Mutually Competing Superprocesses in Independent Random Environments

1. Introduction and Main Results

Superprocesses describe the evolution of large populations undergoing spatial motion and random reproduction. They arise as high-density limits of branching particle systems; see Watanabe [1] and Dawson [2]. Systematic accounts of their construction and properties can be found in Etheridge [3] and Li [4]. For a classical superprocess, the branching property gives a log-Laplace characterization of its transition probabilities. When the reproduction, immigration, or death rates depend on the population itself, this characterization is generally no longer available.

An alternative approach to uniqueness uses the stochastic partial differential equation (SPDE) satisfied by the distribution-function-valued process. Xiong [5] established this connection for super-Brownian motion and proved pathwise uniqueness for the corresponding SPDE. Related stochastic equations for measure-valued processes were studied by Dawson and Li [6]. This approach has subsequently been used for superprocesses with interaction and immigration; see Mytnik and Xiong [7] and Xiong and Yang [8]. Ji et al. [9] constructed mutually interacting superprocesses with migration by particle approximation and established uniqueness through the associated system of distribution-function-valued SPDEs.

Superprocesses with competition describe spatial populations whose growth is limited by competition for resources. Etheridge [10] studied continuum and discrete-space models of a locally regulated population and obtained complementary extinction and survival results. Competition between different populations has also been considered. Evans and Perkins [11] studied pairs of superprocesses with interspecific competition, in which collisions between individuals of different populations may cause deaths in both populations. In the nonspatial setting, Lambert [12] introduced the logistic branching process, in which competition within a population produces a quadratic death rate, see also Berestycki et al. [13]. A general class of continuous-state nonlinear branching processes with population-size-dependent branching rates and competition was studied by Li et al. [14]. These models describe the size of a single population subject to competition among its individuals. Two-way competition between continuous-state populations was studied by Xiong et al. [15]. See Chen et al. [16] for related work.

Random environments provide another source of dependence among individuals. Mytnik [17] constructed superprocesses in random environments as limits of branching particle systems and characterized the limits by a well-posed martingale problem. The environment is white in time and may be correlated in space, and it contributes an additional term to the martingale covariance. Mytnik and Xiong [18] derived a conditional log-Laplace representation through a backward SPDE and obtained local-extinction results under conditions on the spatial covariance.

In this paper, we consider a pair of mutually competing super-Brownian motions in independent, spatially homogeneous Brownian environments. The competition rate of each population may depend on the spatial position and on the current measures of both populations. Let $M_F\left(\mathbb{R}\right)$ be the space of finite nonnegative Borel measures on $\mathbb{R}$, equipped with the topology of weak convergence, and put $E={\left(M_F\left(\mathbb{R}\right)\right)}^2$. We write $\langle \nu ,f\rangle ={\int }_{\mathbb{R}}f\left(x\right)\nu \left(\mathrm{d}x\right)$. For $k\ge 0$, let $C_b^k\left(\mathbb{R}\right)$ be the space of bounded functions with bounded continuous derivatives up to order k, and let $C_c^k\left(\mathbb{R}\right)$ denote its subspace of compactly supported functions. Throughout the paper, all filtrations satisfy the usual conditions. The initial state ${\mathbf{\mu }}_0=({\mathbf{\mu }}_0^1,{\mathbf{\mu }}_0^2)\in E$ is deterministic, and ${\gamma }_1,{\gamma }_2$ are positive constants. We consider a continuous E-valued process $\mathbf{\mu }=({\mathbf{\mu }}^1,{\mathbf{\mu }}^2)$ such that, for every $f,g\in C_c^2\left(\mathbb{R}\right)$,

$$\left\{\begin{array}{c}{\displaystyle M_t^{1,f}=\langle {\mu }_t^1,f\rangle -\langle {\mu }_0^1,f\rangle -{\displaystyle \frac{1}{2}}{\int }_0^t\langle {\mu }_s^1,f^{\prime \prime }\rangle \mathrm{d}s+{\int }_0^t\langle {\mu }_s^1,{\eta }_1(·,{\mu }_s^1,{\mu }_s^2)f\rangle \mathrm{d}s,}\hfill \\ {\displaystyle M_t^{2,g}=\langle {\mu }_t^2,g\rangle -\langle {\mu }_0^2,g\rangle -{\displaystyle \frac{1}{2}}{\int }_0^t\langle {\mu }_s^2,g^{\prime \prime }\rangle \mathrm{d}s+{\int }_0^t\langle {\mu }_s^2,{\eta }_2(·,{\mu }_s^1,{\mu }_s^2)g\rangle \mathrm{d}s}\hfill \end{array}\right.$$

(1)

are continuous square-integrable martingales starting from zero, with

$$\left\{\begin{array}{c}{\displaystyle {\langle M^{1,f}\rangle }_t={\gamma }_1{\int }_0^t\langle {\mu }_s^1,f^2\rangle \mathrm{d}s+{\int }_0^t{\langle {\mu }_s^1,f\rangle }^2\mathrm{d}s,}\hfill \\ {\displaystyle {\langle M^{2,g}\rangle }_t={\gamma }_2{\int }_0^t\langle {\mu }_s^2,g^2\rangle \mathrm{d}s+{\int }_0^t{\langle {\mu }_s^2,g\rangle }^2\mathrm{d}s,}\hfill \\ {\langle M^{1,f},M^{2,g}\rangle }_t=0.\hfill \end{array}\right.$$

(2)

The nonnegative functions ${\eta }_i:\mathbb{R}\times E\to {\mathbb{R}}_{+}$ describe competition deaths. We abbreviate ${\eta }_i(x,{\mu }^1,{\mu }^2)$ to ${\eta }_i(x,\mathbf{\mu })$. Polarization of (2) specifies the covariance for two different test functions in the same population as well. The square in each environmental term is taken after integration against the test function; this is the covariance generated by one Brownian environment acting on the whole population.

We now state the assumptions on the competition rates. For $\nu \in M_F\left(\mathbb{R}\right)$, let $F_{\nu }\left(x\right)=\nu \left((-\infty ,x]\right)$. Fix the smooth weight

$$J\left(x\right)=exp\{-\sqrt{1+x^2}\},{\parallel v\parallel }_J={\int }_{\mathbb{R}}\left|v\left(x\right)\right|J\left(x\right)\mathrm{d}x.$$

Then J is strictly positive and integrable, and $|J^{\prime }|+|J^{\prime \prime }|\le CJ$. For $\mathbf{\mu },\mathbf{\nu }\in E$, set

$$\mathcal{D}(\mathbf{\mu },\mathbf{\nu })={\displaystyle \sum _{i=1}^2}\left(|\langle {\mathbf{\mu }}^i,1\rangle -\langle {\mathbf{\nu }}^i,1\rangle |+{\parallel F_{{\mathbf{\mu }}^i}-F_{{\mathbf{\nu }}^i}\parallel }_J\right).$$

Define the cumulative competition terms by

$${\mathcal{K}}_i(x,\mathbf{\mu })={\int }_{(-\infty ,x]}{\mathbf{\eta }}_i(y,\mathbf{\mu }){\mathbf{\mu }}^i\left(\mathrm{d}y\right),{\mathcal{K}}_i(\infty ,\mathbf{\mu })=\langle {\mathbf{\mu }}^i,{\mathbf{\eta }}_i(·,\mathbf{\mu })\rangle .$$

For $R>0$, let $E_R=\{\mathbf{\mu }\in E:\langle {\mathbf{\mu }}^1,1\rangle +\langle {\mathbf{\mu }}^2,1\rangle \le R\}$.

Assumption 1.1. 

The functions ${\eta }_1,{\eta }_2$ satisfy the following conditions.

(i)

Each ${\eta }_i$ is nonnegative and jointly continuous on $\mathbb{R}\times E$. For every $R>0$, there exists $C_R<\infty$ such that ${\eta }_i(x,\mathbf{\mu })\le C_R$ for $x\in \mathbb{R}$, $\mathbf{\mu }\in E_R$, and $i=1,2$.

(ii)

For every $R>0$, there exists $L_R<\infty$ such that, for $\mathbf{\mu },\mathbf{\nu }\in E_R$,

$$\begin{array}{cc}\hfill {\displaystyle \sum _{i=1}^2}& |{\mathcal{K}}_i(\infty ,\mathbf{\mu })-{\mathcal{K}}_i(\infty ,\mathbf{\nu })|+{\parallel {\mathcal{K}}_i(·,\mathbf{\mu })-{\mathcal{K}}_i(·,\mathbf{\nu })\parallel }_J\le L_R\mathcal{D}(\mathbf{\mu },\mathbf{\nu }).\hfill \end{array}$$

(3)

Only part (i) is needed for existence. Part (ii) is imposed on the cumulative competition terms because these are the drift coefficients in the distribution-function-valued SPDE. Both conditions are local in the total population size; global boundedness of the competition rates is not required.

Theorem 1.2. 

Under Assumption 1.1, for every ${\mathbf{\mu }}_0\in E$, the martingale problem (1)–(2) has a solution and its law on $C({\mathbb{R}}_{+},E)$ is unique. Moreover, for every $T<\infty$ and $p\in \{2,4\}$,

$$\mathbb{E}\left[\underset{0\le t\le T}{sup}{\left(\langle {\mu }_t^1,1\rangle +\langle {\mu }_t^2,1\rangle \right)}^p\right]\le C_{p,T}\left[1+{\left(\langle {\mu }_0^1,1\rangle +\langle {\mu }_0^2,1\rangle \right)}^p\right],$$

(4)

where $C_{p,T}$ depends only on $p,T,{\gamma }_1,{\gamma }_2$. 

