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

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.

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 $$|ti_t$$, $$0\le t\le 1$$ are the coordinate projections from $$C$$ to $$\Bbb R$$.