SlowGuess/NeurIPS_2023_OCR huggingface dataset:
recall that  $\mathcal{A} = A\oplus A + \sum_{k = 1}^{m}G_{k}\otimes G_{k}\in \mathbb{R}^{d^{2}\times d^{2}}$  and  $v = \operatorname {vec}(x_0x_0^\top)\in \mathbb{R}^{d^2}$ . According to the proof A.7 of Theorem 3.5, we obtain that the set of vectors  $\{v,\mathcal{A}v,\dots ,\mathcal{A}^{(d^2 +d - 2) / 2}\}$  are linearly dependent. Because all of these vectors are transferred from vectorizing  $d\times d$  symmetric matrices, thus each of these vectors has only  $(d^2 +d) / 2$  distinct elements and the repetitive elements appear in the same positions in all vectors. Hence, abuse notation a little bit, we can focus on checking those distinct elements in each vector, that is
$$
\left\{\underline {{v}} \mid \underline {{A}} v \right| \dots \left. \mid \underline {{A}} ^ {(d ^ {2} + d - 2) / 2} v \right\}, \tag {44}
$$
where  $\underline{v} \in \mathbb{R}^{(d^2 + d) / 2}$  denotes the vector of deleting the repetitive elements of  $v$ . Since the set of vectors  $\{v, \mathcal{A}v, \ldots, \mathcal{A}^{(d^2 + d - 2) / 2}\}$  are linearly dependent, the set of vectors  $\{\underline{v} | \underline{\mathcal{A}} v| \ldots | \underline{\mathcal{A}}^{(d^2 + d - 2) / 2}v\}$  are linearly dependent, that is
$$
\det \left(\left[ \underline {{v}} \mid \underline {{A}} v \right] \dots \left| \underline {{A}} ^ {\left(d ^ {2} + d - 2\right) / 2} v \right]\right) = 0. \tag {45}
$$
Each entry of  $v$  can be written as a non-zero polynomial function of entries of  $x_0$  since  $v = \operatorname{vec}(x_0 x_0^\top)$ . Each entry of  $\mathcal{A}$  can be written as a non-zero polynomial function of entries of  $A$  and  $G_k$  with  $k = 1, \ldots, m$ , since  $\mathcal{A} = A \oplus A + \sum_{k=1}^{m} G_k \otimes G_k \in \mathbb{R}^{d^2 \times d^2}$ . Hence, the left side of Equation (45) can be expressed as some universal polynomial of the entries of  $x_0$ ,  $A$  and  $G_k$ 's, denotes
$$
p (x _ {0}, A, \{G _ {k} \} _ {k = 1} ^ {m}) = p (x _ {0 1}, \dots , x _ {0 d}, a _ {1 1}, \dots , a _ {d d}, G _ {1, 1 1}, \dots , G _ {1, d d}, \dots , G _ {m, 1 1}, \dots , G _ {m, d d}),
$$
where  $G_{k,ij}$  denotes the  $ij$ -th entry of matrix  $G_{k}$ . Therefore, one concludes that
$$
p (x _ {0}, A, \{G _ {k} \} _ {k = 1} ^ {m}) = 0.
$$
Thus,  $S_{2}$  can be expressed as
$$
S _ {2} := \left\{\left(x _ {0}, A, \left\{G _ {k} \right\} _ {k = 1} ^ {m}\right) \in \mathbb {R} ^ {d + (m + 1) d ^ {2}}: p \left(x _ {0}, A, \left\{G _ {k} \right\} _ {k = 1} ^ {m}\right) = 0 \right\}.
$$
Similar to the calculation of  $p(x_0, A)$  in Equation (43),  $p(x_0, A, \{G_k\}_{k=1}^m)$  can be expressed as a non-zero polynomial function of entries of  $v$  and  $A$ , thus it can also be expressed as a non-zero polynomial function of entries of  $x_0$ ,  $A$  and  $G_k$ 's. Therefore, by Lemma B.1,  $S_2$  has Lebesgue measure zero in  $\mathbb{R}^{d+(m+1)d^2}$ .
We know that  $S \subseteq S_1 \cup S_2$ , let  $\lambda$  be Lebesgue measure on  $\mathbb{R}^{d + (m + 1)d^2}$ , then one has
$$
\lambda (S) \leqslant \lambda (S _ {1}) + \lambda (S _ {2}) = 0.
$$
Thus  $S$  has Lebesgue measure zero in  $\mathbb{R}^{d + (m + 1)d^2}$ .