Remark 1.3. 

Suppose that ${\eta }_i(x,\mathbf{\mu })=c_i+{\sum }_{j=1}^2{a}_{ij}\langle {\mathbf{\mu }}^j,1\rangle$ for $c_i,a_{ij}\ge 0.$ Then the total masses $Z_t^i=\langle {\mu }_t^i,1\rangle$ satisfy

$$\mathrm{d}{Z}_t^i=-Z_t^i\left(c_i+{\displaystyle \sum _{j=1}^2}{a}_{ij}{Z}_t^j\right)\mathrm{d}t+\sqrt{{\gamma }_i{Z}_t^i}\mathrm{d}{\beta }_t^i+Z_t^i\mathrm{d}{B}_t^i,i=1,2,$$

where ${\beta }^1,{\beta }^2,B^1,B^2$ are mutually independent Brownian motions on a possible extension. When $a_{12}=a_{21}=0$, each coordinate, after combining its Brownian terms, is a diffusion case of the nonlinear branching processes studied in [14]. Related two-way competition systems with different noise coefficients are studied in [15].

The proof of Theorem 1.2 follows the particle approximation and SPDE methods in [9], with two points requiring separate treatment. First, population-wide environmental events must be scaled so that they produce the second terms in (2), while individual competition deaths contribute only to the limiting drift. Second, for spatially dependent competition, the total masses do not in general form a closed system of stochastic differential equations. We therefore estimate the differences of the total masses and the distribution functions together, rather than invoking an autonomous mass equation. This also allows the initial measures to have atoms.

The rest of the paper is organized as follows. In Section 2 we construct the particle systems, prove compact containment and tightness, and obtain existence by identifying their limit points. In Section 3 we establish the correspondence with distribution-function-valued SPDEs and prove pathwise uniqueness. The resulting uniqueness in law completes the proof of Theorem 1.2. Throughout the paper, C denotes a finite constant whose value may change from line to line; its dependence on additional parameters is indicated when needed.

2. Particle Approximation and Existence

In this section we construct a sequence of two-type branching particle systems and prove that every subsequential limit solves (1)–(2). Only Assumption 1.1(i) is needed here. Throughout the section the initial measures are deterministic.

For each $n\ge 1$, choose finite configurations

$${\mu }_0^{n,i}={\displaystyle \frac{1}{n}}{\displaystyle \sum _{k=1}^{N_i^n}}{\delta }_{x_k^{n,i}},i=1,2,$$

such that ${\mu }_0^{n,i}\Rightarrow {\mu }_0^i$ in $M_F\left(\mathbb{R}\right)$ as $n\to \infty$. Such configurations can be obtained by discretizing the quantile functions of the normalized initial measures when $\langle {\mu }_0^i,1\rangle >0$; when $\langle {\mu }_0^i,1\rangle =0$, take the empty configuration. In particular, $N_i^n/n\to \langle {\mu }_0^i,1\rangle$ and the initial total masses are uniformly bounded. Each particle has mass $1/n$. Let ${\mathcal{I}}_t^{n,i}$ be the finite set of type-i particles alive at time t, and let

$${\mu }_t^{n,i}={\displaystyle \frac{1}{n}}\sum _{k\in {\mathcal{I}}_t^{n,i}}{\delta }_{X_k^{n,i}\left(t\right)},Z_t^{n,i}=\langle {\mu }_t^{n,i},1\rangle ,Z_t^n=Z_t^{n,1}+Z_t^{n,2}.$$

The evolution is specified as follows.

1.

Spatial motion: Between demographic events, particles move as independent standard Brownian motions. Descendants start at their parent’s location and subsequently use independent Brownian motions.

2.

Branching event: Independently for each type-i particle, at rate $n{\gamma }_i$ it is replaced by either zero or two particles, with probabilities $1/2$ and $1/2$.

3.

Competing event: A type-i particle at x is removed at rate ${\eta }_i(x,{\mu }_t^{n,1},{\mu }_t^{n,2})$. These removals are constructed by thinning independent Poisson random measures.

4.

Random environment: The two types have independent Poisson clocks, each of rate n. At each ringing time, a common sign $\epsilon \in \{-1,1\}$ is chosen with equal probabilities. Independently of this sign, each particle of the corresponding type alive immediately before the event is selected independently with probability $p_n=n^{-1/2}$.

All primitive driving noises and auxiliary marks are mutually independent across mechanisms and types. At each environmental event, the selected particles share the same sign $\epsilon$. The two populations are coupled through the competition rates, which depend on both empirical measures, while events of different types occur at distinct times almost surely. The independent environmental clocks give rise to independent Brownian environments in the limit.

For each n, the particle system is nonexplosive. Indeed, consider the system obtained by suppressing competition deaths. On every bounded time interval, its environmental clocks ring only finitely often, and each event at most doubles the population size. Between environmental events, the population size is dominated by a linear pure-birth process. Thus this system has finitely many particles and jump events on every bounded time interval. Using the same genealogical construction and driving noises, the original system can be coupled with the system without competition so that its particles form a subset of the latter. Whenever the total mass of the dominating system is bounded by R, the total competition rate is at most $nR{C}_R$, by Assumption 1.1(i). Hence competition deaths cannot accumulate in finite time.

2.1. Martingale Decomposition and Moment Estimates

For $f\in C_b^2\left(\mathbb{R}\right)$, by the above particle construction, one can check that

$$\langle {\mu }_t^{n,i},f\rangle =\langle {\mu }_0^{n,i},f\rangle +{\displaystyle \frac{1}{2}}{\int }_0^t\langle {\mu }_s^{n,i},f^{\prime \prime }\rangle \mathrm{d}s-{\int }_0^t\langle {\mu }_s^{n,i},{\eta }_i(·,{\mu }_s^{n,1},{\mu }_s^{n,2})f\rangle \mathrm{d}s+M_t^{n,i,f},$$

(5)

where $M^{n,i,f}$ is a locally square-integrable martingale such that, for $f,g\in C_b^2\left(\mathbb{R}\right)$,

$$\begin{array}{cc}\hfill {\langle M^{n,i,f},M^{n,i,g}\rangle }_t=& {\int }_0^t\left\{{\gamma }_i\langle {\mu }_s^{n,i},fg\rangle +\langle {\mu }_s^{n,i},f\rangle \langle {\mu }_s^{n,i},g\rangle +p_n(1-p_n)\langle {\mu }_s^{n,i},fg\rangle \right\}\mathrm{d}s\hfill \\ \hfill & +{\displaystyle \frac{1}{n}}{\int }_0^t\left\{\langle {\mu }_s^{n,i},f^{\prime }{g}^{\prime }\rangle +\langle {\mu }_s^{n,i},{\eta }_i(·,{\mu }_s^{n,1},{\mu }_s^{n,2})fg\rangle \right\}\mathrm{d}s,\hfill \end{array}$$

(6)

and

$$\begin{array}{c}\hfill {\langle M^{n,1,f},M^{n,2,g}\rangle }_t=0.\end{array}$$

At an environmental event, conditional on the particle configuration $x_1,\dots ,x_N$, the increment of $\langle {\mu }^{n,i},f\rangle$ is $\epsilon n^{-1}{\sum }_{k=1}^N{\xi }_{k}f\left(x_k\right)$, where the ${\xi }_k$ are independent Bernoulli$\left(p_n\right)$ variables, independent of $\epsilon$. This increment has conditional mean zero, and the conditional covariance of the increments corresponding to f and g is

$$p_n^2\langle {\mu }^{n,i},f\rangle \langle {\mu }^{n,i},g\rangle +{\displaystyle \frac{p_n(1-p_n)}{n}}\langle {\mu }^{n,i},fg\rangle .$$

Multiplying by the event rate n and using $n{p}_n^2=1$ gives the second and third terms in (6). The remaining terms arise from spatial motion, branching and competition deaths.

Lemma 2.1. 

For $p=2,4$ and $T>0$, there is a constant $C_{p,T}$, independent of n and of the competition rates, such that

$$\mathbb{E}\left[\underset{0\le t\le T}{sup}{\left(Z_t^n\right)}^p\right]\le C_{p,T}\left\{{\left(Z_0^n\right)}^p+Z_0^n\right\},$$

(7)

where $Z_t^n={\sum }_{i=1}^2\langle {\mu }_t^{n,i},1\rangle$. Let $A_t^{n,i}={\int }_0^t\langle {\mu }_s^{n,i},{\eta }_i(·,{\mu }_s^{n,1},{\mu }_s^{n,2})\rangle \mathrm{d}s.$ Then

$$\mathbb{E}{A}_T^{n,i}\le Z_0^{n,i},\underset{n}{sup}\mathbb{E}\left[{\left(A_T^{n,i}\right)}^2\right]<\infty .$$

(8)

Moreover, for every $f\in C_b^2\left(\mathbb{R}\right)$,

$$\underset{n}{sup}\mathbb{E}\left[\underset{0\le t\le T}{sup}{\left|M_t^{n,i,f}\right|}^4\right]<\infty .$$

(9)

In particular, the martingales in (5) are square integrable. 

Proof. 

