2.1. McKean–Vlasov branching diffusion

Let us introduce first the McKean–Vlasov branching diffusion process by means of a pathwise description. We consider the coefficient functions

\begin{align*} \big(b,\sigma,\gamma,(p_{\ell})_{\ell\in\mathbb{N}}\big) \colon [0,T]\times\mathcal{C}^d\times\mathcal{P}(\mathcal{D}_E) \longrightarrow \mathbb{R}^d\times\mathbb{R}^{d\times d}\times[0,\bar\gamma]\times[0,1]^{\mathbb{N}},\end{align*}

where $\bar \gamma > 0$ is a fixed constant. Namely, b and $\sigma$ are the drift and diffusion coefficients for the movement of each particle, $\gamma$ is the death rate, and $(p_{\ell})_{\ell \in \mathbb{N}}$ is the probability mass function of the progeny distribution. In particular, $p_{\ell}(\!\cdot\!) \in [0,1]$ for each $\ell \in \mathbb{N}$ , and $\sum_{\ell \in \mathbb{N}} p_{\ell}(\!\cdot\!) = 1$ . Let us also define a partition $(I_{\ell}(\!\cdot\!))_{\ell \in \mathbb{N}}$ of [0,1) by $I_{\ell}(\!\cdot\!) \,:\!=\, \big[\sum_{i=0}^{\ell -1}p_i(\!\cdot\!),\sum_{i=0}^{\ell}p_i(\!\cdot\!)\big)$ for each $\ell\in\mathbb{N}$ .

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space with filtration $\mathbb{F} = (\mathcal{F}_t)_{t \ge 0}$ , equipped with an $\mathcal{F}_0$ -measurable E-valued random variable $\xi$ , and a family of mutually independent, $\mathbb{R}^d$ -valued Brownian motions $(W^k)_{k\in\mathbb{K}}$ and Poisson random measures $(Q^k(ds,dz))_{k\in\mathbb{K}}$ on $[0,T] \times [0, \bar \gamma] \times [0,1]$ with Lebesgue intensity measure ${\mathrm{d}} s$ , ${\mathrm{d}} z$ .

We consider first a branching diffusion process with a fixed environment measure $\mu\in \mathcal{P}(\mathcal{D}_E)$ and initial state $\xi$ . It corresponds to an E-valued process $(Z_t)_{t \in [0,T]}$ given by

(4) \begin{equation} Z_t \,:\!=\, \sum_{k\in K_t}\delta_{(k,X^k_t)}, \quad t \in [0,T],\end{equation}

where $K_t$ denotes the collection of all labels of particles alive at time $t \in [0,T]$ and $X^k_t$ corresponds to the position of particle $k\in K_t$ . In particular, we have $Z_0 = \xi = \sum_{k \in K_0} \delta_{(k, X^k_0)}$ . Then, for each $k\in K_t$ , the position $X^k_t$ evolves as a diffusion process characterized by the SDE

(5) \begin{equation} {\mathrm{d}}X^k_t = b\big(t,X^k_{t\wedge\cdot},\mu_t\big)\,{\mathrm{d}} t + \sigma\big(t,X^k_{t\wedge\cdot},\mu_t\big)\,{\mathrm{d}} W^k_t,\end{equation}

where $\mu_t$ is the pushforward measure $\mu_t \,:\!=\, \mu \circ \pi_t^{-1}$ . We also denote by $S_k$ the birth time of particle k. In particular, $S_k = 0$ for each initial particle $k \in K_0$ . Then, each particle k runs a death clock with intensity $\gamma(t, X^k_{t\wedge\cdot}, \mu_t)$ , i.e. its death time $T_k$ is given by

\begin{align*} T_k \,:\!=\, \inf\big\{t>S_k \colon Q^k\big(\{t\}\times\big[0,\gamma\big(t,X^k_{t\wedge\cdot},\mu_t\big)\big] \times[0,1] \big) = 1\big\}.\end{align*}

Let $U_k$ be a random variable satisfying $Q^k \big(\{T_k\} \times [0,\gamma(T_k,X^k_{T_k\wedge\cdot},\mu_{T_k})] \times \{U_k\} \big) =1$ , and note that $U_k$ is uniformly distributed over the interval [0,1]. When $U_k \in I_{\ell}\big(T_k, X^k_{T_k\wedge\cdot}, \mu_{T_k}\big)$ , at time $T_k$ the particle k dies and gives birth to $\ell$ offspring particles $\{k1, \ldots, k \ell \}$ , so that

\begin{equation*} K_{T_k} = (K_{T_k-} \setminus \{k\}) \cup \{k1, \ldots,k \ell\}.\end{equation*}

In particular, the birth time of the offspring particles corresponds to the death time of the parent particle, i.e. $S_{ki}\,:\!=\, T_k$ for $i = 1, \ldots , \ell$ . Further, the offspring particles start from the position of the parent particle, i.e. $X^{ki}_{S_{ki}} = X^k_{T_k-}$ for $i = 1, \ldots, \ell$ . Moreover, for a path-dependent treatment, we define $X^{ki}_t$ as its ancestor position before the birth time, i.e. $X^{ki}_t = X^k_t$ for $t< S_k$ and $i = 1, \ldots , \ell$ .

Then the McKean–Vlasov branching diffusion corresponds to the situation where the environment measure $\mu$ coincides with the law of the branching diffusion process Z, i.e. $\mu = \mathbb{P}\circ Z^{-1}$ , or equivalently, $\mu_t = \mathbb{P} \circ (Z_{t \wedge \cdot})^{-1}$ for all $t\in[0,T]$ .

Alternatively, let us provide a characterization of the McKean–Vlasov branching diffusion described above through a family of SDEs. Denote by $\mathcal{L}$ the infinitesimal generator of the diffusion $(b,\sigma)$ , i.e., for all $(t, \mathrm{x} ,m)\in[0,T]\times\mathcal{C}^d\times\mathcal{P}(\mathcal{D}_E)$ and $f\in C^{2}_{\mathrm{b}}(\mathbb{R}^{d},\mathbb{R})$ ,

\begin{equation*} \mathcal{L} f(t, \mathrm{x}, m) \,:\!=\, \frac{1}{2}\operatorname{Tr}(\sigma\sigma^{\top}(t,\mathrm{x},m)D^2f(\mathrm{x}_t)) + b(t,\mathrm{x},m)\cdot Df(\mathrm{x}_t).\end{equation*}

Then the McKean–Vlasov branching diffusion with initial condition $\xi$ can be characterized as the solution, for all $f=(f^k)_{k\in\mathbb{K}} \in C^2_{\mathrm{b}} (\mathbb{K} \times \mathbb{R}^d, \mathbb{R})$ , $t\in[0,T]$ , to the SDE

(6) \begin{align} \langle Z_t, f \rangle & = \langle\xi ,f\rangle + \int_0^t\sum_{k\in K_s}\mathcal{L}f^k\big(s,X^k_{s\wedge\cdot},\mu_s\big)\,{\mathrm{d}} s \nonumber \\[2pt] & \quad + \int_0^t\sum_{k\in K_s}Df^k(X^k_s)\sigma\big(s,X^k_{s\wedge\cdot},\mu_s\big)\,{\mathrm{d}} W^k_s \nonumber \\[2pt] & \qquad\quad\; \times \int_{(0,t]\times[0,\bar\gamma]\times[0,1]}\sum_{k\in K_{s-}}\sum_{\ell\geq0}\Bigg(\sum_{i=1}^{\ell}f^{ki}-f^k\Bigg) (X^k_s) \nonumber \\[2pt] & \qquad\qquad\qquad \times \mathbf{1}_{[0,\gamma(s,X^k_{s\wedge\cdot},\mu_s)]\times I_{\ell}(s,X^k_{s\wedge\cdot},\mu_s)}(z)\, Q^k({\mathrm{d}} s,{\mathrm{d}} z) \end{align}

with the condition

(7) \begin{equation} \mu_t = \mathbb{P} \circ (Z_{t\wedge \cdot})^{-1} \quad \text{for all }t \in [0,T].\end{equation}

2.2. Strong solution and well-posedness

Let us first consider the strong formulation of the McKean–Vlasov branching diffusion.

Under the following assumptions, we can show the existence and uniqueness of a McKean–Vlasov branching diffusion in the strong sense. Recall that $\mathcal{D}_E$ is equipped with the uniform convergence metric defined in (2) so that the Wasserstein distance $\mathcal{W}_1$ on $\mathcal{P}_1(\mathcal{D}_E)$ is defined with respect to this metric.

Remark 2. Let us provide a simple example of a Lipschitz function on $\mathcal{P}_1(\mathcal{D}_E)$ . Let $f = (f^k)_{k \in \mathbb{K}}\colon \mathbb{K} \times \mathbb{R}^d \to \mathbb{R}$ and assume that there exists $C>0$ such that $|f^k(x)| \le C$ and $|f^k(x)-f^k(y)| \le C|x-y|$ for all $k\in\mathbb{K}$ , $x,y\in\mathbb{R}^d$ . Define $F\colon [0,T]\times \mathcal{P}_1(\mathcal{D}_E) \to \mathbb{R}$ by

\begin{equation*} F(t,m) \,:\!=\, \int_{\mathcal{D}_E} \langle z(t), f\rangle \,m({\mathrm{d}} z). \end{equation*}

Then we can easily check that $| F(t,m^1) - F(t,m^2)| \le C \mathcal{W}_1(m^1_t,m^2_t)$ for all $m^1, m^2 \in \mathcal{P}_1(\mathcal{D}_E)$ . Indeed, given $Z^1_t = \sum_{k\in K_t}\delta_{(k,X^{1,k}_t)}$ and $Z^2_t = \sum_{k \in K'_t} \delta_{(k, X^{2,k}_t)}$ , two arbitrary càdlàg E-valued processes with distributions $m^1$ and $m^2$ respectively, it follows by straightforward computation that

\begin{equation*} |F(t, m^1) - F(t, m^2)| = \Bigg|\mathbb{E}\Bigg[\sum_{k\in K^1_t}f^k(X^{1,k}_t) - \sum_{k\in K^2_t}f^k(X^{2,k}_t)\Bigg]\Bigg| \le C\mathbb{E}[d_E(Z^1_t,Z^2_t)]. \end{equation*}

The proof of Theorem 1 relies on a contraction argument; it is postponed to Section 3.2. Additionally, by using the estimates needed for this contraction argument, we can easily establish a companion stability property for McKean–Vlasov branching diffusion with respect to the initial condition and the coefficient functions. See Appendix A for more details.

2.3. Weak solution and propagation of chaos

As in classical SDE theory, we can consider the notion of weak solution by letting the probability space as well as the Brownian motions and the Poisson random measures be part of the solution. This allows us to establish the existence of McKean–Vlasov branching diffusion under relaxed conditions on the coefficients. Since there is no fixed probability space, we are given a measure $m_0 \in \mathcal{P}(E)$ as an initial condition rather than a random variable $\xi$ with distribution $m_0$ .

Under the following assumption, we can show the existence of a weak solution to the McKean–Vlasov branching diffusion SDE. Recall that $\mathcal{D}_E$ is equipped with the classical Skorokhod metric so that the continuity below has to be understood witb respect to this metric.

Theorem 2 is an immediate consequence of a propagation of chaos property established in Theorem 3. Namely, we consider a large number n of branching diffusion processes interacting through their empirical distribution in the sense of Definition 3, and we show that, when n goes to infinity, it converges to a McKean–Vlasov branching diffusion in the weak sense.

The proof of Theorem 3 is postponed to Section 4.3. It relies on the weak convergence of solutions to appropriate martingale problems, equivalent to the notion of weak solutions.

