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Q1. Let $(\Omega,\mathcal{F},(\mathcal{F}_{t \geq 0}),\mathbb{P})$ be a filtered probability space, and let $(X_t)_{t \geq 0}$ be a progressively measurable real-valued homogeneous Markov process with transition probabilities given by $(P_x^t)_{x \in \mathbb{R},t \geq 0}$ -- that is to say, $P_{X_s(\cdot)}^t(A)$ is a conditional probability of $X_{s+t}^{-1}(A)$ with respect to $\mathcal{F}_s$ (for all $s,t,A$). Suppose also that $\rho:=X_{0\ast}\mathbb{P}$ is stationary. Fix any bounded measurable $f:\mathbb{R} \to \mathbb{R}$; is it the case that

 

$\hspace{5mm} \lim_{T \to \infty} \frac{1}{T} \int_0^T f(X_t(\omega)) \, dt$

 

exists for $\mathbb{P}$-almost all $\omega \in \Omega$?

 

(Please note that we do not assume any kind of continuity of $(X_t)$, but only that it is progressively measurable.)

Q2. Suppose $\rho$ is a stationary probability measure. Does there exist a probability measure $Q$ on the set $\mathcal{M}$ of probability measures on $\mathbb{R}$ (equipped with the usual $\sigma$-algebra, which is known to be standard) such that

 

1. $Q$-almost every $\mu \in \mathcal{M}$ is ergodic;

2. for all $A \in \mathcal{B}(\mathbb{R})$, $\rho(A) = \int_\mathcal{M} \mu(A) \, Q(d\mu)$?

Q1. Let $(\Omega,\mathcal{F},(\mathcal{F}_{t \geq 0}),\mathbb{P})$ be a filtered probability space, and let $(X_t)_{t \geq 0}$ be a progressively measurable real-valued homogeneous Markov process with transition probabilities given by $(P_x^t)_{x \in \mathbb{R},t \geq 0}$ -- that is to say, $P_{X_s(\cdot)}^t(A)$ is a conditional probability of $X_{s+t}^{-1}(A)$ with respect to $\mathcal{F}_s$ (for all $s,t,A$). Suppose also that $\rho:=X_{0\ast}\mathbb{P}$ is stationary. Fix any bounded measurable $f:\mathbb{R} \to \mathbb{R}$; is it the case that

$\hspace{5mm} \lim_{T \to \infty} \frac{1}{T} \int_0^T f(X_t(\omega)) \, dt$

exists for $\mathbb{P}$-almost all $\omega \in \Omega$?

(Please note that we do not assume any kind of continuity of $(X_t)$, but only that it is progressively measurable.)

Q2. Suppose $\rho$ is a stationary probability measure. Does there exist a probability measure $Q$ on the set $\mathcal{M}$ of probability measures on $\mathbb{R}$ (equipped with the usual $\sigma$-algebra, which is known to be standard) such that

1. $Q$-almost every $\mu \in \mathcal{M}$ is ergodic;

2. for all $A \in \mathcal{B}(\mathbb{R})$, $\rho(A) = \int_\mathcal{M} \mu(A) \, Q(d\mu)$?

However, I suspect that the answer to Q1 is no (although I do not have a counterexample!!). More precisely, I suspect that the answer to Q1 is the same as the answer to my question Is it true that all stationary measurable stochastic processes are "measurably stationary"? - and I expect that the answer to that question is no (although again, I do not have a counterexample).

However, I suspect that the answer to Q1 is no (although I do not have a counterexample!!). More precisely, I suspect that the answer to Q1 is the same as the answer to my question Is it true that all stationary measurable stochastic processes are "measurably stationary"? - and I expect that the answer to that question is no (although again, I do not have a counterexample).

Update: I'm pretty sure the answer to Q2 is yes, because I think I can prove it using an ergodic theorem for measurable stochastic semigroups; namely, letting $\rho$ be a stationary probability measure, I think I can first prove that for any bounded measurable $f:\mathbb{R} \to \mathbb{R}$,

$\hspace{5mm} \lim_{T \to \infty} \frac{1}{T}\int_0^T \int_\mathbb{R} \! f(y) P_x^t(dy) \; dt$

exists for $\rho$-almost all $x \in \mathbb{R}$, with the limit (as a function of $x$) being a conditional expectation of $f$ over the probability space $(\mathbb{R},\mathcal{B}(\mathbb{R}),\rho)$ with respect to the $\sigma$-algebra of $\rho$-almost stationary sets. (As mentioned in Kifer's book "Ergodic Theory of Random Transformations", the discrete-time analogue of the above statement can be obtained as a special case of the Chacon-Ornstein ergodic theorem.) Using this fact, it should be possible to prove the ergodic decomposition theorem (by a similar approach as in the proof for deterministic dynamical systems).

However, I suspect that the answer to Q1 is no (although I do not have a counterexample!!). More precisely, I suspect that the answer to Q1 is the same as the answer to my question Is it true that all stationary measurable stochastic processes are "measurably stationary"? - and I expect that the answer to that question is no (although again, I do not have a counterexample).

If the answer to Q1 is no, I wonder whether perhaps it becomes yes in the particular case that $(X_t)_{t \geq 0}$ is a strong Markov process.