2.1. Excursion measure for Brownian motion with drift

In this section we state some well-known results on excursion measures of the Brownian motion with drift. Let $B=(B_t, t\geq0)$ be a standard Brownian motion and let $\beta>0$ be fixed. Let $\theta\in {\mathbb R}$. We consider $B^{(\theta)}=(B^{(\theta)}_t, t\geq 0)$ a Brownian motion with drift $-2\theta$ and scale $\sqrt{2/\beta}$:

(2.1) \begin{equation}B^{(\theta)}_t =\sqrt{\frac{2}{\beta}}\, B_t-2\theta t, \quad t\geq 0.\end{equation}

Consider the minimum process $I^{(\theta)}=(I_t^{(\theta)}, t\geq 0)$ of $B^{(\theta)}$, defined by

\begin{equation*}I^{(\theta)}_t=\min_{u\in [0, t]} B^{(\theta)}_u.\end{equation*}

Let $n^{(\theta)}(de)$ be the excursion measure of the process $B^{(\theta)} - I^{(\theta)}$ above 0 associated with its local time at 0 given by $ -\beta I^{(\theta)}$. This normalization agrees with the one in [Reference Duquesne and Le Gall18] given for $\theta\geq 0$; see the remark below. Let $\sigma=\sigma(e)=\text{inf}\{s>0, \, e(s)=0\}$ and $\zeta=\zeta(e)=\max_{s\in [0, \sigma]}(e_s)$ be the length and the maximum of the excursion e.

Remark 2.1. In this remark, we assume that $\theta>0$ (i.e. the Brownian motion has a negative drift). In the framework of [Reference Duquesne and Le Gall18] (see Section 1.2 therein), $B^{(\theta)}$ is the height process which codes the Brownian continuum random tree (CRT) with branching mechanism $\psi_\theta$ defined by (1.1). It is obtained from the underlying Lévy process $X=(X_t, t\geq 0)$, which in the case of quadratic branching mechanism is the Brownian motion with drift: $X_t=\beta B^{(\theta)}_t=\sqrt{2\beta}\, B_t-2\beta\theta t$ (see the formula (1.7) in [Reference Duquesne and Le Gall18]). According to [Reference Duquesne and Le Gall18, Section 1.1.2], considering the minimum process $I=(I_t, t\geq 0)$, with $I_t=\min_{u\in [0, t]} X_u$, the authors choose the normalization in such a way that $-I$ is the local time at 0 of $X-I$. The choice of the normalization of the local time at 0 of $B^{(\theta)} - I^{(\theta)}$ is justified by the fact that $I=\beta I^{(\theta)}$. Recall the definition of $c_\theta$ in (1.3). Then from Section 3.2.2 and Corollary 1.4.2 in [Reference Duquesne and Le Gall18], we have that for $\theta\geq 0$,

(2.2) \begin{equation}n^{(\theta)}\left[1- {\mathop {\textrm{e}^{-\lambda \sigma}}}\right]=\psi_\theta^{-1}(\lambda), \quad \lambda>0,\end{equation}

and

(2.3) \begin{equation}n^{(\theta)} (\zeta\geq h)=n^{(\theta)} (\zeta> h)=c_\theta(h), \quad h>0.\end{equation}

For $\theta\in {\mathbb R}$, let ${\mathbb P}_\theta^\uparrow(de)$ be the law of $B^{(\theta)} - 2 I^{(\theta)}$. According to Proposition 14 and Theorem 20 in Section VII of [Reference Bertoin10], ${\mathbb P}_\theta^\uparrow(de)$ is the law of $B^{(\theta)}$ conditionally on being positive. For $\theta\in {\mathbb R}$, let $n_{\theta}$ be the excursion measure of $B^{(\theta)}$ outside 0 associated with the local time $L^0=L^0(B^{(\theta)})$. For completeness, we give at the end of this section a proof of the following known result. Let ${\mathcal C}([0, +\infty ))$ be the set of real-valued continuous functions defined on $[0, +\infty )$. Recall that, according to Definition (2.1) of $B^{(\theta)}$, the case $\theta<0$ corresponds to a positive drift for the Brownian motion.

Lemma 2.1. We have for $\theta\in {\mathbb R}$ and $A\in {\mathcal C}([0,+\infty ))$ a measurable subset

(2.4) \begin{equation}n_{\theta}(e\in A)=\frac{\beta}{2} \left[n^{(|\theta|)} (e\in A) + n^{(|\theta|)} ({-}e\in A)+ 2|\theta|{\mathbb P}_\theta^\uparrow ({-}{\textrm{sgn}}(\theta) e \in A)\right].\end{equation}

We also have that

(2.5) \begin{equation}{\mathbb P}_\theta^\uparrow(de)={\mathbb P}_{-\theta}^\uparrow(de)\quad \text{and}\quad n^{(\theta)}(de ) {\textbf 1}_{\{\sigma<+\infty \}}= n^{(|\theta|)}(de),\end{equation}

and for $\theta<0$

(2.6) \begin{equation}n^{(\theta)}(\sigma=+\infty )=2|\theta|\quad \text{and}\quad n^{(\theta)}(de){\textbf 1}_{\{\sigma=+\infty\}} =2|\theta|{\mathbb P}_\theta^\uparrow(de).\end{equation}

Furthermore, if $\theta<0$, then $-\beta I^{(\theta)}_\infty $ is exponentially distributed with parameter $2|\theta|$.

Let $\theta\geq 0$ and let ${\mathcal N}(dh, d \varepsilon, de)=\sum_{i\in {\mathcal I}} \delta(h_i, e_i) (dh,de)$ be a Poisson point measure on ${\mathbb R}_+ \times {\mathcal C}([0, +\infty ))$ with intensity $\beta{\textbf 1}_{\{h\geq 0\}}\, dh \, n^{(\theta)}(de)$. For every $i\in {\mathcal I}$, we set

\begin{equation*}a_i=\sum_{j\in {\mathcal I}}{\textbf 1}_{\{h_j<h_i\}}\sigma(e_j)\quad \mbox{and}\quad b_i=a_i+\sigma (e_i),\end{equation*}

where $\sigma(e_i)$ is the length of excursion $e_i$. For every $t\geq 0$, we set $i_t$ to be the only index $i\in{\mathcal I}$ such that $a_i\leq t<b_i$. Notice that $i_t$ is a.s. well-defined except on a Lebesgue-null set of values of t. We define the process $(Y,J)=((Y_t, J_t), {t\geq 0})$ by

\begin{equation*}Y_t=e_{i_t}(t-a_{i_t})\quad \mbox{and}\quad J_t=h_{i_t}\quad \text{for } t\geq 0,\end{equation*}

with the convention $Y_t=0$ and $J_t=\text{sup}\{J_s, s<t\}$ for t such that $i_t$ is not well-defined. Since $n^{(\theta)}$ is the excursion measure of $B^{(\theta)}- I^{(\theta)}$ above 0 associated to its local time at 0 given by $-\beta I^{(\theta)}$, we deduce the following corollary from excursion theory.

According to [Reference Rogers and Pitman28, Theorem 1], and taking into account the scale $\sqrt{2/\beta}$, the process $B^{(\theta)} - 2 I^{(\theta)}$ is a diffusion on $[0, +\infty )$ with infinitesimal generator

(2.7) \begin{equation}\beta^{-1} \, \partial^2_x + 2|\theta| \coth\!(\beta |\theta|x)\, \partial_x, \quad x\in [0, +\infty ).\end{equation}

Proof of Lemma 2.1. Since $B^{(\theta)} - 2I^{(\theta)}$ and $B^{({-}\theta)} - 2 I^{({-}\theta)}$ have the same distribution (see [Reference Rogers and Pitman28, Theorem 1]), we deduce that ${\mathbb P}_\theta^\uparrow(de)={\mathbb P}_{-\theta}^\uparrow(de)$, which gives the first part of (2.5).

For $\theta, \lambda\in {\mathbb R} $, we set $\varphi_\theta(\lambda)=\psi_\theta(\lambda/\beta)=\beta ^{-1} \lambda^2- 2\theta \lambda$ so that ${\mathbb E}[\!\exp( \lambda B^{(\theta)}_t)]=\exp\!(t \varphi_\theta(\lambda))$. Elementary computations give

\begin{equation*}\int_0^\infty {\mathop {\textrm{e}^{-\lambda x}}} c_\theta(x)^{-1} \,dx={\mathop{\frac{1}{\varphi_\theta(\lambda)}}\nolimits}\quad \text{for all $\lambda>2\beta \max\!(\theta, 0)$}.\end{equation*}

This implies that $1/c_\theta$ is the scale function of $ B^{(\theta)}$; see Theorem 8 in Section VII of [Reference Bertoin10]. Thanks to Theorem 8 and Proposition 15 in Section VII of [Reference Bertoin10], there exists a positive constant $k_\theta$ such that for all $t>0$ and A in the $\sigma$-field ${\mathcal E}_t$ generated by $(e(s), s\leq t)$,

(2.8) \begin{equation}n^{(\theta)} (A, \sigma>t) = k_\theta {\mathbb E}_\theta^\uparrow \left[ c_\theta( e(t)) {\textbf 1}_A \right],\end{equation}

and $ n^{(\theta)} (\zeta>h) = k_\theta c_\theta(h)$ for all $h>0$. We deduce from (2.3) and the latter equality that $k_\theta=1$ for $\theta\geq 0$.

We now prove that $k_\theta=1$ for $\theta<0$ as well. Assume that $\theta<0$. Letting t go to infinity in (2.8) (with A fixed) and using that ${\mathbb P}_\theta^\uparrow(de)$-a.s., $\lim_{t\rightarrow +\infty } e(t)=+\infty$, we deduce that

(2.9) \begin{equation}n^{(\theta)}( A, \sigma=+\infty )=2|\theta| \, k_\theta{\mathbb P}_\theta^\uparrow(A) \quad \text{for all $A\in {\mathcal E}_t$ and all $t\geq 0$,}\end{equation}

and taking for A the whole state space, we get that

(2.10) \begin{equation}n^{(\theta)}(\sigma=+\infty)=2|\theta| k_\theta. \end{equation}

By excursion theory and the chosen normalization, we get that $ - \beta I^{(\theta)}_\infty $ is exponential with parameter $n^{(\theta)}(\sigma=+\infty)$. Since by scaling $ -I^{(\theta)}_\infty$ is also distributed as $\text{inf}\{ B_t - \beta \theta t, \, t\geq 0\}$, we deduce from [Reference Borodin and Salminen12, IV-5-32, p. 70] that $ -I^{(\theta)}_\infty$ is exponential with parameter $2\beta |\theta|$ and thus that $ -\beta I^{(\theta)}_\infty$ is exponential with parameter $2 |\theta|$. This implies that $n^{(\theta)}(\sigma=+\infty)=2|\theta|$, which gives the first part of (2.6); thus, thanks to (2.10), we get $k_\theta=1$. We then use (2.9) with $k_\theta=1$ to get the second part of (2.6).

