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Suppose $X(t)$ is a stationary $d$-dimensional Gaussian diffusion process with initial distribution $X(0)\sim\nu$: $$ X(t) = X(0) + \int_0^t A X(s) ds + W(t), $$ where $A$ is a strictly negative definite $d\times d$ matrix, and $B:d\times d$ is a positive definite matrix, $W(t)$ is a $d$-dimensional Gaussian process with some covariance function. This diffusion process has the stationary distribution $\pi$.

My question is why the following statement is true: for any $\varepsilon>0$ and weakly compact set $K$ of probability measures, there exists $T>0$ depending on $\varepsilon,K$ such that for any $t>T$, \begin{align} \bigg|\mbox{E}\big[h\big(X(t)\big)\big] - \int h d\pi \bigg|<\varepsilon, \ \forall \nu\in K, \quad\quad\quad\quad\mbox{(*)} \end{align} for any continuous function $h$ with compact support on $\mathbb{R}^d$.

I understand that there is a solution for $X(t) = e^{At} X(0) + \int_0^t e^{A(t-s)}dW(s)$. However, I do not understand how the weak compactness enters here. Any comment is welcome. Thank you.

============= Edit: A tentative answer =============

I sketch my tentative answer below, which uses the solution $X(t) = e^{At} X(0) + \int_0^t e^{A(t-s)}dW(s)$. However, my conjecture is that the statement Eq.(*) can be made more general: for any diffusion (not necessarily analytically solvable) that allows for a stationary distribution, if it takes initial distribution in a weakly compact set $K$, then Eq.(*) above would hold. I think as long as this diffusion allows for a weak solution, then my argument below can also be applied to this more general statement.

Let $P_t$ be the distribution of $\int_0^t e^{A(t-s)}dW(s)$. By the weakly compactness of $K$, take an arbitrary finite ($\varepsilon/2$)-cover of $K$: $\{K_1,K_2,...,K_m\}$ such that $\sup_{h\in C_c(\mathbb{R}^d)}|\int h d\nu-\int hd\nu'|<\varepsilon/2$ for any pair of measures $\nu,\nu'\in K_j$ for any $j=1,...,m$. It follows that for such $\nu,\nu'$ $$ \sup_{h\in C_c(\mathbb{R}^d)}\bigg|\int \int h(u+w) dP_t(w)d\nu(u)-\int\int h(u+w)dP_t(w)d\nu'(u)\bigg|<\varepsilon/2. \quad\quad\quad\quad\mbox{(**)} $$ where $C_c(\mathbb{R}^d)$ denotes the space of continuous functions on $\mathbb{R}^d$ with compact support.

Take an arbitrary $\nu_j$ from $K_j$. Since the diffusion is stationary with stationary distribution $\pi$, there exists $T_j$ (depending on $\nu_j$ and $\varepsilon$) such that for any $t>T_j$, $$ \sup_{h\in C_c(\mathbb{R}^d)}\bigg|\int \int h(u+w) dP_t(w)d\nu_j(u)-\int g d\pi\bigg|<\varepsilon/2. \quad\quad\quad\quad\mbox{(***)} $$

Take $T=\max\{T_1,...,T_m\}$ (depending on $\varepsilon$ and $K$). Using $\mbox{(**)},\mbox{(***)}$, we have that for any $t>T$ and $\nu\in K$ (suppose that $\nu\in K_{j_0}$), \begin{align} &\sup_{h\in C_c(\mathbb{R}^d)}\bigg|\int \int h(u+w) dP_t(w)d\nu(u)-\int g d\pi\bigg|\\ &\leq \sup_{h\in C_c(\mathbb{R}^d)}\bigg|\int \int h(u+w) dP_t(w)d\nu(u)-\int \int h(u+w) dP_t(w)d\nu_{j_0}(u)\bigg|\\ &\quad\quad\quad\quad\quad\quad\quad+\sup_{h\in C_c(\mathbb{R}^d)}\bigg|\int \int h(u+w) dP_t(w)d\nu_{j_0}(u)-\int g d\pi\bigg|\\ &<\varepsilon. \end{align}

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