3.1. Technical lemmas

We start by studying the solution to SDE (6) when the environment measure is fixed. More precisely, assume that we are given a deterministic $\mu \in \mathcal{P}_1(\mathcal{D}_E)$ and consider $Z_t=\sum_{k\in K_t} {\delta_{(k,X^k_t)}}$ satisfying, for all $f \in C^2_{\mathrm{b}} (\mathbb{K} \times \mathbb{R}^d, \mathbb{R})$ , $t\in[0,T]$ ,

(10) \begin{align} \langle Z_t, f \rangle & = \langle\xi ,f\rangle + \int_0^t\sum_{k\in K_s}\mathcal{L}f^k(s,X^k_{s\wedge\cdot},\mu_s)\,{\mathrm{d}} s \nonumber \\ & \quad + \int_0^t\sum_{k\in K_s}D\,f^k(X^k_s)\sigma(s,X^k_{s\wedge\cdot},\mu_s)\,{\mathrm{d}} W^k_s \nonumber \\ & \quad + \int_{(0,t]\times [0,\bar \gamma] \times [0,1]}\sum_{k\in K_{s-}}\sum_{ \ell \geq 0} \Bigg(\sum_{i=1}^{\ell}{f^{ki}} - f^k\Bigg)(X^k_s) \nonumber \\ & \qquad\qquad\! \times \mathbf{1}_{[0,\gamma(s,X^k_{s\wedge\cdot},\mu_s)]\times I_{\ell}(s,X^k_{s\wedge\cdot},\mu_s)}(z)\, Q^k({\mathrm{d}} s,{\mathrm{d}} z). \end{align}

By a simple extension of [Reference Claisse4, Proposition 2.1], existence and uniqueness of the strong solution holds for the above SDE.

Proof. The proof of existence and uniqueness relies on the fact that the Lipschitz assumption on b and $\sigma$ ensures the existence and uniqueness of the strong solution to SDE (5) characterizing the movement of each particle. Then we can construct the process step by step by considering the successive branching times. It remains to rule out explosion: this follows from the boundedness assumption on $\gamma$ and $\sum_{\ell\geq0} \ell p_\ell$ , which ensures that the population size has finite moments (see below) and thus remains finite. We refer to [Reference Claisse4, Proposition 2.1] for more details.

Let us turn now to the moment estimate. Denote by $\bar{N}_t\,:\!=\,\sum_{k\in\mathbb{K}}\sup\nolimits_{0\leq t\leq T}\mathbf{1}_{k\in K_t}$ the total number of particles having been alive in the time interval [0, T], and observe that it can be viewed as a modified branching process where particles give birth to new ones without dying. Then

\begin{equation*} \bar{N}_t = \#K_0 + \int_{(0,t]\times[0,\bar \gamma]\times[0,1]}\sum_{k\in K_{s-}}\sum_{\ell\geq0}\ell \mathbf{1}_{I_\ell(s,X^k_{s\wedge\cdot},\mu_s)\times[0,\gamma(s,X^k_{s\wedge\cdot},\mu_s)]}(z)\,Q^k({\mathrm{d}} s,{\mathrm{d}} z). \end{equation*}

We introduce the stopping times $\tau_n\,:\!=\,\inf\{t\geq0\colon\bar{N}_t\geq n\}$ . Then

\begin{align*} \mathbb{E}[\bar{N}_{T\wedge\tau_n}] & = \mathbb{E}[\#K_0] + \mathbb{E}\bigg[\int_0^{T\wedge\tau_n}\sum_{k\in K_t}\sum_{\ell\geq0} \ell p_\ell\big(t,X^k_{t\wedge\cdot},\mu_t\big)\gamma\big(t,X^k_{t\wedge\cdot},\mu_t\big)\,\unicode{x1E6D} \bigg] \\[2pt] & \leq \mathbb{E}[\#K_0] + \bar\gamma M\mathbb{E}\bigg[\int_0^{T\wedge\tau_n}\#{K_t}\,{\mathrm{d}} t\bigg] \leq \mathbb{E}[\#K_0] + \bar\gamma M\mathbb{E}\bigg[\int_0^T\bar{N}_{t\wedge\tau_n}\,{\mathrm{d}} t\bigg]. \end{align*}

By Grönwall’s lemma, we deduce that $\mathbb{E}[\bar{N}_{T\wedge\tau_n}] \leq \mathbb{E}[\#K_0]{\mathrm{e}}^{\bar\gamma M T}$ . The conclusion follows immediately from Fatou’s lemma.

We now provide a stability result for the branching diffusion solution to SDE (10) with respect to the environment measure. It is the key to the contraction argument used in the proof of Theorem 1 and the main technical difficulty. As a preliminary result, we first establish the following technical lemma.

Proof. Fix $(t,\mathrm{x},\mathrm{x}',m,m')\in [0,T]\times\mathcal{C}^d\times\mathcal{C}^d\times \mathcal{P}_1(\mathcal{D}_E)\times\mathcal{P}_1(\mathcal{D}_E)$ and consider the two probability measures $P \,:\!=\,(p_\ell(t,\mathrm{x},m))_{\ell\in\mathbb{N}}$ and $P'\,:\!=\,(p_\ell(t,\mathrm{x}',m'))_{\ell\in\mathbb{N}}$ , which belong to $\mathcal{P}_1(\mathbb{N})$ in view of Assumption 1. Denote the cumulative distribution functions of P and P ^′ as F and F ^′ respectively, and observe that the generalized inverses are given, for all $u\in[0,1]$ , as $F^{-1}(u) = \sum_{\ell\geq0}\ell\mathbf{1}_{I_\ell}(u)$ and $(F')^{-1}(u) = \sum_{\ell\geq0}\ell\mathbf{1}_{I'_{\ell}}(u)$ , where $I_\ell\,:\!=\,I_\ell(t,\mathrm{x},m)$ and $I'_{\ell}\,:\!=\,I_{\ell}(t,\mathrm{x}',m')$ for $\ell\in\mathbb{N}$ . Using the well-known representation of Wasserstein metric for one-dimensional distributions, we have

\begin{equation*} \mathcal{W}_1(P,P') = \int_0^1|F^{-1}(u)-(F')^{-1}(u)|\,{\mathrm{d}} u = \int_0^1\sum_{\ell,\ell'\geq0}|\ell-\ell'|\mathbf{1}_{I_\ell\cap I'_{\ell'}}(u)\,{\mathrm{d}} u. \end{equation*}

In addition, by a change of variable, we also have that

\begin{align*} \mathcal{W}_1(P,P') = \sum_{j\geq0}|F(j)-F'(j)| & = \sum_{j\geq0}\bigg|\sum_{\ell\leq j}p_\ell(t,\mathrm{x},m) - \sum_{\ell\leq j}p_\ell(t,\mathrm{x}',m')\bigg| \\[2pt] & \leq \sum_{j\geq0}\sum_{\ell>j}|p_\ell(t,\mathrm{x},m)-p_\ell(t,\mathrm{x}',m')| \\[2pt] & = \sum_{\ell\geq 0}\ell|p_\ell(t,\mathrm{x},m)-p_\ell(t,\mathrm{x}',m')|. \end{align*}

The conclusion follows immediately.

Lemma 3. Let $\mu^1,\mu^2\in\mathcal{P}_1(\mathcal{D}_E)$ . Denote by $Z^i$ the solution to SDE (10) given $\mu^i$ , $Z^i_0=\xi$ for $i=1,2$ respectively. Then there exist constants $c_d > 0$ , $c_w> 0$ depending on L, $\bar \gamma$ , M, and M^′ such that, for all $T' \in [0,T]$ satisfying $T'+\sqrt{T'}<1/c_d$ ,

\begin{equation*} \mathbb{E}[d_{T'}(Z^1,Z^2)] \leq \frac{c_w(T'+\sqrt{T'})}{1-c_d(T'+\sqrt{T'})} \mathbb{E}[\#K_0]{\mathrm{e}}^{\bar\gamma M T'}\mathcal{W}_1\big(\mu^1_{T'},\mu^2_{T'}\big). \end{equation*}

Proof. Fix some $T'\in[0,T]$ to be determined later. For $t\in[0,T']$ , consider the representations $Z^1_t = \sum_{k\in K^1_t}\delta_{(k,X^{1,k}_t)}$ and $Z^2_t = \sum_{k\in K^2_t}\delta_{(k,X^{2,k}_t)}$ . Let us write $K^{1\cap2}_t \,:\!=\, K^{1}_t \cap K^{2}_t$ and $K^{1\triangle2}_t \,:\!=\, K^{1}_t \triangle K^{2}_t$ . Recall that

(11) \begin{equation} d_{T'}(Z^1,Z^2) = \sup\nolimits_{0\le t\le T'}\Bigg\{\#K^{1\triangle 2}_t + \sum_{k\in K^{1\cap2}_t}\big|X^{1,k}_t-X^{2,k}_t\big|\wedge1\Bigg\}. \end{equation}

Step 1

We start by estimating the first term in (11), counting the number of particles with distinct labels in $Z^1$ and $Z^2$ . For simplicity, for $i = 1,2$ , $k\in\mathbb{K},$ and $t\in[0,T]$ , we write $\gamma^{i,k}_t \,:\!=\, \gamma(t,X^{i,k}_{t\wedge\cdot},\mu^i_t)$ , $I^{i,k}_{\ell,t} \,:\!=\, I_\ell(t,X^{i,k}_{t\wedge\cdot},\mu^i_t)$ , and $p^{i,k}_{\ell,t}\,:\!=\,p_\ell(t,X^{i,k}_{t\wedge\cdot},\mu^i_t)$ . We begin with the following observation:

(12) \begin{equation} \#K^{1\triangle 2}_t\leq J^1_t+J^2_t+J^3_t+J^4_t+J^5_t, \end{equation}

where

\begin{align*} J^1_t & = \int_{(0,t]\times[0,\bar\gamma]\times[0,1]}\sum_{k\in K^1_{s-}\setminus K^2_{s-}}\sum_{\ell\geq0} \ell\mathbf{1}_{I^{1,k}_{\ell,s}\times[0,\gamma^{1,k}_s]}(z)\,Q^k({\mathrm{d}} s,{\mathrm{d}} z), \\[2pt] J^2_t & = \int_{(0,t]\times[0,\bar\gamma]\times[0,1]}\sum_{k\in K^2_{s-}\setminus K^1_{s-}}\sum_{\ell\geq0} \ell\mathbf{1}_{I^{2,k}_{\ell,s}\times[0,\gamma^{2,k}_s]}(z)\,Q^k({\mathrm{d}} s,{\mathrm{d}} z), \\[2pt] J^3_t & = \int_{(0,t]\times[0,\bar\gamma]\times[0,1]}\sum_{k\in K^{1\cap 2}_{s-}}\mathbf{1}_{\gamma^{1,k}_s>\gamma^{2,k}_s} \sum_{\ell\geq0}(\ell+1)\mathbf{1}_{I^{1,k}_{\ell,s}\times (\gamma^{2,k}_s, \gamma^{1,k}_s]}(z)\,Q^k({\mathrm{d}} s,{\mathrm{d}} z), \\[2pt] J^4_t & = \int_{(0,t]\times[0,\bar\gamma]\times[0,1]}\sum_{k\in K^{1\cap 2}_{s-}}\mathbf{1}_{\gamma^{1,k}_s<\gamma^{2,k}_s} \sum_{\ell\geq0}(\ell+1)\mathbf{1}_{I^{2,k}_{\ell,s}\times (\gamma^{1,k}_s,\gamma^{2,k}_s]}(z)\,Q^k({\mathrm{d}} s,{\mathrm{d}} z), \\[2pt] J^5_t & = \int_{(0,t]\times[0,\bar\gamma]\times[0,1]}\sum_{k\in K^{1\cap 2}_{s-}}\sum_{\ell,\ell'\geq0}|\ell-\ell'| \mathbf{1}_{I^{1,k}_{\ell,s}\cap I^{2,k}_{\ell',s}\times[0,\gamma^{1,k}_s\wedge\gamma^{2,k}_s]}(z)\,Q^k({\mathrm{d}} s,{\mathrm{d}} z). \end{align*}

To obtain this estimate, we distinguish between particles born from a parent in the common set $K^{1\cap2}_t$ or not. In the latter case, we assume that all newly born particles also lie in the set $K^{1\triangle 2}_t$ . Then $J^1_t$ and $J^2_t$ correspond to the case when parents are from $K^1_t\setminus K^2_t$ and $K^2_t\setminus K^1_t$ respectively. In the former case, when parents are from $K^{1\cap2}_t$ , the terms $J^3_t$ and $J^4_t$ take care of the case when a particle in one tree branches while the other particle sharing the same label does not. The last term $J_t^5$ concerns particles in two trees sharing a common label branching simultaneously, but giving birth to different numbers of progeny.

