It implies in particular that $\bar{\nu}_x^1 \le C_H \cdot \check{\nu}_{x_\zeta}^1$, where the time $t_H$ and the constants $y/2$ and $m_M$ are slightly relaxed:

$$ \check{\nu}_{x_\zeta}^1 (dx'_H) := \mathbb{P}_{x_\zeta}^{(J:d)} \left( \pi_J(X^{(d)}(2t_H)) \in dx'_H ; 2t_H < T_{y/4}^{J,(d)} \wedge T_{2m_M}^{(J.3|d)} \right). $$

Thanks in addition to (7.86), (7.87), (7.89), we obtain as an intermediate step the following inequality, which holds for any $d \in [J, \infty]$ and any $x, x_\zeta \in E^{(d)}$:

$$ \mathbb{P}_x \left( X^{(d)} (U_H^{(d)}) \in dx', U_H^{(d)} < \infty \right) \le 2C_H \int_{\mathcal{X}_J} \check{\nu}_{x_\zeta}^1 (dx'_H) \bar{\nu}_{x'_H}^2 (dx'). \quad (7.90) $$

To go back to an upper-bound in terms of the original process, we want again to exploit Proposition 7.3.2. So we need to again ensure upper-bounds on the third moments for the last components for which we lost the information. We exploit the representation given in Proposition 7.3.1, firstly on the time-interval $[0, 2t_H]$. Since $x_\zeta \in E^{(d)}$, we have $M_3^{(F|d)}(0) \le m/y$. Thanks to Lemma 7.3.5, we can thus define $m_H > 0$ such that the following inequality holds a.s. on the event $\{2t_H < T_{y/4}^{J,(d)}\}$, for any $d \in [J, \infty]$ and any $x_\zeta \in E^{(d)}$:

$$ \mathbb{P}_{x_\zeta}^{(J:d)} \left( T_{m_H}^{(F.3|d)} \le 2t_H \mid \mathcal{F}^{(J)} \right) \le \frac{1}{2}. $$

With the fact that $\tau_\partial^{(d)} > T_{y/4}^{J,(d)}$, it implies that $\check{\nu}_{x_\zeta}^1 \le 2\bar{\nu}_{x_\zeta}^1$, where:

$$ \check{\nu}_{x_\zeta}^1 (dx_H) := \mathbb{P}_{x_\zeta}^{(J:d)} \left( \pi_J(X^{(d)}(2t_H)) \in dx'_H ; 2t_H < \tau_\partial^{(d)} \wedge T_{2m_M}^{(J.3|d)} \wedge T_{m_H}^{(F.3|d)} \right). \quad (7.91) $$

Recalling that the formula $\check{\nu}_{\pi_J(x_H)}^2 = \nu_{x_H}^2$ produces a well-defined quantity, it implies that $\int_{\mathcal{X}_J} \check{\nu}_{x_\zeta}^1 (dx'_H) \bar{\nu}_{x'_H}^2 (dx') = \int_{\mathcal{X}_d} \hat{\nu}_{x_\zeta}^1 (dx_H) \nu_{x_H}^2 (dx')$, where:

$$ \hat{\nu}_{x_\zeta}^1 (dx_H) := \mathbb{P}_{x_\zeta}^{(J:d)} \left( X^{(d)}(2t_H) \in dx_H ; 2t_H < \tau_\partial^{(d)} \wedge T_{2m_M}^{(J.3|d)} \wedge T_{m_H}^{(F.3|d)} \right). \quad (7.92) $$

Note that this measure $\hat{\nu}_{x_\zeta}^1$ is supported on the set $\{x_H \in \mathcal{X}_d ; M_3^{(F|d)}(x_H) \le m_H\}$. Thanks again to Lemma 7.3.5, we can thus define $m_F \ge m_H$ such that the following inequality holds a.s. for any $d \in [J, \infty]$ and any $x_H \in \mathcal{X}_d$ such that $M_3^{(F|d)}(x_H) \le m_H$:

$$ \mathbb{P}_{x_H}^{(J:d)} \left( T_{m_F}^{(F.3|d)} < t_B \wedge \tau_0^{(J:d)} \mid \mathcal{F}^{(J)} \right) \le \frac{1}{2}. $$