Let ${\overline{N}}_t^{n,i}$ denote the number of type-i particles in the coupled system without competition, and set ${\overline{Z}}_t^n={\displaystyle \frac{1}{n}}{\sum }_{i=1}^2{\overline{N}}_t^{n,i}.$ By the coupling, ${\overline{Z}}_0^n=Z_0^n$ and $0\le Z_t^n\le {\overline{Z}}_t^n,$ for $t\ge 0$. Suppose that the current type-i mass is $z_i=N_i/n$, and write $z=z_1+z_2$. At an environmental event of type i, the increment of the total mass is $\epsilon D_i$, where $D_i=K_i/n,$$K_i\sim Bin(N_i,p_n)$, $p_n=n^{-1/2},$ and $\epsilon$ is an independent symmetric random sign. Consequently,

$$n\mathbb{E}{D}_i^2={\displaystyle \frac{N_i{p}_n(1-p_n)+N_i^2{p}_n^2}{n}}=p_n(1-p_n)z_i+z_i^2.$$

Similarly, one can check that

$$\begin{array}{c}\hfill n\mathbb{E}{D}_i^4\le {\displaystyle \frac{C}{n}}(z_i^4+z_i),\end{array}$$

(10)

Let ${\mathcal{L}}_n$ denote the generator of ${(n^{-1}{\overline{N}}_t^{n,1},n^{-1}{\overline{N}}_t^{n,2})}_{t\ge 0}$. Writing $z=z_1+z_2$, we use ${\mathcal{L}}_n\left(z^p\right)$ to denote its action on the function $(z_1,z_2)\mapsto {(z_1+z_2)}^p$. By the symmetry of the branching and environmental increments, we have ${\mathcal{L}}_{n}z=0.$ For the second moment,

$$\begin{array}{cc}\hfill {\mathcal{L}}_n\left(z^2\right)& ={\displaystyle \sum _{i=1}^2}{\displaystyle \frac{n^2{\gamma }_i{z}_i}{2}}\left[{\left(z+{\displaystyle \frac{1}{n}}\right)}^2+{\left(z-{\displaystyle \frac{1}{n}}\right)}^2-2z^2\right]\hfill \\ \hfill & +{\displaystyle \sum _{i=1}^2}{\displaystyle \frac{n}{2}}\mathbb{E}\left[{(z+D_i)}^2+{(z-D_i)}^2-2z^2\right]\le C(z^2+z).\hfill \end{array}$$

Similarly, one can check that ${\mathcal{L}}_n\left(z^4\right)\le C(z^4+z).$ For $R>Z_0^n$, define ${\overline{\tau }}_R^n=inf\{t\ge 0:{\overline{Z}}_t^n\ge R\}.$ Hence $0\le {\overline{Z}}_{t\wedge {\overline{\tau }}_R^n}^n\le 2R+{\displaystyle \frac{1}{n}}.$ Since ${\mathcal{L}}_{n}z=0$, the stopped process is a bounded martingale, and therefore $\mathbb{E}{\overline{Z}}_{t\wedge {\overline{\tau }}_R^n}^n=Z_0^n.$ For $p\in \{2,4\}$, by Dynkin’s formula,

$$\begin{array}{cc}\hfill \mathbb{E}\left[{\left({\overline{Z}}_{t\wedge {\overline{\tau }}_R^n}^n\right)}^p\right]& \le {\left(Z_0^n\right)}^p+C_p\mathbb{E}{\int }_0^{t\wedge {\overline{\tau }}_R^n}\left[{\left({\overline{Z}}_s^n\right)}^p+{\overline{Z}}_s^n\right]\mathrm{d}s\hfill \\ \hfill & \le {\left(Z_0^n\right)}^p+C_{p}t{Z}_0^n+C_p{\int }_0^t\mathbb{E}\left[{\left({\overline{Z}}_{s\wedge {\overline{\tau }}_R^n}^n\right)}^p\right]\mathrm{d}s.\hfill \end{array}$$

By Gronwall’s inequality, we have

$$\mathbb{E}\left[{\left({\overline{Z}}_{T\wedge {\overline{\tau }}_R^n}^n\right)}^p\right]\le C_{p,T}\left[{\left(Z_0^n\right)}^p+Z_0^n\right].$$

Recall that $Z_t^n\le {\overline{Z}}_t^n$ for $t\ge 0$. Then (7) follows by Doob’s inequality to the stopped martingale and ${\overline{\tau }}_R^n\uparrow \infty$ almost surely as $R\to \infty$. Since ${sup}_n{Z}_0^n<\infty$, it follows that ${sup}_n\mathbb{E}{sup}_{t\le T}{\left(Z_t^n\right)}^p<\infty$ for $p=2,4$. Taking $f=1$ in (5), we obtain

$$Z_t^{n,i}+A_t^{n,i}=Z_0^{n,i}+M_t^{n,i,1}.$$

(11)

Before ${\overline{\tau }}_R^n$, both the total mass and the competition rates are bounded. Thus the stopped martingale is square-integrable, and $\mathbb{E}{Z}_{T\wedge {\overline{\tau }}_R^n}^{n,i}+\mathbb{E}{A}_{T\wedge {\overline{\tau }}_R^n}^{n,i}=Z_0^{n,i}.$ Letting $R\to \infty$ and applying Fatou’s lemma gives $\mathbb{E}{Z}_T^{n,i}+\mathbb{E}{A}_T^{n,i}\le Z_0^{n,i}.$ In particular, $\mathbb{E}{A}_T^{n,i}\le Z_0^{n,i}.$ For $f\in C_b^2\left(\mathbb{R}\right)$, by (6), we have

$$\begin{array}{c}\hfill \mathbb{E}{\langle M^{n,i,f}\rangle }_T\le C_f{\int }_0^T\mathbb{E}\left[Z_s^n+{\left(Z_s^n\right)}^2\right]\mathrm{d}s+{\displaystyle \frac{{\parallel f\parallel }_{\infty }^2}{n}}\mathbb{E}{A}_T^{n,i}\le C_{f,T}.\end{array}$$

Hence $M^{n,i,f}$ is square-integrable and $\mathbb{E}|M_T^{n,i,f}{|}^2=\mathbb{E}{\langle M^{n,i,f}\rangle }_T\le C_{f,T}.$ Using (11) and $Z_T^{n,i}\ge 0$, we have $0\le A_T^{n,i}\le Z_0^{n,i}+\left|M_T^{n,i,1}\right|.$ It follows that

$$\begin{array}{c}\hfill \mathbb{E}\left[{\left(A_T^{n,i}\right)}^2\right]\le 2{\left(Z_0^{n,i}\right)}^2+2\mathbb{E}{\left|M_T^{n,i,1}\right|}^2=2{\left(Z_0^{n,i}\right)}^2+2\mathbb{E}{\langle M^{n,i,1}\rangle }_T\le C_T.\end{array}$$

This proves (8). We next estimate the fourth moments of the martingales. By Cauchy–Schwarz inequality,

$$\begin{array}{cc}\hfill \mathbb{E}\left[{\langle M^{n,i,f}\rangle }_T^2\right]& \le C_f\mathbb{E}\left[{\left({\int }_0^T[Z_s^n+{\left(Z_s^n\right)}^2]\mathrm{d}s\right)}^2\right]+{\displaystyle \frac{C_f}{n^2}}\mathbb{E}\left[{\left(A_T^{n,i}\right)}^2\right]\hfill \\ \hfill & \le C_{f}T{\int }_0^T\mathbb{E}\left[{\left(Z_s^n\right)}^2+{\left(Z_s^n\right)}^4\right]\mathrm{d}s+{\displaystyle \frac{C_f}{n^2}}\mathbb{E}\left[{\left(A_T^{n,i}\right)}^2\right]\le C_{f,T}.\hfill \end{array}$$

The finite-variation terms in (5) are continuous. Then $\mathsf{\Delta }{M}_s^{n,i,f}=\mathsf{\Delta }\langle {\mu }_s^{n,i},f\rangle .$ An ordinary branching event at x produces a jump of absolute size $\left|f\right(x\left)\right|/n$, and a competition death produces a jump $-f\left(x\right)/n$. At an environmental event involving $K_i$ selected particles, the absolute jump is at most ${\parallel f\parallel }_{\infty }{K}_i/n$. Compensating these jumps and using (10), we obtain

$$\begin{array}{cc}\hfill & \mathbb{E}\sum _{0<s\le T}|\mathsf{\Delta }\langle {\mu }_s^{n,i},f\rangle {|}^4=\mathbb{E}\sum _{0<s\le T}{\left|\mathsf{\Delta }{M}_s^{n,i,f}\right|}^4\hfill \\ \hfill & \le {\displaystyle \frac{{\gamma }_i{\parallel f\parallel }_{\infty }^4}{n^2}}{\int }_0^T\mathbb{E}{Z}_s^{n,i}\mathrm{d}s+{\displaystyle \frac{{\parallel f\parallel }_{\infty }^4}{n^3}}\mathbb{E}{A}_T^{n,i}+{\displaystyle \frac{{C\parallel f\parallel }_{\infty }^4}{n}}{\int }_0^T\mathbb{E}\left[{\left(Z_s^n\right)}^4+Z_s^n\right]\mathrm{d}s\hfill \\ \hfill & \le {\displaystyle \frac{C_{f,T}}{n}}.\hfill \end{array}$$

(12)

Finally, the fourth-moment inequality for martingales, one sees that

$$\begin{array}{c}\hfill \mathbb{E}\underset{t\le T}{sup}|M_t^{n,i,f}{|}^4\le C\mathbb{E}\left[{\langle M^{n,i,f}\rangle }_T^2\right]+C\mathbb{E}\sum _{0<s\le T}{\left|\mathsf{\Delta }{M}_s^{n,i,f}\right|}^4\le C_{f,T}.\end{array}$$

