I am interested in what is known about the application of the Krylov-Bogoliubov existence theorem to the time-inhomogeneous case, especially as it relates to an underlying random dynamical system (such as a random pde).

There is some overlap with a few other questions that I could find, but they did not quite address what I am wondering.

A few examples: Proof of Krylov-Bogoliubov theorem

Stationary distribution for time-inhomogeneous Markov process

Proof of Krylov-Bogoliubov theorem

Let X be a polish space and $u_{t}$ be a family of $X$-valued Markov processes on a measurable space with the standard filtration. Let $\beta_{t}, \ \beta_{t}^{*}$ be the semigroups associated with the transition function, $P_{t}(u,\cdot)$. We fix a measure on $X$, $\lambda$, and consider the time-average:

$$\bar{\lambda}(\Gamma)= \frac{1}{t}\int_{0}^{t}(\beta_{s}^{*}\lambda)(\Gamma)ds = \frac{1}{t}\int_{0}^{t}\int_{X}P_{s}(u,\Gamma)\lambda(du)ds$$

Then the Krylov-Bogoliubov theorem gives that if the transition functions are Feller and the family of time-average measures for $t\geq 0$ is tight, then the Markov family has at least one stationary measure.

The proof usually uses the Prokhorov compactness criterion to give weak convergence to a measure that is then shown to be stationary.

So let's say that, for example, a random PDE generates a Feller markov chain that a bounded absorbing set. If the PDE also has a regularizing effect (for instance, maps to a space that is compactly embedded in $X$), this seems to imply the tightness of the time-average measures.

The main issues that I see would be the application of the Prokhorov compactness criterion and the calculation that shows it is stationary. I have not yet looked at the compactness criterion, but I have been looking at proofs that the measure is stationary. These seem to all be given as a homogeneous case, but it seems like it might be adaptable to an inhomogeneous transition function (though the use of homogeneity in changing the bounds on integrals could pose a problem).

I am just curious what is known first.