Let us deal with each term in (12) separately. Firstly, we have

\begin{align*} \mathbb{E}\big[\!\sup\nolimits_{0\leq t\leq T'}J^1_t\big] & = \mathbb{E}\Bigg[\int_{(0,T']\times[0,\bar\gamma]\times[0,1]}\sum_{k\in K^1_{t-}\setminus K^2_{t-}}\sum_{\ell\geq 0} \ell\mathbf{1}_{I^{1,k}_{\ell,t}\times[0,\gamma^{1,k}_t]}(z)\,Q^k({\mathrm{d}} t,{\mathrm{d}} z)\Bigg] \\[2pt] & = \mathbb{E}\Bigg[\int_0^{T'}\sum_{k\in K^1_t\setminus K^2_t}\gamma^{1,k}_t\sum_{\ell\geq 0}\ell p^{1,k}_{\ell,t}\, {\mathrm{d}} t\Bigg] \leq \bar\gamma M\mathbb{E}\bigg[\int_0^{T'}\#(K^1_t\setminus K^2_t)\,{\mathrm{d}} t\bigg]. \end{align*}

Similarly, we have $\mathbb{E}\big[\!\sup\nolimits_{0\leq t\leq T'}J^2_t\big]\leq\bar\gamma M\mathbb{E}\big[ \int_0^{T'}\#(K^2_t\setminus K^1_t)\,{\mathrm{d}} t\big]$ . We deduce that

(13) \begin{equation} \mathbb{E}\big[\!\sup\nolimits_{0\leq t\leq T'}\{J^1_t + J^2_t\}\big] \leq \bar\gamma M\mathbb{E}\bigg[\int_0^{T'}\#K^{1\triangle 2}_t\,{\mathrm{d}} t\bigg]. \end{equation}

In addition, using the same approach,

(14) \begin{equation} \mathbb{E}\big[\!\sup\nolimits_{0\leq t\leq T'}\{J^3_t+J^4_t\}\big] \leq (M+1)\mathbb{E}\Bigg[\int_0^{T'}\sum_{k\in K^{1\cap2}_t}\big|\gamma^{1,k}_t-\gamma^{2,k}_t\big|\,{\mathrm{d}} t\Bigg]. \end{equation}

By the Lipschitz condition on $\gamma$ from Assumption 2, it follows that

(15) \begin{equation} \mathbb{E}\big[\!\sup\nolimits_{0\leq t\leq T'}\{J^3_t+J^4_t\}\big] \leq L(M+1)\mathbb{E}\Bigg[\int_0^{T'}\sum_{k\in K^{1\cap2}_t}\big( \big\|X^{1,k}_{t\wedge\cdot}-X^{2,k}_{t\wedge\cdot}\big\| + \mathcal{W}_1(\mu^1_t,\mu^2_t)\big)\,{\mathrm{d}} t\Bigg]. \end{equation}

Finally, using Lemma 2, we obtain that

(16) \begin{align} \mathbb{E}\big[\!\sup\nolimits_{0\leq t\leq T'}J^5_t\big] & \leq \bar\gamma\mathbb{E}\Bigg[\int_0^{T'}\sum_{k\in K^{1\cap2}_t}\int_0^1\sum_{\ell,\ell'\geq0}|\ell-\ell'| \mathbf{1}_{I^{1,k}_{\ell,t}\cap I^{2,k}_{\ell',t}}(u)\,{\mathrm{d}} u\,{\mathrm{d}} t\Bigg] \nonumber \\[2pt] & \leq \bar\gamma\mathbb{E}\Bigg[\int_0^{T'}\sum_{k\in K^{1\cap2}_t}\sum_{\ell\ge 0} \ell\big|p^{1,k}_{\ell,t}-p^{2,k}_{\ell,t}\big|\,{\mathrm{d}} t\Bigg]. \end{align}

Thus, by the Lipschitz assumption on $(p_\ell)_{\ell\in\mathbb{N}}$ from Assumption 2, we deduce as above that

(17) \begin{equation} \mathbb{E}\big[\!\sup\nolimits_{0\leq t\leq T'}J^5_t\big] \leq \bar\gamma M'\mathbb{E}\Bigg[\int_0^{T'}\sum_{k\in K^{1\cap2}_t} \big(\big\|X^{1,k}_{t\wedge\cdot}-X^{2,k}_{t\wedge\cdot}\big\| + \mathcal{W}_1(\mu^1_t,\mu^2_t)\big)\,{\mathrm{d}} t\Bigg]. \end{equation}

We finish this step by combining (12), (13), (15), and (17) to obtain

(18) \begin{equation} \mathbb{E}\big[\!\sup\nolimits_{0\leq t\leq T'}\#K^{1\triangle 2}_t\big] \leq C\mathbb{E}\Bigg[\int_0^{T'}\Bigg(\#K^{1\triangle 2}_t + \sum_{k\in K^{1\cap2}_t}\big(\big\|X^{1,k}_{t\wedge\cdot}-X^{2,k}_{t\wedge\cdot}\big\| + \mathcal{W}_1(\mu^1_t,\mu^2_t)\big)\Bigg)\,{\mathrm{d}} t\Bigg], \end{equation}

where the constant $C>0$ depends on $(\bar \gamma,L,M,M')$ .

Step 2

Now we turn to the second term in (11) measuring the distance between particles with common labels in $Z_t^1$ and $Z_t^2$ . For simplicity, for $i = 1,2$ , $k\in\mathbb{K}$ , and $t\in[0,T]$ we write $b^{i,k}_t \,:\!=\, b(t,X^{i,k}_{t\wedge\cdot},\mu^i_t)$ and $\sigma^{i,k}_t \,:\!=\, \sigma(t,X^{i,k}_{t\wedge\cdot},\mu^i_t)$ . We observe first that, for any $k\in K^{1\cap2}_t$ ,

\begin{equation*} X^{1,k}_t-X^{2,k}_t = X^{1,k}_{\bar{S}^k} - X^{2,k}_{\bar{S}^k} + \int_0^t\mathbf{1}_{k\in K^{1\cap2}_s}\big(b^{1,k}_s - b^{2,k}_s\big)\,{\mathrm{d}} s + \int_0^t\mathbf{1}_{k\in K^{1\cap2}_s}\big(\sigma^{1,k}_s - \sigma^{2,k}_s\big)\,{\mathrm{d}} W^k_s, \end{equation*}

where $\bar{S}^k=\max(S^{1,k},S^{2,k})$ is the maximum of both times of birth of particle k, so that $k\in K^{1\cap2}_s$ whenever $s\in[\bar{S}^k,t]$ . It follows that

(19) \begin{align} \mathbf{1}_{k\in K^{1\cap2}_t}\big\|X^{1,k}_{t\wedge\cdot}-X^{2,k}_{t\wedge\cdot}\big\| & \le \mathbf{1}_{k\in K^{1\cap2}_t}\big\|X^{1,k}_{\bar{S}^k\wedge\cdot}-X^{2,k}_{\bar{S}^k\wedge\cdot}\big\| + \int_0^t\mathbf{1}_{k\in K^{1\cap2}_s}\big|b^{1,k}_s - b^{2,k}_s\big|\,{\mathrm{d}} s \nonumber \\[2pt] & \quad + \sup\nolimits_{0\le r\le t}\bigg|\int_0^r\mathbf{1}_{k\in K^{1\cap2}_s}\big(\sigma^{1,k}_s - \sigma^{2,k}_s\big)\, {\mathrm{d}} W^k_s\bigg|. \end{align}

Let us deal first with the last two terms on the right-hand side of (19). By using the Lipschitz condition on b and $\sigma$ from Assumption 2, we have, for all $k\in\mathbb{K}$ ,

\begin{equation*} \mathbb{E}\bigg[\int_0^{t}\mathbf{1}_{k\in K^{1\cap2}_s}\big|b^{1,k}_s - b^{2,k}_s\big|\,{\mathrm{d}} s\bigg] \leq L\mathbb{E}\bigg[\int_0^{t}\mathbf{1}_{k\in K^{1\cap2}_s}\big(\big\|X^{1,k}_{s\wedge\cdot}-X^{2,k}_{s\wedge\cdot}\big\| + \mathcal{W}_1(\mu^1_s,\mu^2_s)\big)\,{\mathrm{d}} s\bigg], \end{equation*}

as well as

\begin{align*} & \mathbb{E}\bigg[\sup\nolimits_{0\leq r\leq t}\bigg|\int_0^r\mathbf{1}_{k\in K^{1\cap2}_s} \big(\sigma^{1,k}_s-\sigma^{2,k}_s\big)\,{\mathrm{d}} W^k_s\bigg|\bigg] \\[3pt] & \qquad \leq C_1\mathbb{E}\bigg[\bigg(\int_0^{t}\mathbf{1}_{k\in K^{1\cap2}_s} \big|\sigma^{1,k}_s - \sigma^{2,k}_s\big|^2\,{\mathrm{d}} s\bigg)^{{1}/{2}}\bigg] \\[3pt] & \qquad \leq C_1\sqrt{t}\mathbb{E}\big[\!\sup\nolimits_{0\leq s\leq t}\mathbf{1}_{k\in K^{1\cap2}_s} \big|\sigma^{1,k}_s - \sigma^{2,k}_s\big|\big] \\[3pt] & \qquad \leq C_1L\sqrt{t}\mathbb{E}\big[\!\sup\nolimits_{0\leq s\leq t}\mathbf{1}_{k\in K^{1\cap2}_s} \big(\big\|X^{1,k}_{s\wedge\cdot}-X^{2,k}_{s\wedge\cdot}\big\| + \mathcal{W}_1(\mu^1_s,\mu^2_s)\big)\big], \end{align*}

where the first inequality follows from the Burkholder–Davis–Gundy inequality. Regarding the first term on the right-hand side of (19), we distinguish between particles whose parents are in the common set or not. In the latter case, we use the fact that $\big\|X^{1,k}_{\bar{S}^k\wedge\cdot}-X^{2,k}_{\bar{S}^k\wedge\cdot}\big\|\le 1$ so that

\begin{align*} \mathbb{E}\Bigg[\sum_{k\in\mathbb{K}}\sup\nolimits_{0\le t\le T'}\mathbf{1}_{k\in K^{1\cap2}_t,k^-\in K^{1\triangle 2}_{\bar{S}^k-}} \big\|X^{1,k}_{\bar{S}^k\wedge\cdot}-X^{2,k}_{\bar{S}^k\wedge\cdot}\big\|\Bigg] & \le \mathbb{E}\Bigg[\sum_{k\in\mathbb{K}}\sup\nolimits_{0\le t\le T'} \mathbf{1}_{k\in K^{1}_t\cup K^{2}_t,k^-\in K^{1\triangle 2}_{\bar{S}^k-}}\Bigg] \\[2pt] & = \mathbb{E}\big[J^1_{T'} + J_{T'}^2\big] \le \bar\gamma M\mathbb{E}\bigg[\int_0^{T'}\#K^{1\triangle 2}_t\,{\mathrm{d}} t\bigg], \end{align*}

