## APPENDIX B. PROOF OF LEMMA 5.2

The recursive equality for $\xi_n$ derives from the fundamental theorem of calculus. Let $n \ge 1$,

$$ \frac{\xi_n (E_n, (\hat{\mathcal{V}}_i, 0 \le i \le n))}{\xi_{n-1} (E_{n-1}, (\hat{\mathcal{V}}_i, 0 \le i \le n-1))} = \frac{\psi_\phi (Y_{\hat{T}_{n-1}}, \chi_{\hat{T}_{n-1}})}{\psi_\phi (Y_{\hat{T}_n}, \chi_{\hat{T}_n})} \exp \left( \int_{\hat{T}_{n-1}}^{\hat{T}_n} \frac{\mathcal{G}\psi_\phi (Y_s, \chi_s)}{\psi_\phi (Y_s, \chi_s)} \mathrm{d}s \right). $$

We remark that $\mathcal{G}$ and $G$ introduced in (2.3) and (2.4), verify for all $(x_e, \nu, t) \in \mathbb{W} \times \mathbb{R}_+$

$$ \frac{[\mathcal{G} - G] \psi_\phi(x_e, \nu, t)}{\psi_\phi(x_e, \nu, t)} = \hat{\tau}_{\text{tot}}(x_e, \nu, t) - \int_{\mathcal{X}} B(x, \nu, t)\nu(dx), $$

where $\hat{\tau}_{\text{tot}}$ is defined in (2.10). Then, using derivative chain rule we get that

$$ G(\ln \circ \psi_\phi)(x_e, \nu, t) = \frac{G\psi_\phi(x_e, \nu, t)}{\psi_\phi(x_e, \nu, t)}. $$

Between successive branching events, the evolution of the process is purely deterministic and we can apply the fundamental theorem of calculus:

$$ \int_{\hat{T}_{n-1}}^{\hat{T}_n} G(\ln \circ \psi_\phi)(Y_s, \chi_s, s) \mathrm{d}s = \ln \left( \psi_\phi \left( Y_{\hat{T}_{n-}}, \chi_{\hat{T}_{n-}}, \hat{T}_n \right) \right) - \ln \left( \psi_\phi \left( Y_{\hat{T}_{n-1}}, \chi_{\hat{T}_{n-1}}, \hat{T}_{n-1} \right) \right). $$

We thus have

$$ \exp \left( \int_{\hat{T}_{n-1}}^{\hat{T}_n} \frac{\mathcal{G}\psi_\phi(Y_s, \chi_s, s)}{\psi_\phi(Y_s, \chi_s, s)} \mathrm{d}s \right) = \frac{\psi_\phi(Y_{\hat{T}_{n-}}, \chi_{\hat{T}_{n-}}, \hat{T}_n)}{\psi_\phi(Y_{\hat{T}_{n-1}}, \chi_{\hat{T}_{n-1}}, \hat{T}_{n-1})} \exp \left( \int_{\hat{T}_{n-1}}^{\hat{T}_n} \lambda(Y_s, \chi_s, s) \mathrm{d}s \right), $$

that concludes the proof.

## APPENDIX C. PROOF OF LEMMA 5.4

To establish this lemma, we follow the conceptual decomposition of the spine process, first introduced by Lyons, Pemantle and Peres [44]. In the following, $\psi_\phi \equiv 1$. We consider the process $(\mathring{\chi}_t, t \ge 0)$ describing all the individuals outside the spine and its associated marginal process $\chi_t$, defined for all $t \ge 0$ by:

$$ \mathring{\chi}_t := \bar{\chi}_t - \delta_{(E_t, Y_t)}, \quad \text{and} \quad \chi_t := \chi_t - \delta_{Y_t}. $$

The random set $\mathring{\mathcal{G}}(t)$ of labels of individuals outside the spine living at time $t$ is defined by

$$ \mathring{\mathcal{G}}(t) = \left\{ u \in \mathcal{U} \setminus \{E_t\} : \int_{\mathcal{U} \times \mathcal{X}} \mathbb{1}_{\{v=u\}} \bar{\chi}_t(dv, dx) > 0 \right\}. $$

We also introduce for all $t \ge 0$,

$$ \dot{\mathcal{W}}_t := \sum_{u \in \mathring{\mathcal{G}}(t)} \exp \left( - \int_0^t \Lambda(X_s^u, \mathring{\chi}_s + \delta_{Y_s}, s) \mathrm{d}s \right), $$

where for all $(x, \nu, t) \in \mathbb{W} \times \mathbb{R}_+$, $\Lambda(x, \nu, t) := B(x, \nu, t)(m(x, \nu, t) - 1)$.

We remark that, under Assumption D, for all $t \ge 0$

$$ \dot{\mathcal{W}}_t \le \widehat{\mathcal{W}}_t \le \dot{\mathcal{W}}_t + 1, \qquad (C.1) $$

where $\widehat{\mathcal{W}}_t$ is the process introduced in (5.18).

We first establish the non-degenerate case, and suppose that $\sum_{k \ge 1} k \log(k) \bar{p}_k < +\infty$. We show that the process $(\dot{\mathcal{W}}_t, t \ge 0)$ does not explode and is almost surely bounded at infinity, following proof of Proposition 5 in [3]. To this end, we express it as a solution of a SDE, where we distinguish the contribution of the spinal individual from that of the other individuals.
$$ \bigotimes_{i=1}^{k} (\mathbb{R}_{+} \times \mathcal{U} \times \mathbb{N} \times \mathcal{X}^{n_i}), \text{ such that} $$

$$ \mathbb{E}_{\bar{z}} \left[ \mathbb{1}_{\{T_{\text{Exp}} > t, \mathbb{G}(t) \neq \emptyset\}} H(U_t, (\bar{\nu}_s, s \le t)) \right] = \sum_{k \ge 0} \sum_{a \in \mathcal{A}_k(\bar{z})} \sum_{e \in \mathbb{G}_k(a)} \mathbb{E}_{\bar{z}} \left[ \mathbb{E} \left[ \mathbb{1}_{\{T_{k+1} > t\}} | \mathcal{F}_k \right] \mathbb{1}_{\{T_k \le t\}} F_{k,a}^{t,e} (\mathcal{V}_i, 0 \le i \le k) \prod_{i=0}^{k} \mathbb{1}_{\{\hat{A}_i = a_i\}} \right]. $$

We apply Lemma 5.1 with the non-negative measurable function $F$ defined for all sequence $(v_i, 0 \le i \le k) \in \bigotimes_{i=1}^{k} (\mathbb{R}_{+} \times \mathcal{U} \times \mathbb{N} \times \mathcal{X}^{n_i})$, where for all $0 \le i \le k$, $v_i := (t_i, a_i, y_i)$, by

$$ F(v_i, 0 \le i \le k) := \mathbb{P} \left( S_{k+1} > t - t_k \bigg| \bigcap_{0 \le i \le k} \{ \mathcal{V}_i = v_i \} \right) \mathbb{1}_{t_k \le t} F_{k,a}^{t,e}(v_i, 0 \le i \le k), $$

where $S_{k+1}$ is the random variable giving the $(k+1)$-th inter-arrival time of jumps in the original process. Note that $S_{k+1}$ follows the same probability law as $T_{k+1} - T_k$. We thus get, for all $e \in \mathbb{G}_k(a)$

$$ \mathbb{E}_{\bar{z}} \left[ \mathbb{1}_{\{\mathbb{G}(t) \neq \emptyset\} \cap \{T_{\text{Exp}} > t\}} H(U_t, (\bar{\nu}_s, s \le t)) \right] = \langle z, \psi(\cdot, z) \rangle \sum_{k \ge 0} \sum_{a \in \mathcal{A}_k(\bar{z})} \sum_{e \in \mathbb{G}_k(a)} \mathbb{E}_{\bar{z}} \left[ \mathbb{1}_{\{E_k=e\}} \right. \\ \left. \times \xi_k (E_k, (\hat{\mathcal{V}}_i, 0 \le i \le k)) F(\hat{\mathcal{V}}_i, 0 \le i \le k) \prod_{i=0}^{k} \mathbb{1}_{\{\hat{A}_i=a_i\}} \right]. \quad (5.11) $$

Then, using that $\lambda$ introduced in (5.3) verifies $\lambda := \hat{\tau}_{\text{tot}} - \tau$,

$$ \mathbb{P} \left( S_{k+1} > t - \hat{T}_k \bigg| \bigcap_{0 \le i \le k} \{ \hat{\mathcal{V}}_i = v_i \} \right) \mathbb{1}_{\{\hat{T}_k \le t\}} = e^{\int_{\hat{T}_k}^{t} \lambda(Y_s, \chi_s) ds} e^{-\int_{\hat{T}_k}^{t} \hat{\tau}_{\text{tot}}(Y_s, \chi_s) ds} \mathbb{1}_{\{\hat{T}_k \le t\}} \\ = e^{\int_{\hat{T}_k}^{t} \lambda(Y_s, \chi_s) ds} \mathbb{E} \left[ \mathbb{1}_{\{\hat{T}_{k+1} > t\}} | \hat{\mathcal{F}}_k \right] \mathbb{1}_{\{\hat{T}_k \le t\}}. $$

We recall that on the event $\{\hat{T}_{k+1} > t\}$, there is no jump in the interval $(\hat{T}_k, t]$, thus, applying Lemma 5.2 on this interval we get

$$ \xi_k (E_k, (\hat{\mathcal{V}}_i, 0 \le i \le k)) \mathbb{P} \left( S_k > t - \hat{T}_k \bigg| \bigcap_{0 \le i \le k} \{ \hat{\mathcal{V}}_i = v_i \} \right) \mathbb{1}_{\{\hat{T}_k \le t\}} = \frac{1}{\psi(Y_t, \chi_t)} \exp \left( \int_0^t \frac{\mathcal{G}\psi(Y_s, \chi_s)}{\psi(Y_s, \chi_s)} ds \right) \mathbb{E} \left[ \mathbb{1}_{\{\hat{T}_k \le t < \hat{T}_{k+1}\}} | \hat{\mathcal{F}}_k \right]. \quad (5.12) $$

On the event $\{\hat{T}_k \le t < \hat{T}_{k+1}\}$, $E_k = E(t)$ almost surely, then using (5.11) and (5.12) gives

$$ \mathbb{E}_{\bar{z}} \left[ \mathbb{1}_{\{T_{\text{Exp}} > t, \mathbb{G}(t) \neq \emptyset\}} H(U_t, (\bar{\nu}_s, s \le t)) \right] = \langle z, \psi(\cdot, z) \rangle \sum_{k \ge 0} \sum_{a \in \mathcal{A}_k(\bar{z})} \sum_{e \in \mathbb{G}_k(a)} \mathbb{E}_{\bar{z}} \left[ \mathbb{1}_{\{E(t)=e\}} \right. \\ \left. \times \frac{\mathbb{1}_{\{\hat{T}_k \le t < \hat{T}_{k+1}\}}}{\psi(Y_t, \chi_t)} \exp \left( \int_0^t \frac{\mathcal{G}\psi(Y_s, \chi_s)}{\psi(Y_s, \chi_s)} ds \right) F_{k,a}^{t,e} (\hat{\mathcal{V}}_i, 0 \le i \le k) \prod_{i=0}^{k} \mathbb{1}_{\{\hat{A}_i=a_i\}} \right]. $$
Let $\hat{E} = \mathcal{U} \times \mathbb{R}_+ \times \mathbb{N} \times [0, 1]^N$ and $\hat{E}^* = \mathbb{R}_+ \times \mathbb{N} \times [0, 1]^N \times \mathbb{N}$. Let $\hat{Q}(ds, du, dr, dk, d\theta)$ and $\hat{Q}^*(ds, dr, dk, d\theta, dj)$ be two independent Poisson point measures on $\mathbb{R}_+ \times \hat{E}$ and $\mathbb{R}_+ \times \hat{E}^*$ with respective intensity $dsC_{\mathcal{U}}(du)drC_{\mathbb{N}}(dk)d\hat{\theta}$ and $dsC_{\mathcal{U}}(du)drC_{\mathbb{N}}(dk)d\hat{\theta}^*C_{\mathbb{N}}(dj)$, where $C_{\mathcal{U}}(du)$ and $C_{\mathbb{N}}(dk)$ denote the counting measures respectively on $\mathcal{U}$ and $\mathbb{N}$. We denote by $(\hat{F}_t^*, t \ge 0)$ the canonical filtration associated with $\hat{Q}^*$. We introduce $\hat{\theta} := (\hat{\theta}_i, i \in \mathbb{N}^*)$ and $\hat{\theta}^* := (\hat{\theta}_i^*, i \in \mathbb{N}^*)$ two vectors of independent uniform random variables on $[0, 1]$. The random vectors giving the traits of the offspring are denoted $(\hat{F}_{i,n}(x, \chi, t, \hat{\theta}_i), i \le n)$ outside the spine and $(\hat{F}_{i,n}^*(x, \chi, t, \hat{\theta}_i^*), i \le n)$ for the spinal individual.