Thus (9) holds. This completes the proof. □

2.2. Tightness of the Empirical Measures

To establish compact containment, we next estimate the mass outside large compact sets. Choose $\chi \in C_b^2\left(\mathbb{R}\right)$ such that $0\le \chi \le 1$, $\chi =0$ on $[-1,1]$ and $\chi =1$ on ${[-2,2]}^c$, and put ${\chi }_R\left(x\right)=\chi (x/R)$. Then $\parallel {\chi }_R^{\prime \prime }{\parallel }_{\infty }\le C/R^2$. For $a>0$, apply optional stopping to (5) at

$${\tau }_a=inf\{t\ge 0:\langle {\mu }_t^{n,i},{\chi }_R\rangle >a\}\wedge T.$$

The competition contribution is nonpositive, and the martingale is square integrable by Lemma 2.1. As $\mathbb{E}{Z}_s^{n,i}\le Z_0^{n,i}$, it follows that

$$\mathbb{P}\left(\underset{t\le T}{sup}\langle {\mu }_t^{n,i},{\chi }_R\rangle >a\right)\le {\displaystyle \frac{1}{a}}\left\{\langle {\mu }_0^{n,i},{\chi }_R\rangle +{\displaystyle \frac{CT}{R^2}}{Z}_0^{n,i}\right\}.$$

(13)

The weak convergence of the initial measures implies their uniform tightness. The right-hand side therefore tends to zero as $R\to \infty$, uniformly in n.

Proposition 2.2. 

For every $T>0$, the laws of ${\mu }^n=({\mu }^{n,1},{\mu }^{n,2})$ are tight in $D\left(\right[0,T],E)$, and every limit is supported on $C\left(\right[0,T],E)$. 

Proof. 

We first verify compact containment. For $ϵ>0$, by (7), one can choose L large enough such that ${sup}_n\mathbb{P}({sup}_{t\le T}{Z}_t^n>L)<ϵ/2$. Choose any sequence $a_j\downarrow 0$. By (13), proper $R_j\uparrow \infty$ can be chosen such that

$$\underset{n}{sup}\mathbb{P}\left(\underset{t\le T}{sup}\langle {\mu }_t^{n,i},{\chi }_{R_j}\rangle >a_j\right)<ϵ2^{-j-2},i=1,2.$$

The set

$$K=\left\{({\nu }_1,{\nu }_2):\langle {\nu }_1+{\nu }_2,1\rangle \le L,\langle {\nu }_i,{\chi }_{R_j}\rangle \le a_j\mathrm{for}\mathrm{all}i=1,2,j\ge 1\right\}$$

is closed and uniformly tight, with uniformly bounded total masses. It is therefore compact in $M_F{\left(\mathbb{R}\right)}^2$. The union bound shows that ${\mu }_t^n\in K$ for all $t\le T$ with probability at least $1-ϵ$, uniformly in n. Fix $R>{sup}_n{Z}_0^n$ and set

$${\sigma }_R^n=inf\{t\ge 0:Z_t^n>R\}.$$

By Assumption 1.1(i), the drift and bracket densities in (5) and (6) are bounded in absolute value by $C_{f,R}$ on $[0,{\sigma }_R^n)$, uniformly in n. For stopping times $S_n\le T$ and deterministic ${\theta }_n\in [0,\delta ]$, where $0<\delta \le 1$, we have

$$0\le (S_n+{\theta }_n)\wedge {\sigma }_R^n-S_n\wedge {\sigma }_R^n\le \delta .$$

Since the bracket density is bounded before ${\sigma }_R^n$, optional sampling and the martingale isometry give

$$\begin{array}{cc}\hfill & \mathbb{E}{\left|M_{(S_n+{\theta }_n)\wedge {\sigma }_R^n}^{n,i,f}-M_{S_n\wedge {\sigma }_R^n}^{n,i,f}\right|}^2=\mathbb{E}\left[{\langle M^{n,i,f}\rangle }_{(S_n+{\theta }_n)\wedge {\sigma }_R^n}-{\langle M^{n,i,f}\rangle }_{S_n\wedge {\sigma }_R^n}\right]\le C_{f,R}\delta .\hfill \end{array}$$

The drift density is also bounded before ${\sigma }_R^n$. Thus (5) yields

$$\begin{array}{cc}\hfill & \mathbb{E}{\left|\langle {\mu }_{(S_n+{\theta }_n)\wedge {\sigma }_R^n}^{n,i},f\rangle -\langle {\mu }_{S_n\wedge {\sigma }_R^n}^{n,i},f\rangle \right|}^2\hfill \\ \hfill & \le 2\mathbb{E}{\left|{\int }_{S_n\wedge {\sigma }_R^n}^{(S_n+{\theta }_n)\wedge {\sigma }_R^n}\left[{\displaystyle \frac{1}{2}}\langle {\mu }_s^{n,i},f^{\prime \prime }\rangle -\langle {\mu }_s^{n,i},{\eta }_i(·,{\mu }_s^{n,1},{\mu }_s^{n,2})f\rangle \right]\mathrm{d}s\right|}^2+2C_{f,R}\delta \hfill \\ \hfill & \le 2C_{f,R}^2{\delta }^2+2C_{f,R}\delta \le C_{f,R}(\delta +{\delta }^2),\hfill \end{array}$$

where the constant is enlarged in the last inequality. By (7) and Markov’s inequality,

$$\underset{n}{sup}\mathbb{P}({\sigma }_R^n\le T+1)\le {\displaystyle \frac{1}{R^4}}\underset{n}{sup}\mathbb{E}\underset{s\le T+1}{sup}{\left(Z_s^n\right)}^4\le {\displaystyle \frac{C_T}{R^4}}.$$

Consequently, Chebyshev’s inequality gives, for every $ϵ>0$,

$$\mathbb{P}\left(\left|\langle {\mu }_{S_n+{\theta }_n}^{n,i},f\rangle -\langle {\mu }_{S_n}^{n,i},f\rangle \right|>ϵ\right)\le {\displaystyle \frac{C_T}{R^4}}+{\displaystyle \frac{C_{f,R}}{{ϵ}^2}}(\delta +{\delta }^2).$$

Letting $\delta \downarrow 0$ and then $R\to \infty$ proves Aldous’ condition for $\langle {\mu }^{n,i},f\rangle$. Applying this to a countable separating family in $C_c^{\infty }\left(\mathbb{R}\right)$ together with the constant function 1, and using compact containment, we obtain tightness in $D([0,T],M_F{\left(\mathbb{R}\right)}^2)$; see Ethier and Kurtz [19] and Jacod and Shiryaev [20]. For each f in this family, Markov’s inequality and (12) give

$$\begin{array}{cc}\hfill \mathbb{P}\left(\underset{0<s\le T}{sup}\left|\mathsf{\Delta }\langle {\mu }_s^{n,i},f\rangle \right|>ϵ\right)& \le {\displaystyle \frac{1}{{ϵ}^4}}\mathbb{E}\sum _{0<s\le T}{\left|\mathsf{\Delta }\langle {\mu }_s^{n,i},f\rangle \right|}^4\le {\displaystyle \frac{C_{f,T}}{n{ϵ}^4}}⟶0.\hfill \end{array}$$

Thus the limits are continuous. Since the family separates finite measures and compact containment holds in the weak topology, every measure-valued limit is continuous. □

2.3. Identification of the Limit

Proposition 2.3. 

Under Assumption 1.1(i), for every ${\mathbf{\mu }}_0\in E$, the martingale problem (1)–(2) admits a continuous solution satisfying (4). 

Proof. 

By Proposition 2.2, there exists a subsequence ${\mathbf{\mu }}^{n_k}$ converging in distribution to a continuous process $\mathbf{\mu }$ in $D\left(\right[0,T],E)$. The convergence holds almost surely in the Skorokhod topology and hence uniformly on $[0,T]$, since the limiting process $\mathbf{\mu }$ has continuous paths. For $f\in C_b\left(\mathbb{R}\right)$ and $i=1,2$, the map

$$({\nu }_1,{\nu }_2)⟼\langle {\nu }_i,{\eta }_i(·,{\nu }_1,{\nu }_2)f\rangle$$

is continuous in the weak topology. Indeed, if $({\nu }_1^k,{\nu }_2^k)\Rightarrow ({\nu }_1,{\nu }_2)$, the measures are uniformly tight and have bounded masses. By Assumption 1.1(i), the rates ${\eta }_i(·,{\nu }_1^k,{\nu }_2^k)$ are uniformly bounded and converge uniformly on compact subsets of $\mathbb{R}$ to ${\eta }_i(·,{\nu }_1,{\nu }_2)$. The assertion follows from weak convergence and uniform tightness. Since ${\mathbf{\mu }}^{n_k}\to \mathbf{\mu }$ almost surely, uniformly on $[0,T]$, the ranges of these processes are almost surely contained in a common compact subset of E. The preceding map is therefore uniformly continuous on this set. Consequently,

$$\underset{t\le T}{sup}\left|\langle {\mu }_t^{n,i},{\eta }_i(·,{\mu }_t^n)f\rangle -\langle {\mu }_t^i,{\eta }_i(·,{\mathbf{\mu }}_t)f\rangle \right|⟶0\mathrm{almost}\mathrm{surely}.$$

The same argument applies to the other drift and bracket integrands. In particular, the total masses of all these paths are bounded on $[0,T]$ for almost every sample point. The additional terms in (6) converge uniformly to zero: this follows from $p_n\to 0$, $n^{-1}\to 0$, and local boundedness of the rates.