B.2 The identifiability condition stated in Theorem 3.4 is generic
We will show that the identifiability condition stated in Theorem 3.4 is generic.
Proposition B.2. Let
$$
S := \left\{\left(x _ {0}, A, G\right) \in \mathbb {R} ^ {d + d ^ {2} + d m}: \text {c o n d i t i o n (1 4) i n T h e o r e m 3 . 4 i s v i o l a t e d} \right\},
$$
then  $S$  has Lebesgue measure zero in  $\mathbb{R}^{d + d^2 + dm}$ .
Proof. Suppose  $(x_0, A, G) \in S$ , then  $(x_0, A, G)$  satisfies
$$
\operatorname {r a n k} \left( \right.\left[ \right. x _ {0} \mid A x _ {0} \left. \right| \dots \mid A ^ {d - 1} x _ {0} \mid H _ {. 1} \mid A H _ {. 1} \mid \dots \mid A ^ {d - 1} H _ {. 1} \mid \dots \mid H _ {. d} \mid A H _ {. d} \mid \dots \mid A ^ {d - 1} H _ {. d} \left. \right]\left. \right) <   d,
$$
recall that  $H \coloneqq GG^{T}$ , and  $H_{,j}$  stands for the  $j$ -th column vector of matrix  $H$ , for all  $j = 1, \dots, d$ .
$$
S ^ {\prime} := \left\{\left(x _ {0}, A, G\right) \in \mathbb {R} ^ {d + d ^ {2} + d m}: \operatorname {r a n k} \left(\left[ x _ {0} \mid A x _ {0} \right| \dots \left| A ^ {d - 1} x _ {0} \right]\right) <   d \right\},
$$
one observes that  $S \subseteq S'$ . According to the proof of Proposition B.1, it is readily checked that  $S'$  has Lebesgue measure zero in  $\mathbb{R}^{d + d^2 + dm}$ . Thus,  $S$  has Lebesgue measure zero in  $\mathbb{R}^{d + d^2 + dm}$ .
C Simulation settings
We present the true underlying system parameters along with the initial states of the SDEs employed in the simulation experiments. We randomly generate the true system parameters that satisfy or violate the corresponding identifiability conditions.
For the SDE (1):
1. identifiable case: satisfy condition (14) stated in Theorem 3.4:
$$
x _ {0} ^ {\mathrm {i d}} = \left[ \begin{array}{c} 1. 8 7 \\ - 0. 9 8 \end{array} \right], A ^ {\mathrm {i d}} = \left[ \begin{array}{c c} 1. 7 6 & - 0. 1 \\ 0. 9 8 & 0 \end{array} \right], G ^ {\mathrm {i d}} = \left[ \begin{array}{c c} - 0. 1 1 & - 0. 1 4 \\ - 0. 2 9 & - 0. 2 2 \end{array} \right];
$$
2. unidentifiable case: violate condition (14):
$$
x _ {0} ^ {\mathrm {u n}} = \left[ \begin{array}{c} 1 \\ - 1 \end{array} \right], A ^ {\mathrm {u n}} = \left[ \begin{array}{c c} 1 & 2 \\ 1 & 0 \end{array} \right], G ^ {\mathrm {u n}} = \left[ \begin{array}{c c} 0. 1 1 & 0. 2 2 \\ - 0. 1 1 & - 0. 2 2 \end{array} \right].