We localize the process $(\hat{W}_t, t \ge 0)$ to avoid explosion. Let $T^m := \inf\{t \ge 0 : \text{Card}(\hat{\mathbb{G}}(t)) \ge m\}$ be the first time at which the population outside the spine includes more than $m$ individuals for $m \ge 1$. We remark that the branching events impact the process $(\hat{W}_t, t \le T^m)$ only through the change in the number of individuals. Furthermore, as $\psi_\phi \equiv 1$, every individual in $\hat{\mathbb{G}}(t)$ behaves like the individuals in the original branching process. Then, for all $t \le T^m$

$$ \begin{aligned} \hat{W}_t = \hat{W}_0 & - \int_0^t \sum_{u \in \hat{\mathbb{G}}(s)} \Lambda(X_s^u, \chi_s, s) e^{-\int_0^s \Lambda(X_w^u, \chi_w, w) dw} ds \\ & + \int_0^t \int_{\hat{E}} \mathbb{1}_{\{u \in \hat{\mathbb{G}}(s)\}} \mathbb{1}_{\{r \le B_k(X_{s-}^u, \chi_{s-}, s)\}} (k-1) e^{-\int_0^s \Lambda(X_w^u, \chi_w, w) dw} \hat{Q}(ds, du, dr, dk, d\hat{\theta}) \\ & + \int_0^t \int_{\hat{E}^*} \mathbb{1}_{\{r \le k B_k(X_{s-}^u, \chi_{s-}, s)\}} (k-1) e^{-\int_0^s \Lambda(Y_w, \chi_w, w) dw} \hat{Q}^*(ds, dr, dk, d\hat{\theta}, dj). \end{aligned} $$

The first line describes the dynamics of the function between branching events. The second line describes the branching events outside the spine. The last one is associated with the contribution of the spinal individual that is here a source of immigration, where the individual chosen to be the new spinal individual is removed from the process. Introducing $\tilde{Q}(ds, du, dr, dk, d\hat{\theta})$ the compensated measure of $\hat{Q}(ds, du, dr, dk, d\hat{\theta})$, we get

$$ \begin{aligned} \hat{W}_t = \hat{W}_0 &+ \int_0^t \int_{\hat{E}} \mathbb{1}_{\{u \in \hat{\mathbb{G}}(s)\}} \mathbb{1}_{\{r \le B_k(X_{s-}^u, \chi_{s-}, s)\}} (k-1) e^{-\int_0^s \Lambda(X_w^u, \chi_w, w) dw} \tilde{Q}(ds, du, dr, dk, d\hat{\theta}) \\ &+ \int_0^t \int_{\hat{E}^*} \mathbb{1}_{\{r \le k B_k(X_{s-}^u, \chi_{s-}, s)\}} (k-1) e^{-\int_0^s \Lambda(Y_w, \chi_w, w) dw} \hat{Q}^*(ds, dr, dk, d\hat{\theta}). \end{aligned} $$

Remark that this last equality only holds for constant $\psi_\phi$. Then, conditionally on $\hat{\mathcal{F}}_\infty^*$, $(\hat{W}_{t \wedge T^m}, t \ge 0)$ is a submartingale and we have, for all $t \le T^m$

$$ \mathbb{E}_{\bar{z}}[\hat{W}_t | \hat{\mathcal{F}}_t^*] = \mathbb{E}_{\bar{z}}[\hat{W}_0] + \int_0^t \int_{\hat{E}^*} \mathbb{1}_{\{r \le k B_k(X_{s-}^u, \chi_{s-}, s)\}} (k-1) e^{-\int_0^s \Lambda(Y_w, \chi_w, w) dw} \hat{Q}^*(ds, dr, dk, d\hat{\theta}). \quad (C.2) $$

Using that $\mathbb{E}_{\bar{z}}[\hat{W}_0] = \langle z, 1 \rangle - 1$, along with bounds in Assumption D and in (C.2), we get

$$ \mathbb{E}_{\bar{z}}[\hat{W}_{t \wedge T^m} | \hat{\mathcal{F}}_\infty^*] \le \langle z, 1 \rangle + \int_0^t \int_{\hat{E}^*} \mathbb{1}_{\{r \le k \bar{B} \bar{p}_k\}} (k-1) e^{-cs} \hat{Q}^* (ds, dr, dk, d\hat{\theta}), $$

where $\bar{B} = \sup_{\rho \in \mathbb{W} \times \mathbb{R}_+} B(\rho) < +\infty$. Then there exist two independent families of random variables $(\bar{S}_i, i \ge 0)$ and $(\bar{N}_i, i \ge 0)$, where the $\bar{S}_i$ are independent exponential random variables of parameter $\bar{B}$, and the $\bar{N}_i$ are independent random variables distributed as
Reconstructing the right-hand side and using the fact that the spine process does not extinct, we have

$$
\begin{align*}
\mathbb{E}_{\bar{z}} \left[ 1_{\{T_{\text{Exp}} > t, \mathcal{G}(t) \neq \emptyset\}} H(U_t, (\bar{\nu}_s, s \le t)) \right] &= \\
&\quad \langle z, \psi(\cdot, z) \rangle \mathbb{E}_{\bar{z}} \left[ 1_{\{\hat{T}_{\text{Exp}} > t\}} \frac{p_{E_t}(\bar{\chi}_t)}{\psi(Y_t, \chi_t)} \exp \left( \int_0^t \frac{\mathcal{G}\psi(Y_s, \chi_s)}{\psi(Y_s, \chi_s)} ds \right) H(E_t, (\bar{\chi}_s, s \le t)) \right],
\end{align*}
$$

that concludes the proof.

<figure><img src="image_6.png" /></figure>

5.3. **Proofs of Section 3.** In this section we derive the results on the limiting martingale and a $L \log L$ criterion.

*Proof of Proposition 3.1.* Let $\psi \in \mathcal{D}$ satisfying Assumption 2, we first show for all $t \ge 0$, the integrability of the random variable $W_t(\psi)$ introduced in (3.1). As Assumption 1 is satisfied, we can apply Corollary 2.5 for any $t \ge 0$ to the positive function $f$ on $\mathbb{D}([0, t], \mathcal{X} \times \bar{\mathcal{V}})$, such that for all $u \in \mathcal{G}(t)$,

$$
f((X_s^u, \bar{\nu}_s), 0 \le s \le t) := \exp \left( - \int_0^t \frac{\mathcal{G}\psi(X_s^u, \nu_s)}{\psi(X_s^u, \nu_s)} ds \right).
$$

Corollary 2.5 applied to the function $f$, ensures that for any initial condition $\bar{z}$,

$$
\mathbb{E}_{\bar{z}} \left[ \sum_{u \in \mathcal{G}(t)} \exp \left( - \int_{0}^{t} \frac{\mathcal{G} \psi (X_{s}^{u}, \nu_{s})}{\psi(X_{s}^{u}, \nu_{s})} ds \right) \psi(X_{t}^{u}, \nu_{t}) \right] = \langle z, \psi(\cdot, z) \rangle. \quad (5.13)
$$

This last identity guarantees the integrability of $W_t(\psi)$.

Now, for all $r, t \in \mathbb{R}_+^*$, such that $t \ge r$, we decompose the individuals alive in the population at time $t$ according to their ancestors at time $r$. We then have

$$
\begin{align*}
\mathbb{E}_{\bar{z}} [W_t(\psi) | \mathcal{F}_r] &= \sum_{v \in \mathcal{G}(r)} \exp \left( - \int_0^r \frac{\mathcal{G}\psi(X_s^u, \nu_s)}{\psi(X_s^u, \nu_s)} ds \right) \\
&\quad \times \mathbb{E}_{\bar{z}} \left[ \sum_{u \in \mathcal{G}(t), \text{s.t. } v \le u} \exp \left( - \int_r^t \frac{\mathcal{G}\psi(X_s^u, \nu_s)}{\psi(X_s^u, \nu_s)} ds \right) \psi(X_t^u, \nu_t) \middle| \mathcal{F}_r \right].
\end{align*}
$$

Now, in order to establish that the conditional expectation is equal to $\psi(X_r^v, \nu_r)$ for all $v \in \mathcal{G}(r)$, we apply Corollary 2.5 to the positive functions $f_v$ on $\mathcal{U} \times \mathbb{D}([0, t], \bar{\mathcal{V}})$, such that

$$
f_v(u, (\bar{\nu}_s, 0 \le s \le t)) := \mathbb{1}_{v \le u} \exp \left( - \int_r^t \frac{\mathcal{G}\psi(X_s^u, \nu_s)}{\psi(X_s^u, \nu_s)} ds \right)
$$

and use the Markov property. We get that for all $v \in \mathcal{G}(r)$

$$
\mathbb{E} \left[ \sum_{\substack{u \in \mathcal{G}(t), \\ \text{s.t. } v \le u}} \exp \left( - \int_r^t \frac{\mathcal{G} \psi (X_s^u, \nu_s)}{\psi(X_s^u, \nu_s)} ds \right) \psi (X_t^u, \nu_t) \middle| \mathcal{F}_r \right] = \langle \nu_r, \psi(\cdot, \nu_r) \rangle \mathbb{E}_{\nu_r} [1_{\{v \le E_t\}}].
$$

Finally using the new spinal individual distribution (2.9), we get

$$
\mathbb{E} [W_t(\psi) | \mathcal{F}_r] = \sum_{v \in \mathcal{G}(r)} \exp \left( - \int_0^r \frac{\mathcal{G} \psi (X_s^u, \nu_s)}{\psi (X_s^u, \nu_s)} ds \right) \psi (X_r^v, \nu_r).
$$

That concludes the proof.

<figure><img src="image_7.png" /></figure>

To establish Theorem 3.4, we need the following lemma from measure theory.
$\mathbb{P}(\overline{N}_i = k - 1) = k\bar{p}_k/(\sum_{k \ge 0} k\bar{p}_k)$, such that

$$ \mathbb{E}_{\bar{z}} \left[ \hat{W}_{t \wedge T^m} | \hat{\mathcal{F}}_{\infty}^* \right] \le \langle z, 1 \rangle + \sum_{i \ge 0} \overline{N}_i e^{-c \bar{T}_i}, $$

where $\bar{T}_i := \sum_{k=0}^i \bar{S}_k$. Note that the assumption $\sum_{k \ge 1} k \log(k) \bar{p}_k < +\infty$ ensures that the law of the random variables $\overline{N}_i$ is well defined. The conclusion follows the steps from proof of Proposition 5 in [3]: using the Borel Cantelli lemma, $\sum_{k \ge 1} k \log(k) \bar{p}_k < +\infty$, ensures that $\limsup_{i \to \infty} \ln(\overline{N}_i)/i = 0$ a.s.. Thus the series $(\sum_{i \ge 0} \overline{N}_i e^{-c \bar{T}_i})$ is almost surely finite. Letting $m \to \infty$ and using the upper bound of Assumption D, we get that $T^m \to \infty$ almost surely. By Fatou's lemma, we then have $\sup_{t \ge 0} \mathbb{E}_{\bar{z}} [\hat{W}_t | \hat{\mathcal{F}}_{\infty}^*] < \infty$. The quenched submartingale $(\hat{W}_{t \wedge T^m}, t \ge 0)$ converges a.s. to a finite random variable. Thus $\limsup_{t \to +\infty} \hat{W}_t < +\infty$ almost surely and we conclude the proof using (C.1).