where $k^-$ denotes the parent of particle k, and the last inequality follows from (13). Note that $\sum_{k\in\mathbb{K}}\sup\nolimits_{0\le t\le T'}\mathbf{1}_{k\in K^{1}_t\cup K^{2}_t,k^-\in K^{1\triangle 2}_{\bar{S}^k-}}$ corresponds to the total number of particles born from a parent in the distinct set $K^{1\triangle 2}$ before time T ^′. Alternatively, when the parent belongs to the common set, we have

\begin{align*} & \mathbb{E}\Bigg[\sum_{k\in\mathbb{K}}\sup\nolimits_{0\le t\le T'}\mathbf{1}_{k\in K^{1\cap2}_t,k^-\in K^{1\cap 2}_{\bar{S}^k-}} \big\|X^{1,k}_{\bar{S}^k\wedge\cdot}-X^{2,k}_{\bar{S}^k\wedge\cdot}\big\|\Bigg] \\[2pt] & = \mathbb{E}\Bigg[\int_{(0,T']\times[0,\bar\gamma]\times[0,1]}\sum_{k\in K^{1\cap2}_{t-}}\sum_{\ell,\ell'\geq0} (\ell\wedge \ell')\mathbf{1}_{I^{1,k}_{\ell,t}\cap I^{2,k}_{\ell',t}\times[0,\gamma^{1,k}_t\wedge\gamma^{2,k}_t]}(z) \big\|X^{1,k}_{t\wedge\cdot}-X^{2,k}_{t\wedge\cdot}\big\|\,Q^k({\mathrm{d}} t,{\mathrm{d}} z)\Bigg] \\[2pt] & \leq \bar\gamma\mathbb{E}\Bigg[\int_0^{T'}\sum_{k\in K^{1\cap2}_t}\sum_{\ell,\ell'\geq0} \ell\,\big|I^{1,k}_{\ell,t}\cap I^{2,k}_{\ell',t}\big|\big\|X^{1,k}_{t\wedge\cdot}-X^{2,k}_{t\wedge\cdot}\big\|\, {\mathrm{d}} t\Bigg] \\[2pt] & = \bar\gamma\mathbb{E}\Bigg[\int_0^{T'}\sum_{k\in K^{1\cap2}_t}\sum_{\ell\geq0}\ell p^{1,k}_{\ell,t} \big\|X^{1,k}_{t\wedge\cdot}-X^{2,k}_{t\wedge\cdot}\big\|\,{\mathrm{d}} t\Bigg] \nonumber \\[2pt] & \leq \bar\gamma M\mathbb{E}\Bigg[\int_0^{T'}\sum_{k\in K^{1\cap2}_t} \big\|X^{1,k}_{t\wedge\cdot}-X^{2,k}_{t\wedge\cdot}\big\|\Bigg]. \end{align*}

Taking the supremum successively over $t\in[0,T']$ , the sum over $k\in\mathbb{K}$ , and the expectation in (19), we conclude by combining the inequalities above that

(20) \begin{align} & \mathbb{E}\bigg[\sum_{k\in\mathbb{K}}\sup\nolimits_{0\leq t\leq T'}\mathbf{1}_{k\in K^{1\cap2}_t} \big\|X^{1,k}_{t\wedge\cdot}-X^{2,k}_{t\wedge\cdot}\big\|\bigg] \nonumber \\[2pt] & \qquad \le C\sqrt{T'}\mathbb{E}\Bigg[\sum_{k\in\mathbb{K}}\sup\nolimits_{0\leq t\leq T'}\mathbf{1}_{k\in K^{1\cap2}_t} \big(\big\|X^{1,k}_{t\wedge\cdot}-X^{2,k}_{t\wedge\cdot}\big\| + \mathcal{W}_1(\mu^1_t,\mu^2_t)\big)\Bigg] \nonumber \\[2pt] & \qquad\quad + C\mathbb{E}\Bigg[\int_0^{T'}\Bigg(\#K^{1\triangle 2}_t + \sum_{k\in K^{1\cap2}_t}\big(\big\|X^{1,k}_{t\wedge\cdot}-X^{2,k}_{t\wedge\cdot}\big\| + \mathcal{W}_1\big(\mu^1_t,\mu^2_t\big)\big)\Bigg)\,{\mathrm{d}} t\Bigg], \end{align}

where the constant $C>0$ depends on $(\bar \gamma,L,M)$ .

Step 3

We write $\bar{d}_{T'}(Z^1,Z^2) \,:\!=\, \sup\nolimits_{0\leq t\leq T'}\#K^{1\triangle 2}_t + \sum_{k\in\mathbb{K}}\sup\nolimits_{0\leq t\leq T'}\mathbf{1}_{k\in K^{1\cap2}_t}\big\|X^{1,k}_{t\wedge\cdot}-X^{2,k}_{t\wedge\cdot}\big\|$ . Observe that

\begin{equation*} \mathbb{E}\Bigg[\int_0^{T'}\Bigg(\#K^{1\triangle 2}_t + \sum_{k\in K^{1\cap2}_t}\big\|X^{1,k}_{t\wedge\cdot}-X^{2,k}_{t\wedge\cdot}\big\|\Bigg)\,{\mathrm{d}} t\Bigg] \le T'\mathbb{E}[\bar{d}_{T'}(Z^1,Z^2)], \end{equation*}

and

\begin{align*} \mathbb{E}\Bigg[\int_0^{T'}\sum_{k\in K^{1\cap2}_t}\mathcal{W}_1\big(\mu^1_t,\mu^2_t\big)\,{\mathrm{d}} t\Bigg] & \le T'\mathbb{E}\Bigg[\sum_{k\in\mathbb{K}}\sup\nolimits_{0\leq t\leq T'}\mathbf{1}_{k\in K^{1\cap2}_t} \mathcal{W}_1\big(\mu^1_{t},\mu^2_{t}\big)\Bigg] \\[2pt] & \le T'\mathbb{E}[\#K_0]{\mathrm{e}}^{\bar\gamma M T'}\mathcal{W}_1\big(\mu^1_{T'},\mu^2_{T'}\big), \end{align*}

where the last inequality follows from Lemma 1 and $\mathcal{W}_1(\mu^1_{t},\mu^2_{t}) \le \mathcal{W}_1(\mu^1_{T'},\mu^2_{T'})$ for $t\in[0,T']$ . Combining (18) and (20) and using both inequalities above, we can find some positive constants $c_d>0$ and $c_w>0$ depending on $(\bar\gamma,L,M,M')$ such that

\begin{equation*} \mathbb{E}[\bar{d}_{T'}(Z^1,Z^2)] \le c_d(T'+\sqrt{T'})\mathbb{E}[\bar{d}_{T'}(Z^1,Z^2)] + c_w(T'+\sqrt{T'})\mathbb{E}[\#K_0]{\mathrm{e}}^{\bar\gamma M T'}\mathcal{W}_1(\mu^1_{T'},\mu^2_{T'}). \end{equation*}

Choosing $T'>0$ small enough to have $1-c_d(T'+\sqrt{T'})>0$ , we deduce that

\begin{equation*} \mathbb{E}[\bar{d}_{T'}(Z^1,Z^2)] \leq \frac{c_w(T'+\sqrt{T'})}{1-c_d(T'+\sqrt{T'})} \mathbb{E}[\#K_0]{\mathrm{e}}^{\bar\gamma M T'}\mathcal{W}_1\big(\mu^1_{T'},\mu^2_{T'}\big). \end{equation*}

The conclusion follows immediately by observing that $d_{T'}(Z^1,Z^2)\le \bar{d}_{T'}(Z^1,Z^2)$ .

3.2. Proof of Theorem 1

Proof of Theorem 1. We first outline the strategy of the proof. We initially establish the existence and uniqueness of the SDE (6) with (7) on a small interval [0, T ^′] by constructing a contraction mapping. We then shift the time window to [T ^′, 2T ^′] and so forth to establish the global well-posedness.

Step 2

In this step, we prove that there exists $T'\in[0,T]$ such that the equation $\mu_{T'} = \Psi_{T'}(\mu_{T'})$ admits a unique solution. To this end, we aim to show that $\Psi_{T'}$ is a contraction mapping when T’ is sufficiently small. Let us arbitrarily pick $\mu^1_{T'},\mu^2_{T'}\in\mathcal{P}_1(\mathcal{D}_E)$ . Denote by $Z^i$ the unique solution to the SDE (10) given $\mu^i_{T'}$ for $i=1,2$ . Notice first that

\begin{equation*} \mathcal{W}_1\big(\Psi_{T'}\big(\mu^1_{T'}\big),\Psi_{T'}\big(\mu^2_{T'}\big)\big) \leq \mathbb{E}\big[d\big(Z^1_{T'\wedge\cdot},Z^2_{T'\wedge\cdot}\big)\big] = \mathbb{E}[d_{T'}(Z^1,Z^2)]. \end{equation*}

In view of Lemma 3, there exist constants $c_d,c_w>0$ such that, for any $T'>0$ small enough to have $1-c_d(T'+\sqrt{T'})>0$ ,

\begin{equation*} \mathbb{E}[d_{T'}(Z^1,Z^2)] \leq \frac{c_w(T'+\sqrt{T'})}{1-c_d(T'+\sqrt{T'})} \mathbb{E}[\#K_0]{\mathrm{e}}^{\bar\gamma M T}\mathcal{W}_1\big(\mu^1_{T'},\mu^2_{T'}\big), \end{equation*}

where we also used the trivial inequality ${\mathrm{e}}^{\bar\gamma M T'} \le {\mathrm{e}}^{\bar\gamma M T}$ . Without loss of generality, we can take T’ even smaller to ensure that

\begin{equation*} T'+\sqrt{T'} < \frac{1}{c_d+c_w\mathbb{E}[\#K_0]{\mathrm{e}}^{\bar\gamma M T}}, \end{equation*}

so that

(21) \begin{equation} 0 < \kappa \,:\!=\, \frac{c_w(T'+\sqrt{T'})}{1-c_d(T'+\sqrt{T'})}\mathbb{E}[\#K_0]{\mathrm{e}}^{\bar\gamma M T} < 1. \end{equation}

It follows that $\mathcal{W}_1\big(\Psi_{T'}\big(\mu^1_{T'}\big),\Psi_{T'}\big(\mu^2_{T'}\big)\big) \leq \kappa\mathcal{W}_1\big(\mu^1_{T'},\mu^2_{T'}\big)$ . In other words, $\Psi_{T'}$ is a contraction mapping on $(\mathcal{P}_1(\mathcal{D}_E),\mathcal{W}_1)$ . In particular, the uniqueness of the solution to the SDE (6) with (7) on [0, T ^′] follows immediately. Regarding existence, it suffices to observe that the sequence of iterations $(\Psi^{(n)}_{T'}(\mu_{T'}))_{n\in\mathbb{N}}$ converges as a Cauchy sequence in $(\mathcal{P}_1(\mathcal{D}_E),\mathcal{W}_1)$ , which is complete since $\mathcal{D}_E$ equipped with the uniform topology is complete (though not separable).