It entails that $\nu_{x_H}^2 \le 2\bar{\nu}_{x_H}^2$, where:

$$ \bar{\nu}_{x_H}^2 (dx') := \mathbb{P}_{x_H}^{(J:d)} \left( X^{(d)} (\tau_0^{(J:d)}) \in dx' ; \tau_0^{(J:d)} < t_B \wedge \tau_\partial^{(d)} \wedge T_{m_M}^{(J.3|d)} \wedge T_{m_F}^{(F.3|d)} \right). $$
$$ (7.93) $$

Since $M_3^{(d)}(x) \le M_3^{(J:d)}(x) + M_3^{(F|d)}(x)$, and thanks to the Markov property at time $t_H$, recalling (7.90), (7.91), (7.92) and (7.93), the following inequality holds with $V^{(d)}$ as defined in (7.82) and $\hat{m} := [2m_M + m_H] \lor [m_M + m_F]$, for any $x, x_\zeta \in E^{(d)}$:

$$ \mathbb{P}_x \left( X^{(d)} (U_H^{(d)}) \in \mathrm{d}x', \ U_H^{(d)} < \infty \right) \\ \le 8C_H \cdot \mathbb{P}_{x_\zeta}^{(J:d)} \left( X^{(d)} (V^{(d)}) \in \mathrm{d}x'; \ V^{(d)} < \tau_\partial^{(d)} \wedge T_{\hat{m}}^{(3|d)} \right). $$

We can then relate to the original law $\mathbb{P}_{x_\zeta}$ thanks to Proposition 7.3.2 with an additional factor $C_G > 0$ such that the following inequality holds for any $x, x_\zeta \in E^{(d)}$:

$$ \mathbb{P}_x \left( X^{(d)} (U_H^{(d)}) \in \mathrm{d}x', \ U_H^{(d)} < \infty \right) \\ \le 8C_H \cdot C_G \cdot \mathbb{P}_{x_\zeta} \left( X^{(d)} (V^{(d)}) \in \mathrm{d}x'; \ V^{(d)} < \tau_\partial^{(d)} \right). \quad (7.94) $$

To conclude, we consider general initial conditions $\bar{x} \in \mathcal{X}_d$ and $\bar{x}_\zeta \in \mathcal{D}_3$ and define:

$$ \nu_x^E (\mathrm{d}x') := \mathbb{P}_x \left( X^{(d)} (\tau_E^{(d)}) \in \mathrm{d}x'; \ \tau_E^{(d)} < \frac{1}{3} \wedge \tau_\partial^{(d)} \right). $$

We deduce from (7.75) that $\nu_{\bar{x}_\zeta}^E (E^{(d)}) \ge c_\zeta$ while $\nu_{\bar{x}}^E (E^{(d)}) \le 1$. Thanks (twice) to the strong Markov property at time $\tau_E^{(d)}$ and by virtue of the definitions of $U_H^{(d)}$ and $V^{(d)}$ in (7.81) and (7.82), (7.94) is extended as follows for any $d \in [J, \infty]$ to any $\bar{x} \in \mathcal{X}_d$ and $\bar{x}_\zeta \in \mathcal{D}_3$:

$$ \begin{aligned} & \mathbb{P}_{\bar{x}} \left( X^{(d)} (U_H^{(d)}) \in \mathrm{d}x', \ U_H^{(d)} < \infty \right) \\ &= \int_{E^{(d)}} \nu_{\bar{x}}^E (\mathrm{d}x) \mathbb{P}_x \left( X^{(d)} (U_H^{(d)}) \in \mathrm{d}x', \ U_H^{(d)} < \infty \right) \\ &\le 8C_H \cdot C_G \cdot \frac{\nu_{\bar{x}}^E (E^{(d)})}{\nu_{\bar{x}_\zeta}^E (E^{(d)})} \int_{E^{(d)}} \nu_{\bar{x}_\zeta}^E (dx_\zeta) \mathbb{P}_{x_\zeta} \left( X^{(d)} (V^{(d)}) \in \mathrm{d}x'; \ V^{(d)} < \tau_\partial^{(d)} \right) \\ &\le c_V \cdot \mathbb{P}_{\bar{x}_\zeta} \left( X^{(d)} (V^{(d)}) \in \mathrm{d}x'; \ V^{(d)} < \tau_\partial^{(d)} \right), \end{aligned} $$

where $c_V = 8C_H C_G / c_\zeta > 0$. This concludes the proof of Theorem 7.5.1. $\square$

