Let $\omega \in \Omega$, $0 \le s < t < \infty$, $A \in \mathcal{F}_s$, $F, g \in \mathcal{C}_K^2(\mathbb{R}, \mathbb{R})$ be fixed.

**Step 2. Property (C.3) satisfied except on a $\mathbb{P}_\nu^{\text{FV}}$-null event $N_1(s, t, A, F, g) \in \mathcal{F}_{t^*}$.**
As $\mathcal{L}_{\text{FV}}F_g \in \mathcal{C}_b^2(\mathbb{R}, \mathbb{R})$, the random variable $M_t^{F_g} - M_s^{F_g}$ is bounded. Note that,

$$ 
\begin{align*}
\int_{\Omega} [M_t^{F_g}(\tilde{\omega}) - M_s^{F_g}(\tilde{\omega})] \mathbb{1}_A(\tilde{\omega}) \mathbb{P}_{\omega}(d\tilde{\omega}) &= \mathbb{E}_{\mathbb{Q}_{\omega}(d\tilde{\omega})} \left( [M_t^{F_g} - M_s^{F_g}] \circ \theta_{t^*}(\tilde{\omega}) \mathbb{1}_{\theta_{t^*}^{-1}A}(\tilde{\omega}) \right) \\
&= \mathbb{E}_{\mathbb{P}_\nu^{\text{FV}}} \left( [M_t^{F_g} - M_s^{F_g}] \circ \theta_{t^*} \mathbb{1}_{\theta_{t^*}^{-1}A} \middle| \mathcal{F}_{t^*} \right) (\omega) \\
&= \mathbb{E}_{\mathbb{P}_\nu^{\text{FV}}} \left[ \mathbb{E}_{\mathbb{P}_\nu^{\text{FV}}} \left( [M_t^{F_g} - M_s^{F_g}] \circ \theta_{t^*} \mathbb{1}_{\theta_{t^*}^{-1}A} \middle| \mathcal{F}_{t^*+s} \right) \middle| \mathcal{F}_{t^*} \right] (\omega) \\
&= \mathbb{E}_{\mathbb{P}_\nu^{\text{FV}}} \left[ \mathbb{1}_{\theta_{t^*}^{-1}A} \mathbb{E}_{\mathbb{P}_\nu^{\text{FV}}} \left( [M_t^{F_g} - M_s^{F_g}] \circ \theta_{t^*} \middle| \mathcal{F}_{t^*+s} \right) \middle| \mathcal{F}_{t^*} \right] (\omega) \\
&= 0,
\end{align*}
$$

where the last equality follows from martingale property (8). This chain of equalities shows that the random variable $\omega \mapsto \int_A [M_t^{F_g}(\tilde{\omega}) - M_s^{F_g}(\tilde{\omega})] \mathbb{P}_{\omega}(d\tilde{\omega})$ is null except on a $\mathbb{P}_\nu^{\text{FV}}$-null event $N_1(s, t, A, F, g) \in \mathcal{F}_{t^*}$ which depends on $s, t, A, F$ and $g$.

**Step 3. Property (C.3) satisfied except on a $\mathbb{P}_\nu^{\text{FV}}$-null event $N_2(s, t, F, g) \in \mathcal{F}_{t^*}$.** We consider a countable subcollection $\mathcal{E}$ of $\mathcal{F}_s$ which generates $\mathcal{F}_s$ [30, Definition 3.17 of Section 5.3 (p 306)] and a $\mathbb{P}_\nu$-null event $N_2(s, t, F, g) \in \mathcal{F}_{t^*}$ such that for $\omega \in \Omega \setminus N_2(s, t, F, g)$,

$$ \forall A \in \mathcal{E}, \quad \int_A [M_t^{F_g}(\tilde{\omega}) - M_s^{F_g}(\tilde{\omega})] \mathbb{P}_{\omega}(d\tilde{\omega}) = 0. $$

