**Background of the question.**

One of the problems that arises with Wiener-driven (and more general Lévy-driven) models of noisy systems is that, due to the extremely rapid decay of tails of infinitely divisible distributions, the theoretical long-term predictions of the model may be affected by the almost-sure occurrence of arbitrarily extreme freak events that have no physical meaning for the problems intended to be addressed by the model. One way to get round this is to work explicitly with the long-but-finite-time predictions of the model; but another way round this is to work with a *bounded-noise model that approximates driving by a Wiener process*.

One common bounded-noise approximation is *dichotomous Markov noise* ([1], see also [2]):

 - in place of the driving $\,\circ \,dW_t$ from a Wiener process $(W_t)_{t \in \mathbb{R}}$, we instead have the driving $\sqrt{\lambda}(-1)^{N_{\lambda t}} dt$ for some (large) $\lambda>0$, where $N_0$ is odd or even with equal probability and $(N_\tau-N_0)_{\tau \in \mathbb{R}}$ is a Poisson process of intensity $1$ independent of $N_0$.

However, I suspect it might sometimes be of interest to have a bounded-noise driving process whose regularity properties are "essentially the same" as those of the Wiener process. The question below formalises this.

[1] Van Den Broeck, C. On the relation between white shot noise, Gaussian white noise, and the dichotomic Markov process. *J Stat Phys* **31**, 467–483 (1983). https://doi.org/10.1007/BF01019494  
[2] https://math.stackexchange.com/questions/3643865/
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**The question.**

Some notations:

 - Given a closed connected set $G \subset \mathbb{R}$ with $0 \in G$, write $C_0(G,\mathbb{R})$ for the set of continuous functions $\omega \colon G \to \mathbb{R}$ with $\omega(0)=0$. We equip $C_0(G,\mathbb{R})$ with the topology of uniform convergence on bounded sets.
 - For each closed connected $G \ni 0$, define the probability measure $\mathbb{P}^G$ on $C_0(G,\mathbb{R})$ to be the Wiener measure.
 - Given two closed connected sets $G_2 \supset G_1 \ni 0$, let $\,\pi_{G_1}^{G_2} \colon C_0(G_2,\mathbb{R}) \to C_0(G_1,\mathbb{R})\,$ be the restriction mapping $\pi_{G_1}^{G_2}(\omega) = \omega|_{G_1}$.

Now fix $\Delta>0$ and $M>0$, and introduce the further notations:

 - For each closed connected $G \ni 0$, let $A_{\Delta,M}^G$ be the set of all $\omega \in C_0(G,\mathbb{R})$ for which
$$ |s-t| \leq \Delta \ \Longrightarrow \ |\omega(s)-\omega(t)| \leq M. $$
 - For each compact connected $G \ni 0$, define the probability measure $\mathbb{P}_{\Delta,M}^G$ on $C_0(G,\mathbb{R})$ by
$$ \mathbb{P}_{\Delta,M}^G = \mathbb{P}^G(\,\cdot\,|A_{\Delta,M}^G). $$

Obviously this definition of $\mathbb{P}_{\Delta,M}^G$ does not extend to non-compact $G$, since for non-compact $G$ we have $\mathbb{P}^G(A_{\Delta,M}^G)=0$.

>> Does there exist a probability measure $\mathbb{P}_{\Delta,M}^\mathbb{R}$ on $C_0(\mathbb{R},\mathbb{R})$ such that for each compact connected $G \ni 0$, for any sequences $s_n,t_n \!\uparrow\! \infty$, we have
$$ \pi_G^{[-s_n,t_n]}\mathbb{P}_{\Delta,M}^{[-s_n,t_n]} \to \pi_G^\mathbb{R}\mathbb{P}_{\Delta,M}^\mathbb{R} \text{ weakly as } n \to \infty \, ? $$
If so, is it the case that $\mathbb{P}_{\Delta,M}^\mathbb{R}$-almost every $\omega \colon \mathbb{R} \to \mathbb{R}$ is not locally $\frac{1}{2}$-Hölder-continuous but is locally $\alpha$-Hölder-continuous for all $\alpha \in (0,\frac{1}{2})$?