Let $\theta<0$. Let $t>0$ and $A\in {\mathcal E}_t$. We have

\begin{align*}n^{(\theta)} (A, \sigma>t)&= {\mathbb E}_\theta^\uparrow \left[ c_\theta( e(t)) {\textbf 1}_A \right]\\&= 2|\theta| {\mathbb P}_\theta^\uparrow(A) + {\mathbb E}_{|\theta|}^\uparrow \left[c_{|\theta|}(h)( e(t)) {\textbf 1}_A\right]\\& = n^{(\theta)} (A, \sigma=+\infty ) + n^{(|\theta|)} (A, \sigma>t),\end{align*}

where, for the first equality, we used (2.8) and $k_\theta=1$; for the second, we used the fact that $c_{\theta}(h)=2|\theta| + c_{|\theta|}(h)$ for all $h>0$, by (1.3), and that ${\mathbb P}_\theta^\uparrow(de)={\mathbb P}_{-\theta}^\uparrow(de)$; and for the last, we used (2.8) with $|\theta|$ instead of $\theta$ and $k_{|\theta|}=1$, as well as (2.9). This implies the last part of (2.5) for $\theta<0$. We also deduce from (2.2) that $n^{(\theta)} (\sigma=+\infty )=0$ for $\theta\geq 0$. Thus the second part of (2.5) holds also for $\theta\geq 0$ and thus for $\theta\in {\mathbb R}$.

We shall now prove (2.4). Recall that $n_\theta$ is the excursion measure of $B^{(\theta)}$ outside 0 associated with the local time $L^0=L^0(B^{(\theta)})$, $n^{(\theta)}(de)$ is the excursion measure of $B^{(\theta)}- I^{(\theta)}$ above 0, and $n^{({-}\theta)}(de)$ is the excursion measure of $B ^{({-}\theta)}- I^{({-}\theta)}$ above 0. Notice that $B ^{({-}\theta)}- I^{({-}\theta)}$ is distributed as $-( B^{(\theta)}- M^{(\theta)})$, where $M^{(\theta)} =(M_t^{(\theta)}, t\geq 0)$, defined by $M^{(\theta)}_t=\text{sup}_{u\in [0, t]} B^{(\theta)}_u$, is the maximum process, and thus $n^{({-}\theta)}(d({-}e))$ is the excursion measure of $B ^{(\theta)}- M^{(\theta)}$ below 0. According to [Reference Bertoin9, p. 334] (where the statement is for $\beta=2$, but can clearly be adapted to $\beta>0$ using a scaling in time), we get that

1. (i) $n^{(\theta)}(de)$, the excursion measure of $B^{(\theta)}- I^{(\theta)}$ above 0, is equal, up to a multiplicative constant due to the choice of the normalization of the local times, to ${\textbf 1}_{\{e>0\}} n_\theta(de)$;

2. (ii) $n^{({-}\theta)}(d({-}e))$, the excursion measure of $B ^{(\theta)}- M^{(\theta)}$ below 0, is equal, up to a multiplicative constant due to the choice of the normalization of the local times, to ${\textbf 1}_{\{e>0\}} n_\theta(de)$.

Thus, we have, for some positive constants $a_\theta$ and $b_\theta$, that

\begin{equation*}n_\theta(de)=a_\theta n^{(\theta)}(de) + b_\theta n^{({-}\theta)}(d({-}e)).\end{equation*}

Thanks to (2.5) and (2.6), we find that (2.4) will be proved once we prove that $a_\theta=b_\theta=\beta/2$.

Let us assume for simplicity that $\theta\geq 0$ (the argument is similar for $\theta\leq 0$). By excursion theory, $L^0_\infty $ is exponential with parameter $n_\theta(\sigma=+\infty )= b_\theta n^{({-}\theta)}(\sigma=+\infty )=2\theta b_\theta$. According to [Reference Borodin and Salminen12, V-3-11, p. 90], $L^0_\infty $ is exponential with parameter $\beta \theta$ (use the fact that $(L^0_t, t\geq 0)$ is distributed as $(L^0_{2t/\beta} (W), t\geq 0)$, the local time at 0 of the Brownian motion $W=(W_t=B_t - \beta\theta t, t\geq 0)$). This gives $b_\theta=\beta/2$.

We now prove that $a_\theta=\beta/2$. Let $T= \text{sup}\{B^{(\theta)}_t, t\geq 0\}$. We have

\begin{align*}{\mathbb P}(T<a) & ={\mathbb E}\left[{\mathop {\textrm{e}^{- n_{\theta}(\zeta\geq a, e>0)\, L^0_\infty }}}\right]={\mathbb E}\left[{\mathop {\textrm{e}^{- a_\theta\, n^{(\theta)} (\zeta\geq a)\,L^0_\infty }}}\right]\\[4pt] & = {\mathbb E}\left[{\mathop {\textrm{e}^{- a_\theta c_\theta(a) \, L^0_\infty }}}\right]= \frac{\beta\theta }{\beta\theta+ a_\theta c_\theta(a)},\end{align*}

where we used that $L^0_\infty $ is exponential with parameter $\beta \theta$ for the last equality. Since by scaling T is also distributed as $\text{sup}\{ B_t - \beta \theta t, \, t\geq 0\}$, we deduce from [Reference Borodin and Salminen12, IV-5-32, p. 70] that T is exponential with parameter $2\beta \theta$. This gives ${\mathbb P}(T<a)=1 - {\mathop {\textrm{e}^{- 2\beta\theta a}}}$. Using (1.3), we deduce that $a_\theta=\frac{\beta}{2}$. This ends the proof of the lemma.

2.2. Real trees

The study of real trees was originally motivated by algebraic and geometric problems; see in particular the survey [Reference Dress, Moulton and Terhalle15]. Real trees were first used to study CRTs in [Reference Evans, Pitman and Winter21]; see also [Reference Evans20].

Notice that a real tree is a length space as defined in [Reference Burago, Burago and Ivanov13]. A rooted real tree is given by $({\textbf t}, d_{\textbf t}, \partial_{\textbf t})$, where $\partial=\partial_{\textbf t}$ is a distinguished vertex of ${\textbf t}$, which will be called the root. We remark that the set $\{\partial\}$ is a rooted tree that contains only the root.

Let ${\textbf t}$ be a compact rooted real tree and let $x,y\in{\textbf t}$. We denote by $[\![ x,y]\!]$ the range of the map $f_{x,y}$ described in Definition 2.1. We also set $[\![ x,y[\![=[\![ x,y]\!]\setminus\{y\}$. We define the out-degree of x, denoted by $k_{\textbf t}(x)$, as the number of connected components of ${\textbf t}\setminus\{x\}$ that do not contain the root. If $k_{\textbf t}(x)=0$ (resp. $k_{\textbf t}(x)>1$), then x is called a leaf (resp. a branching point). A tree is said to be binary if the out-degree of its vertices belongs to $\{0,1,2\}$. The skeleton of the tree ${\textbf t}$ is the set $\textrm{sk}({\textbf t})$ of points of ${\textbf t}$ that are not leaves. Notice that ${\textrm{cl}}\; ( \textrm{sk}({\textbf t}))={\textbf t}$, where ${\textrm{cl}}\;(A)$ denote the closure of A.

We denote by ${\textbf t}_x$ the sub-tree of ${\textbf t}$ above x, i.e.,

\begin{equation*}{\textbf t}_x=\{y\in{\textbf t},\ x\in [\![\partial,y]\!]\},\end{equation*}

rooted at x. We say that x is an ancestor of y, and write $x\preccurlyeq y$, if $y\in {\textbf t}_x$. We write $x\prec y$ if furthermore $x\neq y$. Notice that $\preccurlyeq$ is a partial order on ${\textbf t}$. We denote by $x\wedge y$ the most recent common ancestor (MRCA) of x and y in ${\textbf t}$, i.e. the unique vertex of ${\textbf t}$ such that $[\![\partial,x]\!]\cap[\![\partial,y]\!]=[\![\partial, x\wedge y]\!]$.

We denote by $h_{\textbf t}(x)=d_{\textbf t}(\partial,x)$ the height of the vertex x in the tree ${\textbf t}$ and by $H({\textbf t})$ the height of the tree ${\textbf t}$:

\begin{equation*}H({\textbf t})=\max\{h_{\textbf t}(x),\ x\in{\textbf t}\}.\end{equation*}

Recall that ${\textbf t}$ is a compact rooted real tree; let $({\textbf t}_i, {i\in I})$ be a family of rooted trees, and $(x_i, {i\in I})$ a family of vertices of ${\textbf t}$. Let ${\textbf t}_i^\circ={\textbf t}_i\setminus\{\partial_{{\textbf t}_i}\}$. We define the tree ${\textbf t}\circledast_{i\in I} ({\textbf t}_i,x_i)$ obtained by grafting the trees ${\textbf t}_i$ onto the tree ${\textbf t}$ at points $x_i$ by

\begin{align*} {\textbf t}\circledast_{i\in I}({\textbf t}_i,x_i) & ={\textbf t}\sqcup\left(\bigsqcup_{i\in I} {\textbf t}_i^\circ\right),\\[4pt] d_{{\textbf t}\circledast_{i\in I} ({\textbf t}_i,x_i)}(y,y^{\prime}) & =\begin{cases}d_{\textbf t}(y,y^{\prime}) & \mbox{if }y,y^{\prime}\in {\textbf t},\\[3pt] d_{{\textbf t}_i}(y,y^{\prime}) & \mbox{if }y,y^{\prime}\in {\textbf t}_i^\circ,\\[3pt] d_{\textbf t}(y,x_i)+d_{{\textbf t}_i}(\partial_{{\textbf t}_i},y^{\prime}) & \mbox{if } y\in{\textbf t}\mbox{ and }y^{\prime}\in{\textbf t}_i^\circ,\\[3pt] d_{{\textbf t}_i}(y,\partial_{{\textbf t}_i})+d_{\textbf t}(x_i,x_j)& \\[3pt] \qquad+d_{{\textbf t}_j}(\partial_{{\textbf t}_j},y^{\prime})& \mbox{if } y\in{\textbf t}_i^\circ\mbox{ and }y^{\prime}\in{\textbf t}_j^\circ\mbox{ with }i\ne j,\end{cases}\\[3pt] \partial_{{\textbf t}\circledast_{i\in I}({\textbf t}_i,x_i)}& =\partial_{\textbf t},\end{align*}

where $A\sqcup B$ denotes the disjoint union of the sets A and B. Notice that ${\textbf t}\circledast_{i\in I}({\textbf t}_i,x_i)$ might not be compact.

We say that two rooted real trees ${\textbf t}$ and ${\textbf t}^{\prime}$ are equivalent (and we write ${\textbf t}\sim{\textbf t}'$) if there exists a root-preserving isometry that maps ${\textbf t}$ onto ${\textbf t}'$. We denote by ${\mathbb T}$ the set of equivalence classes of compact rooted real trees. The metric space $({\mathbb T},d_{GH})$, with the so-called Gromov–Hausdorff distance $d_{GH}$, is Polish; see [Reference Evans, Pitman and Winter21]. This allows us to define random real trees.

2.3. Coding a compact real tree by a function and the Brownian CRT

Let ${\mathcal E}$ be the set of continuous functions $g:[0,+\infty)\longrightarrow [0,+\infty)$ with compact support and such that $g(0) = 0$. For $g\in {\mathcal E}$, we set $\sigma(g)=\text{sup} \{x, \, g(x)>0\}$. Let $g\in {\mathcal E}$, and assume that $\sigma(g)>0$, i.e. g is not identically zero. For every $s, t \geq 0$, we set

\begin{equation*}m_g(s, t) = \text{inf} _{r\in [s\wedge t,s\vee t ]}g(r)\end{equation*}

and

(2.11) \begin{equation}d_g(s, t) = g(s) + g(t) - 2m_g(s, t ).\end{equation}

It is easy to check that $d_g$ is a pseudo-metric on $[0,+\infty)$. We then say that s and t are equivalent if and only if $d_g(s, t) = 0$, and we denote by $T_g$ the associated quotient space. We keep the notation $d_g$ for the induced distance on $T_g$. Then the metric space $(T_g, d_g)$ is a compact real tree; see [Reference Duquesne and Le Gall19]. We denote by $p_g$ the canonical projection from $[0, +\infty )$ to $T_g$. We will view $(T_g, d_g)$ as a rooted real tree with root $\partial = p_g(0)$. We will call $(T_g,d_g)$ the real tree coded by g, and conversely we will say that g is a contour function of the tree $T_g$. We denote by F the application that associates with a function $g\in{\mathcal E}$ the equivalence class of the tree $T_g$.