Therefore, the two measures

$$ v_{\omega}^{+}(A) := \int_{A} [M_{t}^{F_{g}} - M_{s}^{F_{g}}]^{+}(\tilde{\omega}) \mathbb{P}_{\omega}(d\tilde{\omega}) \quad \text{and} \quad v_{\omega}^{-}(A) := \int_{A} [M_{t}^{F_{g}} - M_{s}^{F_{g}}]^{-}(\tilde{\omega}) \mathbb{P}_{\omega}(d\tilde{\omega}), $$

coincide on $\mathcal{E}$, hence on $\mathcal{F}_s$. Therefore, for $\omega \in \Omega \setminus N_2(s, t, F, g)$, we have proved that for all $A \in \mathcal{F}_s$, $\mathbb{E}_{\mathbb{P}_{\omega}}(\mathbb{1}_A[M_t^{F_g} - M_s^{F_g}]) = 0$.

**Step 4. Property (C.3) satisfied except on a $\mathbb{P}_\nu^{\text{FV}}$-null event $N_3(F, g) \in \mathcal{F}_{t^*}$.** We may set now, the $\mathbb{P}_\nu^{\text{FV}}$-null event

$$ N_3(F, g) := \bigcup_{\substack{s, t \in \mathbb{Q} \\ 0 \le s < t < \infty}} N_2(s, t, F, g). $$

Due to the boundedness and continuity of $t \mapsto M_t^{F_g}$, it follows from the dominated convergence theorem that for $\omega \in \Omega \setminus N_3(F, g)$

$$ \forall s < t, \forall A \in \mathcal{F}_s, \quad \mathbb{E}_{\mathbb{P}_{\omega}}(\mathbb{1}_A[M_t^{F_g} - M_s^{F_g}]) = 0, $$

in other words, for all $\omega \in \Omega \setminus N_3(F, g)$, $(M_t^{F_g}(\tilde{\omega}))_{t \ge 0}$ is a $(\mathcal{F}_s, \mathbb{P}_{\omega}(d\tilde{\omega}))$-martingale.
**Step 5. Conclusion.** Now we define the $\mathbb{P}_\nu^{\text{FV}}$-null event

$$N_4 := \bigcup_{F,g \in \mathcal{B}} N_3(F,g)$$

From the Step 4, we have for all $s \le t$,

$$\forall \omega \in \Omega \setminus N_4, \quad \forall A \in \mathcal{F}_s, \quad \forall F, g \in \mathcal{B}, \quad \mathbb{E}_{\mathbb{P}_\omega} [\mathbb{1}_A (M_t^{F_g} - M_s^{F_g})] = 0.$$

From Step 0, for all $F, g \in \mathcal{C}_K^2(\mathbb{R}, \mathbb{R})$, there exist two sequences $(F_n)_{n \in \mathbb{N}}$, $(g_n)_{n \in \mathbb{N}} \in \mathcal{B}^\mathbb{N}$ such that

$$F_n \xrightarrow{\|\cdot\|_{W_0^{2,\infty}}} F, \quad g_n \xrightarrow{\|\cdot\|_{W_0^{2,\infty}}} g, \quad \text{and} \quad \mathcal{L}_{\text{FV}}(F_n)_{g_n} \xrightarrow{\|\cdot\|_{\infty}} \mathcal{A}F_g.$$

By the dominated convergence theorem, we deduce that for all $\omega \in \Omega$, $s \le t$, and $A \in \mathcal{F}_s$,

$$\mathbb{E}_{\mathbb{P}_\omega} [\mathbb{1}_A (M_t^{F_g} - M_s^{F_g})] = \lim_{n \to +\infty} \mathbb{E}_{\mathbb{P}_\omega} [\mathbb{1}_A (M_t^{(F_n)_{gn}} - M_s^{(F_n)_{gn}})] = 0.$$

which concludes the proof. $\square$

## C.2 Proof of Lemma 5.2

By abuse of notation, we note $h(\omega) = h(\omega)$. We want to prove that for all $0 \le s \le t$, for all $\mathcal{F}_s$-measurable bounded random variable $Z$,

$$\mathbb{E}_{\mathbb{P}_\nu^{\text{FV}}(d\tilde{\omega})} ([M_t(\tilde{\omega}) - M_s(\tilde{\omega})] Z(\tilde{\omega})) = 0.$$