<figure><img src="image_13.png" /></figure>

## 7.6 Concluding the proof of Theorem 2.4.1

We plan to exploit Theorem 2.5.3 and ensure Assumption (AF) (recall Sect. 2.5). Firstly, the sets $\mathcal{D}_\ell^{(d)}$ satisfy Assumption (A0) (recall (7.1)). Thanks to Theorem 7.4.7, Assumption (A1) holds true for the reference measure $\zeta^{(d)}$ defined by (7.59). Thanks to (A1) and [15, Lemma 3.0.2], $\rho_S$ is upper-bounded by some real number $\tilde{\rho}_S$ that
only depends on the constants involved in (A1), and can thus be chosen independent of $d$. In order to satisfy $\rho > \rho_S$, we set $\rho := 2\tilde{\rho}_S$. For this choice of $\rho$, we deduce Assumption (A2) as a consequence of Theorem 7.2.1 (recall also Remark 7.2.2), where the complementary to the transitory domain can be taken as the proposed set $E^{(d)}(m, y, 1)$ for some $m, y > 0$ independent of $d \in [1, \infty]$, or simply $D_L^{(d)}$ with $L \ge \frac{1}{y}$ (we choose $\eta = 1$ for simplicity).

When $d = \infty$ we infer Assumption (A3$_F$) thanks to Theorem 7.5.1 (regardless of the transitory domain that is chosen), for this choice of $\rho$ and $\zeta^{(\infty)}$, noting that $\zeta^{(\infty)}$ is supported on $D_3^{(\infty)}$. Assumption (AF) thus holds true for $d = \infty$. Thanks to Theorem 2.5.3, we then conclude the proof that the semi-group displays QSC and that the Q-process exists on $\mathcal{X}_\infty$.

In addition, recall that $\rho_0^{(\infty)} = -\log[P_{v(\infty)}(1 < \tau_\partial^{(\infty)})]$ (see Remark 2.1.2). Since $\cup_\ell D_\ell^{(\infty)} = \mathcal{X}_\infty$ (recall (7.1)), there exists $\ell$ such that $v^{(\infty)}(D_\ell^{(\infty)}) > 0$. With a slight adaptation of the proof of Lemma 7.2.5, we deduce that $P_x^{(\infty)}(\tau_\partial^{(\infty)} \le 1)$ is uniformly bounded away from 0 for any $x \in D_\ell^{(\infty)}$. This concludes that $\rho_0^{(\infty)} > 0$.

As noted in [14, Subsection 6.1], the statement of Assumption (A3$_F$) is actually required for a single value of $\epsilon$ defined in terms of the parameters involved in Assumptions (A1) and (A2). Since these parameters can be chosen uniformly in $d$, the corresponding value of $\epsilon$ can also be chosen uniformly so that the required properties in Assumption (AF) holds uniformly for any $d$ larger than a given threshold (see the definition of $J$ in (7.77)). In addition and for the same reason, all parameter involved in the convergences can be chosen independently of $d$ sufficiently large. There are indeed intricate yet explicit relations between all these parameters introduced in [14] and the corresponding assumptions. This concludes the proof of Theorem 2.4.1. $\square$

## Appendix A Two representations for the same process

In this Section A, we justify more precisely (see Proposition A.0.1) than we did in Remark 1.2.2 that the process we consider is exactly the same as defined notably in [2], though the representation has been adapted for our purposes.