Step 3

After establishing existence and uniqueness on [0, T ^′] in Step 2, we now aim to extend these properties to a larger interval $[0,T'+\delta T]$ for some $\delta T>0$ to be determined. We write $T''\,:\!=\,T'+\delta T$ with $\delta T\le T-T'$ . Fix $\mu^1_{T''},\mu^2_{T''}\in\mathcal{P}_1(\mathcal{D}_E)$ such that $\mu^1_{T'}=\mu^2_{T'}=\mu_{T'}$ , where $\mu_{T'}$ is the unique solution to $\Psi_{T'}(\mu_{T'})=\mu_{T'}$ constructed in Step 2. Let us denote by $Z^i\in\mathcal{D}_E$ the solution to (10) given $\mu^i_{T''}$ for $i=1,2$ . In particular, it holds that $Z^1_{T'\wedge\cdot} = Z^2_{T'\wedge\cdot}$ by uniqueness in Lemma 1. It follows that

\begin{equation*} \mathcal{W}_1\big(\Psi_{T''}\big(\mu^1_{T''}\big),\Psi_{T''}\big(\mu^2_{T''}\big)\big) \leq \mathbb{E}\big[d\big(Z^1_{T''\wedge\cdot},Z^2_{T''\wedge\cdot}\big)\big] = \mathbb{E}\big[\!\sup\nolimits_{T'\leq t\leq T''}d_E(Z^1_t,Z^2_t)\big]. \end{equation*}

By a straightforward extension of Lemma 3, shifting the origin of time from 0 to T ^′, we deduce that, for any $\delta T\in (0,T-T']$ such that $1-c_d(\delta T+\sqrt{\delta T})>0$ ,

\begin{equation*} \mathcal{W}_1\big(\Psi_{T''}\big(\mu^1_{T''}\big),\Psi_{T''}\big(\mu^2_{T''}\big)\big) \leq \frac{c_w(\delta T+\sqrt{\delta T})}{1-c_d(\delta T+\sqrt{\delta T})} \mathbb{E}[\#K_{T'}]{\mathrm{e}}^{\bar\gamma M\delta T}\mathcal{W}_1\big(\mu^1_{T''},\mu^2_{T''}\big), \end{equation*}

where $\# K_{T'}=\langle Z^1_{T'}, \mathbf{1}{}\rangle = \langle Z^2_{T'}, \mathbf{1}{}\rangle$ . In addition, Lemma 1 ensures that $\mathbb{E}[\#K_{T'}] \leq \mathbb{E}[\#K_0]{\mathrm{e}}^{\bar\gamma M T'}$ . It follows that

\begin{equation*} \mathcal{W}_1\big(\Psi_{T''}\big(\mu^1_{T''}\big),\Psi_{T''}\big(\mu^2_{T''}\big)\big) \leq \frac{c_w(\delta T+\sqrt{\delta T})}{1-c_d(\delta T+\sqrt{\delta T})}\mathbb{E}[\#K_0] {\mathrm{e}}^{\bar\gamma MT}\mathcal{W}_1\big(\mu^1_{T''},\mu^2_{T''}\big), \end{equation*}

where we used the trivial inequality ${\mathrm{e}}^{\bar\gamma MT''} \le {\mathrm{e}}^{\bar\gamma MT}$ . Thus we can repeat the calculation from Step 2 and choose $\delta T = \min(T',T-T')$ so that $\mathcal{W}_1\big(\Psi_{T''}\big(\mu^1_{T''}\big),\Psi_{T''}\big(\mu^2_{T''}\big)\big) \leq \kappa\mathcal{W}_1\big(\mu^1_{T''},\mu^2_{T''}\big)$ , where $0<\kappa<1$ is given by (21). It follows that the SDE (6) with (7) admits a unique solution on $[0,T'+\delta T]$ . We conclude the proof by repeating the above procedure finitely many times in order to cover the whole interval [0, T].

Remark 6. We use a classical contraction argument to obtain the existence and uniqueness of the McKean–Vlasov branching diffusion. However, unlike most of the literature, we work with the metric $\mathcal{W}_1$ instead of $\mathcal{W}_2$ and it appears that our contraction argument fails otherwise. Indeed, working along the lines of Lemma 3 to dominate $\mathbb{E}[d_T(Z^1,Z^2)^2]$ , we come by a term proportional to $\mathcal{W}_2\big(\mu^1_T, \mu^2_T\big)$ on the right-hand side which cannot be dominated by $\mathcal{W}_2\big(\mu^1_T, \mu^2_T\big)^2$ .

4.1. Martingale problem formulation

As in classical SDE theory, the notion of a weak solution can be equivalently formulated by means of an appropriate martingale problem defined on the canonical space.

4.1.1. Martingale problem for n interacting branching diffusions

Let us first consider the n interacting branching diffusion processes given in Definition 3. We introduce the corresponding canonical space $\bar\Omega_n \,:\!=\, \mathcal{D}_E^n$ , equipped with the canonical process $(Z^{1}, \ldots, Z^{n})$ and the canonical filtration $\bar{\mathbb{F}}^n = (\bar{\mathcal{F}}^n_t)_{t \in [0,T]}$ defined by $\bar{\mathcal{F}}^n_t \,:\!=\, \sigma(Z^1_s, \ldots, Z^n_s, s \le t)$ . Let us also consider the empirical measure $\mu^n_t \,:\!=\, (1/n)\sum_{i=1}^n\delta_{Z^i_{t \wedge \cdot}}$ . Further, given $\Phi^n \in C^2_{\mathrm{b}}(\mathbb{R}^n,\mathbb{R})$ and $\bar{\varphi} = (\varphi_1, \ldots, \varphi_n)$ , where $\varphi_i = (\varphi_i^k)_{k\in \mathbb{K}} \in C^2_{\mathrm{b}}(\mathbb{K}\times\mathbb{R}^d,\mathbb{R})$ for each $i =1, \ldots, n$ , we define $\Phi^n_{\bar{\varphi}}\colon E^n \longrightarrow \mathbb{R}$ by $\Phi^n_{\bar{\varphi}}(e_1, \ldots, e_n) \,:\!=\,\Phi^n(\langle e_1,\varphi_1\rangle, \ldots, \langle e_n, \varphi_n \rangle)$ , as well as, for all $t \in [0,T]$ , $i=1,\ldots, n$ , and $\bar \omega = (\omega^1, \ldots, \omega^n) \in \bar \Omega_n$ with $\omega^i_t = \sum_{k \in K^i_t} \delta_{(k,\mathrm{x}^{i,k}_t)}$ ,

\begin{align*} \mathcal{H}^n_i\Phi^n_{\bar{\varphi}}(t,\bar\omega) & \,:\!=\, \frac{1}{2}\partial^2_{ii}\Phi^n_{\bar{\varphi}}(\bar\omega_t)\sum_{k\in K^i_t} \big|D\varphi^k_i\big(\mathrm{x}^{i,k}_t\big)\sigma\big(t,\mathrm{x}^{i,k}_{t\wedge\cdot},m^n_t\big)\big|^2 \\[2pt] & \quad + \partial_{i}\Phi^n_{\bar{\varphi}}(\bar\omega_t)\sum_{k\in K^i_t} \mathcal{L}\varphi^k_i\big(t,\mathrm{x}^{i,k}_{t\wedge \cdot},m^n_t\big) \\[2pt] & \quad + \sum_{k\in K^i_t}\gamma\big(t,\mathrm{x}^{i,k}_{t\wedge\cdot},m^n_t\big)\Bigg( \sum_{\ell\ge0}\Phi^n_{\bar{\varphi}}\Bigg(\omega^1_t,\ldots,\omega^i_{t-}-\delta_{(k,\mathrm{x}^{i,k}_{t-})} \\[2pt] & \qquad\qquad\qquad\qquad\qquad + \sum_{j=1}^{\ell}\delta_{(kj,\mathrm{x}^{i,k}_{t-})},\ldots,\omega^n_t\Bigg) p_{\ell}\big(t,\mathrm{x}^{i,k}_{t\wedge\cdot}, m^n_t\big) - \Phi^n_{\bar{\varphi}}(\bar\omega_{t-})\Bigg),\end{align*}

where $m^n_t \,:\!=\, (1/n)\sum_{j=1}^n\delta_{\omega^j_{t\wedge\cdot}}$ and $\partial_{i}\Phi^n$ (resp. $\partial^2_{ii}\Phi^n$ ) respresents the first (resp. second) partial derivative of $\Phi^n$ with respect to the ith coordinate. Let us also define the process $\mathcal{M}^{\Phi^n_{\bar{\varphi}}} = \big(\mathcal{M}^{\Phi^n_{\bar{\varphi}}}_t\big)_{t \in [0,T]}$ on $\bar \Omega_n$ by $\mathcal{M}^{\Phi^n_{\bar{\varphi}}}_t \,:\!=\, \Phi^n_{\bar{\varphi}}\big(Z^1_t,\ldots,Z^n_t\big) -\int_0^t\sum_{i=1}^n\mathcal{H}^n_i\Phi^n_{\bar{\varphi}}\big(s,Z^1_{s\wedge\cdot},\ldots,Z^n_{s\wedge\cdot}\big)\,{\mathrm{d}} s$ .

Definition 4. A solution to the martingale problem for n interacting branching diffusions with initial condition $m_0 \in \mathcal{P}(E)$ is a probability measure $\bar{\mathbb{P}}^n$ on $\bar \Omega_n$ satisfying

1. (i) $\bar{\mathbb{P}}^n \circ \big(Z^1_0, \ldots, Z^n_0\big)^{-1} = m_0 \times \cdots \times m_0$ ;

2. (ii) $\mathcal{M}^{\Phi^n_{\bar{\varphi}}}$ is a $(\bar{\mathbb{P}}^n, \bar{\mathbb{F}}^n)$ -martingale for all $\Phi^n \in C^2_{\mathrm{b}}(\mathbb{R}^n,\mathbb{R})$ and $\bar{\varphi} = (\varphi_1,\ldots,\varphi_n)$ such that $\varphi_i \in C^2_{\mathrm{b}}(\mathbb{K}\times\mathbb{R}^d,\mathbb{R})$ for each $i =1,\ldots,n$ .

Proposition 1. Let Assumption 1 hold and $m_0\in\mathcal{P}_1(E)$ . Then the following assertions hold:

1. (i) Let $\alpha_n = (\Omega^n,\mathcal{F}^n,\mathbb{F}^n,\mathbb{P}^n,(Z^{i})_{i=1,\ldots,n}, (W^{i,k},Q^{i,k})_{k\in\mathbb{K},i=1,\ldots,n})$ be a weak solution in the sense of Definition 3. Then the probability $\bar{\mathbb{P}}^n \,:\!=\, \mathbb{P}^n \circ (Z^{1}, \ldots, Z^{n})^{-1}$ solves the corresponding martingale problem for n interacting branching diffusions.

2. (ii) Let $\bar{\mathbb{P}}^n$ be a solution to the martingale problem for n interacting branching diffusions. Then there exist, in a possibly enlarged space, Brownian motions and Poisson random measures which, together with the canonical space $\big(\bar\Omega_n,\bar{\mathcal{F}}^n_T,\bar{\mathbb{F}}^n,\bar{\mathbb{P}}^n\big)$ , give a weak solution in the sense of Definition 3.

3. (iii) If we assume further that Assumption 3 holds, then there exists a solution to the martingale problem for n interacting branching diffusions in the sense of Definition 4.

Proof. The arguments are rather classical. Let us provide a sketch of the proof for completeness. First, to construct a solution to the martingale problem from a weak solution, it suffices to apply Itô’s formula as in [Reference Claisse4, Proposition 3.2]. Next, given a solution to the martingale problem $\bar{\mathbb{P}}^n$ , it is also classical to construct, in a possibly enlarged space, Brownian motions and Poisson random measures to represent the process by stochastic integrations; see, e.g., [Reference Kurtz23, Theorem 2.3]. This provides a weak solution. Finally, notice that the n-interacting-branching-diffusions solution to (9) is actually a standard branching diffusion process, in the sense that it does not exhibit nonlinearity in the sense of McKean. As a consequence, we can apply classical weak convergence techniques as in [Reference Stroock and Varadhan29, Chapter 6.1] to deduce existence of a weak solution provided that the coefficients are bounded continuous.

