Continuous version of conditional probability distributions $( \mathcal{L}(X_t | \mathcal{G}) )_{t \geq 0}$ if $(X_t)_{t \geq 0}$ is continuous?

Question

Let me first explain the setup:

Let $(X_t)_{t \geq 0}$ be a stochastic process on some probability space $(\Omega,\mathcal{F},P)$ with values in a complete and separable metric space $E$ (e.g. $E = \mathbb{R}$) and let $\mathcal{G}$ be a sub-$\sigma$-field of $\mathcal{F}$. The conditional probability distribution $\mathcal{L}(X_t | \mathcal{G})$ can then be seen a as a random variable on $(\Omega,\mathcal{G})$ with values in $\mathcal{P}(E)$, which denotes the set of Borel probability measures on $E$. Then $\mathcal{P}(E)$ can be endowed with a metric that metrizes the weak convergence (also often called narrow convergence) of Borel probability measures on E, turning $\mathcal{P}(E)$ into a complete and separable metric space.

The question I have is about the regularization of the regular conditional probability distribution is the following:

If we now suppose that the paths $[0,\infty) \ni t \mapsto X_t(\omega) \in E$ are continuous for each $\omega \in \Omega$, can we choose versions/modifications of the collection of regular conditional probability distributions $(\mathcal{L}(X_t | \mathcal{G}))_{t \geq 0}$, such that the mappings $[0, \infty) \ni t \mapsto \mathcal{L}(X_t | \mathcal{G})(\omega) \in \mathcal{P}(E)$ become continuous for each $\omega \in \Omega$ ? Does this hold in this generality or do we need additional assumptions ?

If $(X_t)_{t \geq 0}$ is for example a Brownian motion, then the above desired regularisation would follow from Kolmogorov's continuity theorem. But I have not found any results for general continuous processes.

Thanks a lot in advance!

Answer 1

For simplicity take $E=\Bbb R$ and the time interval to be $[0,1]$, and think of $X=(X_t)_{0\le t\le 1}$ as a random element of $C=C([0,1]\to\Bbb R)$, a Polish space. We then have a regular conditional distribution of $X$ given $\mathcal G$, call it $Q=Q(\omega,B)$, $\omega\in\Omega, B\in\mathcal B(C)$. And the induced "marginal conditional distribution" $A\mapsto Q(\omega, \{\pi_t\in A\})$ will be weakly continuous in $t$. Here $\pi_t$, $0\le t\le 1$ are the coordinate projections from $C$ to $\Bbb R$.