$$
For the SDE (2):
1. identifiable case: satisfy both A1 and A2 stated in Theorem 3.5:
$$
x _ {0} ^ {\mathrm {i d}} = \left[ \begin{array}{c} 1. 8 7 \\ - 0. 9 8 \end{array} \right], A ^ {\mathrm {i d}} = \left[ \begin{array}{c c} 1. 7 6 & - 0. 1 \\ 0. 9 8 & 0 \end{array} \right], G _ {1} ^ {\mathrm {i d}} = \left[ \begin{array}{c c} - 0. 1 1 & - 0. 1 4 \\ - 0. 2 9 & - 0. 2 2 \end{array} \right], G _ {2} ^ {\mathrm {i d}} = \left[ \begin{array}{c c} - 0. 1 7 & 0. 5 9 \\ 0. 8 1 & 0. 1 8 \end{array} \right];
$$
2. unidentifiable case1: violate A1 satisfy A2:
$$
x _ {0} ^ {\mathrm {u n - A l}} = \left[ \begin{array}{c} 1 \\ 1 \end{array} \right], \quad A ^ {\mathrm {u n - A l}} = \left[ \begin{array}{c c} 2 & 1 \\ 3 & 0 \end{array} \right], \quad G _ {1} ^ {\mathrm {u n - A l}} = \left[ \begin{array}{c c} - 0. 1 1 & - 0. 1 4 \\ - 0. 2 9 & - 0. 2 2 \end{array} \right], \quad G _ {2} ^ {\mathrm {u n - A l}} = \left[ \begin{array}{c c} - 0. 1 7 & 0. 5 9 \\ 0. 8 1 & 0. 1 8 \end{array} \right];
$$
3. unidentifiable case2: satisfy A1 violate A2:
$$
x _ {0} ^ {\mathrm {u n - A 2}} = \left[ \begin{array}{c} 1 \\ - 1 \end{array} \right], A ^ {\mathrm {u n - A 2}} = \left[ \begin{array}{c c} 1 & - 2 \\ - 1 & 0 \end{array} \right], G _ {1} ^ {\mathrm {u n - A 2}} = \left[ \begin{array}{c c} - 0. 3 & 0. 4 \\ - 0. 7 & 0. 2 \end{array} \right], G _ {2} ^ {\mathrm {u n - A 2}} = \left[ \begin{array}{c c} 0. 8 & 0. 2 \\ - 0. 2 & - 0. 4 \end{array} \right].
$$
We have discussed in Section 4 that we use MLE method to estimate the system parameters from discrete observations sampled from the corresponding SDEs. Specifically, the negative log-likelihood function was minimized using the 'scipy opti meize minimize' library in Python.
For all of our experiments, we initialized the parameter values as the true parameters plus 2. In the case of the SDE (1), we utilized the 'trust-constr' method with the hyper-parameter  $\mathrm{gtol} = 1\mathrm{e - }3$  and  $\mathrm{xtol} = 1\mathrm{e - }3$ . On the other hand, for the SDE (2), we applied the BFGS' method and set the hyper-parameter  $\mathrm{gtol} = 1\mathrm{e - }2$ . The selection of the optimization method and the corresponding hyper-parameters was determined through a series of experiments aimed at identifying the most suitable configuration.
D Examples for reliable causal inference for linear SDEs
D.1 Example for reliable causal inference for the SDE (1)
This example is inspired by [17, Example 5.4]. Recall that the SDE (1) is defined as
$$
d X _ {t} = A X _ {t} d t + G d W _ {t}, \quad X _ {0} = x _ {0},
$$
where  $0 \leqslant t < \infty$ ,  $A \in \mathbb{R}^{d \times d}$  and  $G \in \mathbb{R}^{d \times m}$  are constant matrices,  $W$  is an  $m$ -dimensional standard Brownian motion. Let  $X(t; x_0, A, G)$  denote the solution to the SDE (1). Let  $\tilde{A} \in \mathbb{R}^{d \times d}$  and  $\tilde{G} \in \mathbb{R}^{d \times m}$  define the following SDE:
$$
d \tilde {X} _ {t} = \tilde {A} \tilde {X} _ {t} d t + \tilde {G} d W _ {t}, \quad \tilde {X} _ {0} = x _ {0},
$$
$$
X (\because ; x _ {0}, A, G) \stackrel {{\mathrm {d}}} {{=}} \tilde {X} (\because ; x _ {0}, \tilde {A}, \tilde {G}).