Let us recall from (1.1) for any $d \in [2, \infty]$ the definition of our focal process $X^{(d)}$ as follows for any $i \in [0, d]$:

$$dX_i^{(d)}(t) = b_i^{(d)}(X^{(d)}(t)) dt + dN_i^{(d)}(t), \quad (A1)$$

where the vectorial function $b$ encodes the drift term as follows for any $x \in \mathcal{X}_d$ and $i \in [0, d]$:

$$b_i^{(d)}(x) := \alpha \cdot \left( \sum_{j \in [0, d]} j \cdot x_j - i \right) \cdot x_i + \lambda \cdot \left( x_{i-1} - \mathbf{1}_{\{i < d\}} x_i \right),$$
while the martingale process $N_i^{(d)}$, which starts at 0 for $t = 0$, is expressed as follows:

$$ 
\begin{aligned}
dN_i^{(d)}(t) &:= \sum_{j=0}^{d} (\delta_{ij} - X_i^{(d)}(t)) \cdot \sqrt{X_j^{(d)}(t)} dW_j(t) \\
&= \sqrt{X_i^{(d)}(t)} dW_i(t) - X_i^{(d)}(t) dW_{(d)}(t).
\end{aligned} 
\quad (A2) $$

in terms of a family $(W_i)_{i \in \mathbb{Z}_+}$ of mutually independent Brownian motions and an aggregated martingale $W_{(d)}$ with the following definition:

$$ dW_{(d)}(t) := \sum_{j=0}^{d} \sqrt{X_j^{(d)}(t)} dW_j(t), \quad W_{(d)}(0) = 0. $$

On the other hand, we next consider the more classical way that Muller's ratchet diffusion $\hat{X}$ is defined, notably in [2]:

$$ d\hat{X}_i^{(d)}(t) = b_i^{(d)}(\hat{X}^{(d)}(t)) dt + d\hat{N}_i^{(d)}(t), $$

where the martingale term $\hat{N}_i^{(d)}$ is now expressed as follows:

$$ d\hat{N}_i^{(d)}(t) = \sum_{\{j \neq i\}} \sqrt{\hat{X}_i^{(d)}(t) \cdot \hat{X}_j^{(d)}(t)} dW_{i,j}(t), \quad \hat{N}_i^{(d)}(0) = 0. $$

in terms of a family $(W^{i,j})_{i,j}$ of independent Brownian motions, extended to any $i \neq j$ by the symmetry property $W_{i,j}(t) = W_{j,i}(t)$.

For the infinite-dimensional case, we recall the definition of $\mathcal{X}^\eta$ in 2.10 as the set of probabilities on $\mathbb{N}$ with finite $\eta$-th moment ($\eta = 6$ being considered for simplicity in our proofs). The upcoming Proposition A.0.1 summarises the result of this Section A and notably implies Proposition 2.3.1.

**Proposition A.0.1** The processes $(X_i^{(d)})$ and $(\hat{X}_i^{(d)})$ share the same law. Existence and uniqueness in law holds, for processes on $\mathcal{X}_d$ for any $d \in \mathbb{N}$, and, for the case $d = \infty$, on $\mathcal{X}^\eta$ for any $\eta > 2$. For any $d$, the associated infinitesimal generator takes the following nondivergence form, for any $u \in C^2(\mathcal{D}_\ell^{(d)})$ and $x \in \mathcal{D}_\ell^{(d)}$:

$$ \mathcal{L}^{(d)} u(x) = \frac{1}{2} \sum_{i,j=1}^{d} \sigma_{i,j}^{(d)}(x) \partial_{i,j}^2 u(x) + \sum_{i=1}^{d} b_i^{(d)}(x) \partial_i u(x), $$

where the state-dependent diffusion matrix $\sigma^{(d)}$ is expressed as follows:

$$ \sigma_{i,j}^{(d)}(x) := \begin{cases} -x_i \cdot x_j & \text{if } i \neq j, \\ x_i \cdot (1 - x_i) & \text{if } i = j, \end{cases} $$
and the state-dependent drift term $b^{(d)}$ is expressed as follows:

$$b_i^{(d)}(x) := \alpha \cdot (M_1^{(d)}(x) - i) \cdot x_i + \lambda \cdot (x_{i-1} - \mathbf{1}_{\{i < d\}} \cdot x_i).$$