It follows from (5) that $M^{n,i,f}$ converges uniformly on $[0,T]$, almost surely, to the process $M^{i,f}$ defined in (1). Similarly, by (6), uniformly on $[0,T]$, we have

$${\langle M^{n,i,f},M^{n,i,g}\rangle }_t⟶{\int }_0^t\left\{{\gamma }_i\langle {\mu }_s^i,fg\rangle +\langle {\mu }_s^i,f\rangle \langle {\mu }_s^i,g\rangle \right\}\mathrm{d}s.$$

The martingales have uniformly bounded fourth moments by (9). Their products are therefore uniformly integrable. The bracket processes have uniformly bounded second moments by the proof of Lemma 2.1. These bounds justify passage to the limit in the martingale identities for

$$M^{n,i,f},M^{n,i,f}{M}^{n,i,g}-\langle M^{n,i,f},M^{n,i,g}\rangle ,M^{n,1,f}{M}^{n,2,g}.$$

Consequently, the limit $\mathbf{\mu }$ solves (1)–(2) on $[0,T]$ with respect to its natural filtration. The moment bound (4) follows from Lemma 2.1 and Fatou’s lemma. A diagonal argument yields a solution on $[0,\infty )$. □

3. Distribution-Function-Valued SPDEs

In this section we assume Assumption 1.1. We first identify the SPDE associated with the martingale problem. We then prove pathwise uniqueness by estimating simultaneously the distribution functions and the total masses. Finally, we deduce uniqueness in law for the martingale problem.

3.1. The SPDE Representation

For an E-valued process $\mathbf{\mu }$, set

$$u_t^i\left(x\right)={\mu }_t^i\left((-\infty ,x]\right),Z_t^i={\mu }_t^i\left(\mathbb{R}\right)=u_t^i(\infty ),i=1,2.$$

Let $W^1,W^2$ be independent Gaussian white noises on ${\mathbb{R}}_{+}\times {\mathbb{R}}_{+}$ with intensity $\mathrm{d}s\mathrm{d}a$, and let $B^1,B^2$ be independent standard Brownian motions, independent also of $W^1,W^2$. All four noises are understood relative to the same filtration. The associated SPDE is

$$\begin{array}{c}\hfill \mathrm{d}{u}_t^i\left(x\right)={\displaystyle \frac{1}{2}}\mathsf{\Delta }{u}_t^i\left(x\right)\mathrm{d}t-{\mathcal{K}}_i(x,{\mathbf{\mu }}_t)\mathrm{d}t+u_t^i\left(x\right)\mathrm{d}{B}_t^i+\sqrt{{\mathbf{\gamma }}_i}{\int }_0^{\infty }{\mathbf{1}}_{\{a\le u_t^i\left(x\right)\}}{W}^i(\mathrm{d}t,\mathrm{d}a)\end{array}$$

with $i=1,2.$ The equation is interpreted in the distributional sense. More precisely, a solution in the class considered here consists of an adapted continuous E-valued process $\mathbf{\mu }$ and its distribution functions $u^i$ such that, for every $\varphi \in C_c^{\infty }\left(\mathbb{R}\right)$,

$$\begin{array}{cc}\hfill (u_t^i,\varphi )=& (u_0^i,\varphi )+{\displaystyle \frac{1}{2}}{\int }_0^t(u_s^i,{\varphi }^{\prime \prime })\mathrm{d}s-{\int }_0^t({\mathcal{K}}_i(·,{\mathbf{\mu }}_s),\mathbf{\varphi })\mathrm{d}s\hfill \\ \hfill & +\sqrt{{\gamma }_i}{\int }_0^t{\int }_0^{\infty }\left({\int }_{\mathbb{R}}\varphi \left(x\right){\mathbf{1}}_{\{a\le u_s^i\left(x\right)\}}\mathrm{d}x\right)W^i(\mathrm{d}s,\mathrm{d}a)+{\int }_0^t(u_s^i,\varphi )\mathrm{d}{B}_s^i.\hfill \end{array}$$

(14)

where $(v,\varphi )={\int }_{\mathbb{R}}v\left(x\right)\varphi \left(x\right)\mathrm{d}x$.

Lemma 3.1. 

Let $\mathbf{\mu }$ be a solution to (1)–(2). Then, for all $f,g\in C_b^2\left(\mathbb{R}\right)$, the processes defined in (1) are continuous square-integrable martingales satisfying (2). Moreover, $\mathbf{\mu }$ satisfies (4) and, for every $T>0$,

$$\mathbb{E}{\int }_0^T{\mathcal{K}}_i(\infty ,{\mathbf{\mu }}_s)\mathrm{d}s\le Z_0^i,i=1,2.$$

Proof. 

Let ${\tau }_R=inf\{t\ge 0:Z_t^1+Z_t^2\ge R\}$. Choose $\chi \in C_c^{\infty }\left(\mathbb{R}\right)$ with $0\le \chi \le 1$ and $\chi =1$ on $[-1,1]$, and set ${\chi }_r\left(x\right)=\chi (x/r)$. For $f\in C_b^2\left(\mathbb{R}\right)$, apply the martingale problem to $f{\chi }_r$ and stop at ${\tau }_R$. The stopped paths have uniformly tight ranges on compact time intervals. Together with Assumption 1.1(i), dominated convergence and the martingale isometry allow us to let $r\to \infty$. Thus (1)–(2) remain valid for test functions in $C_b^2\left(\mathbb{R}\right)$, with all processes stopped at ${\tau }_R$. In particular, for $f=1$, we have

$$\begin{array}{cc}\hfill Z_t^i& =Z_0^i-{\int }_0^t{\mathcal{K}}_i(\infty ,{\mathbf{\mu }}_s)\mathrm{d}s+M_t^{i,1},\hfill \end{array}$$

where ${\langle M^{i,1}\rangle }_t={\int }_0^t[{\gamma }_i{Z}_s^i+{\left(Z_s^i\right)}^2]\mathrm{d}s$ and $\langle M^{1,1},M^{2,1}\rangle =0$. Let $Z=Z^1+Z^2$. Its finite variation part is nonpositive, and $\mathrm{d}{\langle M^{1,1}+M^{2,1}\rangle }_s\le C(Z_s+Z_s^2)\mathrm{d}s$. Let $M=M^{1,1}+M^{2,1}$. Since the competition terms are nonnegative and $\mathrm{d}{\langle M\rangle }_s\le C(Z_s+Z_s^2)\mathrm{d}s$, By Itô’s formula, for $p\in \{2,4\}$,

$$Z_{t\wedge {\tau }_R}^p\le Z_0^p+p{\int }_0^{t\wedge {\tau }_R}{Z}_s^{p-1}\mathrm{d}{M}_s+C{\int }_0^{t\wedge {\tau }_R}(Z_s^{p-1}+Z_s^p)\mathrm{d}s.$$

By the Burkholder–Davis–Gundy and Young inequalities,

$$\begin{array}{cc}\hfill p\mathbb{E}\underset{r\le t}{sup}\left|{\int }_0^{r\wedge {\tau }_R}{Z}_s^{p-1}\mathrm{d}{M}_s\right|& \le C\mathbb{E}{\left[{\int }_0^{t\wedge {\tau }_R}(Z_s^{2p-1}+Z_s^{2p})\mathrm{d}s\right]}^{1/2}\hfill \\ \hfill & \le C\mathbb{E}\left[{\left(\underset{r\le t}{sup}{Z}_{r\wedge {\tau }_R}^p\right)}^{1/2}{\left({\int }_0^{t\wedge {\tau }_R}(Z_s^{p-1}+Z_s^p)\mathrm{d}s\right)}^{1/2}\right]\hfill \\ \hfill & \le {\displaystyle \frac{1}{2}}\mathbb{E}\underset{r\le t}{sup}{Z}_{r\wedge {\tau }_R}^p+C\mathbb{E}{\int }_0^{t\wedge {\tau }_R}(Z_s^{p-1}+Z_s^p)\mathrm{d}s.\hfill \end{array}$$

Using $z^{p-1}\le 1+z^p$ for $z\ge 0$, we obtain

$$\mathbb{E}\underset{r\le t}{sup}{Z}_{r\wedge {\tau }_R}^p\le Z_0^p+{\displaystyle \frac{1}{2}}\mathbb{E}\underset{r\le t}{sup}{Z}_{r\wedge {\tau }_R}^p+C{\int }_0^t\left(1+\mathbb{E}\underset{r\le s}{sup}{Z}_{r\wedge {\tau }_R}^p\right)\mathrm{d}s,$$

where C is independent of R. Absorbing the supremum term into the left-hand side and applying Gronwall’s inequality, one can get a bound uniform in R. Since Z has continuous paths, ${\tau }_R\uparrow \infty$ almost surely, and Fatou’s lemma, (4) follows. Taking expectations in the stopped mass identity and applying monotone convergence to the competition integral, one sees that

$$\mathbb{E}{\int }_0^T{\mathcal{K}}_i(\infty ,{\mathbf{\mu }}_s)\mathrm{d}s\le Z_0^i,i=1,2.$$

Finally, by (4), $M^{i,f}$ is square-integrable for every $f\in C_b^2\left(\mathbb{R}\right)$. Letting $R\to \infty$ in the stopped identities therefore establishes (1)–(2) for all $f,g\in C_b^2\left(\mathbb{R}\right)$. □

Proposition 3.2. 