$$
Then under our proposed identifiability condition stated in Theorem 3.4, we have shown that the generator of the SDE (1) is identifiable, i.e.,  $(A,GG^{\top}) = (\tilde{A},\tilde{G}\tilde{G}^{\top})$ . Till now, we have shown that under our proposed identifiability conditions, the observational distribution  $\xrightarrow{\text{identity}}$  the generator of the observational SDE. Then we will show that the post-intervention distribution is also identifiable. For notational simplicity, we consider intervention on the first coordinate, making the intervention  $X_{t}^{1} = \xi$  and  $\tilde{X}_t^1 = \xi$  for  $0\leqslant t < \infty$ . It will suffice to show equality of the distributions of the non-intervened coordinates (i.e.,  $X_{t}^{(-1)}$  and  $\tilde{X}_t^{(-1)}$ , note the superscripts do not denote reciprocals, but denote the  $(d - 1)$ -coordinates without the first coordinate). Express the matrices of  $A$  and  $G$  in blocks
$$
A = \left[ \begin{array}{c c} A _ {1 1} & A _ {1 2} \\ A _ {2 1} & A _ {2 2} \end{array} \right], \quad G = \left[ \begin{array}{c} G _ {1} \\ G _ {2} \end{array} \right],
$$
where  $A_{11} \in \mathbb{R}^{1 \times 1}$ ,  $A_{22} \in \mathbb{R}^{(d-1) \times (d-1)}$ ,  $G_1 \in \mathbb{R}^{1 \times m}$  and  $G_2 \in \mathbb{R}^{(d-1) \times m}$ . Also, consider corresponding expressions of matrices  $\tilde{A}$  and  $\tilde{G}$ . By making intervention  $X_t^1 = \xi$ , one obtains the post-intervention process of the first SDE satisfies:
$$
d X _ {t} ^ {(- 1)} = \left(A _ {2 1} \xi + A _ {2 2} X _ {t} ^ {(- 1)}\right) d t + G _ {2} d W _ {t}, \quad X _ {0} ^ {(- 1)} = x _ {0} ^ {(- 1)},
$$
which is a multivariate Ornstein-Uhlenbeck process, according to [59, Corollary 1], this process is a Gaussian process, assuming  $A_{22}$  is invertible, then the mean vector can be described as
$$
E \left[ X _ {t} ^ {(- 1)} \right] = e ^ {A _ {2 2} t} x _ {0} ^ {(- 1)} - \left(I - e ^ {A _ {2 2} t}\right) A _ {2 2} ^ {- 1} A _ {2 1} \xi ,
$$
and based on [59, Theorem 2], the cross-covariance can be described as
$$
\begin{array}{l} V \big (X _ {t + h} ^ {(- 1)}, X _ {t} ^ {(- 1)} \big): = \mathbb {E} \big \{\big (X _ {t + h} ^ {(- 1)} - \mathbb {E} [ X _ {t + h} ^ {(- 1)} ] \big) \big (X _ {t} ^ {(- 1)} - \mathbb {E} [ X _ {t} ^ {(- 1)} ] \big) ^ {\top} \big \} \\ = \int_ {0} ^ {t} e ^ {A _ {2 2} (t + h - s)} G _ {2} G _ {2} ^ {\top} e ^ {A _ {2 2} ^ {\top} (t - s)} d s. \\ \end{array}
$$
Similarly, one can obtain that the mean vector and cross-covariance of the distribution of the post-intervention process of the second SDE by making intervention  $\hat{X}_t^1 = \xi$  satisfy:
$$
E \left[ \tilde {X} _ {t} ^ {(- 1)} \right] = e ^ {\tilde {A} _ {2 2} t} x _ {0} ^ {(- 1)} - \left(I - e ^ {\tilde {A} _ {2 2} t}\right) \tilde {A} _ {2 2} ^ {- 1} \tilde {A} _ {2 1} \xi ,
$$
$$
\begin{array}{l} V (\tilde {X} _ {t + h} ^ {(- 1)}, \tilde {X} _ {t} ^ {(- 1)}): = \mathbb {E} \{(\tilde {X} _ {t + h} ^ {(- 1)} - \mathbb {E} [ \tilde {X} _ {t + h} ^ {(- 1)} ]) (\tilde {X} _ {t} ^ {(- 1)} - \mathbb {E} [ \tilde {X} _ {t} ^ {(- 1)} ]) ^ {\top} \} \\ = \int_ {0} ^ {t} e ^ {\tilde {A} _ {2 2} (t + h - s)} \tilde {G} _ {2} \tilde {G} _ {2} ^ {\top} e ^ {\tilde {A} _ {2 2} ^ {\top} (t - s)} d s \\ \end{array}
$$
Then we will show that  $E[X_{t}^{(-1)}] = E[\tilde{X}_{t}^{(-1)}]$ , and  $V(X_{t + h}^{(-1)}, X_{t}^{(-1)}) = V(\tilde{X}_{t + h}^{(-1)}, \tilde{X}_{t}^{(-1)})$  for all  $0 \leqslant t, h < \infty$ . Recall that we have shown  $(A, GG^{\top}) = (\tilde{A}, \tilde{G}\tilde{G}^{\top})$ , thus,  $A_{22} = \tilde{A}_{22}$  and  $A_{21} = \tilde{A}_{21}$ , then it is readily checked that  $E[X_{t}^{(-1)}] = E[\tilde{X}_{t}^{(-1)}]$  for all  $0 \leqslant t < \infty$ .