We handle now the degenerated case, and suppose that $\sum_{k \ge 1} k \log(k) \underline{p}_k = +\infty$. We denote $(\hat{T}_n^*, n \ge 0)$ the sequence of jumps times of the spinal individual, given by the measure $\hat{Q}^*$, and $(\hat{N}_n^*, n \ge 0)$ the sequence giving the number of children at each branching event of the spine. For all $n \ge 0$, we remark that

$$ \hat{W}_{\hat{T}_n^*} = \hat{W}_{\hat{T}_n^* -} + (\hat{N}_n^* - 1) \exp \left( - \int_0^{\hat{T}_n^*} \Lambda(Y_s, \chi_s, s) ds \right). $$

Using that $\hat{W}^*$ is almost surely non-negative, and using the upper bound in Assumption D, we obtain

$$ \hat{W}_{\hat{T}_n^*} \ge (\hat{N}_n^* - 1) \exp(-C \hat{T}_n^*). \quad (C.3) $$

Assumption C with $\psi_\phi \equiv 1$ implies that for all $\rho \in \mathcal{X} \times \mathcal{V} \times \mathbb{R}_+$, $\sum_{k \ge 1} k p_k(\rho) < +\infty$. Thus $\sum_{k \ge 1} k \underline{p}_k < +\infty$ and we introduce $\underline{N}$ the random variable of law $(k \underline{p}_k/(\sum_{k \ge 1} k \underline{p}_k), k \ge 0)$. The criterion $\sum_{k \ge 1} k \log(k) \underline{p}_k = +\infty$ thus ensures that for all $K > 0$

$$ \frac{\mathbb{E}[\log(\underline{N})]}{K} = +\infty. $$

Thus, we get for all $K > 0$,

$$ \sum_{n \ge 1} \mathbb{P} \left( \frac{\log(\hat{N}_n^*)}{n} \ge K \middle| \hat{\mathcal{F}}_{\hat{T}_n^*}^* \right) \ge \sum_{n \ge 1} \mathbb{P} \left( \frac{\log(\underline{N})}{K} \ge n \right) = +\infty. $$

We can apply the conditional second Borel-Cantelli lemma, see Theorem 4.3.4 in [20]. For all $K > 0$,

$$ \limsup_{n \to \infty} \hat{N}_n^* e^{-Kn} = +\infty \quad \text{a.s.} \quad (C.4) $$

Then using Markov's inequality we get

$$ \mathbb{P} \left( \hat{T}_n^* \ge nK \right) \le \frac{\mathbb{E} \left[ \sum_{i=1}^n \hat{T}_i^* - \hat{T}_{i-1}^* \right]}{nK}. $$

The upper bound in Assumption D ensures that the rate of every inter-arrival time of jumps is lower than $\bar{B}$. Thus we obtain for all $K > 0$,

$$ \mathbb{P} \left( \hat{T}_n^* \ge nK \right) \le \frac{\bar{B}}{K}. \quad (C.5) $$

Thus, combining (C.4) and (C.5) into (C.3), we get that $\limsup_{n \to \infty} \hat{W}_{\hat{T}_n^*} = +\infty$ almost surely. Notice that Assumption D ensures that the jumps times $(\hat{T}_n^*, n \ge 0)$ almost surely degenerate at infinity, thus $\limsup_{t \to \infty} \hat{W}_t = +\infty$ and relation (C.1) concludes the proof.
**Lemma 5.4.** Let $(\Omega, \mathcal{F}, \mu)$ be a probability space and let $\hat{\mu}$ be a finite non negative measure on $\Omega$. Let $(\mathcal{F}_t, 0 \le t)$ be increasing $\sigma$-fields such that $\sigma\left(\bigcup_{0 \le t} \mathcal{F}_t\right) = \mathcal{F}$, and $\hat{\mu}_t, \mu_t$ be the restrictions of $\hat{\mu}$ and $\mu$ to $\mathcal{F}_t$. Suppose that there exists a non-negative $\mathcal{F}_t$-martingale $(W_t, 0 \le t)$ such that for all $t \ge 0$

$$ \frac{d\hat{\mu}_t}{d\mu_t} = W_t. $$

Then, denoting $W := \limsup_{t \to +\infty} W_t$, we have the following dichotomy:

(1) $\int W d\mu = \int W_0 d\mu$ if and only if $W < +\infty$ $\hat{\mu}$-a.s.

(2) $W = 0$ $\mu$-a.s. if and only if $W = +\infty$ $\hat{\mu}$-a.s.

*Proof.* We refer to [2, 45] for the proof of this result in discrete time. The extension to continuous time changes of measure uses Kolmogorov's extension theorem, see Durrett Appendix A [21] for further details. $\square$

We now state the following lemma, that is the dual proposition of Theorem 3.4.

**Lemma 5.5.** Let $\psi \in \mathcal{D}$ satisfying Assumption 7, let $(\hat{W}_t(\psi), t \le \hat{T}_{Exp})$ be the $\hat{\mathcal{F}}_t$-adapted process such that, for all $t \le \hat{T}_{Exp}$,

$$ \hat{W}_t(\psi) := \sum_{u \in \hat{\mathcal{G}}(t)} \psi(X_t^u, \chi_t) \exp \left( - \int_0^t \frac{G\psi(X_s^u, \chi_s)}{\psi(X_s^u, \chi_s)} ds \right), \quad (5.14) $$

where $\hat{\mathcal{G}}(t)$ is the set of labels of individuals living in the spinal population at time $t$.

Under Assumption 5, (3.2) implies that, $\hat{T}_{Exp} = \infty$ and $\limsup_{t \to +\infty} \hat{W}_t(\psi) < +\infty$ almost surely.

Under Assumption 6, (3.3) implies that $\limsup_{t \to +\infty} \hat{W}_t(\psi) = +\infty$ almost surely.

This lemma is the innovative part of this proof. It involves the decomposition of the spinal process as a process with immigration where the spinal individual provides new individuals at a $\psi$-biased rate, and the new spine is the new source of immigration. However as the function $\psi$ is not constant, the spinal construction also changes the dynamics of individuals outside the spine, that do not behave as those in the original process $(\nu_t, t \ge 0)$. Assumption 5 is used to control the behavior of the individuals outside the spine to prove the non degeneracy of the limiting martingale. For the second part of this theorem, we use Assumption 6 to establish that the contribution of the offspring without the new spine is enough to ensure the degeneracy.

*Proof of Theorem 3.4.* Theorem 2.1 applied with any function $\psi \in \mathcal{D}$ satisfying Assumption 7 exhibits the martingale $(W_t(\psi), t \ge 0)$ as a Radon-Nikodym derivative. We can thus apply Lemma 5.4 to the spinal change of measure. The martingale $W_t(\psi)$ and its limit $W(\psi)$, introduced in (3.1), verify the following dichotomy:

(1) $\mathbb{E}_{\bar{z}}[W(\psi)] = \mathbb{E}_{\bar{z}}[W_0(\psi)]$ if and only if $\limsup_{t \to +\infty} \hat{W}_t(\psi) < +\infty$ a.s.

(2) $W(\psi) = 0$ a.s. if and only if $\limsup_{t \to +\infty} \hat{W}_t(\psi) = +\infty$ a.s.

In this case, we use (5.13) to obtain that

$$ \mathbb{E}_{\bar{z}}[W_0(\psi)] = \langle z, \psi(\cdot, z) \rangle, $$

and a direct application of Lemma 5.5 concludes the proof. $\square$

*Proof of Lemma 5.5.* Let $\psi \in \mathcal{D}$ satisfying Assumption 7, and introduce for all $k \in \mathbb{N}$,

$$ \bar{p}_n^* := \sup_{(x, \nu) \in \mathbb{W}} \hat{p}_n^*(x, \nu), \quad \underline{p}_n^* := \inf_{(x, \nu) \in \mathbb{W}} \hat{p}_n^*(x, \nu), $$
APPENDIX D. ALGORITHMIC CONSTRUCTION

We propose the efficient algorithm that generates trajectories of the spine process con-
structed with the $\psi_{\phi}$ function introduced in (4.1). We introduce $F_{\text{div}}^{-1}$, $F_{\text{loss}}^{-1}$ and $F_{\text{loss}}^{*-1}$ the
generalized inverse of the cumulative distribution functions of the random variables $\tilde{\Lambda}$, $\tilde{\Theta}$ and
$\Theta$, defined in (4.2) and (4.3). Using the deterministic evolution of the traits and knowing
the branching events, it is easy to recover the traits at all time of the individuals.

We first generate a realization of a Poisson point process of intensity 1 on the interval
[t₁, t₂], starting from n roots following classical algorithms [46]. It returns a list T<sub>div</sub> =
[t₁, T₁, T₂, ...] of increasing times, a list I<sub>div</sub> = [i<sub>1</sub>, i<sub>2</sub>, ...] containing the numbering of the
individuals that branched at these times, a list L<sub>div</sub> = [λ<sub>1</sub>, λ<sub>2</sub>, ...] of fractions of mass at
birth, and the numbering E of the spinal individual. The numbering choice is arbitrary
and we chose to add every new individuals at the end of the hidden list. This computation
is handled by the following function Tree, where i<sub>0</sub> is the initial numbering of the spinal
individual.

**function** $(T_{\text{div}}, I_{\text{div}}, E, L_{\text{div}}) = \text{Tree}(t_1, t_2, n, i_0, F_{\text{div}}^{-1})$
$(T_{\text{div}}, N) = ([t_1], n)$

% Step 1; generating the division times

while $T_{\text{div}}[\text{end}] < t_2$ do
  Generate $u \sim \text{uniform}(0, 1)$
  append $T_{\text{div}}[\text{end}] - \ln(u)/N$ to $T_{\text{div}}$
  $N \leftarrow N + 1$
  ▷ Branching time
  ▷ New population size
end while

if $T_{\text{div}}[\text{end}] > t_2$ then pop $T_{\text{div}}$

% Step 2; generating the fractions at birth and choosing the spinal individual

$(I_{\text{div}}, E) = ([0, \dots, 0], i_0)$
$\triangleright I_{\text{div}}$ of size length$(T_{\text{div}})$

**for** $i \in \{1, \dots, \text{length}(T_{\text{div}}) - 1\}$ **do**
  Generate $I \sim \text{uniform}(\{1, \dots, n + i - 1\})$
  $I_{\text{div}}[i] \leftarrow I$
  Generate $v, q \sim \text{uniform}(0, 1)$
  append $F_{\text{div}}^{-1}(u)$ to $L_{\text{div}}$
  **if** $E = i$ and $p > F_{\text{div}}^{-1}(u)$ **then** $E \leftarrow n + i$
  ▷ Fraction $\lambda$ at birth
  ▷ New spinal individual position

end for
end function

From the list *I*<sub>div</sub>, it is possible to retrieve the lineage of all individuals using any chosen
labeling method. In our case we used the Ulam-Harris-Neveu notations and we propose the
following algorithm to generate the list *U* of labels from the list *I* previously obtained and
the initial size of the population *n*.

**function** U = Labels(I<sub>div</sub>, n)
  N<sub>div</sub> = length(I<sub>div</sub>)
  U = [1, ..., n, 0, ..., 0]
  ▷ U is of size n + N<sub>div</sub>
  **for** i ∈ {1, ..., N<sub>div</sub>} **do**
    (U[I<sub>div</sub>[i]], U[n + i]) ← (U[I<sub>div</sub>[i]]1, U[I<sub>div</sub>[i]]2)
    ▷ U[I<sub>div</sub>[i]] ∈ U
  **end for**
  U ← QuickSort(U)
  ▷ sorting algorithm in O((N<sub>div</sub> + n) log(N<sub>div</sub> + n))

end function
$$ \overline{\hat{K}}_n^*(\cdot) := \sup_{(x,\nu) \in \mathbb{W}} \hat{K}_n^*(x, \nu, \cdot), \quad \text{and} \quad \underline{\hat{K}}_n^*(\cdot) := \inf_{(x,\nu) \in \mathbb{W}} \hat{K}_n^*((x, \nu, \cdot)), $$

where $(\hat{p}_n^*, n \ge 0)$ and $\hat{K}_n^*$ are respectively the law of the number of children and the measure giving the offspring traits for the $\psi$-spine, defined respectively in (2.7) and (2.6).

We first state two technical lemmas, proven in Appendix B.

**Lemma 5.6.** For all $\psi \in \mathcal{D}$ satisfying Assumption 7, the criteria

$$ \sum_{n \ge 1} \overline{\hat{p}}_n^* \int_{\mathcal{X}^n} \sup_{(x, \nu) \in \mathbb{W}} \left[ \log \left( \sum_{i=1}^n \psi(y^i, \nu^+(x, \mathbf{y})) \right) \right] \overline{\hat{K}}_n^*(\mathbf{y}) \mathcal{M}_n(d\mathbf{y}) < +\infty $$

implies that

$$ \limsup_{t \to +\infty} \psi(Y_t, \chi_t) \exp \left( - \int_0^t \frac{\mathcal{G}\psi(Y_s, \chi_s)}{\psi(Y_s, \chi_s)} ds \right) = 0 \quad \text{a.s.} $$

We recall the notation $\nu^+(x, \mathbf{y}) := \nu + \sum_{i=1}^n \delta_{y^i} - \delta_x$, introduced in (2.2).