4.1.2. Martingale problem for McKean–Vlasov branching diffusion

Let us now consider McKean–Vlasov branching diffusion in the sense of Definition 2. In this setting, we introduce two canonical spaces $\tilde \Omega \,:\!=\, \mathcal{D}_E$ and $\bar\Omega \,:\!=\, \mathcal{D}_E\times\mathcal{P}(\mathcal{D}_E)$ . Let us denote by $\tilde Z$ the canonical process on $\tilde \Omega$ and by $\tilde{\mathbb{F}} = (\tilde{\mathcal{F}}_t)_{t \in [0,T]}$ , $\tilde{\mathcal{F}}_t = \sigma(\tilde Z_s,\, s \le t)$ , the corresponding canonical filtration. Similarly, the canonical process on $\bar\Omega$ is denoted by $(\bar Z, \bar \mu)$ and the corresponding canonical filtration $\bar{\mathbb{F}} = (\bar{\mathcal{F}}_t)_{t \in [0,T]}$ is defined by $\bar{\mathcal{F}}_t \,:\!=\, \sigma(\bar Z_s, \bar \mu_s,\, s \le t)$ . Further, given $\Phi\in C^2_{\mathrm{b}}(\mathbb{R},\mathbb{R})$ and $\varphi = (\varphi^k)_{k\in \mathbb{K}} \in C^2_{\mathrm{b}}(\mathbb{K}\times\mathbb{R}^d,\mathbb{R})$ , let us define

\begin{equation*} \Phi_{\varphi}(e) \,:\!=\, \Phi(\langle e,\varphi\rangle) = \Phi\Bigg(\sum_{k\in K}\varphi^k(x^k)\Bigg) \quad \text{for all}\ e\,:\!=\,\sum_{k\in K}\delta_{(k,x^k)}\in E,\end{equation*}

as well as, for $\tilde \omega \in \tilde \Omega$ with $\tilde \omega_t = \sum_{k \in K_t} \delta_{(k, \mathrm{x}^k_t)}$ and $\tilde{m} \in \mathcal{P}(\tilde \Omega)$ ,

\begin{align*} \mathcal{H}\Phi_{\varphi}(t,\tilde\omega,\tilde{m}) & \,:\!=\, \frac{1}{2}\Phi_{\varphi}''(\tilde\omega_t)\sum_{k\in K_t} \big|D\varphi^k\big(\mathrm{x}^k_t\big)\sigma\big(t,\mathrm{x}^k_{t\wedge\cdot},\tilde{m}_t\big)\big|^2 \\[2pt] & \quad + \Phi_{\varphi}'(\tilde\omega_t)\sum_{k\in K_t}\mathcal{L}\varphi^k(t,\mathrm{x}^k_{t\wedge\cdot}, \tilde{m}_t) \\[2pt] & \quad + \sum_{k\in K_t}\gamma\big(t,\mathrm{x}^k_{t\wedge\cdot},\tilde{m}_t\big)\Bigg( \sum_{\ell\ge0}\Phi_{\varphi}\Bigg(\tilde\omega_{t-}-\delta_{(k,\mathrm{x}^k_{t-})} + \sum_{j=1}^{\ell}\delta_{(kj,\mathrm{x}^{k}_{t-})}\Bigg) \\[2pt] & \qquad\qquad\qquad\qquad\qquad \times p_{\ell}\big(t,\mathrm{x}^k_{t\wedge\cdot},\tilde{m}_t\big) - \Phi_{\varphi}(\tilde\omega_{t-})\Bigg).\end{align*}

Let us also define the process $\tilde{\mathcal{M}}^{\Phi_\varphi, \tilde{m}} = \big(\tilde{\mathcal{M}}^{\Phi_\varphi, \tilde{m}}_t\big)_{t \in [0,T]}$ on $\tilde \Omega$ by

\begin{equation*} \tilde{\mathcal{M}}^{\Phi_\varphi, \tilde{m}}_t \,:\!=\, \Phi_{\varphi}(\tilde Z_t) - \int_0^t\mathcal{H}\Phi_{\varphi}\big(s,\tilde Z_{s\wedge\cdot},\tilde{m}_s\big)\,{\mathrm{d}} s.\end{equation*}

Definition 5. A solution to the martingale problem for McKean–Vlasov branching diffusion with initial condition $m_0 \in \mathcal{P}(E)$ is a probability measure $\bar{\mathbb{P}}$ on the canonical space $\bar \Omega$ such that

1. (i) $\bar{\mathbb{P}} \circ \bar Z_0^{-1} = m_0$ ;

2. (ii) $\bar\mu_t = \mathcal{L}^{\bar{\mathbb{P}}}(\bar Z_{t\wedge\cdot}\mid\bar\mu_t)$ for all $t\in[0,T]$ ;

3. (iii) for $\bar{\mathbb{P}}$ -almost every $\bar \omega \in \bar \Omega$ , the process $\tilde{\mathcal{M}}^{\Phi_\varphi,\bar\mu(\bar\omega)}$ is a $(\bar\mu(\bar\omega),\tilde{\mathbb{F}})$ -martingale for all $\Phi\in C^2_{\mathrm{b}}(\mathbb{R},\mathbb{R})$ and $\varphi = (\varphi^k)_{k\in \mathbb{K}} \in C^2_{\mathrm{b}}(\mathbb{K}\times\mathbb{R}^d,\mathbb{R})$ .

Proposition 2. Let Assumption 1 hold and $m_0 \in \mathcal{P}_1(E)$ . The following assertions hold:

1. (i) Given a weak solution $\alpha = \big(\Omega,\mathcal{F},\mathbb{F},\mathbb{P},Z,\mu, (W^k,Q^k)_{k\in\mathbb{K}}\big)$ to the McKean–Vlasov branching diffusion SDE in the sense of Definition 2, the probability $\bar{\mathbb{P}} \,:\!=\, \mathbb{P} \circ (Z, \mu)^{-1}$ provides a solution to the corresponding martingale problem in the sense of Definition 5.

2. (ii) Given a solution $\bar{\mathbb{P}}$ to the martingale problem for McKean–Vlasov branching diffusion in the sense of Definition 5, there exist, in a possibly enlarged space, Brownian motions and Poisson random measures which, together with the canonical space $(\bar\Omega,\bar{\mathcal{F}}_T,\bar{\mathbb{F}},\bar{\mathbb{P}})$ , give a weak solution in the sense of Definition 2.

Proof. The proof is similar to [Reference Djete, Possama and Tan8, Lemma 4.4]. Let $\alpha$ be a weak solution defined as above, and $\bar{\mathbb{P}}\,:\!=\,\mathbb{P}\circ(Z,\mu)^{-1}$ . We aim to prove that $\bar{\mathbb{P}}$ solves the martingale problem for McKean–Vlasov branching diffusion. It is straightforward to check that $\bar{\mathbb{P}} \circ \bar Z^{-1}_0 = m_0$ and $\bar\mu_t = \mathcal{L}^{\bar{\mathbb{P}}}(\bar Z_{t\wedge\cdot}\mid\bar\mu_t)$ . Then, given $h\colon\mathcal{D}_E\to\mathbb{R}$ and $g\colon\mathcal{P}(\mathcal{D}_E)\to\mathbb{R}$ arbitrary bounded measurable maps, we observe that, for any $0\le s\le t\le T$ ,

\begin{align*} \mathbb{E}^{\mathbb{P}}\big[h(Z_{s\wedge\cdot})g(\mu)\big(\tilde{\mathcal{M}}^{\Phi_\varphi,\mu}_t - \tilde{\mathcal{M}}^{\Phi_\varphi,\mu}_s\big)(Z)\big] & = \mathbb{E}^{\bar{\mathbb{P}}}\big[h(\bar Z_{s\wedge\cdot})g(\bar\mu) \big(\tilde{\mathcal{M}}^{\Phi_\varphi,\bar\mu}_t - \tilde{\mathcal{M}}^{\Phi_\varphi,\bar\mu}_s\big)(\bar Z)\big] \\[3pt] & = \mathbb{E}^{\bar{\mathbb{P}}}\big[\mathbb{E}^{\bar{\mathbb{P}}}\big[ h(\bar Z_{s\wedge\cdot})\big(\tilde{\mathcal{M}}^{\Phi_\varphi,\bar\mu}_t - \tilde{\mathcal{M}}^{\Phi_\varphi,\bar\mu}_s\big)(\bar Z)\mid\bar\mu\big]g(\bar\mu)\big] \\[3pt] & = \mathbb{E}^{\bar{\mathbb{P}}}\big[\mathbb{E}^{\bar\mu}\big[ h(\tilde Z_{s\wedge\cdot})\big(\tilde{\mathcal{M}}^{\Phi_\varphi,\bar\mu}_t - \tilde{\mathcal{M}}^{\Phi_\varphi,\bar\mu}_s\big)\big]g(\bar\mu)\big], \end{align*}

where the last equality comes from $\bar\mu = \mathcal{L}^{\bar{\mathbb{P}}}(\bar Z\mid\bar\mu)$ . In addition, it follows easily from Itô’s formula that $\mathbb{E}^{\mathbb{P}}\big[h(Z_{s\wedge\cdot})g(\mu)\big(\tilde{\mathcal{M}}^{\Phi_\varphi, \mu}_t - \tilde{\mathcal{M}}^{\Phi_\varphi,\mu}_s\big)(Z)\big]=0$ . By the arbitrariness of g, we deduce that $\mathbb{E}^{\bar\mu(\bar\omega)}\big[h(\tilde Z_{s\wedge\cdot})\big(\tilde{\mathcal{M}}^{\Phi_\varphi,\bar\mu(\bar\omega)}_t - \tilde{\mathcal{M}}^{\Phi_\varphi,\bar\mu(\bar\omega)}_s\big)\big] = 0$ , $\bar{\mathbb{P}}$ -almost surely (a.s.). We conclude, by the arbitrariness of h, that $\tilde{\mathcal{M}}^{\Phi_\varphi,\mu(\bar\omega)}$ is a $(\bar\mu(\bar\omega),\tilde{\mathbb{F}})$ -martingale for $\bar{\mathbb{P}}$ -almost every $\bar\omega\in\bar\Omega$ .

Conversely, by similar arguments to Proposition 1, if we have a solution to the martingale problem, we can construct, in a possibly enlarged space, Brownian motions and Poisson random measures to obtain a weak solution to the McKean–Vlasov branching diffusion SDE.

4.2. Tightness of n interacting branching diffusions

Let $(\alpha_n)_{n \ge 1}$ be a sequence of n interacting branching diffusions in the sense of Definition 3, i.e. $\alpha_n = \big(\Omega^n, \mathcal{F}^n, \mathbb{F}^n, \mathbb{P}^n, (Z^{i})_{i = 1, \ldots, n}, (W^{i,k}, Q^{i,k})_{k\in\mathbb{K}, i = 1, \ldots, n} \big)$ is a weak solution to the SDE (9) with initial condition $m_0 \in \mathcal{P}(E)$ . We consider the corresponding sequence of probability measures $(\bar{\mathbb{P}}^n)_{n \ge 1}$ on the canonical space $\bar \Omega = \mathcal{D}_E \times \mathcal{P}(\mathcal{D}_E)$ , given by

\begin{equation*} \bar{\mathbb{P}}^n \,:\!=\, \frac1n\sum_{i=1}^n\mathbb{P}^n \circ (Z^{i},\mu^n)^{-1}, \quad \text{where}\ \mu^n \,:\!=\, \frac1n \sum_{i=1}^n \delta_{Z^{i}}.\end{equation*}

The goal of this section is to establish the tightness of this sequence. To this end, we first provide an estimate for the number of particles in each of the n branching diffusion processes.