Every solution of (1)–(2) admits, on an extension of its probability space, a representation (14) with mutually independent noises $W^1,W^2,B^1,B^2$. Conversely, every solution of (14) in the class specified above solves (1)–(2). In this representation,

$$Z_t^i=Z_0^i-{\int }_0^t{\mathcal{K}}_i(\infty ,{\mathbf{\mu }}_s)\mathrm{d}s+\sqrt{{\mathbf{\gamma }}_i}{\int }_0^t{\int }_0^{\infty }{\mathbf{1}}_{\{a\le Z_s^i\}}{W}^i(\mathrm{d}s,\mathrm{d}a)+{\int }_0^t{Z}_s^i\mathrm{d}{B}_s^i.$$

(15)

Proof. 

Suppose first that $\mathbf{\mu }$ is a solution to (1)–(2). For $\varphi \in C_c^{\infty }\left(\mathbb{R}\right)$, let $\tilde{\varphi }\left(y\right)={\int }_y^{\infty }\varphi \left(x\right)\mathrm{d}x.$ By $\tilde{\varphi }\in C_b^2\left(\mathbb{R}\right)$, Lemma 3.1 and Fubini’s theorem, it gives

$$\begin{array}{cc}\hfill (u_t^i,\varphi )& =\langle {\mu }_t^i,\tilde{\varphi }\rangle =\langle {\mu }_0^i,\tilde{\varphi }\rangle +{\displaystyle \frac{1}{2}}{\int }_0^t\langle {\mu }_s^i,{\tilde{\varphi }}^{\prime \prime }\rangle \mathrm{d}s-{\int }_0^t\langle {\mu }_s^i,{\eta }_i(·,{\mathbf{\mu }}_s)\tilde{\mathbf{\varphi }}\rangle \mathrm{d}s+M_t^{i,\tilde{\mathbf{\varphi }}}\hfill \\ \hfill & =(u_0^i,\varphi )+{\displaystyle \frac{1}{2}}{\int }_0^t(u_s^i,{\varphi }^{\prime \prime })\mathrm{d}s-{\int }_0^t({\mathcal{K}}_i(·,{\mathbf{\mu }}_s),\mathbf{\varphi })\mathrm{d}s+M_t^{i,\tilde{\mathbf{\varphi }}}.\hfill \end{array}$$

Let $N_t^i\left(\varphi \right)=M_t^{i,\tilde{\varphi }}$. For $\varphi ,\psi \in C_c^{\infty }\left(\mathbb{R}\right)$, we have

$$\begin{array}{cc}\hfill \langle {\mu }_s^i,\tilde{\varphi }\tilde{\psi }\rangle & ={\int }_{{\mathbb{R}}^2}\varphi \left(x\right)\psi \left(y\right)u_s^i(x\wedge y)\mathrm{d}x\mathrm{d}y\hfill \\ \hfill & ={\int }_0^{\infty }\left({\int }_{\mathbb{R}}\varphi \left(x\right){\mathbf{1}}_{\{a\le u_s^i\left(x\right)\}}\mathrm{d}x\right)\left({\int }_{\mathbb{R}}\psi \left(y\right){\mathbf{1}}_{\{a\le u_s^i\left(y\right)\}}\mathrm{d}y\right)\mathrm{d}a.\hfill \end{array}$$

Consequently,

$$\begin{array}{cc}\hfill {\langle N^i\left(\varphi \right),N^i\left(\psi \right)\rangle }_t& ={\gamma }_i{\int }_0^t{\int }_0^{\infty }\left({\int }_{\mathbb{R}}\varphi \left(x\right){\mathbf{1}}_{\{a\le u_s^i\left(x\right)\}}\mathrm{d}x\right)\left({\int }_{\mathbb{R}}\psi \left(y\right){\mathbf{1}}_{\{a\le u_s^i\left(y\right)\}}\mathrm{d}y\right)\mathrm{d}a\mathrm{d}s\hfill \\ \hfill & +{\int }_0^t(u_s^i,\varphi )(u_s^i,\psi )\mathrm{d}s,\hfill \end{array}$$

(16)

and ${\langle N^1\left(\varphi \right),N^2\left(\psi \right)\rangle }_t=0.$ Let $U={(L^2\left({\mathbb{R}}_{+}\right)\oplus \mathbb{R})}^2$ and $H=L^2(\mathbb{R},J\left(x\right)\mathrm{d}x)$. Define $Q\left(u\right):U\to H^2$ by

$${\left[Q\left(u\right)(h_1,b_1,h_2,b_2)\right]}_i\left(x\right)=\sqrt{{\gamma }_i}{\int }_0^{u^i\left(x\right)}{h}_i\left(a\right)\mathrm{d}a+b_i{u}^i\left(x\right),i=1,2.$$

The map Q is continuous in the Hilbert–Schmidt norm, and

$${\parallel Q\left(u\right)\parallel }_{\mathrm{HS}}^2={\displaystyle \sum _{i=1}^2}{\int }_{\mathbb{R}}\left[{\gamma }_i{u}^i\left(x\right)+{\left(u^i\left(x\right)\right)}^2\right]J\left(x\right)\mathrm{d}x\le C{\displaystyle \sum _{i=1}^2}\left[u^i(\infty )+{\left(u^i(\infty )\right)}^2\right].$$

Thus $Q\left(u_s\right)$ is predictable and, by Lemma 3.1, $\mathbb{E}{\int }_0^T{\parallel Q\left(u_s\right)\parallel }_{\mathrm{HS}}^2\mathrm{d}s<\infty .$ Since $(u^i,\varphi )={\langle u^i,\varphi /J\rangle }_H$, (16) gives the covariance density $Q\left(u_s\right)Q{\left(u_s\right)}^{*}$ of the joint martingale family. Moreover, $\{\varphi /J:\varphi \in C_c^{\infty }\left(\mathbb{R}\right)\}$ is dense in H. By Ondreját ([22], Theorem 2 and Corollary 6), on an extension of the probability space, there exist Gaussian white noises $W^1,W^2$ and standard Brownian motions $B^1,B^2$, all mutually independent, such that

$$\begin{array}{cc}\hfill N_t^i\left(\varphi \right)& =\sqrt{{\gamma }_i}{\int }_0^t{\int }_0^{\infty }\left({\int }_{\mathbb{R}}\varphi \left(x\right){\mathbf{1}}_{\{a\le u_s^i\left(x\right)\}}\mathrm{d}x\right)W^i(\mathrm{d}s,\mathrm{d}a)+{\int }_0^t(u_s^i,\varphi )\mathrm{d}{B}_s^i.\hfill \end{array}$$

Substitution into the preceding identity proves (14).

Conversely, suppose that $\mathbf{\mu }$ satisfies (14). Define the generalized inverse by ${\left(u_s^i\right)}^{-1}\left(a\right)=inf\{x\in \mathbb{R}:u_s^i\left(x\right)>a\}$ for $0<a<Z_s^i.$ For every bounded Borel function h, ${\int }_0^{Z_s^i}h\left({\left(u_s^i\right)}^{-1}\left(a\right)\right)\mathrm{d}a=\langle {\mu }_s^i,h\rangle .$ Moreover, for $f\in C_c^{\infty }\left(\mathbb{R}\right)$ and almost every $a\in (0,Z_s^i)$, we have $-{\int }_{\mathbb{R}}{f}^{\prime }\left(x\right){\mathbf{1}}_{\{a\le u_s^i\left(x\right)\}}\mathrm{d}x=f\left({\left(u_s^i\right)}^{-1}\left(a\right)\right).$ Taking $\varphi =-f^{\prime }$ in (14) and integrating by parts, we obtain

$$\begin{array}{c}\hfill \langle {\mu }_t^i,f\rangle =-(u_t^i,f^{\prime })=\langle {\mu }_0^i,f\rangle +{\displaystyle \frac{1}{2}}{\int }_0^t\langle {\mu }_s^i,f^{\prime \prime }\rangle \mathrm{d}s-{\int }_0^t\langle {\mu }_s^i,{\eta }_i(·,{\mathbf{\mu }}_s)f\rangle \mathrm{d}s+M_t^{i,f},\end{array}$$

where

$$M_t^{i,f}=\sqrt{{\gamma }_i}{\int }_0^t{\int }_0^{Z_s^i}f\left({\left(u_s^i\right)}^{-1}\left(a\right)\right)W^i(\mathrm{d}s,\mathrm{d}a)+{\int }_0^t\langle {\mu }_s^i,f\rangle \mathrm{d}{B}_s^i.$$

(17)

By the independence of the driving noises, we have

$$\begin{array}{cc}\hfill {\langle M^{i,f},M^{i,g}\rangle }_t& ={\gamma }_i{\int }_0^t{\int }_0^{Z_s^i}f\left({\left(u_s^i\right)}^{-1}\left(a\right)\right)g\left({\left(u_s^i\right)}^{-1}\left(a\right)\right)\mathrm{d}a\mathrm{d}s+{\int }_0^t\langle {\mu }_s^i,f\rangle \langle {\mu }_s^i,g\rangle \mathrm{d}s\hfill \\ \hfill & ={\int }_0^t\left[{\gamma }_i\langle {\mu }_s^i,fg\rangle +\langle {\mu }_s^i,f\rangle \langle {\mu }_s^i,g\rangle \right]\mathrm{d}s,\hfill \end{array}$$

and ${\langle M^{1,f},M^{2,g}\rangle }_t=0.$ After stopping at ${\tau }_R$, smooth approximation extends these identities to $f,g\in C_c^2\left(\mathbb{R}\right)$. The argument of Lemma 3.1 then gives (4) and shows that the local martingales are square-integrable. Hence $\mathbf{\mu }$ solves (1)–(2).