**Lemma 5.7.** For all $\psi \in \mathcal{D}$ satisfying Assumptions 6 and 7, the criteria

$$ \sum_{n \ge 1} \underline{\hat{p}}_n^* \int_{\mathcal{X}^n} \inf_{(x, \nu) \in \mathbb{W}} \left[ \log \left( \sum_{i=1}^n \psi(y^i, \nu^+(x, \mathbf{y})) \right) \right] \underline{\hat{K}}_n^*(\mathbf{y}) \mathcal{M}_n(d\mathbf{y}) = +\infty $$

implies that for any sequence of integers $(j_n, n \ge 2)$ such that $1 \le j_n \le n$,

$$ \sum_{n \ge 2} \underline{\hat{p}}_n^* \int_{\mathcal{X}^n} \inf_{(x, \nu) \in \mathbb{W}} \left[ \log \left( \sum_{\substack{i=1 \\ i \ne j_n}}^n \psi(y^i, \nu^+(x, \mathbf{y})) \right) \right] \underline{\hat{K}}_n^*(\mathbf{y}) \mathcal{M}_n(d\mathbf{y}) = +\infty. $$

To establish Lemma 5.5, we follow the conceptual decomposition of the spine process, first introduced by Lyons, Pemantle and Peres [45], that is, viewing the spine as an immigration source into a process without a spine. We consider the $\psi$-spinal process $(\tilde{\chi}_t, t \ge 0)$ describing all the individuals outside the spine and its associated marginal process $\tilde{\chi}_t$, defined for all $t \ge 0$ by:

$$ \tilde{\chi}_t := \bar{\chi}_t - \delta_{(E_t, Y_t)}, \quad \text{and} \quad \tilde{\chi}_t := \chi_t - \delta_{Y_t}. $$

We denote by $\tilde{\mathcal{G}}(t)$ the random set of labels of non-spinal individuals living at time $t$

$$ \tilde{\mathcal{G}}(t) = \left\{ u \in \mathcal{U} \setminus \{E_t\} : \int_{\mathcal{U} \times \mathcal{X}} \mathbb{1}_{\{v=u\}} \bar{\chi}_t(dv, dx) > 0 \right\}. $$

We also introduce for all $t \ge 0$,

$$ \dot{W}_t(\psi) := \sum_{u \in \tilde{\mathcal{G}}(t)} \psi(X_t^u, \chi_t) \exp \left( - \int_0^t \Lambda(X_s^u, \chi_s) ds \right), $$

where for all $x, \nu \in \mathbb{W}$,

$$ \Lambda(x, \nu) := \frac{\mathcal{G}\psi(x, \nu)}{\psi(x, \nu)}. $$

• We handle first the **degenerated case**, and suppose that Assumption 6 holds and that

$$ \sum_{n \ge 1} \underline{\hat{p}}_n^* \int_{\mathcal{X}^n} \inf_{(x, \nu) \in \mathbb{W}} \left[ \log \left( \sum_{i=1}^n \psi(y^i, \nu^+(x, \mathbf{y})) \right) \right] \underline{\hat{K}}_n^*(\mathbf{y}) \mathcal{M}_n(d\mathbf{y}) = +\infty. \quad (5.15) $$

We notice that, almost surely for all $t \ge 0$, the martingale $\tilde{W}_t(\psi)$ introduced in (5.14) verifies

$$ \dot{W}_t(\psi) \le \tilde{W}_t(\psi). \quad (5.16) $$
The last line of this algorithm lists the labels of individuals in increasing order, grouping siblings together. The label of the spinal individual is thus $U[E]$. To compute a trajectory of the spinal process on the interval $[0, T]$, we need to establish the time of jumps and their outcomes. Using the deterministic evolution of the traits and knowing the branching events, it is easy to recover the traits at all time for the individuals. Furthermore, depending on the statistics that one want to evaluate on the system, it might not be necessary to compute the traits of all individuals. For that reason we propose an algorithmic construction of a trajectory of the spinal process, returning the list of jumps times, events and labels of the individuals living at time $T$. This algorithm also distinguishes the spinal individual in the population by returning the label of the spinal individual in the living ones at time $T$. To simplify we will denote $PPP_{[t_1,t_2]}(c(\cdot))$ the list of times given by a Poisson point process of intensity $c(\cdot)$ on the interval $[t_1, t_2]$, computed using Lewis' thinning algorithm [39].

<caption>**Algorithm 1** Simulation algorithm of the jumps times until time $T$</caption>

**Input:** model parameters: $d, F_{\text{div}}^{-1}, F_{\text{loss}}^{-1}, F_{\text{loss}}^{*-1}, T$, initial condition: $z = [x^1, \dots, x^n]$.

**Output:** $(T_{\text{div}}, I_{\text{div}}, E, L_{\text{div}}, T_{\text{loss}}, T_{\text{loss}}^*, I_{\text{loss}}, L_{\text{loss}}, L_{\text{loss}}^*)$

% Initializing the spine

Generate $u \sim \text{uniform}(0, 1)$

$i_0 = 1$

while $\sum_{i=1}^{i_0} x^i / (\sum_{i=1}^{n} x^i) < u$ do
$i_0 \leftarrow i_0 + 1$

end while

% Generating binary spinal tree on division events

$(T_{\text{div}}, I_{\text{div}}, E, L_{\text{div}}) = \text{Tree}(0, T, n, i_0, F_{\text{div}}^{-1})$

% Generating loss events for the spine and individuals outside the spine

$(T_{\text{loss}}, T_{\text{loss}}^*, I_{\text{loss}}) = ([\cdot], [\cdot], [\cdot])$

for $i \in \{1, \dots, \text{length}(T_{\text{div}})\}$ do

% For the individuals outside the spine

Generate $P \sim \text{PPP}_{[T_{\text{div}}[i-1], T_{\text{div}}[i]]} (K_{\text{loss}}(n + i - 1)^2 d(\cdot))$ ▷ Times of loss events

append $P$ to $T_{\text{loss}}$

Generate $I_u \sim (\text{uniform}(\{1, \dots, n + i - 1\})^{\text{length}(T_{\text{loss}}[i])})$ ▷ Distributing loss events

append $I_u$ to $I_{\text{loss}}$

% For the spine

Generate $P^* \sim \text{PPP}_{[T_{\text{div}}[i-1], T_{\text{div}}[i]]} ((n + i - 1)d(\cdot))$ ▷ Times of loss events

append $P^*$ to $T_{\text{loss}}^*$

end for

Generate $L_u \sim (\text{uniform}(0, 1)^{\text{length}(T_{\text{loss}})})$ ▷ Generating fractions lost

$L_{\text{loss}} = F_{\text{loss}}^{-1}(L_u)$

Generate $L_u^* \sim (\text{uniform}(0, 1)^{\text{length}(T_{\text{loss}}^*)})$ ▷ Generating fractions lost

$L_{\text{loss}}^* = F_{\text{loss}}^{*-1}(L_u^*)$

The output tuple $(T_{\text{div}}, I_{\text{div}}, E, L_{\text{div}}, T_{\text{loss}}, T_{\text{loss}}^*, I_{\text{loss}}, L_{\text{loss}}, L_{\text{loss}}^*)$ is the minimal information needed to construct the spinal process introduced in Section 4. The list $U$ of labels of the individuals can be computed with the function $\text{Labels}(I_{\text{div}}, n)$. Notice that a lot of operations can be parallelized in this algorithm, unlike in the classical Lewis' algorithm. Moreover, depending on the statistic that one wants to compute on this process, the algorithm can be further simplified. For example, if one takes interest in the total biomass of the population, the indexes of the individuals that branched are not necessary.
We denote $(\hat{T}_k^*, k \ge 0)$ the sequence of random jumps times of the spine, and $(\hat{N}_k^*, k \ge 0)$ the sequence of random variables giving the number of children at each branching event of the spine. For every $k \ge 0$, $(X_k^i, 1 \le i \le \hat{N}_k^*)$ is the random vector giving the types of the children of the spine, among them the trait of the new spine is denoted $Y_{\hat{T}_k^*}$. We also introduce the sequence of filtrations $(\mathcal{F}_k, k \ge 1)$ such that for all $k \ge 0$:

$$ \mathcal{F}_k := \hat{\mathcal{F}}_{\hat{T}_{k+1}^*}. \qquad (5.17) $$

where $\hat{\mathcal{F}}_t$ is the canonical filtration of the spinal process up to time $t$. Thus, $\mathcal{F}_k$ corresponds to the information on the process until the time of the $(k+1)$-th jump. We notice that

$$ \begin{aligned} \dot{\mathcal{W}}_{\hat{T}_k^*} = \dot{\mathcal{W}}_{\hat{T}_k^*} - &+ \sum_{i=1}^{\hat{N}_k^*} \psi(\hat{X}_k^i, \chi_{\hat{T}_k^*}) \exp \left( - \int_0^{\hat{T}_k^*} \Lambda(X_s^i, \chi_s) \, ds \right) \\ &- \psi(Y_{\hat{T}_k^*}, \chi_{\hat{T}_k^*}) \exp \left( - \int_0^{\hat{T}_k^*} \Lambda(Y_s, \chi_s) \, ds \right). \end{aligned} $$

Using that for all $t \ge 0$, $\dot{\mathcal{W}}_t$ is almost surely non-negative, and using the upper bound of $\Lambda$ in Assumption 7, we obtain

$$ \begin{aligned} \dot{\mathcal{W}}_{\hat{T}_k^*} &\ge \left( \sum_{i=1}^{\hat{N}_k^*} \psi(\hat{X}_k^i, \chi_{\hat{T}_k^*}) - \psi(Y_{\hat{T}_k^*}, \chi_{\hat{T}_k^*}) \right) \exp(-C\hat{T}_k^*) \\ &\ge \left( \sum_{i=1}^{\hat{N}_k^*} \psi(\hat{X}_k^i, \chi_{\hat{T}_k^*}) - \max_{1 \le j \le n} \psi(X_k^j, \chi_{\hat{T}_k^*}) \right) \exp(-C\hat{T}_k^*). \end{aligned} \qquad (5.18) $$

Let us introduce for all $K > 0$ and all $k \ge 1$, the event

$$ B_k^K := \left\{ \log \left( \sum_{i=1}^{\hat{N}_k^*} \psi(\hat{X}_k^i, \chi_{\hat{T}_k^*}) - \max_{1 \le j \le k} \psi(X_k^j, \chi_{\hat{T}_k^*}) \right) \ge kK \right\} \in \mathcal{F}_n. $$

We will now establish that

$$ \sum_{k \ge 1} \mathbb{P} \left( B_k^K \middle| \mathcal{F}_{k-1} \right) = +\infty. $$

First notice that

$$ \sum_{k \ge 1} \mathbb{P} (B_k^K | \mathcal{F}_{k-1}) = \sum_{k \ge 1} \sum_{n \ge 1} \hat{p}_n^* (Y_{\hat{T}_k^*}, \chi_{\hat{T}_k^*}) \int_{\mathcal{X}^n} \hat{K}_n^* (Y_{\hat{T}_k^*}, \chi_{\hat{T}_k^*}, \mathbf{y}) \times \mathbb{1} \left\{ \log \left( \sum_{i=1}^n \psi(y^i, \chi_{\hat{T}_k^*}) - \max_{1 \le j \le n} \psi(y^j, \chi_{\hat{T}_k^*}) \right) \ge kK \right\} \mathcal{M}_n(\mathrm{d}\mathbf{y}). $$

Notice that the indicator function is always zero for $n = 1$. Taking the infimum over $\mathbb{W}$, we get that

$$ \sum_{n \ge 1} \mathbb{P} (B_k^K | \mathcal{F}_{k-1}) \ge \sum_{n \ge 2} \hat{p}_n^* \int_{\mathcal{X}^n} \hat{K}_n^* (\mathbf{y}) \times \sum_{k \ge 1} \mathbb{1} \left\{ \inf_{(x, \nu) \in \mathbb{W}} \left( \log \left( \sum_{i=1}^n \psi(y^i, \nu^+(x, \mathbf{y})) - \max_{1 \le j \le n} \psi(y^j, \nu^+(x, \mathbf{y})) \right) \right) \ge kK \right\} \mathcal{M}_n(\mathrm{d}\mathbf{y}). $$
ACKNOWLEDGMENTS

I would like to thank L. Coquille, A. Marguet and C. Smadi for their advises and com-
ments during this work. I thank V. Bansaye for stimulating discussions on the subject of
this paper and S. Billiard for fruitful discussions on the Yule model. I acknowledge par-
tial support by the Chair "Modélisation Mathématique et Biodiversité" of VEOLIA-Ecole
Polytechnique-MNHN-F.X. This work is supported by the French National Research Agency
in the framework of the "France 2030" program (ANR-15-IDEX-0002) and by the LabEx
PERSYVAL-Lab (ANR-11-LABX-0025-01).

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Finally, using that $\sum_{n \ge 2} \underline{p}_n^* \int_{\mathcal{X}^n} \hat{\underline{K}}_n^*(\mathbf{y}) \mathcal{M}_n(\mathrm{d}\mathbf{y}) \le 1$ and that, for all $A \in \mathbb{R}$,

$$ \sum_{k \ge 0} 1_{\{A \ge kK\}} \ge -1 + \frac{A}{K}, $$

we get

$$ \sum_{k \ge 1} \mathbb{P} (B_k^K | \mathcal{F}_{k-1}) \ge -1 + \frac{1}{K} \sum_{n \ge 2} \underline{p}_n^* \int_{\mathcal{X}^n} \hat{\underline{K}}_n^*(\mathbf{y}) \\ \times \inf_{(x, \nu) \in \mathbb{W}} \left( \log \left( \sum_{i=1}^n \psi(y^i, \nu^+(x, \mathbf{y})) - \max_{1 \le j \le n} \psi(y^j, \nu^+(x, \mathbf{y})) \right) \right) \mathcal{M}_n(\mathrm{d}\mathbf{y}). $$

Lemma 5.7 with (5.15) ensures that the lower bound is infinite and therefore

$$ \sum_{k \ge 1} \mathbb{P} (B_k^K | \mathcal{F}_{k-1}) = +\infty. $$

We can thus apply the conditional second Borel-Cantelli lemma, see Theorem 4.3.4 in [21]. For all $K > 0$,

$$ \limsup_{k \to \infty} \left( \sum_{i=1}^{\hat{N}_k^*} \psi(\hat{X}_k^i, \chi_{\hat{T}_k^*}) - \psi(Y_{\hat{T}_k^*}, \chi_{\hat{T}_k^*}) \right) e^{-Kk} = +\infty \quad \text{a.s.} $$

Furthermore, the bounds on the branching rate in Assumption 7 ensure that $\hat{T}_k^*$ grows linearly almost surely to infinity as $k$ tends to infinity. Thus, using (5.18) we get that $\limsup_{t \to \infty} \hat{W}_t(\psi) = +\infty$ and relation (5.16) concludes the proof.

