I have a problem setup that reduces to finding the stationary state of a discrete time continuous state stationary Markov process. Is that state unique, as it is in the discrete state case?
Specifically, given the stationary Markov process: $$f_{i+1}(x) = \int \mathcal{L}(x|y) f_i(y) \operatorname{d}y,$$ where $\mathcal{L}(x|y)$ is positive semi-definite and satisfies $\int \mathcal{L}(x|y) \operatorname{d} x = 1\ \forall y$, does the stationary state: $$f(x) = \int \mathcal{L}(x|y) f(y) \operatorname{d}y$$ have all of these properties:
- exists,
- unique,
- positive semi-definite, and
- the eigenfunction that has the largest eigenvalue?
The last two are satisfied if a continuous version of the Perron–Frobenius theorem can be shown.
Is it enough to say that the process can be approximated arbitrarily well by a sequence of discrete time discrete state processes defined by a sequence of partitions of $x$ with norm $\rightarrow 0$?
Add: does this qualify as a Harris chain (I'm having trouble understanding the notation in the Wikipedia article)? Both the Wikipedia article and this Quora question reference section 6.8 of "Probability" by Durrett, but I have no way of accessing that work right now.
Conjecture: uniqueness requires irriducibility of $\mathcal{L}(x|y)$, and convergence requires either: the set which defines the domain of $f_i$ is closed, or at least one measure of central tendency for each dimension must converge to somewhere within the domain. These are separable requirements (e.g. $\mathcal{L}(x|y)=\delta(x-y)$ holds all distributions stationary [maximal non-uniqueness], $\mathcal{L}(x|y)= \frac{1}{\sqrt{2\pi}\sigma}\exp\left(-\frac{[x-my-\mu]^2}{2\sigma^2}\right)$ for $x,y\in(-\infty,\infty)$ converges iff $|m| < 1$ [reparameterizing to $x'\in (0,1)$ shows that it converges to a delta function on the boundary]).