$$
G G ^ {\top} = \left[ \begin{array}{c c} G _ {1} G _ {1} ^ {\top} & G _ {1} G _ {2} ^ {\top} \\ G _ {2} G _ {1} ^ {\top} & G _ {2} G _ {2} ^ {\top} \end{array} \right] = \tilde {G} \tilde {G} ^ {\top},
$$
thus,  $G_2G_2^\top = \tilde{G}_2\tilde{G}_2^\top$ , then it is readily checked that  $V(X_{t + h}^{(-1)},X_t^{(-1)}) = V(\tilde{X}_{t + h}^{(-1)},\tilde{X}_t^{(-1)})$  for all  $0\leqslant t,h < \infty$ . Since both of these two post-intervention processes are Gaussian processes, according to Lemma 3.2, the distributions of these two post-intervention processes are the same. That is, the post-intervention distribution is identifiable.
D.2 Example for reliable causal inference for the SDE (2)
Recall that the SDE (2) is defined as
$$
d X _ {t} = A X _ {t} d t + \sum_ {k = 1} ^ {m} G _ {k} X _ {t} d W _ {k, t}, \quad X _ {0} = x _ {0},
$$
where  $0 \leqslant t < \infty$ ,  $A, G_k \in \mathbb{R}^{d \times d}$  for  $k = 1, \ldots, m$  are some constant matrices,  $W := \{W_t = [W_{1,t}, \ldots, W_{m,t}]^\top : 0 \leqslant t < \infty\}$  is an  $m$ -dimensional standard Brownian motion. Let  $X(t; x_0, A, \{G_k\}_{k=1}^m)$  denote the solution to the SDE (2). Let  $\tilde{A}, \tilde{G}_k \in \mathbb{R}^{d \times d}$  for  $k = 1, \ldots, m$  define the following SDE:
$$
d \tilde {X} _ {t} = \tilde {A} \tilde {X} _ {t} d t + \sum_ {k = 1} ^ {m} \tilde {G} _ {k} \tilde {X} _ {t} d W _ {k, t}, \quad \tilde {X} _ {0} = x _ {0},
$$
$$
X (\because ; x _ {0}, A, \{G _ {k} \} _ {k = 1} ^ {m}) \stackrel {{\mathrm {d}}} {=} \tilde {X} (\because ; x _ {0}, \tilde {A}, \{\tilde {G} _ {k} \} _ {k = 1} ^ {m}).
$$
Then under our proposed identifiability condition stated in Theorem 3.5, we have shown that the generator of the SDE (2) is identifiable, i.e.,  $(A,\sum_{k = 1}^{m}G_{k}xx^{\top}G_{k}^{\top}) = (\tilde{A},\sum_{k = 1}^{m}\tilde{G}_{k}xx^{\top}\tilde{G}_{k}^{\top})$  for all  $x\in \mathbb{R}^d$ . Till now, we have shown that under our proposed identifiability conditions, the observational distribution  $\xrightarrow{\mathrm{identity}}$  the generator of the observational SDE. Then we aim to show that the post-intervention distribution is also identifiable. For notational simplicity, we consider intervention on the first coordinate, making the intervention  $X_{t}^{1} = \xi$  and  $\tilde{X}_t^1 = \xi$  for  $0\leqslant t < \infty$ . It will suffice to show equality of the distributions of the non-intervened coordinates (i.e.,  $X_{t}^{(-1)}$  and  $\tilde{X}_t^{(-1)})$ . Express the matrices of  $A$  and  $G_{k}$  for  $k = 1,\dots ,m$  in blocks
$$
A = \left[ \begin{array}{c c} A _ {1 1} & A _ {1 2} \\ A _ {2 1} & A _ {2 2} \end{array} \right], G _ {k} = \left[ \begin{array}{c c} G _ {k, 1 1} & G _ {k, 1 2} \\ G _ {k, 2 1} & G _ {k, 2 2} \end{array} \right],
$$
where  $A_{11}, G_{k,11} \in \mathbb{R}^{1 \times 1}$ ,  $A_{22}, G_{k,22} \in \mathbb{R}^{(d-1) \times (d-1)}$ . Also consider corresponding expressions of matrices  $\tilde{A}$  and  $\tilde{G}_k$  for  $k = 1, \ldots, m$ . By making intervention  $X_t^1 = \xi$ , one obtains the post-intervention process of the first SDE satisfies:
$$
d X _ {t} ^ {(- 1)} = (A _ {2 1} \xi + A _ {2 2} X _ {t} ^ {(- 1)}) d t + \sum_ {k = 1} ^ {m} (G _ {k, 2 1} \xi + G _ {k, 2 2} X _ {t} ^ {(- 1)}) d W _ {k, t}, \quad X _ {0} ^ {(- 1)} = x _ {0} ^ {(- 1)}.