Conversely, every rooted compact real tree (T, d) can be coded by a continuous function g (up to a root-preserving isometry); see [Reference Duquesne16].

We define the Brownian CRT, $\tau=F (e)$, as the (equivalence class of the) tree coded by the positive excursion e under $n^{(\theta)}$; see Section 2.1. And we define the measure ${\mathbb N}^{(\theta)}$ on ${\mathbb T}$ as the ‘distribution’ of $\tau$, i.e. the pushforward of the measure $n^{(\theta)}$ by the application F. Notice that $H(\tau)=\zeta(e)$.

Let e have ‘distribution’ $n^{(\theta)}(de)$, and let $(\Lambda_s^a, s\geq 0, a\geq 0)$ be the local time of e at time s and level a. Then we define the local time measure of $\tau$ at level $a\geq 0$, denoted by $\ell_a(dx)$, as the pushforward of the measure $d\Lambda_s^a$ by the map F; see [Reference Duquesne and Le Gall19, Theorem 4.2]. We shall define $\ell_a$ for $a\in {\mathbb R}$ by setting $\ell_a=0$ for $a\in {\mathbb R}\setminus [0, H(\tau)]$.

2.4. Trees with one semi-infinite branch

The goal of this section is to describe the genealogical tree of a stationary CB with immigration (restricted to the population that appeared before time 0). For this purpose, we add an immortal individual living from $-\infty$ to 0, which will be the spine of the genealogical tree (i.e. the semi-infinite branch) and will be represented by the half straight line $({-}\infty,0]$; see Figure 2. Since we are interested in the genealogical tree, we do not record the population generated by the immortal individual after time 0. The distinguished vertex in the tree will be the point 0; in the terminology of Section 2.2, this would be the root of the tree. In what follows, however, in accordance with the natural intuition, we will speak of it as the distinguished leaf. In the same spirit, we will give another definition for the height of a vertex in such a tree in order to allow negative heights.

Figure 2: An example of a tree with a semi-infinite branch.

2.4.1. Forests

A forest ${\textbf f}$ is a family $((h_i,{\textbf t}_i), \, i\in I)$ of points of ${\mathbb R}\times {\mathbb T}$. Using an immediate extension of the grafting procedure, for an interval $\mathfrak{I}\subset {\mathbb R}$ we define the real tree

(2.12) \begin{equation}{\textbf f}_\mathfrak{I}=\mathfrak{I}\circledast_{i\in I, h_i\in \mathfrak{I}}({\textbf t}_i, h_i).\end{equation}

For $i\in I$, let us denote by $d_i$ the distance on the tree ${\textbf t}_i$ and by ${\textbf t}_i^\circ={\textbf t}_i\setminus\{\partial_{{\textbf t}_i}\}$ the tree ${\textbf t}_i$ without its root. The distance on ${\textbf f}_\mathfrak{I}$ is then defined, for $x,y\in{\textbf f}_\mathfrak{I}$, by

\begin{equation*}d_{\textbf f}(x,y)= \left\{\begin{array}{l@{\quad}l}d_i(x,y) & \text{ if }x,y\in{\textbf t}_i^\circ,\\[4pt] h_{{\textbf t}_i}(x)+|h_i-h_j|+h_{{\textbf t}_j}(y) & \text{ if } x\in{\textbf t}_i^\circ,\ y\in{\textbf t}_j^\circ\mbox{ with }i\ne j,\\[4pt] |x-h_j|+h_{{\textbf t}_j}(y) & \text{ if } x\in\sqcup_{i\in I}{\textbf t}_i^\circ,\ y\in{\textbf t}_j^\circ,\\[4pt] |x-y| & \text{ if } x,y \in\sqcup_{i\in I}{\textbf t}_i^\circ.\end{array}\right.\end{equation*}

Let us recall the following lemma (see [Reference Abraham and Delmas3]).

Lemma 2.2. Let $\mathfrak{I}\subset {\mathbb R}$ be a closed interval. If, for every $a,b\in\mathfrak{I}$ such that $a<b$, and every $\varepsilon>0$, the set $\{i\in I,\ h_i\in[a,b],\ H({\textbf t}_i)>\varepsilon\}$ is finite, then the tree ${\textbf f}_\mathfrak{I}$ is a complete locally compact length space.

2.4.2. Trees with one semi-infinite branch

Definition 2.2. We define ${\mathbb T}_1$ as the set of forests ${\textbf f}=((h_i,{\textbf t}_i), \, i\in I)$ such that

• • for every $i\in I$, $h_i\leq 0$;

• • for every $a<b$ and every $\varepsilon>0$, the set $\{i\in I,\ h_i\in[a,b],\ H({\textbf t}_i)>\varepsilon\}$ is finite.

The following corollary, which is an elementary consequence of Lemma 2.2, associates with a forest ${\textbf f}\in{\mathbb T}_1$ a complete and locally compact real tree.

Corollary 2.2. Let ${\textbf f}=((h_i,{\textbf t}_i), \, i\in I)\in {\mathbb T}_1$. Then the tree ${\textbf f}_{({-}\infty,0]}$ defined by (2.12) is a complete and locally compact real tree.

Conversely, let $({\textbf t},d_{\textbf t}, \rho_0)$ be a complete and locally compact rooted real tree. We denote by ${\mathcal S}({\textbf t})$ the set of vertices $x\in{\textbf t}$ such that at least one of the connected components of ${\textbf t}\setminus\{x\}$ that do not contain $\rho_0$ is unbounded. If ${\mathcal S}({\textbf t})$ is not empty, then it is a tree which contains $\rho_0$. We say that ${\textbf t}$ has a unique semi-infinite branch if ${\mathcal S}({\textbf t})$ is non-empty and has no branching point. We let $({\textbf t}_i^\circ,i\in I)$ denote the connected components of ${\textbf t}\setminus{\mathcal S}({\textbf t})$. For every $i\in I$, we let $x_i$ denote the unique point of ${\mathcal S}({\textbf t})$ such that $\text{inf}\{d_{\textbf t}(x_i,y),\ y\in{\textbf t}_i^\circ\}=0$, and

\begin{equation*}{\textbf t}_i={\textbf t}_i^\circ\cup\{x_i\},\qquad h_i=-d(\rho_0,x_i).\end{equation*}

We shall say that $x_i$ is the root of ${\textbf t}_i$. Notice first that $({\textbf t}_i, d_{\textbf t}, x_i)$ is a bounded rooted tree. It is also compact, since, according to the Hopf–Rinow theorem (see [Reference Burago, Burago and Ivanov13, Theorem 2.5.26]), it is a bounded closed subset of a complete locally compact length space. Thus it belongs to ${\mathbb T}$.

The family ${\textbf f}=((h_i,{\textbf t}_i), \, i\in I)$ is therefore a forest with $h_i<0$. To check that it belongs to ${\mathbb T}_1$, we need to prove that the second condition in Definition 2.2 is satisfied, which is a direct consequence of the fact that the tree ${\textbf f}_{[a,b]}$ is locally compact.

We can therefore identify the set ${\mathbb T}_1$ with the set of (equivalence classes of) complete locally compact rooted real trees with a unique semi-infinite branch. We can follow [Reference Abraham, Delmas and Hoscheit4] to endow ${\mathbb T}_1$ with a Gromov–Hausdorff-type distance for which ${\mathbb T}_1$ is a Polish space.

We extend the partial order defined for trees in ${\mathbb T}$ to forests in ${\mathbb T}_1$, with the idea that the distinguished leaf $\rho_0=0$ is at the tip of the semi-infinite branch. Let ${\textbf f}=(h_i,{\textbf t}_i)_{i\in I}\in{\mathbb T}_1$ and write ${\textbf t}={\textbf f}_{({-}\infty,0]}$, viewed as a real tree rooted at $\rho_0=0$ (with a unique semi-infinite branch ${\mathcal S}({\textbf t})=({-}\infty,0]$). For $x,y\in {\textbf t}$, we set $x\preccurlyeq y$ if either $x,y\in S({\textbf t})$ and $d_{\textbf f}(x,\rho_0) \geq d_{\textbf f}(y, \rho_0)$, or $x,y\in {\textbf t}_i$ for some $i\in I$ and $x\preccurlyeq y$ (with the partial order for the rooted compact real tree ${\textbf t}_i$), or $x\in {\mathcal S}({\textbf t})$ and $y\in {\textbf t}_i$ for some $i\in I$ and $d_{\textbf f}(x,\rho_0)\geq |h_i|$. We write $x\prec y$ if furthermore $x\neq y$. We define $x\wedge y$, the MRCA of $x,y\in {\textbf t}$, as x if $x\preccurlyeq y$, as $x \wedge y$ if $x,y\in {\textbf t}_i$ for some $i\in I$ (with the MRCA for the rooted compact real tree ${\textbf t}_i$), and as $h_i\wedge h_j$ if $x\in {\textbf t}_i$ and $y\in {\textbf t} _j$ for some $i\neq j$. We define the height of a vertex $x\in {\textbf t}$ as

\begin{equation*}h_{\textbf f}(x)= d_{\textbf f}(x, \rho_0 \wedge x)- d_{\textbf f}(\rho_0, \rho_0 \wedge x).\end{equation*}

Notice that the definition of the height function $h_{\textbf f}$ for a forest ${\textbf f}=(h_i,{\textbf t}_i)_{i\in I}\in {\mathbb T}_1$ is different from the height function of the tree ${\textbf t}={\textbf f}_{({-}\infty,0]}$ viewed as a tree in ${\mathbb T}$, as in the former case the root $\rho_0$ is viewed as a distinguished vertex above the semi-infinite branch (all elements of this semi-infinite branch have negative heights for $h_{\textbf f}$, whereas all the heights are nonnegative for $h_{\textbf t}$).

2.4.4. Genealogical tree of an extant population

For a tree ${\textbf t}\in {\mathbb T}$ or ${\textbf t}\in {\mathbb T}_1$ (recall that we identify a forest ${\textbf f}\in{\mathbb T}_1$ with the tree ${\textbf t}={\textbf f}_{({-}\infty,0]}$, with a different definition for the height function) and $h\geq 0$, we define ${\mathcal Z}_h({\textbf t})=\{x\in {\textbf t}, \,h_{\textbf t}(x)=h\}$ to be the set of vertices of ${\textbf t}$ at level h, also called the extant population at time h, and we define the genealogical tree of the vertices of ${\textbf t}$ at level h by

(2.13) \begin{equation}{\mathcal G}_h({\textbf t})=\{x\in {\textbf t}; \, \exists y\in {\mathcal Z}_h({\textbf t}) \text{ such that } x\preccurlyeq y\}.\end{equation}

For ${\textbf f}\in{\mathbb T}_1$, we write ${\mathcal G}_h({\textbf f})$ for ${\mathcal G}_h({\textbf f}_{({-}\infty,0]})$.

3.1. Construction of a tree from a point measure

Let ${\mathcal A}=\sum_{i\in {\mathcal I}}\delta_{(x_i,\zeta_i)}$ be a point process on ${\mathbb R}^*\times [0,+\infty)$ satisfying (i) and (ii) from Definition 3.1. We shall associate with this ancestral process a genealogical tree. Informally the genealogical tree is constructed as follows. We view this process as a sequence of vertical segments in ${\mathbb R}^2$, the tips of the segments being the $x_i$ and their lengths being the $\zeta_i$. We then attach the bottom of each segment such that $x_i>0$ (resp. $x_i<0$) to the first left (resp. first right) longer segment, or to the half-line $\{0\}\times ({-}\infty,0]$ if such a segment does not exist. This gives a (non-compact, unrooted) real tree that may not be complete. See Figure 1 for an example.