**Remark A.0.2** Recall that our representation of the martingale term corresponds to the idea that internal demographic fluctuations $W^i$ specific to each subpopulation level $i$ get compensated thanks to the process $W$. On the other hand, the classical representation is associated to the principles of Moran models, where demographic fluctuations are generated by choosing one parent at random whose offspring replaces another individual also chosen at random. By distinguishing these contributions in terms of the two involved types $(i, j)$ (be it the parental type or the killed type), we can derive the contribution $\sqrt{\hat{X}_i^{(d)}(t) \cdot \hat{X}_j^{(d)}(t)} dW_{i,j}(t)$ to the fluctuations. It explains why $W_{i,j} = -W_{j,i}$ and how the previous term can be interpreted as a random flux, which does not need to be compensated.

**Proof** : According to [20, Theorem 2.1], the existence and the uniqueness in law of solutions hold for the system A1 of SDEs (for both finite and infinite $d$) provided that the drift term $b_i^{(d)}$ satisfies the following three conditions:

* $b_i(x) \ge 0$ uniformly in $x \in \mathcal{X}_d$,

* $\sum_{i \in [0, d]} b_i(x) = 0$,

* there exists a matrix $(q_{ij})_{i,j \in [0, d]}$ such that $q_{ij} \ge 0$ and $\sup_{k \in [0, d]} \sum_{i=0}^d q_{ik} < +\infty$ for every $i$ and $j$ of $[0, d]$, and such that the following inequality holds for any $x$ and $x'$ of $\mathcal{X}_d$ ($i \in [0, d]$):

$$|b_i^{(d)}(x) - b_i^{(d)}(x')| \le \sum_{\{j \in S\}} q_{ij} \cdot |x_j - x'_j|.$$

The first two properties are satisfied for any $d \in [2, \infty]$. For finite $d \in \mathbb{N}$, the following estimate on the drift term holds for any $i \in [0, d]$ and $x, x' \in \mathcal{X}_d$:

$$|b_i(x) - b_i(x')| \le (2\alpha \cdot d + \lambda) \cdot \sum_{j=0}^{d} |x_j - x'_j|.$$

The corresponding choice of $q_{ij} \equiv 2\alpha \cdot d + \lambda$ ensures for any $d \in \mathbb{N}$ the existence and uniqueness in law for the system A1 of SDEs. Thanks to [2, Theorem 3] (which relies on the above result of [20]), existence and uniqueness in law also hold on the set $\mathcal{X}^\eta$ for any $\eta > 2$. The associated infinitesimal generator is expressed in [20, Theorem 1.1] as stated in Proposition A.0.1.

For completeness and to check that the laws of $\mathcal{N}_i^{(d)}(t)$ and $\hat{\mathcal{N}}_i^{(d)}(t)$ actually coincide, we provide next the computations of the quadratic variations of these martingale terms. We first consider the quadratic variation of each component $i$ for the process $\mathcal{N}_i^{(d)}$:

$$d\langle \mathcal{N}_i^{(d)} \rangle_t = X_i^{(d)}(t) d\langle W_i \rangle_t + (X_i^{(d)}(t))^2 d\langle W_{(d)} \rangle_t - 2\sqrt{X_i^{(d)}(t)} \cdot X_i^{(d)}(t) d\langle W_i, W_{(d)} \rangle_t \\ = X_i^{(d)}(t) \cdot (1 - X_i^{(d)}(t)) dt,$$

<figure><img src="image_14.png" /></figure>
then for the process $\hat{\mathcal{N}}_i^{(d)}$:

$$ 
\begin{aligned}
d\langle \hat{\mathcal{N}}_i^{(d)} \rangle_t &= \sum_{\{j \neq i\}} \hat{X}_i^{(d)}(t) \cdot \hat{X}_j^{(d)}(t) \, dt \\
&= \hat{X}_i^{(d)}(t) \cdot (1 - \hat{X}_i^{(d)}(t)) \, dt.
\end{aligned}
$$

