Let $\vartheta(t)$ be a real Ergodic stationary Markov process solution to the stochastic differential equation $$ \mathrm{d}\vartheta(t)=A_1(\vartheta(t),t)\mathrm{d}t+A_2(\vartheta(t),t)\mathrm{d}W(t) $$ where $A_1,A_2$ are unknown $\sigma$-finite measurable functions and $W(t)$ is the standard Brownian motion satisfying the usual conditions. Assume that there exists a finite constant $C$ such that $$ |A_1(x)-A_1(y)|+|A_2(x)-A_2(y)|\leq C|x-y| $$ for any $x,y\in\mathbb{R}$ and that $\vartheta(t)$ is $\rho$-mixing with exponentially decaying $\rho$-mixing coefficient $\rho_t(\cdot)$ and strong mixing coefficient $\alpha(\ell)\leq\rho(\ell)$ for any integer $\ell$. Moreover, assume that the conditional density $p_{n}(x\mid y)$ of $\vartheta(t+n)$ given $\vartheta(t)$ is continuous in the arguments $(x,y)$ and bounded by a constant $c$ independent of $n$ and that the unconditional density $p(x)$ is continuous, bounded and time-invariant.
From Jensen's inequality, we know that the conditional operator $\mathbb{E}(\vartheta(t)\mid\mathcal{F}_t)$ is a contraction mapping, where $\mathcal{F}_t$ denotes the usual filtration equipped on the underlying probability space. My question is if, provided that $\vartheta(t)\neq\mathbb{E}\vartheta$, the inequality becomes strict? That is, $$ \varphi(\mathbb{E}(\vartheta(t)\mid\mathcal{F}_t))<\mathbb{E}(\varphi(\vartheta(t))\mid\mathcal{F}_t),\quad\vartheta(t)\neq\mathbb{E}\vartheta $$ for $\varphi$ a convex function.
To the best of my knowledge, I couldn't find any theory that connects ergodicity to strict contractions for real stationary ergodic Markov processes, or any stochastic process for matter. Any help in the form of tips, proofs, contradictions or references are welcome, thank you in advance!
For some additional context, the model 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. From an economic perspective, it can be useful to have a quantitative measure on whether a process is mean reverting. From the above mentioned paper, it is achieved to test whether the process $\vartheta(t)$ is stationary, yet if this implies that the process in expectation is pulled back to its unconditional mean is not discussed.