• Now we treat the **non-degenerated** case, and suppose that Assumption 5 holds and that

$$ \sum_{n \ge 1} \bar{\underline{p}}_n^* < +\infty, \quad \sup_{n \ge 1} \int_{\mathcal{X}^n} \bar{\hat{K}}_n^*(\mathbf{y}) \mathcal{M}_n(\mathrm{d}\mathbf{y}) < +\infty, \qquad (5.19) $$

and

$$ \sum_{n \ge 1} \bar{\underline{p}}_n^* \int_{\mathcal{X}^n} \sup_{(x, \nu) \in \mathbb{W}} \left[ \log \left( \sum_{i=1}^n \psi(y^i, \nu^+(x, \mathbf{y}), t) \right) \right] \bar{\hat{K}}_n^*(\mathbf{y}) \mathcal{M}_n(\mathrm{d}\mathbf{y}) < +\infty. \qquad (5.20) $$

We notice that

$$ \hat{W}_t(\psi) = \hat{W}_t(\psi) - \psi(Y_t, \chi_t) e^{-\int_0^t \Lambda(Y_s, \chi_s) ds}. \qquad (5.21) $$

In the following we prove that $\limsup_{t \to +\infty} \hat{W}_t(\psi) < +\infty$ almost surely, and use Lemma 5.6 to conclude the proof.

Let $\hat{E} = \mathcal{U} \times \mathbb{R}_+ \times \mathbb{N} \times \mathcal{X}^\mathbb{N}$ and $\hat{E}^* = \mathbb{R}_+ \times \mathbb{N} \times \mathcal{X}^\mathbb{N}$. We introduce $\hat{Q}(ds, du, dr, dn, d\mathbf{y})$ and $\hat{Q}^*(ds, dr, dn, d\mathbf{y})$ two independent Poisson point measures on $\mathbb{R}_+ \times \hat{E}$ and $\mathbb{R}_+ \times \hat{E}^*$ with respective intensity $ds(\sum_{v \in \mathcal{U}} \delta_v(du))dr \sum_{i \ge 1} \delta_i(dn)M_i(d\mathbf{y})$ outside the spine and $dsdr \sum_{i \ge 1} \delta_i(dn)M_i(d\mathbf{y})$ for the spine. We denote by $(\hat{\mathcal{F}}_t^*, t \ge 0)$ the canonical filtration associated with $\hat{Q}^*$. Recall that the set of labels of the living individuals in the population outside the spine is denoted by $\mathbb{G}(t)$ and the random index of the new spine, introduced in (2.9), is denoted by $J(x, \nu, \mathbf{y})$.

We localize the process $(\hat{W}_t(\psi), t \ge 0)$ to avoid explosion and use Itô's formula to explicit $\hat{W}_t(\psi)$. Let $\hat{T}^m$ be the $m$-th branching event of the spinal process for $m \ge 1$.
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For all $T \le \hat{T}^m$,

$$ 
\begin{aligned}
\dot{W}_T(\psi) = \dot{W}_0 - \int_0^T I_t dt + \int_0^T \int_{\hat{E}} \hat{I}_t \hat{Q} (dt, du, dr, dn, dy) \\
+ \int_0^T \int_{\hat{E}^*} \left\{ \sum_{i=1}^n \mathbb{1}_{\{i \ne J(Y_{t-}, \chi_{t-}, y)\}} \psi(y^i, \chi_t^+(Y_t, y)) \exp\left(-\int_0^t \Lambda(Y_s, \chi_s) ds\right) \right. \\
\left. + \sum_{v \in \hat{G}_{t-}} \left[ \psi(X_t^v, \chi_t^+(Y_t, y)) - \psi(X_t^v, \chi_{t-}) \right] \exp\left(-\int_0^t \Lambda(X_s^v, \chi_s) ds\right) \right\} \\
\times \mathbb{1}_{\{r \le \hat{B}_n^*(Y_{t-}, \chi_{t-}) \hat{K}_n^*(Y_{t-}, \chi_{t-}, y)\}} \hat{Q}^*(dt, dr, dn, dy)
\end{aligned} 
\quad (5.22) $$

where

$$ I_t := \sum_{u \in \hat{G}_t} [G\psi - G\psi](X_t^u, \chi_t) e^{-\int_0^t \Lambda(X_s^u, \chi_s) ds}, $$

and

$$ 
\begin{aligned}
\hat{I}_t := & \left\{ \left[ \sum_{i=1}^n \psi(y^i, \chi_t^+(X_t^u, y)) - \psi(X_t^u, \chi_t^+(X_t^u, y)) \right] \exp\left(-\int_0^t \Lambda(X_s^u, \chi_s) ds\right) \right. \\
& \left. + \sum_{v \in \hat{G}_{t-}} \left[ \psi(X_t^v, \chi_t^+(X_t^u, y)) - \psi(X_t^v, \chi_{t-}) \right] \exp\left(-\int_0^t \Lambda(X_s^v, \chi_s) ds\right) \right\} \\
& \times \mathbb{1}_{\{u \in \hat{G}_t\}} \mathbb{1}_{\{r \le \hat{B}_n(Y_t, X_t^u, \chi_{t-}) \hat{K}_n(Y_t, X_t^u, \chi_{t-}, y)\}}.
\end{aligned} 
\quad (5.23) $$

$I_t$ describes the deterministic evolution of types between the time of jump, $\hat{I}_t$ corresponds to the jumps outside the spine, and the last integral in (5.22) is the contribution from the spinal individual.

Using the expression of the operator $\mathcal{G}$ and Assumption 5 we have for all $t > 0$, all $u \in \hat{G}_t$

$$ 
\begin{aligned}
I_t = & \sum_{u \in \hat{G}_t} \exp\left(-\int_0^t \Lambda(X_s^u, \chi_s) ds\right) \sum_{n \ge 0} B_n(X_t^u, \chi_t) \\
& \times \int_{\mathcal{X}^n} \left[ \sum_{i=1}^n \psi(y^i, \chi_t^+(X_t^u, y)) - \psi(X_t^u, \chi_t^+(X_t^u, y)) \right] K_n(X_t^u, \chi_t, y) \mathcal{M}_n(\mathrm{d}y).
\end{aligned} 
$$

Notice that Assumption 5 implies that, for all $(x, (x_e, \nu), t) \in \mathcal{X} \times \mathbb{W}$,

$$ \hat{B}_n(x_e, x, \nu) \hat{K}_n(x_e, x, \nu, y) = B_n(x, \nu) K_n(x, \nu, y). $$

Thus using Assumption 5 in (5.23) we get

$$ \int_0^T \int_{\hat{E}} \hat{I}_t \, dt \, dr \sum_{i \ge 1} \delta_i(dn) \mathcal{M}_i(\mathrm{d}\mathbf{y}) = \int_0^T I_t \, dt. $$

Thus, using Assumption 5 we get, for all $T \le \hat{T}^m$

$$ 
\begin{aligned}
\mathbb{E} \left[ \dot{W}_T(\psi) \mid \hat{\mathcal{F}}_T^* \right] = \mathbb{E} \left[ \dot{W}_0(\psi) \right] \\
+ \int_0^T \int_{\hat{E}^*} \sum_{i=1}^n \mathbb{1}_{\{i \ne J(Y_{t-}, \chi_{t-}, t, y)\}} \psi(y^i, \chi_t^+(Y_t, y)) \exp \left( -\int_0^t \Lambda(Y_s, \chi_s) \mathrm{d}s \right) \\
\times \mathbb{1}_{\{r \le \hat{B}_n^*(Y_{t-}, \chi_{t-}) \hat{K}_n^*(Y_{t-}, \chi_{t-}, y)\}} \hat{Q}^*(\mathrm{d}t, \mathrm{d}r, \mathrm{d}n, \mathrm{d}\mathbf{y}).
\end{aligned} 
$$
Using the positivity of $\psi$ we get a majorant by adding the new spine in the sum. We then take the supremum over $\mathbb{W}$ in the indicator function and notice that for all $n \ge 0$, all $(x, \nu) \in \mathbb{W}$,

$$ \hat{B}_n^*(x, \nu) = \hat{p}_n^*(x, \nu) \sum_{k \ge 0} \hat{B}_k^*(x, \nu) \le \overline{\hat{p}_n^* \tau}, $$

where we used Assumption 7 for the last inequality. Putting these steps together and using the lower bound for $G\psi/\psi$ in Assumption 7 we get:

$$ \mathbb{E} \left[ \left| \dot{W}_T(\psi) \right| \widehat{\mathcal{F}}_T^* \right] \le \mathbb{E} \left[ \left| \dot{W}_0(\psi) \right| \right] + \int_0^T \int_{\hat{E}^*} \mathbb{1}_{\{r \le \bar{\tau} \overline{\hat{p}}_n^* \overline{\hat{K}}_n^*(\mathbf{y})\}} \times \exp \left\{ \sup_{(x, \nu) \in \mathbb{W}} \left( \log \left( \sum_{i=1}^k \psi(y^i, \nu_+(x, \mathbf{y})) \right) \right) - ct \right\} \hat{Q}^*(\mathrm{d}t, \mathrm{d}r, \mathrm{d}n, \mathrm{d}\mathbf{y}) . \quad (5.24) $$

Finally we use (5.19) to get an upper bound of the term in the indicator

$$ \bar{\tau} \overline{\hat{p}}_n^* \overline{\hat{K}}_n^*(\mathbf{y}) \le \bar{\tau} \left( \sum_{i \ge 1} \overline{\hat{p}}_i^* \right) \left( \sup_{i \ge 1} \int_{\mathcal{X}^i} \overline{\hat{K}}_i^*(\mathbf{y}) \mathcal{M}_k(\mathrm{d}\mathbf{y}) \right) \left( \frac{\overline{\hat{p}}_n^*}{\sum_{i \ge 1} \overline{\hat{p}}_i^*} \right) \left( \frac{\overline{\hat{K}}_n^*(\mathbf{y})}{\int_{\mathcal{X}^n} \overline{\hat{K}}_n^*(\mathbf{w}) \mathcal{M}_n(\mathrm{d}\mathbf{w})} \right), $$

and introduce the sequences of independent random variables $(S_k, k \ge 1)$ of exponential law of parameter $\bar{\tau} \left( \sum_{i \ge 1} \overline{\hat{p}}_i^* \right) \left( \sup_{i \ge 1} \int_{\mathcal{X}^i} \overline{\hat{K}}_i^*(\mathbf{y}) \mathcal{M}_i(\mathrm{d}\mathbf{y}) \right)$. We also introduce the independent sequence of couples of random variables $((N_k, \mathbf{Y}_k), k \ge 1)$ such that for all $k$, $N_k$ follows the law $(\overline{\hat{p}}_k^* / \sum_{i \ge 1} \overline{\hat{p}}_i^*)$ and $\mathbf{Y}_k | N_k$ follows the law $\overline{\hat{K}}_{N_k}^*(\mathbf{y}) / \int_{\mathcal{X}^n} \overline{\hat{K}}_{N_k}^*(\mathbf{w}) \mathcal{M}_{N_k}(\mathrm{d}\mathbf{w})$. Thus we can write (5.24) with a compound Poisson process

$$ \mathbb{E} \left[ \left| \dot{W}_T(\psi) \right| \widehat{\mathcal{F}}_T^* \right] \le \mathbb{E} \left[ \left| \dot{W}_0(\psi) \right| \right] + \sum_{k \ge 1} \exp \left\{ \sup_{(x, \nu) \in \mathbb{W}} \left( \log \left( \sum_{j=1}^{N_k} \psi \left( Y_k^j, \nu_+(x, \mathbf{Y}_k) \right) \right) \right) - c \sum_{j=1}^k S_j \right\}. $$

In order to show that the series is almost surely finite, we introduce the following sequence of events: for every $K > 0$ and every $k \ge 0$

$$ B_k^K := \left\{ \sup_{(x, \nu) \in \mathbb{W}} \left( \log \left( \sum_{j=1}^{N_k} \psi \left( Y_k^j, \nu_+(x, \mathbf{Y}_k) \right) \right) \right) \ge kK \right\}. $$