Figure 3: An example of the distance d(u, v) defined in (3.2).

Let us turn to a more formal definition. Let us define ${\mathcal I}^{\text{d}}=\{i\in{\mathcal I},\ x_i \gt 0\}$ and ${\mathcal I} ^{\text{g}}=\{i\in{\mathcal I},\ x_i \lt 0\}= {\mathcal I}\setminus{\mathcal I}^{\text{d}}$. We also set ${\mathcal I}_0={\mathcal I}\sqcup \{0\}$, $x_0=0$, and $\zeta_0=+\infty$. For every $i\in{\mathcal I}_0$, we denote by $S_i=\{x_i\}\times({-}\zeta_i,0]$ the vertical segment in ${\mathbb R}^2$ that links the points $(x_i,0)$ and $(x_i,-\zeta_i)$. Notice that we omit the lowest point of the vertical segment. Finally we define

(3.1) \begin{equation}\mathfrak{T}=\bigsqcup_{i\in{\mathcal I}_0}S_i.\end{equation}

We now define a distance on $\mathfrak{T}$. We first define the distance between leaves of $\mathfrak{T}$, i.e. points $(x_i,0)$ with $i\in{\mathcal I}_0$, then extend it to every point of $\mathfrak{T}$. For $i,j\in{\mathcal I}_0$ such that $x_i<x_j$, we set

(3.2) \begin{equation}d((x_i,0),(x_j,0))=2 \max\{\zeta_k, \,x_k \in J(x_i,x_j)\},\end{equation}

where, for $x<y$, $J(x,y)= (x,y] $ (resp. [x, y), resp. $[x,y]\backslash \{0\}$) if $x\geq 0$ (resp. $y\leq 0$, resp. $x<0$ and $y> 0$), with the convention $\max\emptyset=0$. For $u=(x_i,a)\in S_i$ and $v=(x_j,b)\in S_j$, we set

(3.3) \begin{equation}d(u,v)=|a-b|{\textbf 1}_{\{x_i=x_j\}}+ (|a-r|+|b-r|){\textbf 1}_{\{x_i\neq x_j\}},\end{equation}

with $r=\frac{1}{2}d((x_i, 0), (x_j, 0))$. See Figure 3 for an example. It is easy to verify that d is a distance on $\mathfrak{T}$. Notice that $\mathfrak{T}$ is not compact, in particular because of the infinite half-line attached to (0, 0). In order to stick to the framework of Section 2.4, we will consider the origin (0, 0) to be the distinguished point in $\mathfrak{T}$, located at height $h=0$.

Finally, we define $\mathfrak{T}({\mathcal A})$, with the metric d, as the completion of the metric space $(\mathfrak{T},d)$.

Remark 3.1. For every $i\in{\mathcal I}^{\text{d}}$, we set $i_\ell$ to be the index in ${\mathcal I}_0$ such that

\begin{equation*}x_{i_\ell}=\max\{x_j,\ 0\leq x_j<x_i\mbox{ and }\zeta_j>\zeta_i\}.\end{equation*}

Notice that $i_\ell$ is well-defined, since there are only a finite number of indices $j\in{\mathcal I}_0$ such that $x_j \in [0, x_i)$ and $\zeta_j>\zeta_i$. Similarly, for $i\in{\mathcal I}^{\text{g}}$, we set $i_r$ to be the index in ${\mathcal I}_0$ such that

\begin{equation*}x_{i_r}=\min\{x_j,\ x_i<x_j\leq 0\mbox{ and }\zeta_j>\zeta_i\}.\end{equation*}

The distance d identifies the point $(x_i,-\zeta_i)$ (which does not belong to $\mathfrak{T}$ by definition) with the point $(x_{i_\ell},-\zeta_i)$ if $x_i>0$ and with the point $(x_{i_r},-\zeta_i)$ if $x_i<0$, as illustrated on the right-hand side of Figure 4.

Figure 4: The Brownian motions with drift, the ancestral process, and the associated genealogical tree.

We shall call ${\mathfrak{T}}({\mathcal A})$ the tree associated with the ancestral process ${\mathcal A}$.

Proof. By construction of $(\mathfrak{T},d)$ (see (3.1), (3.2), and (3.3)), it is easy to check that $\mathfrak{T}$ is connected and that d satisfies the so-called four-points condition (see Lemma 3.12 in [Reference Evans20]). To conclude, use the fact that those two conditions characterize real trees (see Theorem 3.40 in [Reference Evans20]). This implies that both $(\mathfrak{T}, d)$ and its completion are real trees. By construction of $\mathfrak{T}$, it is easy to check that $\mathfrak{T}({\mathcal A})$ has a unique semi-infinite branch.

Let us now prove that $\mathfrak{T}({\mathcal A})$ is locally compact. Let $(y_n, n\in {\mathbb N})$ be a bounded sequence of $\mathfrak{T}$.

On one hand, let us assume that there exists $i\in {\mathcal I}_0$ and a subsequence $(y_{n_k}, k\in {\mathbb N})$ such that $y_{n_k}$ belongs to $S_i=\{x_i\}\times ({-}\zeta_i, 0]$. Since, for $i\in {\mathcal I}$, there exists a unique $j\in {\mathcal I}_0$ such that $S_i\cup \{(x_j,-\zeta_i)\}$ is compact in $(\mathfrak{T},d)$ (see Remark 3.1), and since, for $i=0$, $S_0=\{0\}\times ({-}\infty,0]$, we deduce that the bounded sequence $(y_{n_k}, k\in {\mathbb N})$ has an accumulation point in $S_i\cup \{(x_j,-\zeta_i)\}$ if $i\in {\mathcal I}$ or in $\{0\}\times ({-}\infty , 0]$ if $i=0$.

On the other hand, let us assume that for all $i\in {\mathcal I}_0$ the sets $\{n, \, y_n\in S_i\}$ are finite. For $n\in {\mathbb N}$, let $i_n$ be uniquely defined by $y_n\in S_{i_n}$. Since $(y_n, n\in {\mathbb N})$ is bounded, we deduce from Conditions (ii)–(iii) in Definition 3.1 that the sequence $(x_{i_n}, n\in {\mathbb N})$ is bounded in ${\mathbb R}$. In particular, there is a subsequence $(x_{i_{n_k}}, k\in {\mathbb N})$ that converges to a limit, say a. Without loss of generality, we can assume that the subsequence is non-decreasing. We deduce from Condition (ii) in Definition 3.1 that $\lim_{\varepsilon\downarrow 0} \max\{ \zeta_i, a-\varepsilon<x_i<a\}=0$. This implies, thanks to Definition (3.2), that $(\{x_{i_{n_k}}\}\times\{0\}, k\in {\mathbb N})$ is Cauchy in $\mathfrak{T}$ and, using (ii) again, that $\lim_{k\rightarrow+\infty } \zeta_{i_{n_k}}=0$. Then we use the fact that

\begin{equation*}d(y_{n_k}, y_{n_{k^{\prime}}})\leq \zeta_{i_{n_k}} + \zeta_{i_{n_{k^{\prime}}}}+ d((x_{n_k}, 0), (x_{n_{k^{\prime}}}, 0))\end{equation*}

to conclude that the sequence $(y_{n_k}, k\in {\mathbb N})$ is Cauchy in $\mathfrak{T}$.

We deduce that every bounded sequence in $\mathfrak{T}$ has a Cauchy subsequence. This proves that $\mathfrak{T}({\mathcal A})$, the completion of $\mathfrak{T}$, is locally compact.

3.2. The ancestral process of the Brownian forest

Let $\theta\geq 0$. Let ${\mathcal N}(dh,d\varepsilon,de)=\sum_{i\in I}\delta_{(h_i,\varepsilon_i,e_i)}(dh,d\varepsilon,de)$ be, under ${\mathbb P}^ {(\theta)}$, a Poisson point measure on ${\mathbb R}\times\{-1,1\}\times{\mathcal E}$ with intensity $\beta dh\, (\delta_{-1}(d\varepsilon)+\delta_{1}(d\varepsilon))\,n^{(\theta)}(de)$, and let ${\mathcal F}^{(\theta)} =((h_i,\tau_i), \, i\in I)$ be the associated Brownian forest where $\tau_i=T_{e_i}$ is the tree associated with the excursion $e_i$; see Section 2.3. As explained in Section 3.2.4, this Brownian forest models the evolution of a stationary population directed by the branching mechanism $\psi_\theta$ defined in (1.1).

We want to describe the genealogical tree of the extant population at some fixed time, say 0. The genealogical tree under consideration is then ${\mathcal G}_0({\mathcal F}^{(\theta)})$ as defined by (2.13). To describe the distribution of this tree, we use an ancestral process as described in the previous subsection. We first construct a contour process $(B_t,t\in{\mathbb R})$ (obtained by the concatenation of two independent Brownian motions distributed as $B^{(\theta)}$) which codes for the tree ${\mathcal F}^{(\theta)}_{({-}\infty,0]}$ (see Section 2.4 for the notation). The supplementary variables $\varepsilon_i$ are needed at this point to determine whether the tree ${\textbf t}_i$ is located on the left or on the right of the infinite spine. The atoms of the ancestral process are then the pairs formed by the points of growth of the local time at 0 of B and the depth of the associated excursion of B below 0.

3.2.1. Construction of the contour process

Let $\theta\geq 0$. Set ${\mathcal I}=\{i\in I; \, h_i<0\}$.

For every $i\in {\mathcal I}$, we set

\begin{equation*}a_i=\sum_{j\in {\mathcal I}}{\textbf 1}_{\{\varepsilon _j=\varepsilon_i\}}{\textbf 1}_{\{h_j<h_i\}}\sigma(e_j)\quad \mbox{and}\quad b_i=a_i+\sigma (e_i),\end{equation*}

where we recall that $\sigma(e_i)$ is the length of the excursion $e_i$. For every $t\geq 0$, we denote by $i_t^{\text{d}}$ (resp. $i_{t}^{\text{g}}$) the only index $i\in{\mathcal I}$ such that $\varepsilon_i=1$ (resp. $\varepsilon_i=-1$) and $a_i\leq t<b_i$. Notice that $i_t^{\text{d}}$ and $i_{t}^{\text{g}}$ are a.s. well-defined except on a Lebesgue-null set of values of t. We set $B^{\text{d}}=(B^{\text{d}}_t, {t\geq 0})$ and $B^{\text{g}}=(B^{\text{g}}_t, {t\geq 0})$, where, for $t\geq 0$,

\begin{equation*}B^{\text{d}}_t=h_{i_t^{\text{d}}}+e_{i_t^{\text{d}}}(t-a_{i_t^{\text{d}}})\quad \mbox{and}\quad B^{\text{g}}_t=h_{i_{t}^{\text{g}} }+e_{i_{t}^{\text{g}}}(\sigma(e_{i_{t}^{\text{g}}})-(t-a_{i_{ t}^{\text{g}}})).\end{equation*}

We deduce from Corollary 2.1 that the processes $B^\textrm{d}$ and $B^\textrm{g}$ are two independent Brownian motions distributed as $B^{(\theta)}$. We define the process $B=(B_t,t\in{\mathbb R})$ by $B_t=B^{\text{d}}_t{\textbf 1}_{\{t>0\}} + B^{\text{g}}_{-t}{\textbf 1}_{\{t<0\}} $. By construction, the process B indeed codes for the tree ${\mathcal F}^{(\theta)}_{({-}\infty,0]}$.