We secondly consider the cross-variations for $i \neq j$, which yields for the process $M^{(d)}$:

$$ 
\begin{aligned}
d\langle \mathcal{N}_i^{(d)}, \mathcal{N}_j^{(d)} \rangle_t \\
&= \sqrt{X_i^{(d)}(t) \cdot X_j^{(d)}(t)} d\langle W_i, W_j \rangle_t - \sqrt{X_i^{(d)}(t) \cdot X_j^{(d)}(t)} d\langle W_i, W_{(d)} \rangle_t \\
&\quad - X_i^{(d)}(t) \cdot \sqrt{X_j^{(d)}(t)} d\langle W_{(d)}, W_j \rangle_t + X_i^{(d)}(t) \cdot X_j^{(d)}(t) d\langle W_{(d)} \rangle_t \\
&= \left[ 0 - \sqrt{X_i^{(d)}(t) \cdot X_j^{(d)}(t) \cdot \sqrt{X_i^{(d)}(t) - X_i^{(d)}(t)}} \right. \\
&\qquad \left. \cdot \sqrt{X_j^{(d)}(t) \cdot \sqrt{X_j^{(d)}(t) + X_i^{(d)}(t) \cdot X_j^{(d)}(t)}} \right] dt, \\
&= -X_i^{(d)}(t) \cdot X_j^{(d)}(t) dt,
\end{aligned}
$$

then yields for the process $\hat{M}^{(d)}$:

$$ 
\begin{aligned}
d\langle \hat{\mathcal{N}}_i^{(d)}, \hat{\mathcal{N}}_j^{(d)} \rangle_t &= - \sum_{\{k \neq i\}} \sum_{\{l \neq j\}} \sqrt{\hat{X}_i^{(d)}(t) \cdot \hat{X}_k^{(d)}(t) \cdot \hat{X}_j^{(d)}(t) \cdot \hat{X}_l^{(d)}(t)} d\langle W_{i,k}, W_{l,j} \rangle_t \\
&= -\hat{X}_i^{(d)}(t) \cdot \hat{X}_j^{(d)}(t) dt,
\end{aligned}
$$

since $d\langle W_{i,k}, W_{l,j} \rangle_t = \mathbf{1}_{\{i=l, j=k\}} dt$.

Since the quadratic variations are expressed in the same ways for the solutions $X^{(d)}$ and $\hat{X}^{(d)}$ to the systems respectively A1 and A2, we conclude thanks to [20, Theorem 1.1] that the laws of these two processes coincide. This concludes the proof of Proposition A.0.1. □

## Appendix B Justification for the boundedness of the moments

In this Section B, we derive from the proof of [2, Theorem 3] the local boundedness of the moment processes, as stated in this last Proposition B.0.1.

**Proposition B.0.1** The process $M_k^{(\infty)}$ is a.s. locally upper-bounded for any $k \ge 1$, any $\alpha, \lambda > 0$ and any $x \in \mathcal{X}_\infty \cap \mathcal{X}^k$ under both $\mathbb{P}_x$ and $\mathbb{P}_x^{(J:d)}$, whatever $J \in \mathbb{N}$.

*Proof:* Without loss of generality, we assume that $k \ge 3$ and consider any $x \in \mathcal{X}^k$, any $\lambda > 0$ and any $t > 0$. The conclusion of the proof of [2, Proposition 2.2] can
be restated as the following upper-bound, which holds in the case where $\alpha = 0$ (as indicated with the notation $\mathbb{P}^{[0]}$):

$$ \mathbb{P}_x^{[0]} \left( \sup_{\{s \le t\}} M_k^{(\infty)}(s) < \infty \right) = 1. \qquad (\text{B1}) $$