Using the law of the couple of random variables $((N_k, \mathbf{Y}_k), k \ge 1)$ we get that

$$ \sum_{k \ge 1} \mathbb{P}(B_k^K) = \sum_{k \ge 1} \sum_{n \ge 1} \frac{\overline{\hat{p}}_n^*}{\sum_{i \ge 1} \overline{\hat{p}}_i^*} \times \int_{\mathcal{X}^n} \mathbb{1}_{\left\{ \sup_{(x, \nu) \in \mathbb{W}} \left( \log \left( \sum_{j=1}^n \psi(y^j, \nu_+(x, \mathbf{y})) \right) \right) \ge kK \right\}} \frac{\overline{\hat{K}}_n^*(\mathbf{y}) \mathcal{M}_n(\mathrm{d}\mathbf{y})}{\int_{\mathcal{X}^n} \overline{\hat{K}}_n^*(\mathbf{w}) \mathcal{M}_n(\mathrm{d}\mathbf{w})}. $$

Using that $\int_{\mathcal{X}^n} \overline{\hat{K}}_n^*(\mathbf{w}) \mathcal{M}_n(\mathrm{d}\mathbf{w}) \ge 1$, $\sum_{i \ge 1} \overline{\hat{p}}_i^* \ge 1$ and $\sum_{k \ge 1} \mathbb{1}_{A \ge kK} \le 0 \lor (A/K)$, we get

$$ \sum_{k \ge 1} \mathbb{P}(B_k^K) \le 0 \lor \left( \frac{1}{K} \sum_{k \ge 1} \overline{\hat{p}}_k^* \int_{\mathcal{X}^n} \sup_{(x, \nu) \in \mathbb{W}} \left( \log \left( \sum_{j=1}^k \psi(y^j, \nu_+(x, \mathbf{y})) \right) \right) \overline{\hat{K}}_k^*(\mathbf{y}) \mathcal{M}_n(\mathrm{d}\mathbf{y}) \right). $$
We now use the Borel Cantelli lemma -see Theorem 2.3.1 in [21]- with (5.20) to ensure that

$$ \limsup_{n \to \infty} \frac{\sup_{(x, \nu) \in \mathbb{W}} \left( \log \left( \sum_{j=1}^{N_n} \psi \left( Y_n^j, \nu_+ (x, \mathbf{Y}_n) \right) \right) \right)}{n} = 0 \quad \text{a.s.} $$

Notice that and (5.19) ensures that, asymptotically, $\sum_{j=1}^k S_j$ grows linearly with $k$. Thus

$$ \sum_{k \ge 1} \exp \left\{ \sup_{(x, \nu) \in \mathbb{W}} \left( \log \left( \sum_{j=1}^{N_k} \psi \left( Y_k^j, \nu_+ (x, \mathbf{Y}_k) \right) \right) \right) - c \sum_{j=1}^k S_j \right\} < \infty \quad \text{a.s.} $$

The upper bound for the branching rate in Assumption 7 ensures that $\hat{T}^m \to \infty$ almost surely as $m$ tends to infinity, and Fatou's lemma gives that $\sup_{t \ge 0} \mathbb{E}_{\bar{z}}[\hat{W}_t(\psi)|\hat{\mathcal{F}}_\infty^*] < \infty$. Thus, the quenched submartingale $(\hat{W}_{t \wedge \hat{T}^m}(\psi), t \ge 0)$ converges almost surely to a finite random variable and

$$ \limsup_{t \to +\infty} \hat{W}_t(\psi) < +\infty \quad \text{a.s.} $$

We conclude the proof using (5.21) and Lemma 5.6. $\square$

## APPENDIX A. PROOF OF PROPOSITION 2.4

The spine process $((E_t, \bar{\chi}_t), t \ge 0)$ defined by Dynamics 2.2 and 2.3 can be rigorously expressed as the solution of a SDE driven by a multivariate point measure.

We recall that the offspring traits at birth $\mathbf{y} = (y^1, \dots, y^n)$ of an individual of trait $x$ in a population $\nu$ at time $t$ is given by the law $K_n(x, \nu, \mathbf{y})\mathcal{M}_n(\mathrm{d}\mathbf{y})$. Let $\hat{E} = \mathbb{R}_+ \times \mathcal{U} \times \mathbb{R}_+ \times \mathbb{N} \times \mathcal{X}^\mathbb{N}$ and $\hat{E}^* = \mathbb{R}_+ \times \mathbb{R}_+ \times \mathbb{N} \times \mathcal{X}^\mathbb{N}$. Let $\hat{Q}(\mathrm{d}s, \mathrm{d}u, \mathrm{d}r, \mathrm{d}n, \mathrm{d}\mathbf{y})$ and $\hat{Q}^*(\mathrm{d}s, \mathrm{d}r, \mathrm{d}n, \mathrm{d}\mathbf{y})$ be two independent Poisson point measures on $\hat{E}$ and $\hat{E}^*$ with respective intensity $\mathrm{d}s(\sum_{v \in \mathcal{U}} \delta_v(\mathrm{d}u))\mathrm{d}r \sum_{i \ge 1} \delta_i(\mathrm{d}n)\mathcal{M}_i(\mathrm{d}\mathbf{y})$ outside the spine and $\mathrm{d}s\mathrm{d}r \sum_{i \ge 1} \delta_i(\mathrm{d}n)\mathcal{M}_i(\mathrm{d}\mathbf{y})$ for the spine. We denote by $(\hat{\mathcal{F}}_t, t \ge 0)$ the canonical filtration associated with these Poisson point measures. The $\hat{\mathcal{F}}_t$-adapted set of labels of the living individuals in the population outside the spine is denoted $\hat{\mathcal{G}}(t)$. Finally we recall the notation $J(x, \nu, \mathbf{y})$, introduced in (2.9), for the random index of the new spine after the branching of a spine of trait $x$ in a population $\nu$ at time $t$ to an offspring $\mathbf{y}$.

Let $(e, \bar{z}) \in \bar{\mathbb{W}}$. Under Assumptions 1 and 3, the process $((E_t, \bar{\chi}_t), t \ge 0)$ is the unique $\hat{\mathcal{F}}_t$-adapted solution, for every function $g \in C^1(\mathcal{U} \times \mathcal{U} \times \mathcal{X}, \mathbb{R})$ and $t \ge 0$, of the following equation

$$ \begin{aligned} \langle \bar{\chi}_t, g(E_t, \cdot) \rangle := \langle \bar{z}, g(e, \cdot) \rangle &+ \int_0^t \int_{\mathcal{U} \times \mathcal{X}} \frac{\partial g}{\partial x}(E_s, u, x) \cdot \mu(x, \chi_s) \bar{\chi}_s(\mathrm{d}u, \mathrm{d}x) \mathrm{d}s \\ &+ \int_{\hat{E}} \left[ \sum_{i=1}^n g(E_{s-}, u i, \mathbf{y}) - g(E_{s-}, u, X_{s-}^u) \right] \\ & \times \mathbb{1}_{\{u \in \hat{\mathcal{G}}_{s-}\}} \mathbb{1}_{\{r \le \hat{B}_n(Y_{s-}, X_{s-}^u, \chi_{s-}) \hat{K}_n(Y_{s-}, X_{s-}^u, \chi_{s-}, \mathbf{y})\}} \hat{Q}(\mathrm{d}s, \mathrm{d}u, \mathrm{d}r, \mathrm{d}n, \mathrm{d}\mathbf{y}) \\ &+ \int_{\hat{E}^*} \left[ \sum_{i=1}^n g(E_{s-}J(Y_{s-}, \chi_{s-}, \mathbf{y}), E_{s-}J(Y_{s-}, \chi_{s-}, \mathbf{y}), \mathbf{y}) - g(E_{s-}, E_{s-}, Y_{s-}) \right] \\ & \times \mathbb{1}_{\{r \le \hat{B}_k^*(Y_{s-}, \chi_{s-}) \hat{K}_k^*(Y_{s-}, \chi_{s-}, \mathbf{y})\}} \hat{Q}^*(\mathrm{d}s, \mathrm{d}r, \mathrm{d}n, \mathrm{d}\mathbf{y}). \end{aligned} \quad (A.1) $$

This assertion is shown following the same computations than for proof of Proposition 1.1 and [46], using the positivity of the $\psi$ function. Assumption 3 leads to the same bound as Assumption 1.3 in the case of a spine process.
Now we establish the expression of the generator of the marginal spine process. We recall the notation $\nu^+(x, \mathbf{y}) := \nu + \sum_{i=1}^n \delta_{y^i} - \delta_x$, introduced in (2.2). Taking expectations of (A.1) for the marginal process on $\mathbb{W}$, we derive the non-homogeneous infinitesimal operator of $((Y_t, \chi_t), t \ge 0)$, following steps in [25]. It is given by the operator $\hat{L}_\psi$, defined for every $F \in \mathcal{D}$, introduced in (2.1), and $(x_e, \nu) \in \mathbb{W}$, by

$$ \hat{L}_\psi F(x_e, \nu) := GF(x_e, \nu) \\ + \sum_{n \ge 0} \left\{ \hat{B}_n^*(x_e, \nu) \int_{\mathcal{X}^n} \left[ \frac{\sum_{i=1}^n F\psi(y^i, \nu^+(x_e, \mathbf{y}))}{\sum_{j=1}^n \psi(y^j, \nu^+(x_e, \mathbf{y}))} - F(x_e, \nu) \right] \hat{K}_n^*(x_e, \nu, \mathbf{y}) \mathcal{M}_n(\mathrm{d}\mathbf{y}) \right. \\ \left. + \int_{\mathcal{X}} \hat{B}_n(x, \nu) \int_{\mathcal{X}^n} \left[ F(x_e, \nu^+(x, \mathbf{y})) - F(x_e, \nu) \right] \hat{K}_n(x, \nu, \mathbf{y}) \mathcal{M}_n(\mathrm{d}\mathbf{y})\nu(dx) \right. \\ \left. - \hat{B}_n(x_e, \nu) \int_{\mathcal{X}^n} \left[ F(x_e, \nu^+(x_e, \mathbf{y})) - F(x_e, \nu) \right] \hat{K}_n(x_e, \nu, \mathbf{y}) \mathcal{M}_n(\mathrm{d}\mathbf{y}) \right\}. $$

The first line gives the dynamical evolution between branching events. The second is related to spinal branching events and the choice of a new spinal individual among the offspring population. The last two lines describe the branching events outside the spine, for all individuals but the spinal one. Using that $G[\psi F] = FG\psi + \psi GF$, we get

$$ \frac{\mathcal{G}[\psi F]}{\psi}(\cdot) - \frac{\mathcal{G}\psi}{\psi}F(\cdot) = GF(\cdot) + \sum_{n \ge 0} \frac{B(\cdot)}{\psi(\cdot)}\hat{T}_n(\cdot), $$

where the jump part $\hat{T}_n$ is defined for all $(x_e, \nu) \in \mathbb{W}$ by

$$ \begin{aligned} \hat{T}_n(\rho_e) := & p_n(\rho_e) \int_{\mathcal{X}^n} \sum_{i=1}^n \psi F(y_i, \nu^+(x_e, \mathbf{y})) - \psi F(x_e, \nu^+(x_e, \mathbf{y})) K_n(\rho_e, \mathbf{y}) \mathcal{M}_n(\mathrm{d}\mathbf{y}) \\ & + \int_{\mathcal{X}} p_n(x, \nu) \int_{\mathcal{X}^n} [\psi F(x_e, \nu^+(x, \mathbf{y})) - \psi F(\rho_e)] K_n(x, \nu, \mathbf{y}) \mathcal{M}_n(\mathrm{d}\mathbf{y})\nu(dx) \\ & - p_n(\rho_e) \int_{\mathcal{X}^n} \left[ \sum_{i=1}^n \psi(y_i, \nu^+(x_e, \mathbf{y})) - \psi(x_e, \nu^+(x_e, \mathbf{y})) \right] F(\rho_e) K_n(\rho_e, \mathbf{y}) \mathcal{M}_n(\mathrm{d}\mathbf{y}) \\ & - \int_{\mathcal{X}} p_n(x, \nu) \int_{\mathcal{X}^n} [\psi(x_e, \nu^+(x, \mathbf{y})) - \psi(\rho_e)] F(\rho_e) K_n(x, \nu, \mathbf{y}) \mathcal{M}_n(\mathrm{d}\mathbf{y})\nu(dx), \end{aligned} \quad (A.2) $$

where we used the notation $\rho_e := (x_e, \nu)$. Rearranging the terms in (A.2) we get

$$ \begin{aligned} \hat{T}_n(\rho_e) = & p_n(\rho_e) \int_{\mathcal{X}^n} \sum_{i=1}^n \psi(y_i, \nu^+(x_e, \mathbf{y})) [F(y_i, \nu^+(x_e, \mathbf{y})) - F(\rho_e)] K_n(\rho_e, \mathbf{y}) \mathcal{M}_n(\mathrm{d}\mathbf{y}) \\ & + \int_{\mathcal{X}} p_n(x, \nu) \\ & \times \int_{\mathcal{X}^n} [F(x_e, \nu^+(x, \mathbf{y})) - F(\rho_e)] \psi(x_e, \nu^+(x, \mathbf{y})) K_n(x, \nu, \mathbf{y}) \mathcal{M}_n(\mathrm{d}\mathbf{y})\nu(dx) \\ & - p_n(\rho_e) \int_{\mathcal{X}^n} [F(x_e, \nu^+(x_e, \mathbf{y})) - F(\rho_e)] \psi(x_e, \nu^+(x_e, \mathbf{y})) K_n(\rho_e, \mathbf{y}) \mathcal{M}_n(\mathrm{d}\mathbf{y}). \end{aligned} $$

We conclude the proof using the branching rates introduced in Dynamics 2.2 and 2.3.