3.2.2. The ancestral process

Let $(L_t^{\ell},t\geq 0)$ be the local time at 0 of the process $B^{\ell}$, where $\ell\in\{\text{g}, \text{d}\}$. We denote by $((\alpha_i,\beta_i), \, i\in {\mathcal I}^{\ell})$ the excursion intervals of $B^{\ell}$ below 0, omitting the last infinite excursion if any, and for every $i\in {\mathcal I}^{\ell}$ we set $\zeta_i=-\min\{B_s^{\ell},\ s\in(\alpha_i,\beta_i)\}$.

We consider the point measure on ${\mathbb R}\times {\mathbb R}_+$ defined by

\begin{equation*}{\mathcal A}^{\mathcal N}(du,d\zeta)=\sum_{i\in{\mathcal I}^{\text{d}}}\delta_{(L^{\text{d}}_{\alpha_i}, \zeta_i)}(du,d\zeta)+\sum_{i\in{\mathcal I}^{\text{g}}}\delta_{({-}L^{\text{g}}_{\alpha_i}, \zeta_i)}(du,d\zeta).\end{equation*}

See Figure 4 for a representation of the contour process B, the ancestral process ${\mathcal A}^{\mathcal N}$, and the genealogical tree ${\mathcal G}_0({\mathcal F}^{(\theta)})$. In this figure, the horizontal axis represents the time for Brownian motion on the left-hand figure, whereas it is in the scale of local time for the ancestral process on the two right-hand figures. This will always be the case in the rest of the paper when we are dealing with ancestral processes.

Let $[-E_{\textrm{g}},E_{\textrm{d}}]$ be the closed support of the measure ${\mathcal A}^{\mathcal N}(du,{\mathbb R}_+)$:

\begin{align*}E_{\textrm{d}} & =\text{inf}\{u\geq 0,\ {\mathcal A}([u,+\infty)\times{\mathbb R}_+)=0\}\\\text{and}\quad E_{\textrm{g}} & =\text{inf}\{u\geq 0,\ {\mathcal A}(({-}\infty,-u]\times{\mathbb R}_+)=0\},\end{align*}

with the convention that $\text{inf} \emptyset=+\infty $. Notice that, for $\ell\in\{\text{g},\text{d}\}$, we also have $E_\ell=L_\infty^\ell$. We now give the distribution of the ancestral process ${\mathcal A}^{\mathcal N}$. Recall that $c_\theta$ was defined in (1.3).

Proposition 3.2. Let $\theta\geq 0$. Under ${\mathbb P}^{(\theta)}$, the random variables $E_{\textrm{g}},E_{\textrm{d}}$ are independent and exponentially distributed with parameter $2\theta$ (and mean $1/2\theta$), with the convention that $E_{\textrm{d}}=E_{\textrm{g}}=+\infty $ if $\theta=0$. Under ${\mathbb P}^{(\theta)}$ and conditionally given $(E_{\textrm{g}},E_{\textrm{d}})$, the ancestral process ${\mathcal A}^{\mathcal N}(du, d\zeta)$ is a Poisson point measure with intensity

\begin{equation*}{\textbf 1}_{({-}E_{\textrm{g}},E_{\textrm{d}})}(u)\, du\, |c^{\prime}_\theta(\zeta)|d\zeta.\end{equation*}

Notice that the random measure ${\mathcal A}^{\mathcal N}$ satisfies Conditions (i)–(iii) from Definition 3.1 and is thus indeed an ancestral process.

This result is very similar to Corollary 2 in [Reference Bi and Delmas11]. The main additional ingredient here is the order (given by the u variable), which will be very useful in the simulation.

Proof. Since $B^{\text{d}}$ and $B^{\text{g}}$ are independent with the same distribution, we deduce that $E_{\textrm{g}}$ and $E_{\textrm{d}}$ are independent with the same distribution. Let $\theta>0$. Since $B^{\text{d}}$ is a Brownian motion with drift $-2\theta$, we deduce from Lemma 2.1 that $E_{\textrm{d}}$ is exponential with mean $1/2\theta$. The case $\theta=0$ is immediate.

The excursions below zero of $B^{\text{d}}$ conditionally given $E_d$ are excursions of a Brownian motion $B^{({-}\theta)}$ with drift $2\theta$ (by symmetry with respect to 0) conditioned on being finite, i.e. excursions of a Brownian motion $B^{(\theta)}$ with drift $-2\theta$; see Lemma 2.1. Moreover, by (1.3), $c_\theta$ is exactly the tail distribution of the maximum of an excursion under $n^{(\theta)}$. Standard theory of Brownian excursions gives then the result.

3.2.3. Identification of the trees

Let ${\mathfrak{T}}^{\mathcal N}=\mathfrak{T}({\mathcal A}^{\mathcal N})$ denote the locally compact tree associated with the ancestral process ${\mathcal A}^{\mathcal N}$; see Proposition 3.1. Based on the following proposition, we shall say that the ancestral process ${\mathcal A}^{\mathcal N}$ codes for the genealogical tree of the extant population at time 0 for the forest ${\mathcal F}^{(\theta)}$.

Proposition 3.3. Let $\theta\geq 0$. The trees ${\mathcal G}_0({\mathcal F}^{(\theta)})$ under ${\mathbb P}^{(\theta)}$ and ${\mathfrak{T}}^{\mathcal N}$ belong to the same equivalence class in ${\mathbb T}_1$.

Proof. Let us first remark that the genealogical tree ${\mathcal G}_0({\mathcal F}^{(\theta)})$ can be directly constructed using the process B as described in Figure 5.

Figure 5: The genealogical tree inside the Brownian motions.

More precisely, recall that B is the contour function of the tree ${\mathcal F}^{(\theta)}_{({-}\infty,0]}$. Let us denote by $p_B$ the canonical projection from ${\mathbb R}$ to ${\mathcal F}^{(\theta)}_{({-}\infty,0]}$ as defined in Section 2.4. Recall that $((\alpha_i,\beta_i), \, i\in {\mathcal I}^{\ell})$, with $\ell\in\{\text{g}, \text{d}\}$, are the excursion intervals of $B^\ell$ below 0. Then ${\mathcal G}_0({\mathcal F}^{(\theta)})$ is the smallest complete sub-tree of ${\mathcal F}^{(\theta)}_{({-}\infty,0]}$ that contains the points $(p_B(\alpha_i), i\in {\mathcal I}_{\text{g}}\bigcup{\mathcal I}_{\text{d}} )$ and the semi-infinite branch of ${\mathcal F}^{(\theta)}_{({-}\infty,0]}$.

Let $i,j\in{\mathcal I}$ with $0<\alpha_i<\alpha_j$, for instance. By definition of the tree coded by a function, the distance between $p_B(\alpha_i)$ and $p_B(\alpha_j)$ in ${\mathcal G}_0({\mathcal F}^{(\theta)})$ is given by

\begin{equation*}d(p_B(\alpha_i),p_B(\alpha_j))=-2\min_{u\in [\alpha_i,\alpha_j]}B_u.\end{equation*}

But, by definition of ${\mathcal A}^{\mathcal N}$, we have

\begin{align*}-\min_{u\in [\alpha_i,\alpha_j]}B_u & =\max_{k\in{\mathcal I}\, \alpha_i\leq \alpha_k\lt \alpha_j}\bigl({-}\min_{u\in[\alpha_k,\beta_k]}B_u\bigr)\\ &=\max_{k\in{\mathcal I}\, \alpha_i\leq \alpha_k \lt \alpha_j} \zeta_k.\end{align*}

The other cases $\alpha_j<\alpha_i<0$ and $\alpha _i<0<\alpha_j$ can be handled similarly. We deduce that the distances on a dense subset of leaves of ${\mathcal G}_0({\mathcal F}^{(\theta)})$ and $\mathfrak{T}^{\mathcal N}$ coincide, which implies the result by completeness of the trees.

3.2.4. Local times and other contour processes

Recall $\theta\geq 0$. The Brownian forest ${\mathcal F}^{(\theta)}$ can be viewed as the genealogical tree of a stationary CB process (associated with the branching mechanism $\psi_\theta$ defined in (1.1)); see [Reference Chen and Delmas14]. To be more precise, for every $i\in I$ let $(\ell^{(i)}_a, \, a\geq 0)$ be the local time measures of the tree $\tau_i$. For every $t\in{\mathbb R}$, we consider the measure ${\textbf Z}_t$ on ${\mathcal Z}_t({\mathcal F}^{(\theta)})$ defined by

(3.4) \begin{equation}{\textbf Z}_t(dx)=\sum _{i\in I} {\textbf 1}_{\tau_i}(x)\, \ell_{t-h_i}^{(i)} (dx),\end{equation}

and we write $Z_t={\textbf Z}_t(1)$ for its total mass, which also represents the population size at time t. For $\theta=0$, we have $Z_t=+\infty$ a.s. for every $t\in{\mathbb R}$. For $\theta>0$, the process $(Z_t, {t\geq 0})$ is a stationary Feller diffusion, solution of the stochastic differential equation (1.2); see Corollary 3.3. in [Reference Chen and Delmas14], as well as Theorem 1.2 in [Reference Duquesne17].

In the literature, one also considers the so-called Kesten tree, which is the genealogical tree associated with the Feller diffusion $Z^+$ solving (1.2) for $t\geq 0$ with initial condition $ Z_0^+=0$. It corresponds to the genealogy of a sub-critical branching process started from an infinitesimal individual alive at time 0, conditionally on the non-extinction event. In our setting, the genealogical tree corresponds to ${\mathcal F}^{(\theta)}_{[0, +\infty )}$, and the process $Z^+$ is distributed as ${\textbf Z}^+(1)=({\textbf Z}_t^+(1), t\geq 0)$, with the measure ${\textbf Z}^+_t$ on ${\mathcal Z}_t({\mathcal F}^{(\theta)}_{[0, +\infty )})$ defined by

\begin{equation*}{\textbf Z}^+_t(dx)=\sum _{i\in I} {\textbf 1}_{\{h_i\geq 0\}}\, {\textbf 1}_{\tau_i}(x)\, \ell_{t-h_i}^{(i)} (dx).\end{equation*}

It can also be described using a contour process obtained by the concatenation at infinity of two independent Brownian motions distributed as $B^{(\theta)}$ conditioned to be positive. We use the description given in [Reference Duquesne17], which is valid in the general Lévy case; see also [Reference Athreya, Löhr and Winter7, Section 7.4], which corresponds to the case $\theta=0$.

Let ${\mathcal E}_2$ be the set of continuous nonnegative functions g defined on ${\mathbb R}$ such that $g(0)=0$ and $\lim_{x\rightarrow -\infty } g(x)= \lim_{x\rightarrow +\infty }g(x)=+\infty$. For such a function, we again consider the pseudo-metric $d_g$ defined by (2.11), but for $s,t\in{\mathbb R}$, and with $m_g(s,t)$ replaced by $m_g(s,t)= \text{inf}_{r\not \in [s\wedge t,s\vee t ]} g(r)$ if $st<0$. We define the tree $T^+_g$ as the quotient space on ${\mathbb R}$ induced by this pseudo-metric. We set $p_g$ to be the canonical projection from ${\mathbb R}$ onto $T_g$. For $g\in {\mathcal E}_2$, the triplet $(T^+_g,d_g,p_g(0))$ is a complete locally compact rooted real tree with a unique semi-infinite branch. We continue to call g the contour process of $T^+_g$.

