# Characterization of stationary distribution of diffusion process

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}