Proof. In view of [Reference Sznitman30, Proposition 2.2(ii)], the tightness of $(\bar{\mathbb{P}}^n)_{n \ge 1}$ is equivalent to the tightness of $(\mathbb{E}^{\mathbb{P}^n}[\mu^n])_{n \ge 1}$ , where, for all $\varphi\colon \mathcal{D}_E\to \mathbb{R}$ measurable bounded,

\begin{align*} \big\langle\mathbb{E}^{\mathbb{P}^n}[\mu^n],\varphi\big\rangle \,:\!=\, \mathbb{E}^{\mathbb{P}^n}[\langle\mu^n,\varphi\rangle] = \frac1n\sum_{i=1}^n\mathbb{E}^{\mathbb{P}^n}[\varphi(Z^{i})] = \mathbb{E}^{\mathbb{P}^n}[\varphi(Z^{1})]. \end{align*}

Therefore, it suffices to prove the tightness of $(\mathbb{P}^n \circ (Z^{1})^{-1})_{n \ge 1}$ in $\mathcal{P}(\mathcal{D}_E)$ .

Recall that the process $Z^{1}_t=\sum_{k\in K^{1}_t} \delta_{(k,X^{1,k}_t)}$ takes values in E equipped with the distance $d_E$ defined in (1), and that we consider the associated Skorokhod topology on $\mathcal{D}_E$ . We aim to apply Aldous’ criterion of tightness (see, e.g., [Reference Billingsley2, Theorem 16.10]). Namely, we have to prove the following assertions:

• • For all $\varepsilon>0$ , there exist $n_0\in\mathbb{N}$ and $K>0$ such that

  (22) \begin{equation} \text{for all}\ n \geq n_0,\ \mathbb{P}^n\big(\!\sup\nolimits_{0\leq t\leq T}d_E(Z^{1}_t,e_0) \geq K\big) \leq \varepsilon, \end{equation}
  where $e_0$ is the null measure on $\mathbb{K}\times\mathbb{R} ^d$ .

• • For all $\varepsilon>0$ ,

  (23) \begin{equation} \lim_{\delta\rightarrow0}\sup\nolimits_{{n\ge 1}}\sup\nolimits_{\tau\in\mathcal{T}^n} \mathbb{P}^n\big(d_E(Z^{1}_\tau, Z^{1}_{(\tau+\delta)\wedge T})>\varepsilon\big) = 0, \end{equation}
  where $\mathcal{T}^n$ is the collection of all [0, T]-valued stopping times on $(\Omega^n, \mathcal{F}^n, \mathbb{F}^n)$ .

The first condition (22) follows directly from Lemma 4 and Markov’s inequality by observing that $d_E(Z^{1}_t,e_0)=\# K^{1}_t$ . Regarding the second condition (23), given $\delta >0$ , $n\ge 1$ , and $\tau\in\mathcal{T}^n$ , we write $\tau'\,:\!=\, (\tau + \delta)\wedge T$ and observe that

(24) \begin{align} & d_E(Z^{1}_\tau,Z^{1}_{\tau'}) \nonumber \\[2pt] & \quad \leq \sum_{k\in K^{1}_\tau}\big|X^{1,k}_{\tau'} - X^{1,k}_{\tau}\big| \wedge 1 \nonumber \\[2pt] & \qquad + \int_{(\tau,\tau']\times[0,\bar \gamma] \times [0,1]}\sum_{k\in K^{1}_{s-}}\sum_{\ell \geq0} (\ell +1)\mathbf{1}_{I_{\ell} (s,X^{1,k}_{s\wedge\cdot},\mu^n_{s-})\times[0,\gamma(s,X^{1,k}_{s\wedge\cdot},\mu^n_{s-})]}(z)\, Q^{1,k}({\mathrm{d}} s,{\mathrm{d}} z). \end{align}

The second term corresponds to the number of particles in $\cup_{\tau\le s\le \tau'} K^{1}_s \triangle K^{1}_{\tau'}$ as it counts the total number of particles born or dead between $\tau$ and $\tau'$ . As for the first term, it provides an upper bound for the contribution of particles in the common set $K^{1}_\tau \cap K^{1}_{\tau'}$ to the distance $d_E$ by neglecting the potential death of particles. Here there is a slight abuse of notation as we need to extend the path of particles who have died before time $\tau'$ as solution to SDE (5).

We start by dealing with the first term on the right-hand side of (24). Notice that

\begin{equation*} \mathbb{E}^{\mathbb{P}^n}\Bigg[\sum_{k\in K^{1}_\tau}\!\big|X^{1,k}_{\tau'}-X^{1,k}_\tau\big|\!\Bigg] \leq \mathbb{E}^{\mathbb{P}^n}\Bigg[\sum_{k\in K^{1}_\tau} \! \bigg(\int_\tau^{\tau'}\!\!\big|b\big(s,X^{1,k}_{s\wedge\cdot},\mu^n_s\big)\big|\,{\mathrm{d}} s + \bigg|\int_\tau^{\tau'}\!\!\sigma\big(s,X^{1,k}_{s\wedge\cdot},\mu^n_s\big)\,{\mathrm{d}} W^{1,k}_s\big|\!\bigg)\!\Bigg]. \end{equation*}

On the one hand, we have $\mathbb{E}^{\mathbb{P}^n}\big[\sum_{k\in K^{1}_\tau}\int_\tau^{\tau'}\big|b(s,X^{1,k}_{s\wedge\cdot},\mu^n_s)\big|\,{\mathrm{d}} s\big] \leq \|b\|_\infty\delta\,\mathbb{E}^{\mathbb{P}^n}[\#K^{1}_\tau]$ . On the other hand, it follows from the Burkholder–Davis–Gundy inequality that

\begin{align*} \mathbb{E}^{\mathbb{P}^n}\Bigg[\sum_{k\in K^{1}_\tau}\bigg|\int_\tau^{\tau'}\sigma \big(s,X^{1,k}_{s\wedge\cdot},\mu^n_s\big)\,{\mathrm{d}} W^{1,k}_s\bigg|\Bigg] & = \sum_{k\in\mathbb{K}}\mathbb{E}^{\mathbb{P}^n}\bigg[\bigg|\int_\tau^{\tau'}\mathbf{1}_{k\in K^{1}_\tau} \sigma\big(s,X^{1,k}_{s\wedge\cdot},\mu^n_s\big)\,{\mathrm{d}} W^{1,k}_s\bigg|\bigg] \\[2pt] & \leq \sum_{k\in\mathbb{K}}C_1\mathbb{E}^{\mathbb{P}^n}\bigg[\bigg(\int_\tau^{\tau'} \mathbf{1}_{k\in K^{1}_\tau}\sigma\big(s,X^{1,k}_{s\wedge\cdot},\mu^n_s\big)^2\,{\mathrm{d}} s\bigg)^{1/2}\bigg] \\[2pt] & \leq C_1\|\sigma\|_\infty\sqrt{\delta}\,\mathbb{E}^{\mathbb{P}^n}[\#K^{1}_\tau]. \end{align*}

Combining both the above estimates, we deduce that

\begin{align*} \mathbb{E}^{\mathbb{P}^n}\Bigg[\sum_{k\in K^{1}_\tau}\big|X^{1,k}_{\tau'}-X^{1,k}_\tau\big|\Bigg] \leq C(\delta+\sqrt{\delta})\mathbb{E}^{\mathbb{P}^n}[\!\sup\nolimits_{0\leq t \leq T}\# K^{1}_t]. \end{align*}

As for the second term on the right-hand side of (24), direct computation yields

\begin{align*} & \mathbb{E}^{\mathbb{P}^n}\Bigg[\int_{(\tau,\tau']\times[0,\bar \gamma] \times [0,1]}\sum_{k\in K^{1}_{s-}}\sum_{\ell\geq0} (\ell+1)\mathbf{1}_{I_\ell(s,X^{1,k}_{s\wedge\cdot},\mu^n_{s-})\times[0,\gamma(s,X^{1,k}_{s\wedge\cdot},\mu^n_{s-})]}(z)\, Q^{1,k}({\mathrm{d}} s,{\mathrm{d}} z)\Bigg] \\[2pt] & \qquad\qquad = \mathbb{E}^{\mathbb{P}^n}\Bigg[\int_\tau^{\tau'}\sum_{k\in K^{1}_s} \gamma\big(s,X^{1,k}_{s\wedge\cdot},\mu^n_s\big)\sum_{\ell\geq0}(\ell+1) p_\ell\big(s,X^{1,k}_{s\wedge\cdot},\mu^n_s\big)\,{\mathrm{d}} s\Bigg] \\[2pt] & \qquad\qquad \leq \bar\gamma(M+1)\delta\,\mathbb{E}^{\mathbb{P}^n}[\sup\nolimits_{0\leq t\leq T}\# K^{1}_t]. \end{align*}

It follows that $\mathbb{E}^{\mathbb{P}^n}\big[d_E\big(Z^{1}_\tau,Z^{1}_{(\tau+\delta)\wedge T}\big)\big] \leq C(\delta + \sqrt{\delta})\mathbb{E}^{\mathbb{P}^n}[\sup\nolimits_{0\leq t\leq T}\#K^{1}_t]$ . In light of Lemma 4, we deduce that the second condition (23) holds by Markov’s inequality.

4.3. Proof of Theorem 3

Proof of Theorem 3. Recall that the tightness of $(\mathbb{P}^n \circ (\mu^n)^{-1})_{n \ge 1}$ was established in Proposition 3. More precisely, it was proved that the sequence $\bar{\mathbb{P}}^n \,:\!=\, (1/n) \sum_{i=1}^n \mathbb{P}^n \circ (Z^{i}, \mu^n)^{-1}$ is tight in $\mathcal{P}(\bar \Omega)$ . Let us consider a subsequence, still denoted by $(\bar{\mathbb{P}}^{n})_{n \ge 1}$ , converging weakly to $\bar{\mathbb{P}}$ . We aim to prove that $\bar{\mathbb{P}}$ is a solution to the martingale problem for McKean–Vlasov branching diffusion in the sense of Definition 5. The conclusion then follows from Proposition 2.

First, it is easy to check that $\bar{\mathbb{P}} \circ \bar Z_0^{-1} = m_0$ . Indeed, we have $\mathbb{P}^n \circ (Z^{i}_0)^{-1} = m_0$ for each $i = 1, \ldots, n$ , and so $\bar{\mathbb{P}}^n \circ \bar Z_0^{-1} = m_0$ . Then we can pass to the limit as $\bar Z_0$ is continuous for the Skorokhod topology.

Next, by the definition of $\bar{\mathbb{P}}^{n}$ , it is easy to see that $\mathcal{L}^{\bar{\mathbb{P}}^{n}}(\bar Z_{t\wedge\cdot} \mid \bar\mu_t) = \bar\mu_t$ , $\bar{\mathbb{P}}^{n}$ -a.s. for all $t \in[0,T]$ . Indeed, for all $\phi \in C_{\mathrm{b}}( \mathcal{P}(\tilde \Omega), \mathbb{R})$ and $\psi \in C_{\mathrm{b}}(\mathcal{D}_E, \mathbb{R})$ ,

\begin{align*} \mathbb{E}^{\bar{\mathbb{P}}^{n}}[\psi(\bar Z_{t\wedge \cdot})\phi(\bar\mu_t)] = \frac{1}{n}\sum_{i=1}^{n}\mathbb{E}^{\mathbb{P}^{n}}[\psi(Z^{i}_{t\wedge\cdot})\phi(\mu^{n}_t)] = \mathbb{E}^{\bar{\mathbb{P}}^{n}}[\langle\bar\mu_t, \psi\rangle\phi(\bar\mu_t)]. \end{align*}

It follows by passing to the limit that $\mathbb{E}^{\bar{\mathbb{P}}}[\psi(\bar Z_{t\wedge\cdot})\phi(\bar\mu_t)] = \mathbb{E}^{\bar{\mathbb{P}}}[\langle\bar\mu_t, \psi\rangle\phi(\bar\mu_t)]$ . We conclude that $\mathcal{L}^{\bar{\mathbb{P}}}(\bar Z_{t\wedge\cdot} \mid \bar\mu_t) = \bar\mu_t$ , $\bar{\mathbb{P}}$ -almost everywhere for all $t \in [0,T]$ .