Finally, let ${\chi }_r$ be the cutoffs used in Lemma 3.1. By dominated convergence,

$$\begin{array}{c}\hfill \mathbb{E}{\int }_0^T{\int }_0^{Z_s^i}{\left|{\chi }_r\left({\left(u_s^i\right)}^{-1}\left(a\right)\right)-1\right|}^2\mathrm{d}a\mathrm{d}s=\mathbb{E}{\int }_0^T\langle {\mu }_s^i,{({\chi }_r-1)}^2\rangle \mathrm{d}s⟶0,\end{array}$$

and $\mathbb{E}{\int }_0^T{\left|\langle {\mu }_s^i,{\chi }_r\rangle -Z_s^i\right|}^2\mathrm{d}s\to 0.$ The drift terms converge by Lemma 3.1 and $\parallel {\chi }_r^{\prime \prime }{\parallel }_{\infty }\to 0$. Thus the martingale isometry allows us to let $r\to \infty$ in the identity for $\langle {\mu }_t^i,{\chi }_r\rangle$ and (17), yielding (15) with the same driving noises. □

3.2. Pathwise Uniqueness

Let $(u^1,u^2)$ and $({\widehat{u}}^1,{\widehat{u}}^2)$ be two solutions of (14) with the same initial measures, on the same filtered probability space and driven by the same four noises. Let $\mathbf{\mu },\widehat{\mathbf{\mu }}$ be their measure-valued processes, and set

$$v_t^i=u_t^i-{\widehat{u}}_t^i,z_t^i=Z_t^i-{\widehat{Z}}_t^i,D_t={\displaystyle \sum _{i=1}^2}\left(|z_t^i|+{\parallel v_t^i\parallel }_J\right).$$

For $R>Z_0^1+Z_0^2$, define

$${\tau }_R=inf\{t\ge 0:Z_t^1+Z_t^2\ge R\mathrm{or}{\widehat{Z}}_t^1+{\widehat{Z}}_t^2\ge R\}.$$

The stopping times tend to infinity almost surely. All estimates below are stopped at ${\tau }_R$.

We use the usual smooth approximations to the absolute value. Set $a_0=1$ and $a_k=exp\{-k(k+1)/2\}$ for $k\ge 1$. Choose nonnegative smooth functions ${\psi }_k$, supported in $(a_k,a_{k-1})$, such that ${\int }_{a_k}^{a_{k-1}}{\psi }_k\left(r\right)\mathrm{d}r=1$ and ${\psi }_k\left(r\right)\le {\displaystyle \frac{2}{kr}}.$ Let ${\phi }_k\left(z\right)={\int }_0^{\left|z\right|}{\int }_0^y{\psi }_k\left(r\right)\mathrm{d}r\mathrm{d}y$. Then

$$|{\phi }_k^{\prime }|\le 1,0\le {\phi }_k^{\prime \prime }\left(z\right)\left|z\right|\le {\displaystyle \frac{2}{k}},\left|z\right|-a_{k-1}\le {\phi }_k\left(z\right)\le \left|z\right|.$$

(18)

For $r,\widehat{r}\in [0,R]$, the two noise coefficients have the elementary difference estimates

$${\int }_0^{\infty }|{\mathbf{1}}_{\{a\le r\}}-{\mathbf{1}}_{\{a\le \widehat{r}\}}{|}^2\mathrm{d}a=|r-\widehat{r}|,|r-\widehat{r}{|}^2\le 2R|r-\widehat{r}|.$$

(19)

Lemma 3.3. 

For every $t\le T$, we have ${\sum }_{i=1}^2\mathbb{E}\left|z_{t\wedge {\tau }_R}^i\right|\le L_R{\int }_0^t\mathbb{E}\left[{\mathbf{1}}_{\{s<{\tau }_R\}}{D}_s\right]\mathrm{d}s.$ 

Proof. 

Subtracting the two equations (15), we obtain

$$\begin{array}{cc}\hfill z_{t\wedge {\tau }_R}^i=& -{\int }_0^{t\wedge {\tau }_R}\left[{\mathcal{K}}_i(\infty ,{\mathbf{\mu }}_s)-{\mathcal{K}}_i(\infty ,{\widehat{\mathbf{\mu }}}_s)\right]\mathrm{d}s\hfill \\ \hfill & +\sqrt{{\gamma }_i}{\int }_0^{t\wedge {\tau }_R}{\int }_0^{\infty }\left[{\mathbf{1}}_{\{a\le Z_s^i\}}-{\mathbf{1}}_{\{a\le {\widehat{Z}}_s^i\}}\right]W^i(\mathrm{d}s,\mathrm{d}a)+{\int }_0^{t\wedge {\tau }_R}{z}_s^i\mathrm{d}{B}_s^i.\hfill \end{array}$$

Since $Z_s^i,{\widehat{Z}}_s^i\le R$ for $s<{\tau }_R$ and $|{\phi }_k^{\prime }|\le 1$, the stochastic integrals arising from Itô’s formula are square-integrable martingales. Thus, using $z_0^i=0$ and (19), we have

$$\begin{array}{cc}\hfill \mathbb{E}{\phi }_k\left(z_{t\wedge {\tau }_R}^i\right)=& -\mathbb{E}{\int }_0^{t\wedge {\tau }_R}{\phi }_k^{\prime }\left(z_s^i\right)\left[{\mathcal{K}}_i(\infty ,{\mathbf{\mu }}_s)-{\mathcal{K}}_i(\infty ,{\widehat{\mathbf{\mu }}}_s)\right]\mathrm{d}s\hfill \\ \hfill & +{\displaystyle \frac{{\gamma }_i}{2}}\mathbb{E}{\int }_0^{t\wedge {\tau }_R}{\phi }_k^{\prime \prime }\left(z_s^i\right)\left|z_s^i\right|\mathrm{d}s+{\displaystyle \frac{1}{2}}\mathbb{E}{\int }_0^{t\wedge {\tau }_R}{\phi }_k^{\prime \prime }\left(z_s^i\right){\left(z_s^i\right)}^2\mathrm{d}s.\hfill \end{array}$$

By (18),

$${\displaystyle \frac{1}{2}}{\phi }_k^{\prime \prime }\left(z_s^i\right)\left[{\gamma }_i\left|z_s^i\right|+{\left(z_s^i\right)}^2\right]\le {\displaystyle \frac{{\gamma }_i+\left|z_s^i\right|}{k}}\le {\displaystyle \frac{{\gamma }_i+R}{k}},s<{\tau }_R.$$

Summing over i and applying (3) gives

$$\begin{array}{cc}\hfill {\displaystyle \sum _{i=1}^2}\mathbb{E}{\phi }_k\left(z_{t\wedge {\tau }_R}^i\right)& \le \mathbb{E}{\int }_0^{t\wedge {\tau }_R}{\displaystyle \sum _{i=1}^2}\left|{\mathcal{K}}_i(\infty ,{\mathbf{\mu }}_s)-{\mathcal{K}}_i(\infty ,{\widehat{\mathbf{\mu }}}_s)\right|\mathrm{d}s+{\displaystyle \frac{({\gamma }_1+{\gamma }_2+2R)t}{k}}\hfill \\ \hfill & \le L_R{\int }_0^t\mathbb{E}\left[{\mathbf{1}}_{\{s<{\tau }_R\}}{D}_s\right]\mathrm{d}s+{\displaystyle \frac{({\gamma }_1+{\gamma }_2+2R)t}{k}}.\hfill \end{array}$$

Since $\left|z\right|\le {\phi }_k\left(z\right)+a_{k-1}$, it follows that

$${\displaystyle \sum _{i=1}^2}\mathbb{E}\left|z_{t\wedge {\tau }_R}^i\right|\le L_R{\int }_0^t\mathbb{E}\left[{\mathbf{1}}_{\{s<{\tau }_R\}}{D}_s\right]\mathrm{d}s+{\displaystyle \frac{({\gamma }_1+{\gamma }_2+2R)t}{k}}+2a_{k-1}.$$

Letting $k\to \infty$, the result follows. □

Lemma 3.4. 

There is a constant $C_R<\infty$ such that, for every $t\le T$,

$${\displaystyle \sum _{i=1}^2}\mathbb{E}{\parallel v_{t\wedge {\tau }_R}^i\parallel }_J\le C_R{\int }_0^t\mathbb{E}\left[{\mathbf{1}}_{\{s<{\tau }_R\}}{D}_s\right]\mathrm{d}s.$$

Proof. 