$$
Since this post-intervention process is not a Gaussian process, one cannot explicitly show that the post-intervention distribution is identifiable. Instead, we check the surrogate of the post-intervention distribution, that is the first- and second-order moments of the post-intervention process  $X_{t}^{(-1)}$ . Which denote as  $m(t)^{(-1)} = \mathbb{E}[X_t^{(-1)}]$  and  $P(t)^{(-1)} = \mathbb{E}[X_t^{(-1)}(X_t^{(-1)})^\top ]$  respectively. Then  $m(t)^{(-1)}$  and  $P(t)^{(-1)}$  satisfy the following ODE systems:
$$
\frac {d m (t) ^ {(- 1)}}{d t} = A _ {2 1} \xi + A _ {2 2} m (t) ^ {(- 1)}, \quad m (0) ^ {- 1} = x _ {0} ^ {(- 1)},
$$
$$
\begin{array}{l} \frac {d P (t) ^ {(- 1)}}{d t} = m (t) ^ {(- 1)} \xi^ {\top} A _ {2 1} ^ {\top} + A _ {2 1} \xi (m (t) ^ {(- 1)}) ^ {\top} + P (t) ^ {(- 1)} A _ {2 2} ^ {\top} + A _ {2 2} P (t) ^ {(- 1)} \\ + \sum_ {k = 1} ^ {m} \left(G _ {k, 2 1} \xi \xi^ {\top} G _ {k, 2 1} ^ {\top} + G _ {k, 2 2} m (t) ^ {(- 1)} \xi^ {\top} G _ {k, 2 1} ^ {\top} + G _ {k, 2 1} \xi (m (t) ^ {(- 1)}) ^ {\top} G _ {k, 2 2} ^ {\top} \right. \tag {46} \\ + G _ {k, 2 2} P (t) ^ {(- 1)} G _ {k, 2 2} ^ {\top}), \quad P (0) ^ {(- 1)} = x _ {0} ^ {(- 1)} \left(x _ {0} ^ {(- 1)}\right) ^ {\top}. \\ \end{array}
$$
Similarly, one can obtain the ODE systems describing the  $\tilde{m}(t)^{(-1)}$  and  $\tilde{P}(t)^{(-1)}$ . Then we will show that  $m(t)^{(-1)} = \tilde{m}(t)^{(-1)}$  and  $P(t)^{(-1)} = \tilde{P}(t)^{(-1)}$  for all  $0 \leqslant t < \infty$ . Recall that we have
shown  $A = \tilde{A}$ , thus  $A_{21} = \tilde{A}_{21}$  and  $A_{22} = \tilde{A}_{22}$ , then it is readily checked that  $m(t)^{(-1)} = \tilde{m}(t)^{(-1)}$  for all  $0 \leqslant t < \infty$ .
In the proof of Theorem 3.5, we have shown that
$$
\sum_ {k = 1} ^ {m} G _ {k} P (t) G _ {k} ^ {\top} = \sum_ {k = 1} ^ {m} \tilde {G} _ {k} P (t) \tilde {G} _ {k} ^ {\top}
$$
for all  $0 \leqslant t < \infty$ , where  $P(t) = \mathbb{E}[X_t X_t^\top]$ . Simple calculation shows that
$$
\sum_ {k = 1} ^ {m} G _ {k} P (t) G _ {k} ^ {\top} = \sum_ {k = 1} ^ {m} \left(\left[ \begin{array}{c c} G _ {k, 1 1} & G _ {k, 1 2} \\ G _ {k, 2 1} & G _ {k, 2 2} \end{array} \right] \left[ \begin{array}{c c} \xi \xi^ {\top} & \xi (m (t) ^ {(- 1)}) ^ {\top} \\ m (t) ^ {(- 1)} \xi^ {\top} & P (t) ^ {(- 1)} \end{array} \right] \left[ \begin{array}{c c} G _ {k, 1 1} ^ {\top} & G _ {k, 2 1} ^ {\top} \\ G _ {k, 1 2} ^ {\top} & G _ {k, 2 2} ^ {\top} \end{array} \right]\right),
$$
Then one can get that the  $(2,2)$ -th block entry of the matrix  $\sum_{k=1}^{m} G_k P(t) G_k^\top$  is the same as the  $\sum_{k=1}^{m} (\ldots)$  part in the ODE corresponds to  $P(t)^{(-1)}$  (i.e., Equation (46)), since  $\sum_{k=1}^{m} G_k P(t) G_k^\top = \sum_{k=1}^{m} \tilde{G}_k P(t) \tilde{G}_k^\top$ , then the  $\sum_{k=1}^{m} (\ldots)$  part in the ODEs correspond to both  $P(t)^{(-1)}$  and  $\tilde{P}(t)^{(-1)}$  are the same. Thus, it is readily checked that  $P(t)^{(-1)} = \tilde{P}(t)^{(-1)}$  for all  $0 \leqslant t < \infty$ .