Using [29, Definition 3.2 (iii)] of Section 1] we deduce that

$$\mathbb{E}_{\mathbb{P}_\nu^{\text{FV}}(d\tilde{\omega})} ([M_t(\tilde{\omega}) - M_s(\tilde{\omega})] Z(\tilde{\omega})) = \mathbb{E}_{\mathbb{P}_\nu(d\omega)} [\mathbb{E}_{\mathbb{Q}_\omega(d\tilde{\omega})} ([M_t(\tilde{\omega}) - M_s(\tilde{\omega})] Z(\tilde{\omega}))].$$

Thus, it is sufficient to prove that for $\mathbb{P}_\nu^{\text{FV}}$-almost every $\omega \in \Omega$, $(M_t(\tilde{\omega}))_{0 \le t \le T}$ is a $\mathbb{Q}_\omega(d\tilde{\omega})$-martingale and this is what we propose to establish in the rest of this proof.

For fixed $\omega$, the function $h(\omega) \in \mathcal{C}_b^2(\mathbb{R}, \mathbb{R})$ can be considered as deterministic. We deduce from Theorem C.1 that there exists a $\mathbb{P}_\nu^{\text{FV}}$-null event $N \in \mathcal{F}_{t^*}$ such that for all $\omega \in \Omega \setminus N$, $(M_t^{\text{id}}(h(\omega))(\tilde{\omega}))_{0 \le t \le T}$ is a $\mathbb{P}_\omega(d\tilde{\omega})$-martingale. We deduce from [30, Theorem 3.18 of Section 5.3 (p 307)] that $\mathbb{P}_\nu^{\text{FV}}$-almost every $\omega \in \Omega$,

$$\tilde{\omega}_{|_{[0,t^*]}} = \omega_{|_{[0,t^*]}}, \quad \mathbb{Q}_\omega(d\tilde{\omega})-\text{a.s.} \qquad (C.4)$$

This implies that, $\mathbb{Q}_\omega(d\tilde{\omega})$-almost surely,

$$\mathcal{M}_t(\tilde{\omega}) = M_t^{\text{id}}(h(\omega))(\tilde{\omega}) - M_{t \wedge t^*}^{\text{id}}(h(\omega))(\tilde{\omega}) = \begin{cases} M_{t-t^*}^{\text{id}}(h(\omega))(\theta_{t^*}(\tilde{\omega})) & \text{if } t > t^* \\ 0 & \text{if } t \le t^* \end{cases} \\ = M_{(t-t^*)^+}^{\text{id}}(h(\omega))(\theta_{t^*}(\tilde{\omega}))$$

where $(a)^+$ designates the non-negative part of $a \in \mathbb{R}$. Let $n \in \mathbb{N}^*$ and $0 \le s \le T$. To prove the martingale property for all $\mathcal{F}_s$-measurable bounded random variable $Z$, it is sufficient to prove it on elementary events. Then, we consider a random variable $Z$ of the form

$$Z(\omega) := \mathbb{1}_{\{\omega_{t_1} \in \Gamma_1, \dots, \omega_{t_n} \in \Gamma_n\}}$$
where for all $i \in \{1, \dots, n\}$, $t_i \le s$ and $\Gamma_i \subset \mathcal{M}_1(\mathbb{R})$ measurable. We define

$$ \tilde{Z}(\omega, \tilde{\omega}) := \mathbb{1}_{\{\omega_{t_i} \in \Gamma_i, \forall i \in \{1, \dots, n\} \text{ such that } t_i \le t^*\}} \mathbb{1}_{\{\tilde{\omega}_{t_j} \in \Gamma_j, \forall j \in \{1, \dots, n\} \text{ such that } t_j > t^*\}} $$