Let $\rho$ be a nonnegative smooth function supported in $[-1,1]$ with integral one, and let ${\rho }_{\epsilon }\left(x\right)={\epsilon }^{-1}\rho (x/\epsilon )$, $0<\epsilon \le 1$. Denote spatial convolution by an asterisk. Set

$$v_s^{i,\epsilon }={\rho }_{\epsilon }*v_s^i,{\kappa }_s^i={\mathcal{K}}_i(·,{\mathbf{\mu }}_s)-{\mathcal{K}}_i(·,{\widehat{\mathbf{\mu }}}_s),G_s^i(a,x)={\mathbf{1}}_{\{a\le u_s^i\left(x\right)\}}-{\mathbf{1}}_{\{a\le {\widehat{u}}_s^i\left(x\right)\}}.$$

Apply (14) with the translated mollifier and then Itô’s formula to ${\phi }_k\left(v_{t\wedge {\tau }_R}^{i,\epsilon }\left(x\right)\right)$. Integrating against $J\left(x\right)\mathrm{d}x$ and taking expectations gives

$$\begin{array}{cc}\hfill \mathbb{E}{\int }_{\mathbb{R}}{\phi }_k\left(v_{t\wedge {\tau }_R}^{i,\epsilon }\right)J\mathrm{d}x=& {\displaystyle \frac{1}{2}}\mathbb{E}{\int }_0^{t\wedge {\tau }_R}{\int }_{\mathbb{R}}{\phi }_k^{\prime }\left(v_s^{i,\epsilon }\right)\mathsf{\Delta }{v}_s^{i,\epsilon }J\mathrm{d}x\mathrm{d}s+{\displaystyle \frac{1}{2}}\mathbb{E}{\int }_0^{t\wedge {\tau }_R}{\int }_{\mathbb{R}}{\phi }_k^{\prime \prime }\left(v_s^{i,\epsilon }\right){\left(v_s^{i,\epsilon }\right)}^{2}J\mathrm{d}x\mathrm{d}s\hfill \\ \hfill & -\mathbb{E}{\int }_0^{t\wedge {\tau }_R}{\int }_{\mathbb{R}}{\phi }_k^{\prime }\left(v_s^{i,\epsilon }\right)({\rho }_{\epsilon }*{\kappa }_s^i)J\mathrm{d}x\mathrm{d}s\hfill \\ \hfill & +{\displaystyle \frac{{\gamma }_i}{2}}\mathbb{E}{\int }_0^{t\wedge {\tau }_R}{\int }_{\mathbb{R}}{\phi }_k^{\prime \prime }\left(v_s^{i,\epsilon }\left(x\right)\right)A_s^{i,\epsilon }\left(x\right)J\left(x\right)\mathrm{d}x\mathrm{d}s,\hfill \end{array}$$

(20)

where

$$A_s^{i,\epsilon }\left(x\right)={\int }_0^{\infty }{\left|({\rho }_{\epsilon }*G_s^i(a,·))\left(x\right)\right|}^2\mathrm{d}a.$$

Every integral is integrable for fixed $k,\epsilon ,R,T$. We estimate the four terms on the right of (20). Twice integrating by parts and using convexity,

$$\begin{array}{c}\hfill {\int }_{\mathbb{R}}{\phi }_k^{\prime }\left(v^{i,\epsilon }\right)\mathsf{\Delta }{v}^{i,\epsilon }J\mathrm{d}x={\int }_{\mathbb{R}}{\phi }_k\left(v^{i,\epsilon }\right)J^{\prime \prime }\mathrm{d}x-{\int }_{\mathbb{R}}{\phi }_k^{\prime \prime }\left(v^{i,\epsilon }\right)|{\partial }_x{v}^{i,\epsilon }{|}^{2}J\mathrm{d}x\le C{\int }_{\mathbb{R}}\left|v^{i,\epsilon }\right|J\mathrm{d}x.\end{array}$$

Since $|{\partial }_{x}logJ|\le 1$, we have ${\parallel {\rho }_{\epsilon }*w\parallel }_J\le e{\parallel w\parallel }_J.$ It follows that the first term is bounded by $C\mathbb{E}{\int }_0^{t\wedge {\tau }_R}{\parallel v_s^i\parallel }_J\mathrm{d}s$. The absolute value of the second term is at most $e\mathbb{E}{\int }_0^{t\wedge {\tau }_R}{\parallel {\kappa }_s^i\parallel }_J\mathrm{d}s$. By Jensen’s inequality and (19), we have

$$A_s^{i,\epsilon }\left(x\right)\le ({\rho }_{\epsilon }*|v_s^i\left|\right)\left(x\right).$$

(21)

For each fixed k, we now let $\epsilon \downarrow 0$. At almost every x, both $v_s^{i,\epsilon }\left(x\right)$ and $({\rho }_{\epsilon }*|v_s^i\left|\right)\left(x\right)$ converge to $v_s^i\left(x\right)$ and $|v_s^i\left(x\right)|$, respectively. Because ${\phi }_k^{\prime \prime }$ is bounded for fixed k and $|v_s^i|\le 2R$ before ${\tau }_R$, by dominated convergence and (21), one sees that

$$\underset{\epsilon \downarrow 0}{lim\; sup}\mathbb{E}{\int }_0^{t\wedge {\tau }_R}{\int }_{\mathbb{R}}{\phi }_k^{\prime \prime }\left(v_s^{i,\epsilon }\right)A_s^{i,\epsilon }J\mathrm{d}x\mathrm{d}s\le \mathbb{E}{\int }_0^{t\wedge {\tau }_R}{\int }_{\mathbb{R}}{\phi }_k^{\prime \prime }\left(v_s^i\right)\left|v_s^i\right|J\mathrm{d}x\mathrm{d}s\le {\displaystyle \frac{{2T\parallel J\parallel }_{L^1}}{k}}.$$

The fourth term in (20) is bounded directly by

$${\displaystyle \frac{1}{k}}\mathbb{E}{\int }_0^{t\wedge {\tau }_R}{\parallel v_s^{i,\epsilon }\parallel }_J\mathrm{d}s\le {\displaystyle \frac{e}{k}}\mathbb{E}{\int }_0^{t\wedge {\tau }_R}{\parallel v_s^i\parallel }_J\mathrm{d}s.$$

Thus, after taking $\epsilon \downarrow 0$, summing over i, and using (3), we obtain

$${\displaystyle \sum _{i=1}^2}\mathbb{E}{\int }_{\mathbb{R}}{\phi }_k\left(v_{t\wedge {\tau }_R}^i\right)J\mathrm{d}x\le C_R{\int }_0^t\mathbb{E}\left[{\mathbf{1}}_{\{s<{\tau }_R\}}{D}_s\right]\mathrm{d}s+{\displaystyle \frac{C_{R,T}}{k}}.$$

Then the result follows by letting $k\to \infty$. □

Theorem 3.5. 

Pathwise uniqueness holds for (14). 

Proof. 

Combining Lemmas 3.3 and 3.4, and ${\mathbf{1}}_{\{s<{\tau }_R\}}{D}_s\le D_{s\wedge {\tau }_R}$, we have

$$\mathbb{E}{D}_{t\wedge {\tau }_R}\le C_R{\int }_0^t\mathbb{E}{D}_{s\wedge {\tau }_R}\mathrm{d}s.$$

By Gronwall’s inequality, $D_{t\wedge {\tau }_R}=0$ almost surely for every fixed t. By the right-continuity of the distribution functions, ${\mathbf{\mu }}_{t\wedge {\mathbf{\tau }}_R}={\widehat{\mathbf{\mu }}}_{t\wedge {\mathbf{\tau }}_R}$ almost surely for each $t\ge 0$. Taking rational t and using the weak continuity of both processes, this equality holds simultaneously for all $t\ge 0$. Since ${\tau }_R\uparrow \infty$ almost surely, letting $R\to \infty$ shows that $\mathbf{\mu }$ and $\widehat{\mathbf{\mu }}$ are indistinguishable. □

3.3. Uniqueness for the Martingale Problem

Proof 

(Proof of Theorem 1.2). Existence follows from Proposition 2.3, and the moment bound follows from Lemma 3.1. By Proposition 3.2, every solution admits the SPDE representation (14), for which pathwise uniqueness holds by Theorem 3.5. At each time, future noise increments are independent of the solution filtration up to that time, so these weak solutions satisfy the compatibility condition of Kurtz [21]. The general Yamada–Watanabe theorem therefore yields uniqueness in law, and Proposition 3.2 gives uniqueness for (1)–(2). The result follows. □

4. Conclusions

In this paper, we have studied two competing superprocesses in independent Brownian environments. The competition rates may depend on the spatial position and on the current measures of both populations. By combining particle approximations with an SPDE representation of the distribution functions, we have established the following results:

1.

Under Assumption 1.1 (i), the martingale problem (1)–(2) admits a continuous solution from every pair of finite initial measures.

2.

Under Assumption 1.1, the law of the solution on $C({\mathbb{R}}_{+},E)$ is unique. This follows from the SPDE representation and pathwise uniqueness for the associated system of distribution functions.

3.

Every solution satisfies, for $T>0$ and $p\in \{2,4\}$,

$$\mathbb{E}\left[\underset{0\le t\le T}{sup}{\left(\langle {\mu }_t^1,1\rangle +\langle {\mu }_t^2,1\rangle \right)}^p\right]\le C_{p,T}\left[1+{\left(\langle {\mu }_0^1,1\rangle +\langle {\mu }_0^2,1\rangle \right)}^p\right],$$

where $C_{p,T}$ depends only on $p,T,{\gamma }_1,{\gamma }_2$.

The existence proof requires only continuity, nonnegativity and local boundedness of the competition rates. The additional local Lipschitz condition on the cumulative competition terms is used to prove pathwise uniqueness, through simultaneous estimates of the distribution functions and total masses. In particular, global boundedness of the competition rates is not required.

Further questions include extensions to more than two populations and correlated environmental noises, as well as the long-time behaviour of the competing system.