Though we cannot explicitly show that the post-intervention distribution is identifiable, showing that the first- and second-order moments of the post-intervention process is identifiable can indicate the identification of the post-intervention distribution to a considerable extent.
E Conditions for identifying the generator of a linear SDE with multiplicative noise when its explicit solution is available
Proposition E.1. Let  $x_0 \in \mathbb{R}^d$  be fixed. The generator of the SDE (2) is identifiable from  $x_0$  if the following conditions are satisfied:
C1 rank  $\left([x_0|Ax_0|\ldots |A^{d - 1}x_0|H_{\cdot 1}|AH_{\cdot 1}|\ldots |A^{d - 1}H_{\cdot 1}|\ldots |H_{\cdot d}|AH_{\cdot d}|\ldots |A^{d - 1}H_{\cdot d}]\right) = d$
C2  $\mathrm{rank}([v|\mathcal{A}v|\ldots |\mathcal{A}^{(d^2 +d - 2) / 2}v]) = (d^2 +d) / 2,$
C3  $AG_{k} = G_{k}A$  and  $G_{k}G_{l} = G_{l}G_{k}$  for all  $k,l = 1,\ldots ,m$
where  $H \coloneqq \sum_{k=1}^{m} G_k x_0 x_0^\top G_k^\top$ , and  $H_{j}$  stands for the  $j$ -th column vector of matrix  $H$ , for all  $j = 1, \dots, d$ . And  $\mathcal{A} = A \oplus A + \sum_{k=1}^{m} G_k \otimes G_k \in \mathbb{R}^{d^2 \times d^2}$ ,  $\oplus$  denotes Kronecker sum and  $\otimes$  denotes Kronecker product,  $v$  is a  $d^2$ -dimensional vector defined by  $v \coloneqq \operatorname{vec}(x_0 x_0^\top)$ , where  $\operatorname{vec}(M)$  denotes the vectorization of matrix  $M$ .
Proof. Let  $\tilde{A},\tilde{G}_k\in \mathbb{R}^{d\times d}$  and  $\tilde{A}\tilde{G}_k = \tilde{G}_k\tilde{A}$ ,  $\tilde{G}_k\tilde{G}_l = \tilde{G}_l\tilde{G}_k$  for all  $k,l = 1,\ldots ,m$ , such that  $X(\cdot ;x_0,A,\{G_k\}_{k = 1}^m)\stackrel {\mathrm{d}}{=}X(\cdot ;x_0,\tilde{A},\{\tilde{G}_k\}_{k = 1}^m)$ , we denote as  $X\stackrel {\mathrm{d}}{=}\tilde{X}$ , we will show that under our identifiability condition, for all  $x\in \mathbb{R}^d$ ,  $(A,\sum_{k = 1}^{m}G_{k}xx^{\top}G_{k}^{\top}) = (\tilde{A},\sum_{k = 1}^{m}\tilde{G}_{k}xx^{\top}\tilde{G}_{k}^{\top})$ . By applying the same notations used in the proof of Theorem 3.5, in the following, we denote  $A_{1}\coloneqq A$ ,  $A_{2}\coloneqq \tilde{A}$ ,  $G_{1,k}\coloneqq G_{k}$  and  $G_{2,k}\coloneqq \tilde{G}_k$ , and denote  $X\stackrel {\mathrm{d}}{=}X$  as  $X^{1}\stackrel {\mathrm{d}}{=}X^{2}$ .