By (C.4), $\tilde{Z}(\omega, \tilde{\omega}) = Z(\omega)$, $\mathbb{Q}_{\omega}(d\tilde{\omega})$ - a.s. Therefore, for $\mathbb{P}_{\nu}^{\text{FV}}$-almost every $\omega \in \Omega$,

$$ \begin{align*} & \mathbb{E}_{\mathbb{Q}_{\omega}} ([M_t - M_s] Z) \\ &= \mathbb{E}_{\mathbb{Q}_{\omega}(d\tilde{\omega})} \left( [M_{(t-t^*)}^{\text{id}}(h(\omega))(\theta_{t^*}(\tilde{\omega})) - M_{(s-t^*)}^{\text{id}}(h(\omega))(\theta_{t^*}(\tilde{\omega}))] \tilde{Z}(\omega, \tilde{\omega}) \right) \\ &= \mathbb{1}_{\{\omega_{t_i} \in \Gamma_i, \forall i \in \{1, \dots, n\} \text{ such that } t_i \le t^*\}} \times \\ & \quad \mathbb{E}_{\mathbb{Q}_{\omega}(d\tilde{\omega})} \left( [M_{t-t^*}^{\text{id}}(h(\omega))(\theta_{t^*}(\tilde{\omega})) - M_{s-t^*}^{\text{id}}(h(\omega))(\theta_{t^*}(\tilde{\omega}))] \right. \\ & \qquad \left. \times \mathbb{1}_{\{[\theta_{t^*}(\tilde{\omega})]_{t_j-t^*} \in \Gamma_j, \forall j \in \{1, \dots, n\} \text{ such that } t_j > t^*\}} \right) \\ & \stackrel{(C.2)}{=} \mathbb{1}_{\{\omega_{t_i} \in \Gamma_i, \forall i \in \{1, \dots, n\} \text{ such that } t_i \le t^*\}} \mathbb{E}_{\mathbb{P}_{\omega}(d\tilde{\omega})} \left( [M_{t-t^*}^{\text{id}}(h(\omega))(\hat{\omega}) - M_{s-t^*}^{\text{id}}(h(\omega))(\hat{\omega})] \right. \\ & \qquad \left. \times \mathbb{1}_{\{\hat{\omega}_{t_j-t^*} \in \Gamma_j, \forall j \in \{1, \dots, n\} \text{ such that } t_j > t^*\}} \right) \\ &= 0, \end{align*} $$

using that $(M_{t-t^*}^{\text{id}}(h(\omega))(\hat{\omega}))_{t^* \le t \le T+t^*}$ is a $\mathbb{P}_{\omega}(d\hat{\omega})$-martingale if the internal indicator is non zero. Thus, for $\mathbb{P}_{\nu}^{\text{FV}}$-almost every $\omega \in \Omega$, $(M_t(\tilde{\omega}))_{0 \le t \le T}$ is a $\mathbb{Q}_{\omega}(d\tilde{\omega})$-martingale which completes the first part of this proof. In similar way, we can prove

$$ M_t^{\text{id}^2}(h(\tilde{\omega}))(\tilde{\omega}) - M_{t \wedge t^*}^{\text{id}^2}(h(\tilde{\omega}))(\tilde{\omega}) $$

is a $\mathbb{P}_{\nu}^{\text{FV}}$-martingale. Applying Itô's formula to compute $\langle h(\tilde{\omega}_t), \tilde{\omega}_t \rangle^2$ and comparing it to the previous result, we obtain the announced result. $\square$

**Funding.** This work was partially funded by the Chair "Modélisation Mathématique et Biodiversité" of VEOLIA-Ecole Polytechnique-MNHN-F.X. N.C. was partially funded by the European Union (ERC, SINGER, 101054787). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them.

**Acknowledgments.** We thank the two anonymous referees for their useful comments.

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