Let $B^+=(B^+_t, t\in {\mathbb R})$ be such that $(B^+_t, t\geq 0)$ and $(B^+_{-t}, t\geq 0)$ are independent and distributed as $B^{(\theta)} -2 I^{(\theta)}$, which is a diffusion on ${\mathbb R}_+$ with infinitesimal generator given by (2.7). Thanks to Corollary 2.1, we get that the tree $T^+_{B^+}$ with contour process $B^+$ is distributed as the genealogical tree ${\mathcal F}^{(\theta)}_{[0, +\infty )}$ which is associated to the Feller diffusion ${\textbf Z}^+(1)$ (solution of (1.2) on ${\mathbb R}^+$ with $Z_0=0$).

Let $\theta>0$. It is also immediate to give the contour process of the genealogical tree conditionally on the extinction being at time 0. Recall the tree defined by its contour process with the concatenation at 0 defined in Lemma 2.3. Set $B^-=-B^+$. It is left to the reader to check that the tree $T^-_{B^-}$ with contour process $B^-$ is distributed as the genealogical tree of the Feller diffusion ${\textbf Z}^-(1)=({\textbf Z}^-(1)_t, t\leq 0)$ conditioned to die at time 0 and started with the stationary distribution at $-\infty $ (solution of (1.2) on ${\mathbb R}_-$ with $Z_0=0$), where the measure ${\textbf Z}^-_t$ on ${\mathcal Z}_t({\mathcal F}^{(\theta)}_{({-}\infty , 0]})$ is defined by

\begin{equation*}{\textbf Z}^-_t(dx)=\sum _{i\in I} {\textbf 1}_{\{\zeta_i+h_i < 0\}}\, {\textbf 1}_{\tau_i}(x)\, \ell_{t-h_i}^{(i)} (dx),\end{equation*}

where $\zeta_i$ is the height of the tree $\tau_i$. This result can also be deduced from the reversal property of the Brownian tree; see [Reference Abraham and Delmas3].

4.1. Static simulation

In what follows, S stands for ‘static’. Assume $n\in {\mathbb N}^*$ is fixed. We present a way to simulate $T_n$ under ${\mathbb P}^{(\theta)}$ with $\theta>0$. See Figures 6 and 7 for an illustration for $n=5$.

1. (i) The size of the population on the left (resp. right) of the origin is $E_{\textrm{g}}$ (resp. $E_{\textrm{d}}$), where $E_{\textrm{g}}, E_{\textrm{d}}$ are independent exponential random variables with mean $1/2\theta$. Set $Z_0=E_{\textrm{g}}+E_{\textrm{d}}$ for the total size of the population at time 0. Let $(U_k, k\in {\mathbb N}^*)$ be independent random variables uniformly distributed on [0, 1] and independent of $(E_{\textrm{g}}, E_{\textrm{d}})$. Set $X_{0}=0$ and, for $k\in {\mathbb N}^*$, $X_k=Z_0 U_k - E_{\textrm{g}}$ as well as ${\mathcal X}_k=\{-E_{\textrm{g}},E_{\textrm{d}}, X_0, \ldots, X_k\}$.

2. (ii) For $1\leq k\leq n$, set $X_{k,n}^{\text{g}}=\max \{x\in {\mathcal X}_n, \, x<X_k\}$ and $X_{k,n}^{\text{d}}=\min \{x\in {\mathcal X}_n, \, x>X_k\}$. We also set $I^{\text{S}}_{k}=[X_{k,n}^{\text{g}}, X_k]$ if $X_k>0$ and $I^{\text{S}}_{k}=[X_k, X_{k,n}^{\text{d}}]$ if $X_k<0$.

3. (iii) Conditionally on $(E_{\textrm{g}}, E_{\textrm{d}}, X_1, \ldots, X_n)$, let $(\zeta_{k}^{ \text{S}}, 1\leq k\leq n)$ be independent random variables such that for $1\leq k\leq n$, $\zeta_{k}^{ \text{S}}$ is distributed as $\zeta^*_\delta$ (see (4.1)), with $\delta= |I^{\text{S}}_{k}|$. Consider the tree $\mathfrak{T}^{\text{S}}_n$ corresponding to the ancestral process ${\mathcal A}_n^{\text{S}}=\sum_{k=1}^n \delta_{(X_k, \zeta_{k}^{ \text{S}})}$.

Figure 6: One realization of $E_{\textrm{g}}, E_{\textrm{d}}, X_1, \ldots, X_5$.

Figure 7: One realization of the tree $\mathfrak{T}_5^{\text{S}}$.

This gives an exact simulation of the tree $T_n$, according to the following result.

Proof. Let $B=(B_t, t\in {\mathbb R})$ be the Brownian motion with drift defined in Subsection 3.2.1, and let $(L_t,t\in{\mathbb R})$ be its local time at 0, i.e.

\begin{equation*}L_t=L_t^{\text{d}}{\textbf 1}_{t>0}+L_{-t}^{\text{g}}{\textbf 1}_{t<0}.\end{equation*}

We set $L_\infty=L_\infty^{\text{d}}+L_{\infty}^{\text{g}}$, and we consider independent and identically distributed variables $(S_1,\ldots, S_n)$ distributed according to $dL_s/L_\infty$. We denote by $(S_{(1)},\ldots,S_{(n)})$ the order statistics of $(S_1,\ldots S_n)$, and for every $i\leq n$ we set

\begin{equation*}M_i=\begin{cases}-\min_{u\in [S_{(i)}, S_{(i+1)}\wedge 0]}B_u & \mbox{if }S_{(i)}<0,\\-\min_{u\in [S_{(i-1)}\vee 0, S_{(i)}]}B_u & \mbox{if }S_{(i)}>0.\end{cases}\end{equation*}

We set

\begin{equation*}\mathcal{A}_n=\sum_{1\leq i\leq n}\delta_{(L_{S_{(i)}},\zeta_i)},\end{equation*}

which is an ancestral process (see Definition 3.1), and let $\mathfrak{T}(\mathcal{A}_n)$ be the associated tree. As B is the contour process of the tree $\mathcal{F}_{({-}\infty,0]}$, we get that $T_n$ and $\mathfrak{T}(\mathcal{A}_n)$ are equally distributed.

Moreover, by Proposition 3.2, Proposition 3.3, and standard results on Poisson point processes, we get that $\mathfrak{T}(\mathcal{A}_n)$ and $\mathfrak{T}_n^S$ are also equally distributed.

4.2. Dynamic simulation (I)

We can modify the static simulation of the previous section to provide a natural dynamic construction of the genealogical tree. In what follows, D stands for ‘dynamic’. Let $\theta>0$. We build recursively a family of ancestral processes $({\mathcal A}_n, n\in {\mathbb N})$, with ${\mathcal A}_0^{\text{D}}=0$ and

\begin{equation*}{\mathcal A}_n^{\text{D}}=\sum_{k=1}^n \delta_{(V_k, \zeta_k^{\text{D}})}\end{equation*}

for $n\in {\mathbb N}^*$.

• (i) Let $E_{\textrm{g}}$, $E_{\textrm{d}}$, $(X_n, n\in {\mathbb N})$, and $({\mathcal X}_n, n\in {\mathbb N}^*)$ be defined as in (i) of Section 4.1. For $n\in {\mathbb N} ^*$, set $X_n^{\text{g}}=\max \{x\in {\mathcal X}_n, x<X_n\}$ and $X_n^{\text{d}}=\min \{x\in {\mathcal X}_n, x>X_n\}$. For $n\in {\mathbb N} ^*$ and $\ell\in \{\text{g}, \text{d}\}$, define the interval $I_n^{\ell}= [X_n \wedge X_n^\ell, X_n\vee X_n^\ell]$ and its length $|I_n^{\ell}|=|X_n - X_n^\ell|$.

  We shall consider and check by the induction the following hypothesis: for $n\geq 2$ the random variables $V_1, \ldots, V_{n-1}$ are such that

  (4.2) \begin{equation}X_{(0,n-1)}<V_{(1,n-1)}<X_{(1, n-1)}< \cdots < V_{(n-1, n-1)} < X_{(n-1, n-1)},\end{equation}
  where $(V_{(1, n-1)}, \ldots, V_{(n,n)})$ and $(X_{(0,n-1)}, \ldots, X_{(n-1,n-1)})$ respectively are the order statistics of $(V_1, \ldots, V_{n-1})$ and of $(X_0, \ldots, X_{n-1})$. Notice that (4.2) holds trivially for $n=1$.

  We set ${\mathcal W}_n^{\text{D}}=(E_{\textrm{g}}, E_{\textrm{d}}, X_1, \ldots, X_n, V_1, \ldots, V_{n-1},\zeta^{\text{D}}_{1}, \ldots, \zeta^{\text{D}}_{n-1})$.

• (ii) Assume $n\geq 1$. We work conditionally on ${\mathcal W}^{\text{D}}_n$. On the event $\{X_n^{\text{d}}=E_{\textrm{d}}\}$ set $I_n=I_n^{\text{g}}$ and on the event $\{X_n^{\text{g}}=-E_{\textrm{g}}\}$ set $I_n=I_n^{\text{d}}$. On the event $\{X_n^{\text{d}}=E_{\textrm{d}}\}\bigcup \{X_n^{\text{g}}=-E_{\textrm{g}}\}$, let $V_n$ be uniform on $I_n$, and let $\zeta_{n}^{\text{D}}$ be distributed as $\zeta^*_\delta$ (see (4.1)) with $\delta= |I_n|$.

• (iii) Assume $n\geq 2$ and that (4.2) holds. We work conditionally on ${\mathcal W}^{\text{D}}_n$. On the event $\{-E_{\textrm{g}}<X_n^{\text{g}},\,X_n^{\text{d}}<E_{\textrm{d}}\}$, there exists a unique integer $\kappa_n\in \{1, \ldots, n-1\}$ such that $V_{\kappa_n}\in [ X_n^{\text{g}},\,X_n^{\text{d}}]$. If $X_n\in [X_n^{\text{g}}, V_{\kappa_n})$, set $I_n=I_n^{\text{g}}$; if $X_n\in [V_{\kappa_n}, X_n^{\text{d}}]$, set $I_n=I_n^{\text{d}}$. Then let $V_n$ be uniform on $I_n$, and let $\zeta_{n}^{\text{D}}$ be distributed as $\zeta^*_\delta$, with $\delta= |I_n|$, conditionally on being less than $\zeta^{\text{D}}_{\kappa_n}$.

• (iv) Thanks to (ii) and (iii), notice that (4.2) holds with $n-1$ replaced by n, so that the induction is valid. Set

  \begin{equation*}{\mathcal A}_n^{\text{D}}={\mathcal A}_{n-1}^{\text{D}}+ \delta_{(V_n, \zeta^{\text{D}}_{n})}\end{equation*}
  and consider the tree $\mathfrak{T}_n^{\text{D}}$ corresponding to the ancestral process ${\mathcal A}_n^{\text{D}}$.

See Figures 8 and 9 for an example of $\mathfrak{T}_4^{\text{D}}$ and $\mathfrak{T}_5^{\text{D}}$.

Figure 8: An example of the tree $\mathfrak{T}_4^{\text{D}}$.

Figure 9: An example of the tree $\mathfrak{T}_5^{\text{D}}$. The length of the new branch attached to $V_{5}$ is conditioned to be less than that of the previous branch that was in the considered interval attached to $V_{2}$.

Then we have the following result.