## APPENDIX B. PROOF OF LEMMAS 5.6 AND 5.7

*Proof of Lemma 5.6.* We recall the notations $((\hat{T}_k^*, \hat{N}_k^*, (\hat{X}_k^i, 1 \le i \le \hat{N}_k^*)), k \ge 0)$ for the sequence of jumps times, number of children and type corresponding to branching events of the spine. The
trait of the new spine is denoted $Y_{\hat{T}_k^*}$. We also recall the sequence of filtrations $(\mathcal{F}_k, k \ge 1)$, introduced in (5.17). Using these notations, with the lower bound for $\mathcal{G}\psi/\psi$ in Assumption 7 and the positivity of the function $\psi$, we have for all $k \ge 0$

$$ \log \left( \psi \left( Y_{\hat{T}_k^*}, \chi_{\hat{T}_k^*} \right) \right) - \int_0^{\hat{T}_k^*} \Lambda (Y_s, \chi_s) \, ds \le \log \left( \sum_{i=1}^{\hat{N}_k^*} \psi \left( \hat{X}_k^i, \chi_{\hat{T}_k^*} \right) \right) - c \hat{T}_k^* \quad (B.1) $$

as $Y_{T_k} \in \{X_k^i, 1 \le i \le N_k\}$. We introduce for all $K > 0$ and all $k \ge 1$, the event

$$ B_k^K := \left\{ \log \left( \sum_{i=1}^{\hat{N}_k^*} \psi \left( \hat{X}_k^i, \chi_{\hat{T}_k^*} \right) \right) \ge kK \right\} \in \mathcal{F}_k. $$

First notice that

$$ \sum_{k \ge 1} \mathbb{P} \left( B_k^K \middle| \mathcal{F}_{k-1} \right) = \sum_{k \ge 1} \sum_{n \ge 1} \hat{p}_n^* \left( Y_{\hat{T}_k^{*-}}, \chi_{\hat{T}_k^{*-}} \right) \\ \times \int_{\mathcal{X}^n} \mathbb{1} \left\{ \log \left( \sum_{i=1}^n \psi(y^i, \chi_{\hat{T}_k^*}) \right) \ge kK \right\} \hat{K}_n^* \left( Y_{\hat{T}_k^{*-}}, \chi_{\hat{T}_k^{*-}}, \mathbf{y} \right) \mathcal{M}_n (\mathrm{d}\mathbf{y}). $$

Taking the supremum over $\mathbb{W}$, we get that

$$ \sum_{k \ge 1} \mathbb{P} \left( B_k^K \middle| \mathcal{F}_{k-1} \right) \le \sum_{n \ge 1} \bar{\hat{p}}_n^* \int_{\mathcal{X}^n} \sum_{k \ge 1} \mathbb{1} \left\{ \sup_{(x, \nu) \in \mathbb{W}} \left( \log \left( \sum_{i=1}^n \psi(y^i, \nu^+(x, \mathbf{y})) \right) \right) \ge kK \right\} \bar{\hat{K}}_n^* (\mathbf{y}) \mathcal{M}_n (\mathrm{d}\mathbf{y}). $$

We recall the notation $\nu^+(x, \mathbf{y}) := \nu + \sum_{i=1}^n \delta_{y^i} - \delta_x$, introduced in (2.2). Finally, using that for all $A \in \mathbb{R}$, $\sum_{k \ge 1} \mathbb{1}_{A \ge kK} \le 0 \vee (A/K)$ we get

$$ \sum_{k \ge 1} \mathbb{P} \left( B_k^K \middle| \mathcal{F}_{k-1} \right) \le 0 \vee \left( \frac{1}{K} \sum_{n \ge 1} \bar{\hat{p}}_n^* \int_{\mathcal{X}^n} \sup_{(x, \nu) \in \mathbb{W}} \left[ \log \left( \sum_{i=1}^n \psi(y^i, \nu^+(x, \mathbf{y})) \right) \right] \bar{\hat{K}}_n^* (\mathbf{y}) \mathcal{M}_n (\mathrm{d}\mathbf{y}) \right). $$

Thus (5.20) ensures that

$$ \sum_{k \ge 1} \mathbb{P} \left( B_k^K \middle| \mathcal{F}_{k-1} \right) < +\infty, $$

and the second Borel-Cantelli Lemma- see Theorem 4.3.4 in [21]- gives that for all $K > 0$, eventually

$$ \log \left( \psi \left( Y_{\hat{T}_k^*}, \chi_{\hat{T}_k^*} \right) \right) \le kK \quad \text{a.s.} $$

Furthermore, the bounds on the jump rate in Assumption 7 ensure that $\hat{T}_k^*$ grows linearly almost surely to infinity as $k$ tends to infinity. Thus, using (B.1) we get that almost surely

$$ \limsup_{k \to \infty} \left[ \log \left( \psi \left( Y_{\hat{T}_k^*}, \chi_{\hat{T}_k^*} \right) \right) - \int_0^{\hat{T}_k^*} \Lambda (Y_s, \chi_s) \, ds \right] = -\infty. $$

The fact that $\lim_{k \to \infty} \hat{T}_k^* = +\infty$ almost surely concludes the proof. $\square$
Proof of Lemma 5.7. First, notice that, using a log-sum inequality along with the positivity of the function $\psi$ we get, for all $n \ge 2$, $(x, \nu) \in \mathbb{W}$, $\mathbf{y} \in \mathcal{X}^n$ and $j_n \in \{1, \dots, n\}$

$$ 
\begin{aligned}
\log \left( \sum_{i=1}^{n} \psi(y^i, \nu^+(x, \mathbf{y})) \right) &= \log \left( \sum_{i=1}^{n} \mathbb{1}_{i \neq j_n} \psi(y^i, \nu^+(x, \mathbf{y})) + \psi(y^{j_n}, \nu^+(x, \mathbf{y})) \right) \\
&\le \log(2) + \frac{\psi(y^{j_n}, \nu^+(x, \mathbf{y}))}{\sum_{i=1}^{n} \psi(y^i, \nu^+(x, \mathbf{y}))} \log \left( \psi(y^{j_n}, \nu^+(x, \mathbf{y})) \right) \\
&\quad + \frac{\sum_{i=1}^{n} \mathbb{1}_{i \neq j_n} \psi(y^i, \nu^+(x, \mathbf{y}))}{\sum_i \psi(y^i, \nu^+(x, \mathbf{y}))} \log \left( \sum_{i=1}^{n} \mathbb{1}_{i \neq j_n} \psi(y^i, \nu^+(x, \mathbf{y})) \right).
\end{aligned}
$$

Thus, taking the infimum in the previous inequality and using (3.3), we have for any arbitrary state $(x_0, \nu_0) \in \mathbb{W}$

$$ 
\begin{aligned}
& \sum_{n \ge 2} \hat{p}_n^*(\rho_0) \int_{\mathcal{X}^n} \frac{\psi(y^{j_n}, \nu_0^+(x_0, \mathbf{y}))}{\sum_{i=1}^n \psi(y^i, \nu_0^+(x_0, \mathbf{y}))} \log \left( \psi(y^{j_n}, \nu_0^+(x_0, \mathbf{y})) \right) \hat{K}_n^*(\rho_0, \mathbf{y}) \mathcal{M}_n(\mathrm{d}\mathbf{y}) \\
& \quad + \sum_{n \ge 2} \underline{\hat{p}}_n^* \int_{\mathcal{X}^n} \inf_{(x, \nu) \in \mathbb{W}} \left[ \log \left( \sum_{i \neq j_n} \psi(y^i, \nu^+(x, \mathbf{y})) \right) \right] \hat{K}_n^*(\mathbf{y}) \mathcal{M}_n(\mathrm{d}\mathbf{y}) = +\infty \quad (B.2)
\end{aligned}
$$

where we used the notation $\rho_0 := x_0, \nu_0$. We conclude the proof using Assumption 6 to ensure that the first term in (B.2) is finite. $\square$

## APPENDIX C. ALGORITHMIC CONSTRUCTION

We propose an algorithm that generates trajectories of the $\psi$-spine process introduced in (4.3). We denote by $F_{\mathrm{div}}^{-1}$, $F_{\mathrm{loss}}^{-1}$ and $F_{\mathrm{loss}}^{*-1}$ the generalized inverse of the cumulative distribution functions of the random variables $\hat{\Lambda}, \hat{\Theta}$ and $\Theta$, defined in (4.4) and (4.5). Using the deterministic evolution of the traits and knowing the branching events, it is easy to recover the traits at all time of the individuals.

We first generate a realization of an homogeneous Poisson point process of intensity 1 on the interval $[t_1, t_2]$, starting from $n$ roots following classical algorithms [49]. It returns a list $T_{\mathrm{div}} = [t_1, T_1, T_2, \dots]$ of increasing times, a list $I_{\mathrm{div}} = [i_1, i_2, \dots]$ containing the numbering of the individuals that branched at these times, a list $L_{\mathrm{div}} = [\lambda_1, \lambda_2, \dots]$ of fractions of mass at birth, and the numbering $E$ of the spinal individual. We call numbering a labeling method of the individuals in the population that does not encodes the whole lineage of the individual. The numbering choice is arbitrary and we chose to add every new individuals at the end of the hidden list, thus at each division event a parent is chosen uniformly in the population and its child is added at the end of the list. This computation is handled by the function Tree explained in Figure 5, where $i_0$ is the initial numbering of the spinal individual.

From the list $I_{\mathrm{div}}$, it is possible to retrieve the lineage of all individuals using any chosen labeling method. In our case we used the Ulam-Harris-Neveu notations and the algorithm generating the list $U$ of labels from the list $I$ previously obtained and the initial size of the population $n$ is detailed Figure 6. The last line of this algorithm lists the labels of individuals in increasing order, grouping siblings together. The label of the spinal individual is thus $U[E]$.

To compute a trajectory of the spinal process on the interval $[0, T]$, we need to establish the time of jumps and their outcomes. Using the deterministic evolution of the traits and knowing the branching events, it is easy to recover the traits at all time for the individuals. Furthermore, depending on the statistics that one want to evaluate on the system, it might not be necessary to compute the traits of all individuals. For that reason we propose an algorithmic construction of a trajectory of the spinal process, returning the list of jumps times, events and labels of the individuals living at time $T$. This algorithm, presented in Figure 7 also distinguishes the spinal
function $(T_{\text{div}}, I_{\text{div}}, E, L_{\text{div}}) = \text{Tree}(t_1, t_2, n, i_0, F_{\text{div}}^{-1})$

$(T_{\text{div}}, N) = ([t_1], n)$

% Step 1; generating the division times

while $T_{\text{div}}[\text{end}] < t_2$ do

Generate $u \sim \text{uniform}(0, 1)$

append $T_{\text{div}}[\text{end}] - \ln(u)/N$ to $T_{\text{div}}$

$N \leftarrow N + 1$

▷ Branching time

▷ New population size

end while

if $T_{\text{div}}[\text{end}] > t_2$ then pop $T_{\text{div}}$

% Step 2; generating the fractions at birth and choosing the spinal individual

$(I_{\text{div}}, E) = ([0, \dots, 0], i_0)$

▷ $I_{\text{div}}$ of size length($T_{\text{div}}$)

for $i \in \{1, \dots, \text{length}(T_{\text{div}}) - 1\}$ do

Generate $I \sim \text{uniform}(\{1, \dots, n + i - 1\})$

$I_{\text{div}}[i] \leftarrow I$

▷ Branching individual

Generate $v, q \sim \text{uniform}(0, 1)$

append $F_{\text{div}}^{-1}(u)$ to $L_{\text{div}}$

▷ Fraction $\lambda$ at birth

if $E = i$ and $p > F_{\text{div}}^{-1}(u)$ then $E \leftarrow n + i$

▷ New spinal individual position

end for

end function

<caption>FIGURE 5. Simulation algorithm of the jumps times until time $T$.</caption>

function $U = \text{Labels}(I_{\text{div}}, n)$

$N_{\text{div}} = \text{length}(I_{\text{div}})$

$U = [1, \dots, n, 0, \dots, 0]$

▷ $U$ is of size $n + N_{\text{div}}$

for $i \in \{1, \dots, N_{\text{div}}\}$ do

$(U[I_{\text{div}}[i]], U[n + i])) \leftarrow (U[I_{\text{div}}[i]]1, U[I_{\text{div}}[i]]2)$

▷ $U[I_{\text{div}}[i]] \in \mathcal{U}$

end for

$U \leftarrow \text{QuickSort}(U)$

▷ sorting algorithm in $O((N_{\text{div}} + n) \log(N_{\text{div}} + n))$

end function

<caption>FIGURE 6. Labeling function.</caption>

individual in the population by returning $\text{PPP}_{[t_1, t_2]}(c(\cdot))$ the list of times given by a Poisson point process of intensity $c(\cdot)$ on the interval $[t_1, t_2]$, computed using Lewis' thinning algorithm [40].

