Let $\vartheta(t)$ be a real ergodic continuous Markov chain solution to the stochastic differential equation $$ \mathrm{d}\vartheta(t)=A_1(\vartheta(t),t)\mathrm{d}t+A_2(\vartheta(t),t)\mathrm{d}W(t) $$ for $A_1,A_2$ unknown $\sigma$-finite functions and $W(t)$ the standard Brownian motion satisfying the usual conditions. Define the transition operator $H$ as $$ (H_tg)(a)=\mathbb{E}(g(\vartheta(t))\mid \vartheta_0=a) $$ where $g$ is an arbitrary Borel-measurable function in $\mathbb{R}$. Assume that $H_t$ satisfies $$ |H_s|_{2}=\sup_{g,\mathbb{E}g(X)=0}\frac{\mathbb{E}^{1/2}(H_sg)^2(X)}{\mathbb{E}^{1/2}g^2(X)}<1 $$ for some $s>0$. At last, assume that the conditional density $p_{\ell}(x\mid y)$ of $\vartheta_{t+\ell}$ given $\vartheta_t$ is continuous in the arguments $(x,y)$ and bounded and that the density $p(x)$ is bounded and time-invariant.
Then is the process $\vartheta(t)$ necessary mean-reverting, or a similar type of characteristic that pulls the process to its unconditional mean? And if not, does there exists conditions on the operators $A_1$ and $A_2$ that achieve this?
For some additional information, the operator $|\cdot|_2$ is called the Rosenblatt condition in the book nonparametric techniques in statistical inference. The condition is used in the paper by Fan, Fan and Lv Aggregation of Nonparametric Estimators for Volatility Matrix for their aggregated estimator for the diffusion operator using a kernel estimator, which requires the above conditions, among others, to be satisfied.