Proof. We consider $\sum _{i\in {\mathcal I}}\delta_{(u_i,\zeta_i)}$ the ancestral process associated to the Poisson point measure $\sum_{i\in I}\delta_{(h_i,\varepsilon_i,e_i)}$ defined in Subsection 3.2.2. Let $(X^{\prime\prime}_k, k\in {\mathbb N} ^*)$ be independent uniform random variables on $[-E_{\textrm{g}}, E_{\textrm{d}}]$. Set $X^{\prime\prime}_0=0$. For $n\geq 1$, let us denote by $(X^{\prime\prime}_{(k,n)}, {0\leq k\leq n})$ the order statistic of $(X^{\prime\prime}_0,\ldots,X^{\prime\prime}_n)$.

For every $n\geq 1$ and every $1\leq k\leq n$, we set $i_{k,n}$ to be the index in ${\mathcal I}$ such that

\begin{equation*}\zeta_{i_{k,n}}=\max _{X^{\prime\prime}_{(k-1,n)}\leq u_i < X^{\prime\prime}_{(k,n)}}\zeta_i.\end{equation*}

Note that this index exists, since for every $\varepsilon>0$, the set $\{i\in {\mathcal I},\ \zeta _i>\varepsilon\}$ is a.s. finite. We set $V^{\prime\prime}_{(k,n)}=u_{i_{k,n}}$ and notice that, by standard Poisson point measure properties, $V^{\prime\prime}_{(k,n)}$ is, conditionally given $(X^{\prime\prime}_0,\ldots,X^{\prime\prime}_n)$, uniformly distributed on $[X^{\prime\prime}_{(k-1,n)},X^{\prime\prime}_{(k,n)}]$. We define

\begin{equation*}{\mathcal A}^{\prime\prime}_n=\sum_{k=1}^n \delta_{(V^{\prime\prime}_{(k,n)}, \zeta_{i_{k,n}})}.\end{equation*}

By construction, it is easy to check that the order statistics

\begin{equation*}X^{\prime\prime}_{(0,n)}<V^{\prime\prime}_{(1,n)}<X^{\prime\prime}_{(1, n)}< \cdots < V^{\prime\prime}_{(n, n)} < X^{\prime\prime}_{(n, n)}\end{equation*}

are distributed as

\begin{equation*}X_{(0,n)}<V_{(1,n)}<X_{(1, n)}< \cdots < V_{(n, n)} < X_{(n, n)}.\end{equation*}

For $1\leq k\leq n$, let $j_{k,n}\in \{1, \ldots, n\}$ be the index such that $V_{(k,n)}=V_{j_{k,n}}$. By construction, we then deduce that $(((V_{(k,n)},\zeta^{\text{D}}_{j_{k,n}}), 1\leq k\leq n), n\in {\mathbb N}^*)$ is distributed as $(((V^{\prime\prime}_{(k,n)},\zeta_{i_{k,n}}), 1\leq k\leq n), n\in {\mathbb N}^*)$. This implies that the sequences of ancestral processes $({\mathcal A}^{\prime\prime}_n, n\in {\mathbb N}^*)$ and $({\mathcal A}_n, n\in{\mathbb N}^*)$ have the same distribution. Then we use Proposition 3.3 to get that the sequence of trees $(T^{\prime\prime}_n, n\in {\mathbb N}^*)$, with $T^{\prime\prime}_n$ associated to ${\mathcal A}^{\prime\prime}_n$, is distributed as $(T_n, n\in {\mathbb N}^*)$.

4.3. Dynamic simulation (II)

In a sense, we had to introduce another piece of random information corresponding to the position $V_{n}$ of the largest spine of the sub-tree containing $X_{n}$. The construction in this subsection provides a way to remove this additional information (which is now hidden), but at the expense of possibly exchanging the newly inserted branch with one of its neighbors. In what follows, H stands for ‘hidden’. An example is provided for $\mathfrak{T}^{\text{H}}_4$ and $\mathfrak{T}^{\text{H}}_5$ in Figures 10, 11, and 12.

Figure 10: An example of the tree $\mathfrak{T}^{\text{H}}_4$ with the new individual $X_5$.

Figure 11: An example of the tree $\mathfrak{T}^{\text{H}}_5$ with $\mathfrak{T}^{\text{H}}_4$ given in Figure 10 and the event associated with $p_{\textrm{d}}$ (a new segment is attached to $X_{5}$).

Figure 12: An example of the tree $\mathfrak{T}^{\text{H}}_5$ with $\mathfrak{T}^{\text{H}}_4$ given in Figure 10 and the event associated with $p_{\textrm{g}}$ (the segment previously attached to $X_2$ is now attached to $X_5$, and a new segment is attached to $X_2$).

Let $\theta>0$. We build recursively a family of ancestral processes $({\mathcal A}_n^{\text{H}}, n\in {\mathbb N})$, with ${\mathcal A}_0^{\text{H}}=0$ and

\begin{equation*}{\mathcal A}_n^{\text{H}}=\sum_{k=1}^n \delta_{(X_k, \zeta_{k,n}^{\text{H}})}\end{equation*}

for $n\in {\mathbb N}^*$.

1. (i) Let $E_{\textrm{g}}$, $E_{\textrm{d}}$, $(X_n, n\in {\mathbb N})$, and $({\mathcal X}_n, n\in {\mathbb N}^*)$ be defined as in (i) of Subsection 4.1. For $n\in {\mathbb N} ^*$, set $X_n^{\text{g}}=\max \{x\in {\mathcal X}_n, x<X_n\}$ and $X_n^{\text{d}}=\min \{x\in {\mathcal X}_n, x>X_n\}$. For $n\in {\mathbb N} ^*$ and $\ell\in \{\text{g}, \text{d}\}$, define the interval $I_n^{\ell}= [X_n \wedge X_n^\ell, X_n\vee X_n^\ell]$ and its length $|I_n^{\ell}|=|X_n - X_n^\ell|$.

   We set ${\mathcal W}_n^{\text{H}}=(E_{\textrm{g}}, E_{\textrm{d}}, X_1, \ldots, X_n,\zeta^{\text{H}}_{1, n-1}, \ldots, \zeta^{\text{H}}_{n-1, n-1})$.

2. (ii) Assume $n\geq 1$. On the event $\{X_n^{\text{d}}=E_{\textrm{d}}\}$ set $I_n=I_n^{\text{g}}$, and on the event $\{X_n^{\text{g}}=-E_{\textrm{g}}\}$ set $I_n=I_n^{\text{d}}$. Conditionally on ${\mathcal W}_n^{\text{H}}$, let $\zeta_{n,n}^{\text{H}}$ be distributed as $\zeta^*_\delta$ (see (4.1)) with $\delta= |I_n|$; and for $1\leq k\leq n-1$, set $\zeta^{\text{H}}_{k,n}=\zeta^{\text{H}}_{k,n-1}$.

3. (iii) Assume $n\geq 2$. We work conditionally on ${\mathcal W}_n^{\text{H}}$. We define

   \begin{equation*}p_{\textrm{d}}=\frac{|I_n^{\text{d}}|}{|I_n^{\text{d}}|+|I_n^{\text{g}}|}\quad \text{and}\quad p_{\textrm{g}}=1-p_{\textrm{d}}=\frac{|I_n^{\text{g}}|}{|I_n^{\text{d}}|+|I_n^{\text{g}}|} \cdot\end{equation*}

   1. (a) On the event $\{0\leq X_n^{\text{g}},\,X_n^{\text{d}}<E_{\textrm{d}}\}$, there exists a unique integer $\kappa_n^{\text{d}} \in \{1, \ldots, n-1\}$ such that $X_{\kappa_n^{\text{d}}} = X_n^{\text{d}}$. For $1\leq k\leq n-1$ and $k\neq \kappa_n^{\text{d}}$, set $\zeta^{\text{H}}_{n,k} = \zeta^{\text{H}}_{n-1,k}$. Write

      \begin{equation*}\zeta^{\text{H}}_n=\zeta^{\text{H}}_{n-1, \kappa_n^{\text{d}}}.\end{equation*}
      With probability $p_{\textrm{d}}$, set $\zeta_{n,\kappa_n^{\text{d}}}^{\text{H}}= \zeta^{\text{H}}_{n}$ and let $\zeta_{n,n}^{\text{H}}$ be distributed as $\zeta^*_\delta$, with $\delta= |I_n^{\text{g}}|$, conditionally on being less than $\zeta^{\text{H}}_{n}$.

      With probability $p_{\textrm{g}}$, set $\zeta_{n,n}^{\text{H}}= \zeta^{\text{H}}_{n}$ and let $\zeta^{\text{H}}_{n, \kappa_n^{\text{d}}}$ be distributed as $\zeta^*_\delta$, with $\delta= |I_n^{\text{d}}|$, conditionally on being less than $\zeta^{\text{H}}_{n}$.

   2. (b) On the event $\{-E_{\textrm{g}}< X_n^{\text{g}},\,X_n^{\text{d}}\leq 0\}$, there exists a unique integer $\kappa_n^{\text{g}} \in \{1, \ldots, n-1\}$ such that $X_{\kappa_n^{\text{g}}} = X_n^{\text{g}}$. For $1\leq k\leq n-1$ and $k\neq \kappa_n^{\text{g}}$, set $\zeta^{\text{H}}_{n,k} = \zeta^{\text{H}}_{n-1,k}$. Write

      \begin{equation*}\zeta^{\text{H}}_n=\zeta^{\text{H}}_{n-1, \kappa_n^{\text{g}}}.\end{equation*}
      With probability $p_{\textrm{g}}$, set $\zeta_{n,\kappa_n^{\text{g}}}^{\text{H}}= \zeta^{\text{H}}_{n}$ and let $\zeta_{n,n}^{\text{H}}$ be distributed as $\zeta^*_\delta$, with $\delta= |I_n^{\text{d}}|$, conditionally on being less than $\zeta^{\text{H}}_{n}$.

      With probability $p_{\textrm{d}}$, set $\zeta_{n,n}^{\text{H}}= \zeta^{\text{H}}_{n}$ and let $\zeta^{\text{H}}_{n, \kappa_n^{\text{g}}}$ be distributed as $\zeta^*_\delta$, with $\delta= |I_n^{\text{g}}|$, conditionally on being less than $\zeta^{\text{H}}_{n}$.

4. (iv) Let $\mathfrak{T}^{\text{H}}_n$ be the tree corresponding to the ancestral process

   \begin{equation*}{\mathcal A}_n^{\text{H}}=\sum_{k=1}^n \delta_{(X_k, \zeta^{\text{H}}_{k,n})}.\end{equation*}

We now have the next result.

Proof. The proof is left to the reader. It is in the same spirit as the proof of Lemma 4.2, but here we consider the random variables $((V^{\prime\prime}_{(k,n)}, 1\leq k\leq n), n\in {\mathbb N}^*)$ to be unobserved.

4.4. Simulation of genealogical tree conditional on its maximal height

Let ${\mathcal F}^{(\theta)}=((\tau_i, h_i), \, i\in I)$ be a Brownian forest under ${\mathbb P}^{(\theta)}$. Recall the definition of $A_0$, the time to the MRCA of the population living at time 0, given in (5.3). The goal of this section is to simulate the genealogical tree $T_n$ of n individuals uniformly sampled in the population living at time 0, conditionally given that the time to the MRCA of the whole population is h, i.e. given $A_0=h$.