We consider any $\alpha > 0$ and denote for clarity $\mathbb{P}^{[\alpha]}$ instead of $\mathbb{P}^{(\infty)}$ for the law of the process $X^{(\infty)}$ solution to (1.1). Thanks to [2, Subsection 2.1, proof of Theorem 3], $\mathbb{P}^{[\alpha]}$ is related to $\mathbb{P}^{[0]}$ with a Girsanov transform, namely $\mathbb{P}_x^{[\alpha]}|_{\mathcal{F}_t} = Z_\alpha(t) \cdot \mathbb{P}_x^{[0]}|_{\mathcal{F}_t}$ holds for any $x \in \mathcal{X}_\infty$ and any $t > 0$, which involves the following martingale process $(Z_\alpha(s))_{s \ge 0}$:

$$ \begin{aligned} Z_\alpha(s) &:= \exp \left( -\alpha \mathcal{M}_1^{(\infty)}(s) - (\alpha^2/2) \cdot \int_0^s M_2^{(\infty)}(s) ds \right), \\ &\le \exp \left( -\alpha \cdot [M_1^{(\infty)}(x) + \lambda s] \right). \end{aligned} $$

In the above expression, $\mathcal{M}_1^{(\infty)}(s)$ is the martingale component of $M_1^{(\infty)}(s)$, as stated in Lemma 6.1.5. Since $Z_\alpha(t)$ is uniformly upper-bounded, (B1) entails that $\sup_{\{s \le t\}} M_k^{(\infty)}(s) < \infty$ holds also a.s. under $\mathbb{P}_x^{[\alpha]}$, provided $x \in \mathcal{X}^k$.

We next consider $J \in \mathbb{N}$ and the law $\mathbb{P}_x^{[\alpha],J} = \mathbb{P}_x^{[\alpha],(J:\infty)}$ as stated in Sect. 7.3. Thanks to Proposition 7.3.1, and with the same arguments as above, $\mathbb{P}^{[\alpha],J}$ is related to $\mathbb{P}^{[0]}$ with another Girsanov transform, namely $\mathbb{P}_x^{[\alpha],J}|_{\mathcal{F}_t} = Z_\alpha^J(t) \cdot \mathbb{P}_x^{[0]}|_{\mathcal{F}_t}$ holds for any $x \in \mathcal{X}_\infty$ and any $t > 0$, which involves the following martingale process $(Z_\alpha^J(s))_{s \ge 0}$:

$$ \begin{aligned} Z_\alpha^J(s) &:= \exp \left( -\alpha \mathcal{M}_1^{(J:\infty)}(s) - (\alpha^2/2) \cdot \int_0^s M_2^{(J:\infty)}(s) ds \right), \\ &\le \exp \left( -\alpha \cdot \left[ (M_1^{(J:\infty)}(x) \wedge J) + \lambda s \right] \right). \end{aligned} $$

Similarly, $\mathcal{M}_1^{(J:\infty)}(s)$ is the martingale component of $M_1^{(J:\infty)}(s)$. With the same arguments as for the proof of Lemma 6.1.5, its quadratic variation involves the process $M_2^{(J:\infty)}$, which is the second moment saturated at value $J$, i.e.:

$$ \begin{aligned} M_2^{(J:\infty)}(t) &:= \sum_{i \le J-1} i^2 \cdot X_i^{(\infty)}(t) + J^2 \cdot X_{(J)}^{(\infty)}(t) \\ &= \sum_{i \ge 0} (i \wedge J)^2 \cdot X_i^{(\infty)}(t). \end{aligned} $$

Since $Z_\alpha^J(t)$ is uniformly upper-bounded, (B1) entails that $\sup_{\{s \le t\}} M_k^{(\infty)}(s) < \infty$ holds a.s. under $\mathbb{P}_x^{[\alpha],J}$, provided $x \in \mathcal{X}^k$. This concludes the proof of Proposition B.0.1. □

<figure><img src="image_15.png" /></figure>
**Acknowledgements** We greatly appreciated the constructive feedback from the anonymous reviewers that has helped to strengthen the overall quality of this article.

**Funding** Open access funding provided by Université d'Évry Val-d'Essonne.

**Data availability** No datasets were generated or analysed during the current study.

## Declarations

**Conflict of interest** The authors have no relevant financial or non-financial interests to disclose.

**Open Access** This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

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