We first show that  $H_{1} = H_{2}$  ( $H_{i} \coloneqq \sum_{k=1}^{m} G_{i,k} x_{0} x_{0}^{\top} G_{i,k}^{\top}$ ). Indeed, since  $X^{1}, X^{2}$  have the same distribution, one has
$$
\mathbb {E} [ f \left(X _ {t} ^ {1}\right) ] = \mathbb {E} [ f \left(X _ {t} ^ {2}\right) ] \tag {47}
$$
for all  $0 \leqslant t < \infty$  and  $f \in C^{\infty}(\mathbb{R}^{d})$ . By differentiating (47) at  $t = 0$ , one finds that
$$
\left(\mathcal {L} _ {1} f\right) \left(x _ {0}\right) = \left(\mathcal {L} _ {2} f\right) \left(x _ {0}\right), \tag {48}
$$
$$
\left(\mathcal {L} _ {i} f\right) \left(x _ {0}\right) = \sum_ {k = 1} ^ {d} \sum_ {l = 1} ^ {d} \left(A _ {i}\right) _ {k l} x _ {0 l} \frac {\partial f}{\partial x _ {k}} \left(x _ {0}\right) + \frac {1}{2} \sum_ {k, l = 1} ^ {d} \left(H _ {i}\right) _ {k l} \frac {\partial^ {2} f}{\partial x _ {k} \partial x _ {l}} \left(x _ {0}\right),
$$
where  $(M)_{kl}$  denotes the  $kl$ -entry of matrix  $M$ , and  $x_{0l}$  is the  $l$ -th component of  $x_0$ . Since (48) is true for all  $f$ , by taking
$$
(H _ {1}) _ {p q} = (H _ {2}) _ {p q},
$$
for all  $p, q = 1, \ldots, d$ . As a result,  $H_{1} = H_{2}$ . Let us call this matrix  $H$ . That is
$$
H := H _ {1} = \sum_ {k = 1} ^ {m} G _ {1, k} x _ {0} x _ {0} ^ {\top} G _ {1, k} ^ {\top} = \sum_ {k = 1} ^ {m} G _ {2, k} x _ {0} x _ {0} ^ {\top} G _ {2, k} ^ {\top} = H _ {2}.
$$
$$
A _ {1} A ^ {j - 1} x _ {0} = A _ {2} A ^ {j - 1} x _ {0} \quad \text {f o r a l l} j = 1, 2, \dots , \tag {49}
$$
next, we will derive the relationship between  $A_{i}$  and  $H$ . Under condition C3, the SDE system (2) has an explicit solution (cf. [25]):
$$
X _ {t} := X \left(t; x _ {0}, A, \left\{G _ {k} \right\} _ {k = 1} ^ {m}\right) = \exp \left\{\left(A - \frac {1}{2} \sum_ {k = 1} ^ {m} G _ {k} ^ {2}\right) t + \sum_ {k = 1} ^ {m} G _ {k} W _ {k, t} \right\} x _ {0}, \tag {50}
$$
then the covariance of  $X_{t}$ ,  $P(t,t + h) = \mathbb{E}[X_t X_{t + h}^\top]$  can be calculated as
$$
\begin{array}{l} \mathbb {E} [ X _ {t} X _ {t + h} ^ {\top} ] \\ = \mathbb {E} \left[ \exp \left\{\left(A - \frac {1}{2} \sum_ {k = 1} ^ {m} G _ {k} ^ {2}\right) t + \sum_ {k = 1} ^ {m} G _ {k} W _ {k, t} \right\} x _ {0} x _ {0} ^ {\top} \exp \left\{\left(A ^ {\top} - \frac {1}{2} \sum_ {k = 1} ^ {m} \left(G _ {k} ^ {2}\right) ^ {\top}\right) (t + h) \right. \right. \tag {51} \\ \left. \left. + \sum_ {k = 1} ^ {m} G _ {k} ^ {\top} W _ {k, t + h} \right\} \right] \\ = e ^ {A t} e ^ {- \frac {1}{2} \sum_ {k = 1} ^ {m} G _ {k} ^ {2} t} \mathbb {E} \left[ e ^ {\sum_ {k = 1} ^ {m} G _ {k} W _ {k, t}} x _ {0} x _ {0} ^ {T} e ^ {\sum_ {k = 1} ^ {m} G _ {k} ^ {\top} W _ {k, t + h}} \right] e ^ {- \frac {1}{2} \sum_ {k = 1} ^ {m} (G _ {k} ^ {2}) ^ {\top} (t + h)} e ^ {A ^ {\top} (t + h)}, \\ \end{array}
$$