The output tuple $(T_{\text{div}}, I_{\text{div}}, E, L_{\text{div}}, T_{\text{loss}}, T_{\text{loss}}^{*}, I_{\text{loss}}, L_{\text{loss}}, L_{\text{loss}}^{*})$ is the minimal information needed to construct the spinal process introduced in Section 4. The list $U$ of labels of the individuals can be computed with the function $\text{Labels}(I_{\text{div}}, n)$. Notice that a lot of operations can be parallelized in this algorithm, unlike in the classical Lewis' algorithm. Moreover, depending on the statistic that one wants to compute on this process, the algorithm can be further simplified. For example, if one takes interest in the total biomass of the population, the indexes of the individuals that branched are not necessary.

**Proposition C.1.** Let $T > 0$ and $T_{\text{div}} = (T_{\text{div}}^{i}, 0 \le i \le N_{\text{div}} + 1)$ be a sequence of increasing branching times with $T_{\text{div}}^{0} := 0$ and $T_{\text{div}}^{N_{\text{div}}+1} := T$. The size $\tilde{N}$ of the spinal population starting from $N_0$ individuals is such that

$$ \int_{0}^{T} \hat{N}_{s} ds = (N_{0} + N_{\text{div}})T - \sum_{i=0}^{N_{\text{div}}} T_{\text{div}}^{i}, $$
**Input:** model parameters: $d, F_{\text{div}}^{-1}, F_{\text{loss}}^{-1}, F_{\text{loss}}^{*-1}, T$, initial condition: $z = [x^1, \dots, x^n]$.

**Output:** $(T_{\text{div}}, I_{\text{div}}, E, L_{\text{div}}, T_{\text{loss}}, T_{\text{loss}}^{*}, I_{\text{loss}}, L_{\text{loss}}, L_{\text{loss}}^{*})$

% Initializing the spine

Generate $u \sim \text{uniform}(0, 1)$

$i_0 = 1$

while $\sum_{i=1}^{i_0} x^i / (\sum_{i=1}^{n} x^i) < u$ do
$i_0 \leftarrow i_0 + 1$

end while

% Generating binary spinal tree on division events

$(T_{\text{div}}, I_{\text{div}}, E, L_{\text{div}}) = \text{Tree}(0, T, n, i_0, F_{\text{div}}^{-1})$

% Generating loss events for the spine and individuals outside the spine

$(T_{\text{loss}}, T_{\text{loss}}^{*}, I_{\text{loss}}) = (\cdot, \cdot, \cdot)$

for $i \in \{1, \dots, \text{length}(T_{\text{div}})\}$ do

% For the individuals outside the spine

Generate $P \sim \text{PPP}_{[T_{\text{div}}[i-1], T_{\text{div}}[i]]} (K_{\text{loss}}(n + i - 1)^2 d(\cdot)) \triangleright \text{Times of loss events}$

append $P$ to $T_{\text{loss}}$

Generate $I_u \sim (\text{uniform}(\{1, \dots, n + i - 1\})^{\text{length}(T_{\text{loss}}[i])}) \triangleright \text{Distributing loss events}$
append $I_u$ to $I_{\text{loss}}$

% For the spine

Generate $P^* \sim \text{PPP}_{[T_{\text{div}}[i-1], T_{\text{div}}[i]]} ((n + i - 1)d(\cdot)) \triangleright \text{Times of loss events}$

append $P^*$ to $T_{\text{loss}}^*$

end for

Generate $L_u \sim (\text{uniform}(0, 1)^{\text{length}(T_{\text{loss}})}) \triangleright \text{Generating fractions lost}$

$L_{\text{loss}} = F_{\text{loss}}^{-1}(L_u)$

Generate $L_u^* \sim (\text{uniform}(0, 1)^{\text{length}(T_{\text{loss}}^*)}) \triangleright \text{Generating fractions lost}$

$L_{\text{loss}}^* = F_{\text{loss}}^{*-1}(L_u^*)$

<caption>FIGURE 7. Simulation algorithm of the jumps times until time T</caption>

and

$$ \int_{0}^{T} \hat{N}_{s}^{2} ds = (N_{0} + N_{\text{div}})^{2} T - (2N_{0} - 1) \sum_{i=1}^{N_{\text{div}}} T_{\text{div}}^{i} - 2 \sum_{i=1}^{N_{\text{div}}} i T_{\text{div}}^{i}. $$

*Proof.* The first integral term is directly computed using that

$$ \sum_{i=0}^{N_{\text{div}}} i (T_{\text{div}}^{i+1} - T_{\text{div}}^{i}) = N_{\text{div}} T - \sum_{i=0}^{N_{\text{div}}} T_{\text{div}}^{i}. \qquad (C.1) $$

The second integral term is computed by noticing that

$$ \int_{0}^{T} \hat{N}_{s}^{2} ds = N_{0}^{2} T + 2N_{0} \sum_{i=0}^{N_{\text{div}}} i (T_{\text{div}}^{i+1} - T_{\text{div}}^{i}) + \sum_{i=0}^{N_{\text{div}}} i^{2} (T_{\text{div}}^{i+1} - T_{\text{div}}^{i}), $$

and that

$$ \sum_{i=0}^{N_{\text{div}}} i^2 (T_{\text{div}}^{i+1} - T_{\text{div}}^{i}) = N_{\text{div}}^2 T - \sum_{i=1}^{N_{\text{div}}} (2i - 1) T_{\text{div}}^{i}. \qquad (C.2) $$

Using (C.1) and (C.2) and rearranging the terms concludes the proof. $\square$

**Proposition C.2.** For all $0 \le i \le N_{\text{div}}$, let $(T_{\text{loss}}^{i,j}, 1 \le j \le N_{\text{loss}}^i)$ be the sequence of loss events times in $[T_{\text{div}}^i, T_{\text{div}}^{i+1})$. If there is no loss event in this period of time, $N_{\text{loss}}^i = 0$ and we use the
formalism $T_{loss}^{i,N_{loss}^i} = T_{loss}^{i,1} = T_{div}^{i-1}$. The fraction of lost masses are denoted by $\theta_{i,j}$ and the label of the individual that suffered the loss is denoted $(i, j)$. Then

$$ \int_0^T B_s ds = \sum_{i=0}^{N_{div}} \left[ \frac{B_{T_{div}^i}}{\mu} (e^{\mu(T_{div}^{i+1}-T_{div}^i)} - 1) - \sum_{k=1}^{N_{loss}^i} \frac{1-\theta_{i,k}}{\mu} X_{T_{div}^i}^{(i,k)} (e^{\mu(T_{div}^{i+1}-T_{div}^i)} - e^{\mu(T_{loss}^{i,k}-T_{div}^i)}) \right]. $$

*Proof.* We recall that $T_{div}^{N_{div}+1} = T$, and then

$$ \int_0^T B_s ds = \sum_{i=0}^{N_{div}} \left[ \int_{T_{div}^i}^{T_{loss}^{i,1}} B_s ds + \sum_{j=2}^{N_{loss}^i} \int_{T_{loss}^{i,j}}^{T_{loss}^{i,j}} B_s ds + \int_{T_{loss}^{i,N_{loss}^i}}^{T_{div}^{i+1}} B_s ds \right]. $$

Furthermore, for all $0 \le i \le N_{div}$, all $1 \le j \le N_{loss}^i - 1$ and all time $t \in (T_{loss}^{i,j}, T_{loss}^{i,j+1})$, the total biomass verifies

$$ B_t = \left( B_{T_{div}^i} - \sum_{k=1}^j (1 - \theta_{i,k}) X_{T_{div}^i}^{(i,k)} \right) \exp(\mu(t - T_{div}^i)). $$

Thus we integrate on those intervals to get

$$ \int_0^T B_s ds = \sum_{i=0}^{N_{div}} \left[ \frac{B_{T_{div}^i}}{\mu} e^{-\mu T_{div}^i} (e^{\mu T_{loss}^{i,1}} - e^{-\mu T_{div}^i}) + \sum_{j=2}^{N_{loss}^i} \left( \frac{B_{T_{div}^i}}{\mu} - \sum_{k=1}^{j-1} \frac{1-\theta_{i,k}}{\mu} X_{T_{div}^i}^{(i,k)} \right) e^{-\mu T_{div}^i} (e^{\mu T_{loss}^{i,j}} - e^{-\mu T_{loss}^{i,j-1}}) + \left( \frac{B_{T_{div}^i}}{\mu} - \sum_{k=1}^{N_{loss}^i} \frac{1-\theta_{i,k}}{\mu} X_{T_{div}^i}^{(i,k)} \right) e^{-\mu T_{div}^i} (e^{\mu T_{div}^{i+1}} - e^{-\mu T_{loss}^{i,N_{loss}^i}}) \right]. $$

Notice that for all $0 \le i \le N_{div}$ the sum over $j$ is telescoping for the term in $B_{T_{div}^i}$ and thus

$$ \int_0^T B_s ds = \sum_{i=0}^{N_{div}} \left[ \frac{B_{T_{div}^i}}{\mu} e^{-\mu T_{div}^i} (e^{\mu T_{div}^{i+1}} - e^{-\mu T_{div}^i}) - \sum_{j=2}^{N_{loss}^i} \sum_{k=1}^{j-1} \frac{1-\theta_{i,k}}{\mu} X_{T_{div}^i}^{(i,k)} e^{-\mu T_{div}^i} (e^{\mu T_{loss}^{i,j}} - e^{-\mu T_{loss}^{i,j-1}}) - \sum_{k=1}^{N_{loss}^i} \frac{1-\theta_{i,k}}{\mu} X_{T_{div}^i}^{(i,k)} e^{-\mu T_{div}^i} (e^{\mu T_{div}^{i+1}} - e^{-\mu T_{loss}^{i,N_{loss}^i}}) \right]. $$
Inverting the sums over $j$ and $k$ in the second line and using the fact that the sum over $j$ becomes telescoping we get

$$ \int_0^T B_s ds = \sum_{i=0}^{N_{\text{div}}} \left[ \frac{B_{T_{\text{div}}^i}}{\mu} e^{-\mu T_{\text{div}}^i} (e^{\mu T_{\text{div}}^{i+1}} - e^{\mu T_{\text{div}}^i}) - \sum_{k=1}^{N_{\text{loss}}^i-1} \frac{1-\theta_{i,k}}{\mu} X_{T_{\text{div}}^i}^{(i,k)} e^{-\mu T_{\text{div}}^i} (e^{\mu T_{\text{loss}}^{i,N_{\text{loss}}^i}} - e^{\mu T_{\text{loss}}^{i,k}}) - \sum_{k=1}^{N_{\text{loss}}^i} \frac{1-\theta_{i,k}}{\mu} X_{T_{\text{div}}^i}^{(i,k)} e^{-\mu T_{\text{div}}^i} (e^{\mu T_{\text{div}}^{i+1}} - e^{\mu T_{\text{div}}^{i,N_{\text{loss}}^i}}) \right]. $$

We conclude the proof using the distributive property of the sum and rearranging the terms. □

Notice that in order to compute $B_{T_{\text{div}}^i}$, we need to know the past loss events of each individuals. Thus two approaches are possible: forward or backward. The backward method avoid the storage of the traits of each individuals but is computationally heavy as many calculations are performed multiple times. In fact at every branching event, the past events of the common ancestor are integrated twice. The forward method, on the other hand, avoid these multiplicities but need to store at every branching event the traits of every individual. According to our available RAM, we chose the forward method. This method is also more convenient as the computation of $\Pi_T$ is then straightforward.

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**Acknowledgments.**

I would like to thank L. Coquille, A. Marguet and C. Smadi for their useful advises and comments during this work. I thank V. Bansaye for stimulating discussions on the subject of this paper and S. Billiard for fruitful discussions on the Yule model.

This work is supported by the French National Research Agency in the framework of the "France 2030" program (ANR-15-IDEX-0002) and by the LabEx PERSYVAL-Lab (ANR-11-LABX-0025-01). I acknowledge partial support by the Chair "Modélisation Mathématique et Biodiversité" of VEOLIA-Ecole Polytechnique-MNHN-F.X.

CHARLES MEDOUS, UNIV. GRENOBLE ALPES, CNRS, IF, 38000 GRENOBLE, FRANCE;

UNIV. GRENOBLE ALPES, INRIA, 38000 GRENOBLE, FRANCE

Email address: charles.medous@univ-grenoble-alpes.fr