1D7D9}}(x)|+|v_s^{\unicode{x1D7DA}}(x)|\bigr)J(x){\textrm{d}} x\, {\textrm{d}} s\biggr].\end{align}

By Proposition 4.3 and putting (4.11)–(4.14) together, one can see that

\begin{align*}\mathbb{E}\biggl[\int_{\mathbb{R}}\bigl(|v_t^{\unicode{x1D7D9}}(x)|+|v_t^{\unicode{x1D7DA}}(x)|\bigr)J(x){\textrm{d}} x\biggr]\leq K\mathbb{E}\biggl[\int_0^t\int_{\mathbb{R}}\bigl(|v_s^{\unicode{x1D7D9}}(x)|+|v_s^{\unicode{x1D7DA}}(x)|\bigr)J(x){\textrm{d}} x\, {\textrm{d}} s\biggr].\end{align*}

Then Gronwall’s inequality implies that

\begin{align*}\mathbb{E}\biggl[\int_{\mathbb{R}}\bigl(|v_t^{\unicode{x1D7D9}}(x)|+|v_t^{\unicode{x1D7DA}}(x)|\bigr)J(x){\textrm{d}} x\biggr]=0\end{align*}

for any $t \ge 0$ . Therefore $|v_t^{\unicode{x1D7D9}}(x)|=|v_t^{\unicode{x1D7DA}}(x)|=0$ for any $t\geq 0$ and $x \in \mathbb{R}$ almost surely. Pathwise uniqueness follows.