Let

\begin{equation*}{\mathcal A}(du,d\zeta)=\sum_{j\in {\mathcal I}}\delta_{(u_j,\zeta_j)}(du,d\zeta)\end{equation*}

be the ancestral process of Definition 3.1. Recall the notation $E_{\textrm{g}},E_{\textrm{d}}$ from Subsection 3.2.2. Let $\zeta_{\text{max}} =\text{sup} \{\zeta_j,\ j\in {\mathcal I}\}$ and define the random index $J_0\in {\mathcal I}$ so that $\zeta_{\text{max}} =\zeta_{J_0}$. Note that $J_0$ is well-defined since for every $\varepsilon>0$, the set $\{j\in {\mathcal I}, \zeta_j>\varepsilon\}$ is finite. We set $X=u_{J_0}\in({-}E_{\textrm{g}}, E_{\textrm{d}})$. Note that $\zeta_{\text{max}} $ is distributed as $A_0$.

For $r\in {\mathbb R}$, let $r_+=\max(0,r)$ and $r_-=\max(0, -r)$ be respectively the positive and negative part of r. The proof of the next lemma is postponed to the end of this section.

Let $h>0$ be fixed. For $\delta>0$, let $\zeta^{*,h}_\delta$ be a positive random variable distributed as $\zeta^*_\delta$ conditionally on $\{\zeta^*_\delta\leq h\}$; i.e. for $0\leq u\leq h$,

\begin{equation*}{\mathbb P}(\zeta^{*,h}_{\delta \leq u})={\mathbb P}(\zeta_{\delta}^{*\leq u} m|\zeta_{\delta}^{*\leq h})={ {\textrm{e}^{-\delta(c_\theta(u)-c_\theta(h))}}}.\end{equation*}

Then the static simulation runs as follows.

1. (i) Simulate three independent random variables $E_1,E_2,E_3$ exponentially distributed with parameter $2\theta+c_\theta(h)$, and another independent Bernoulli variable $\xi$ with parameter $1/2$. If $\xi=0$, set $E_{\textrm{g}}=E_1,\ X=E_2,\ E_{\textrm{d}}=E_2+E_3$, and if $\xi=1$, set $E_{\textrm{g}}=E_1+E_2,\ X=-E_2,\ E_{\textrm{d}}=E_3$. Let $X_k$ and ${\mathcal X}_k$ be defined as in (i) of Subsection 4.1 for $1\leq k\leq n$.

2. (ii) Let the intervals $I_k^{\text{S}}$ be defined as in (ii) of Subsection 4.1 for $1\leq k\leq n$.

3. (iii) Conditionally on $(E_{\textrm{g}}, E_{\textrm{d}}, X, X_1, \ldots, X_n)$, let $(\zeta_{k}^{h}, 1\leq k\leq n)$ be independent random variables such that, for $1\leq k\leq n$, $\zeta_{k}^{h}$ is distributed as $\zeta^{*,h}_\delta$ with $\delta= |I^{\text{S}}_{k}|$ if $X \not \in I_k^{\text{S}}$, and $\zeta_{k}^{h}=h$ if $X \in I_k^{\text{S}}$. Consider the tree $\mathfrak{T}^h_n$ corresponding to the ancestral process

   \begin{equation*}{\mathcal A}_n^h=\sum_{k=1} \delta_{(X_k, \zeta_{k}^{h})}.\end{equation*}

The proof of the following result, which relies on Lemma 4.4, is similar to that of Lemma 4.1, and is not reproduced here.

Notice that the height of $\mathfrak{T}_n^h$ is less than or equal to h. When it is strictly less than h, this means that no individual of the oldest family has been sampled.

Proof of Lemma 4.4. By Proposition 3.2, the pair $E=(E_{\textrm{g}},E_{\textrm{d}})$ under ${\mathbb P}^{(\theta)}$ has density

\begin{equation*}f_E(e_{\textrm{g}}, e_{\textrm{d}})=(2\theta)^2{\mathop {\textrm{e}^{-2\theta(e_{\textrm{g}}+e_{\textrm{d}})}}}{\textbf 1}_{\{e_{\textrm{g}}\geq 0, e_{\textrm{d}}\geq 0\}}.\end{equation*}

Moreover, by standard results on Poisson point measures, the conditional density of the pair $(X,\zeta_{\text{max}})$ given $(E_{\textrm{d}},E_{\textrm{g}})=(e_g,e_d)$ exists and is

\begin{align*}f_{X,\zeta_{\text{max}}}^{E=(e_{\textrm{g}},e_{\textrm{d}})}(x,h)&=\frac{1}{e_{\textrm{g}}+e_{\textrm{d}}}{\textbf 1}_{[-e_{\textrm{g}},e_{\textrm{d}}]}(x)\, (e_{\textrm{g}}+e_{\textrm{d}})\, |c^{\prime}_\theta(h)|{\mathop {\textrm{e}^{-c_\theta(h)(e_{\textrm{g}}+e_{\textrm{d}})}}}{\textbf 1}_{\{h\geq 0\}}\\& ={\textbf 1}_{[-e_{\textrm{g}},e_{\textrm{d}}]}(x)\, |c^{\prime}_\theta(h)|{\mathop {\textrm{e}^{-c_\theta(h)(e_{\textrm{g}}+e_{\textrm{d}})}}}{\textbf 1}_{\{h\geq 0\}}.\end{align*}

We deduce that the vector $(E_{\textrm{g}},E_{\textrm{d}},X,\zeta_{\text{max}})$ has density

\begin{equation*}f(e_{\textrm{g}},e_{\textrm{d}},x,h)=(2\theta)^2|c^{\prime}_\theta(h)|{\mathop {\textrm{e}^{-(2\theta+c_\theta(h))(e_{\textrm{g}}+e_{\textrm{d}})}}}{\textbf 1}_{\{ e_{\textrm{g}}\geq 0, \, e_{\textrm{d}}\geq 0, \, -e_{\textrm{g}}\leq x\leq e_{\textrm{d}}, \, h\geq 0\}}\end{equation*}

and that the random variable $\zeta_{\text{max}}$ has density

\begin{align*}f_{\zeta_{\text{max}}}(h)& =\int (2\theta)^2|c^{\prime}_\theta(h)|{\mathop {\textrm{e}^{-(2\theta+c_\theta(h))(e_{\textrm{g}}+e_{\textrm{d}})}}}{\textbf 1}_{\{ e_{\textrm{g}}\geq 0, \, e_{\textrm{d}}\geq 0, \, -e_{\textrm{g}}\leq x\leq e_{\textrm{d}}, \, h\geq 0\}}\,de_{\textrm{g}}\,de_{\textrm{d}}\,dx\\& =(2\theta)^2|c^{\prime}_\theta(h)|\frac{2}{(2\theta+c_\theta(h))^3}\, {\textbf 1}_{\{h\geq 0\}}.\end{align*}

Therefore, the conditional density of the vector $(E_{\textrm{g}},E_{\textrm{d}},X)$ given $\zeta_{\text {max}}=h$ is

\begin{equation*}f_{E,X}^{\zeta_{\text{max}}=h}(e_{\textrm{g}},e_{\textrm{d}},x)=\frac{1}{2}(2\theta+c_\theta(h))^3{\mathop {\textrm{e}^{-(2\theta+c_\theta(h))(e_{\textrm{g}}+e_{\textrm{d}})}}}{\textbf 1}_{\{e_{\textrm{g}}\geq 0, \, e_{\textrm{d}}\geq 0, \, -e_{\textrm{g}}\leq x\leq e_{\textrm{d}}\}}.\end{equation*}

For any nonnegative measurable function $\varphi$, we have

\begin{align*} & {\mathbb E}^{(\theta)}[ \varphi(E_{\textrm{g}}+X_-,|X|,E_{\textrm{d}}-X_+){\textbf 1}_{\{X\geq 0\}} m|\zeta_{\text{max}}=h] \\ & \quad ={\mathbb E}^{(\theta)}[\varphi(E_{\textrm{g}},X,E_{\textrm{d}}-X){\textbf 1}_{\{X\geq 0\}} m|\zeta_{\text{max}}=h]\\ & \quad =\int\varphi(e_{\textrm{g}},x,e_{\textrm{d}}-x)\frac{1}{2}(2\theta+c_\theta(h))^3{\mathop {\textrm{e}^{-(2\theta+c_\theta(h))(e_{\textrm{g}}+e_{\textrm{d}})}}}{\textbf 1}_{\{e_{\textrm{g}}\geq 0, \, e_{\textrm{d}}\geq x\geq 0\}}\,de_{\textrm{g}}\, de_{\textrm{d}}\,dx\\ & \quad =\int\varphi(e_1,e_2,e_3)\frac{1}{2}(2\theta+c_\theta(h))^3{{\textrm{e}^{-(2\theta+c_\theta(h))(e_1+e_2+e_3)}}}{\textbf 1}_{\{e_1\geq 0, \, e_2\geq 0, \, e_3\geq 0\}}\, de_1\,de_2\, de_3,\end{align*}

using an obvious change of variables. Similarly, we get

\begin{multline*}{\mathbb E}^{(\theta)}[ \varphi(E_{\textrm{g}}+X_-,|X|,E_{\textrm{d}}-X_+){\textbf 1}_{\{X<0\}} m|\zeta_{\text{max}}=h]\\={\mathbb E}^{(\theta)}[ \varphi(E_{\textrm{g}}+X_-,|X|,E_{\textrm{d}}-X_+){\textbf 1}_{\{X\geq 0\}} m|\zeta_{\text{max}}=h].\end{multline*}

This proves the lemma.

5.1. Preliminary results

Let $E_{\textrm{g}}$ and $E_{\textrm{d}}$ be two independent exponential random variables with parameter $2\theta$. Let ${\mathcal N}(dz,d\tau)$ be, conditionally given $(E_{\textrm{g}}, E_{\textrm{d}})$, distributed as a Poisson point measure with intensity

\begin{equation*}{\textbf 1}_{[-E_{\textrm{g}}, E_{\textrm{d}}]}(z) \, dz{\mathbb N}^{(\theta)}[d\tau].\end{equation*}

We denote by $(z_i,\tau_i)_{i\in I}$ the atoms of the measure ${\mathcal N}$, so that

\begin{equation*}{\mathcal N}=\sum_{i\in I}\delta_{z_i,\tau_i}.\end{equation*}

We define $\tilde L=(\tilde L_\varepsilon, \varepsilon>0)$ with

\begin{equation*}\tilde L_\varepsilon=\sum_{i\in I} (\zeta_i -\varepsilon)_+,\end{equation*}

where $\zeta_i=H(\tau_i)$ is the height of $\tau_i$. Notice that only a finite number of trees $(\tau_i)_{i\in I}$ have height greater than $\varepsilon$, so the sum in $\tilde L_\varepsilon$ is a.s. finite, and $\tilde L_\varepsilon$ is well-defined. Let $(U_k, k\in {\mathbb N}^*)$ be independent random variables uniformly distributed on [0, 1] and independent of $({\mathcal N}, E_{\textrm{g}}, E_{\textrm{d}})$. We set $ X_{0}=0$, and $ X_k=(E_{\textrm{g}}+E_{\textrm{d}}) U_k - E_{\textrm{g}}$ for $k\in{\mathbb N}^*$. Fix $n\in {\mathbb N}^*$. Let $ X_{(0,n)}\leq \cdots \leq X_{(n,n)}$ be the corresponding order statistic of $( X_0, \ldots, X_n)$. We set $ X_{({-}1, n)}=-E_{\textrm{g}}$ and $ X_{(n+1, n)}=E_{\textrm{d}}$. We define the interval $I_{k,n}=( X_{(k-1, n)}, X_{(k,n)})$ and its length $\Delta_{k,n}= X_{(k,n)} - X_{(k-1, n)}$ for $0\leq k\leq n+1$. We set $\Delta_n=(\Delta_{k,n}, 0\leq k\leq n+1)$. For $1\leq k\leq n$, we define $\tilde \Lambda=(\tilde \Lambda_n, n\in {\mathbb N} ^*)$ by