It remains to prove that $\tilde{\mathcal{M}}^{\Phi_\varphi, \bar \mu(\bar \omega)}$ is a $(\bar \mu(\bar \omega), \tilde{\mathbb{F}})$ -martingale for $\bar{\mathbb{P}}$ -almost every (a.e.) $\bar \omega \in \bar \Omega$ . Equivalently, we aim to show that, for all $0\le s\le t \le T$ and $h\colon\tilde \Omega \to \mathbb{R}$ bounded continuous and $\tilde{\mathcal{F}}_s$ –measurable, $\mathbb{E}^{\bar{\mathbb{P}}}\big[\big\langle\bar\mu, \big(\tilde{\mathcal{M}}^{\Phi_\varphi,\bar\mu}_t-\tilde{\mathcal{M}}^{\Phi_\varphi,\bar\mu}_s\big)h\big\rangle^2\big] = 0$ . We start by observing that $\tilde{\mathcal{M}}^{\Phi_\varphi, \mu^n}_t(Z^{i})$ is a martingale under $\mathbb{P}^n$ with representation

\begin{align*} \tilde{\mathcal{M}}^{\Phi_\varphi, \mu^n}_t(Z^{i}) & = \Phi_{\varphi}(Z^{i}_0) + \int_0^t\Phi_{\varphi}'(Z^{i}_s)\sum_{k\in K^{i}_s}D\varphi^k(X^{i, k}_s) \sigma\big(s,X^{i, k}_{s\wedge\cdot},\mu^n_s\big)\,{\mathrm{d}} W^{i,k}_s \\[2pt] & \quad + \int_{(0,t]\times[0,\bar \gamma] \times [0,1]}\sum_{k\in K^{i}_{s-}} G^{i,k}_s(z)\,\bar{Q}^{i,k}({\mathrm{d}} s,{\mathrm{d}} z), \end{align*}

where $\bar{Q}^{i,k}({\mathrm{d}} s,{\mathrm{d}} z)\,:\!=\,Q^{i,k}({\mathrm{d}} s,{\mathrm{d}} z) - {\mathrm{d}} s\,{\mathrm{d}} z$ is the compensated Poisson random measure, and

\begin{equation*} G^{i,k}_s(z) \,:\!=\, \sum_{\ell\geq0}\Bigg(\Phi_{\varphi}\Bigg(Z^{i}_{s-}+\sum_{j=1}^{\ell}\delta_{(kj,X^{i, k}_s)}\Bigg) - \Phi_{\varphi}(Z^{i}_{s-})\Bigg) \mathbf{1}_{I_\ell(s,X^{i,k}_{s\wedge\cdot},\mu^n_{s-})\times[0,\gamma(s,X^{i,k}_{s\wedge\cdot},\mu^n_{s-})]}(z). \end{equation*}

Since $(W^{i,k},Q^{i,k})_{k\in\mathbb{K}, i=1\ldots,n}$ are independent Brownian motions and Poisson random measures, the martingales $\big(\tilde{\mathcal{M}}^{\Phi_\varphi, \mu^n}_t(Z^{i})\big)_{i=1,\ldots,n}$ are orthogonal under $\mathbb{P}^n$ . It follows that

\begin{align*} \mathbb{E}^{\bar{\mathbb{P}}^n}\big[\big\langle\bar\mu,\big(\tilde{\mathcal{M}}^{\Phi_\varphi,\bar\mu}_t - \tilde{\mathcal{M}}^{\Phi_\varphi,\bar\mu}_s\big)h\big\rangle^2\big] & = \mathbb{E}^{\mathbb{P}^n}\Bigg[\Bigg(\frac{1}{n}\sum_{i=1}^n h(Z^{i}) \big(\tilde{\mathcal{M}}^{\Phi_\varphi,\mu^n}_t(Z^{i}) - \tilde{\mathcal{M}}^{\Phi_\varphi,\mu^n}_s(Z^{i})\big)\Bigg)^2\Bigg] \\[3pt] & = \frac{1}{n^2}\sum_{i=1}^n\mathbb{E}^{\mathbb{P}^n}\big[h(Z^{i})^2 \big(\tilde{\mathcal{M}}^{\Phi_\varphi,\mu^n}_t(Z^{i})-\tilde{\mathcal{M}}^{\Phi_\varphi,\mu^n}_s(Z^{i})\big)^2\big] \\[3pt] & = \frac{1}{n}\mathbb{E}^{\mathbb{P}^n}\big[h(Z^{1})^2 \big(\tilde{\mathcal{M}}^{\Phi_\varphi,\mu^n}_t(Z^{1})-\tilde{\mathcal{M}}^{\Phi_\varphi,\mu^n}_s(Z^{1})\big)^2\big]. \end{align*}

In addition, the increment $\tilde{\mathcal{M}}^{\Phi_\varphi, \mu^n}_t(Z^{1}) - \tilde{\mathcal{M}}^{\Phi_\varphi, \mu^n}_s(Z^{1})$ admits the following quadratic variation:

\begin{equation*} \int_s^t\Phi_{\varphi}'(Z^{1}_r)^2\sum_{k\in K^{1}_r} \big|D\varphi^k(X^{1,k}_r)\sigma(r,X^{1,k}_{r\wedge\cdot},\mu^n_r)\big|^2\,{\mathrm{d}} r + \int_s^t\int_{[0,\bar\gamma]\times[0,1]}\sum_{k\in K^{1}_r}G^{1,k}_r(z)^2\,{\mathrm{d}} z\,{\mathrm{d}} r. \end{equation*}

By Assumption 1, together with the fact that $\Phi\in C^2_{\mathrm{b}}(\mathbb{R})$ and $\varphi \in C^2_{\mathrm{b}}(\mathbb{K}\times\mathbb{R}^d)$ , there exists a constant $C>0$ , which may vary from line to line, such that

\begin{equation*} \int_s^t\Phi_{\varphi}'(Z^{1}_r)^2\sum_{k\in K^{1}_r}\big|D\varphi^k(X^{1,k}_r) \sigma\big(r,X^{1,k}_{r\wedge\cdot},\mu^n_r\big)\big|^2\,{\mathrm{d}} r \leq C\int_s^t \# K^{1}_r\,{\mathrm{d}} r \end{equation*}

and

\begin{align*} & \int_s^t\int_{[0,\bar\gamma]\times[0,1]}\sum_{k\in K^{1}_r}G^{1,k}_r(z)^2\,{\mathrm{d}} z\,{\mathrm{d}} r \\[2pt] & \quad = \int_s^t\sum_{k\in K^{1}_r}\gamma\big(r,X^{1,k}_{r\wedge\cdot},\mu^n_r\big)\sum_{\ell\geq0}\Bigg( \Phi_{\varphi}\Bigg(Z^{1}_r+\sum_{j=1}^{\ell}\delta_{(kj,X^{1,k}_r)}\Bigg) - \Phi_{\varphi}(Z^{1}_r)\Bigg)^2 p_\ell\big(r,X^{1,k}_{r\wedge\cdot},\mu^n_r\big)\,{\mathrm{d}} r \\[2pt] & \quad \leq C\int_s^t \# K^{1}_r\,{\mathrm{d}} r. \end{align*}

Along with Lemma 4, it follows that

\begin{equation*} \mathbb{E}^{\bar{\mathbb{P}}^n}\big[\big\langle\mu,\big(\tilde{\mathcal{M}}^{\Phi_\varphi,\mu}_t - \tilde{\mathcal{M}}^{\Phi_\varphi,\mu}_s\big)h\big\rangle^2\big] \leq \frac{C\|h\|^2(t-s)}{n}\mathbb{E}^{\mathbb{P}^n}[\sup\nolimits_{s\leq r\leq t}\#K^{1}_r] \xrightarrow[n\rightarrow\infty]{} 0. \end{equation*}

Finally, following the arguments in the proof of [Reference Claisse, Ren and Tan6, Lemma 4.14], we can prove, up to the localizing sequence $\tilde{\tau}_{\tilde{n}}\,:\!=\,\inf\{r\ge 0;\, \langle \tilde Z_r,\mathbf{1}\rangle\ge \tilde{n}\}$ , that the following convergence holds: for all $s, t\notin D_{\bar{\mathbb{P}}}$ , a countable subset of [0, T], such that $s\le t$ ,

\begin{align*} \mathbb{E}^{\bar{\mathbb{P}}^n}\big[\big\langle\bar\mu, \big(\tilde{\mathcal{M}}^{\Phi_\varphi,\bar\mu}_{t\wedge\tilde{\tau}_{\tilde{n}}} - \tilde{\mathcal{M}}^{\Phi_\varphi,\bar\mu}_{s\wedge\tilde{\tau}_{\tilde{n}}}\big)h\big\rangle^2\big] \xrightarrow[n\rightarrow\infty]{} \mathbb{E}^{\bar{\mathbb{P}}}\big[\big\langle\bar\mu, \big(\tilde{\mathcal{M}}^{\Phi_\varphi,\bar\mu}_{t\wedge\tilde{\tau}_{\tilde{n}}} - \tilde{\mathcal{M}}^{\Phi_\varphi,\bar\mu}_{s\wedge\tilde{\tau}_{\tilde{n}}}\big)h\big\rangle^2\big]. \end{align*}

Note that $\tilde{\tau}_{\tilde{n}}$ is continuous for the Skorokhod topology in view of [Reference Jacod and Shiryaev19, Proposition VI.2.7]. This proves that, for all $s, t \notin D_{\bar{\mathbb{P}}}$ , $\bar{\mathbb{P}}$ -a.e. $\bar\omega$ , $\big\langle\bar\mu(\bar\omega),\big(\tilde{\mathcal{M}}^{\Phi_\varphi,\bar\mu(\bar\omega)}_{t\wedge\tilde{\tau}_{\tilde{n}}} - \tilde{\mathcal{M}}^{\Phi_\varphi,\bar\mu(\bar\omega)}_{s\wedge\tilde{\tau}_{\tilde{n}}}\big)h\big\rangle = 0$ . Using the right-continuity of $\tilde{\mathcal{M}}^{\Phi_\varphi,\bar\mu(\bar\omega)}$ , we obtain that this equality actually holds for all $0 \le s \le t \le T$ , and thus the stopped process $\tilde{\mathcal{M}}^{\Phi_\varphi,\bar\mu(\bar\omega)}_{\tilde{\tau}_{\tilde{n}}\wedge\cdot}$ is a $(\bar\mu(\bar\omega),\tilde{\mathbb{F}})$ -martingale for $\bar{\mathbb{P}}$ –a.e. $\bar\omega$ . Finally, we observe that, by arguments similar to Lemma 4, $\mathbb{E}^{{\bar\mu(\bar\omega)}}[\sup\nolimits_{0\leq t\leq T}{\langle\tilde Z_t,\mathbf{1}\rangle}] < \infty$ , $\bar{\mathbb{P}}$ -a.s., so that the process $\tilde{\mathcal{M}}^{\Phi_\varphi,\bar\mu(\bar\omega)}$ is uniformly integrable and thus is a $(\bar\mu(\bar\omega),\tilde{\mathbb{F}})$ -martingale. This concludes the proof of Theorem 3.