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 finite-time behaviour (including, for example, local regularity or roughness of paths) is "essentially the same" as for a Wiener process, apart from the very slight (i.e. physically immeasurably small) statistical dependence of future increments upon past increments that is necessary to be able to control the magnitude of increments. 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/

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. Note that the Borel $\sigma$-algebra is given by $$ \mathcal{B}(C_0(G,\mathbb{R})) \ = \ \sigma(\omega \mapsto \omega(t) \, : \, t \in G) \ = \ \sigma(\omega \mapsto \omega(t) \, : \, t \in \mathbb{Q} \cap G). $$
• 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.

Now fix $\Delta>0$ and $M>0$, and for any closed connected $G \ni 0$, let $$ A_G \ := \ \{ \omega \in C_0(G,\mathbb{R}) \, : \, \text{for all $s,t \in G$, } |s-t| \leq \Delta \ \Rightarrow \ |\omega(s)-\omega(t)| \leq M. \} $$ Note that $A_G$ is a closed subset of $C_0(G,\mathbb{R})$.

(A) Is it the case that for each $\tau>0$ there is a probability measure $\tilde{\mathbb{P}}_{[0,\tau]}$ on $A_{[0,\tau]}$ such that for any sequences $s_n,t_n \!\uparrow\! \infty$ and any bounded continuous $g \colon A_{[0,\tau]} \to \mathbb{R}$, $$ \frac{1}{\mathbb{P}_{[-s_n,t_n]}(A_{[-s_n,t_n]})} \int_{A_{[-s_n,t_n]}} g(\omega|_{[0,\tau]}) \, \mathbb{P}_{[-s_n,t_n]}(d\omega) \, \to \, \int_{A_{[0,\tau]}} g \, d\tilde{\mathbb{P}}_{[0,\tau]} $$ as $n \to \infty$?

If so: (B) Is $\tilde{\mathbb{P}}_{[0,\tau]}$ absolutely continuous with respect to $\left.\mathbb{P}_{[0,\tau]}\right|_{\mathcal{B}(A_{[0,\tau]})}$? (C) If we imagine fixing $\Delta > 0$ but allowing $M$ to vary, do we have that for every bounded continuous $g \colon C_0([0,\tau],\mathbb{R}) \to \mathbb{R}$, $$ \int_{A_{[0,\tau]}} g \, d\tilde{\mathbb{P}}_{[0,\tau]} \to \int_{C_0([0,\tau],\mathbb{R})} g \, d\mathbb{P}_{[0,\tau]} \ \text{ as } M \to \infty \, \text{?} $$

Physical meaning:

The stochastic process $(\omega \mapsto \omega(t))_{t \in [0,\tau]}$ over the probability space $(A_{[0,\tau]},\mathcal{B}(A_{[0,\tau]}),\tilde{\mathbb{P}}_{[0,\tau]})$ represents a "bounded-noise driving process" on $[0,\tau]$, where hopefully, for a given value of $\Delta$, if $M$ is sufficiently large then this process approximates a Wiener process on $[0,\tau]$. Now if this measure $\tilde{\mathbb{P}}_{[0,\tau]}$ does exist for all $\tau>0$, then obviously the intervals $[0,\tau]$ can be generalised to any compact intervals $G \ni 0$. It is then not hard to show that one can define a probability measure $\tilde{\mathbb{P}}$ on $A_\mathbb{R}$ whose natural projection onto $A_G$ coincides with $\tilde{\mathbb{P}}_G$ for any compact interval $G \ni 0$. (See immediately below for a proof.) The stochastic process $(\omega \mapsto \omega(t))_{t \in \mathbb{R}}$ over the probability space $(A_\mathbb{R},\mathcal{B}(A_\mathbb{R}),\tilde{\mathbb{P}})$ intuitively represents "a bi-infinite-time Wiener process conditioned on the event that a jump of size $M$ never takes place within a time-interval of width $\Delta$".

Note that the absolute continuity in the second question implies in particular that $\tilde{\mathbb{P}}$-almost every sample path has the same local regularity properties (e.g. local $\alpha$-Hölder-continuity for all $\alpha \in (0,\frac{1}{2})$ but not for $\alpha=\frac{1}{2}$) as the generic sample paths of a Wiener process.

Construction of $\tilde{\mathbb{P}}$ given the measures $\tilde{\mathbb{P}}_G$.

First note that the family of probability measures $\tilde{\mathbb{P}}_G$ across compact intervals $G \ni 0$ is consistent: for compact intervals $G_2 \supset G_1 \ni 0$, to verify that $\tilde{\mathbb{P}}_{G_2}$ projects onto $\tilde{\mathbb{P}}_{G_1}$, it is sufficient to verify that $\tilde{\mathbb{P}}_{G_1}$ agrees with the projection of $\tilde{\mathbb{P}}_{G_2}$ regarding the integral of every bounded continuous functions $g_1 \colon A_{G_1} \to \mathbb{R}$; but for any such $g_1$, the map $g_2 \colon A_{G_2} \to \mathbb{R}$ sending $\omega \mapsto g_1(\omega|_{G_1})$ is a bounded continuous function and for any compact interval $G_3 \supset G_2$, for any $\omega \in A_{G_3}$ we have $g_2(\omega|_{G_2})=g_1(\omega|_{G_1})$. Hence we have consistency, and so we can apply the Kolmogorov extension theorem to obtain a probability measure $\tilde{\tilde{\mathbb{P}}}$ on $(\mathbb{R}^\mathbb{R},\mathcal{B}(\mathbb{R})^{\otimes \mathbb{R}})$ whose natural projection onto $\mathbb{R}^G$ agrees with $\tilde{\mathbb{P}}_G(\,\cdot\,\cap A_G)$ for each compact interval $G \ni 0$. For $\tilde{\tilde{\mathbb{P}}}$-almost every $\omega \colon \mathbb{R} \to \mathbb{R}$, we have that $\omega(0)=0$ and the restriction $\omega|_\mathbb{Q}$ is uniformly continuous on bounded sets, and hence there exists $\kappa(\omega) \in A_\mathbb{R}$ agreeing with $\omega$ on $\mathbb{Q}$. So take $\tilde{\mathbb{P}}$ to be the image measure of $\tilde{\tilde{\mathbb{P}}}$ under $\kappa(\cdot)$.

First thought on tackling the question. My strong intuition is that for each $E \in \mathcal{B}(A_{[0,\tau]})$, there exists a number $p(E) \in [0,1]$ such that for any sequences $s_n,t_n \!\uparrow\! \infty$, $$ \frac{1}{\mathbb{P}_{[-s_n,t_n]}(A_{[-s_n,t_n]})} \int_{A_{[-s_n,t_n]}} \mathbf{1}_E(\omega|_{[0,\tau]}) \, \mathbb{P}_{[-s_n,t_n]}(d\omega) \, \to \, p(E) $$ as $n \to \infty$. If this is correct, then the Vitali-Hahn-Saks theorem implies that $p(\cdot)$ is a probability measure, and so we can simply take $\tilde{\mathbb{P}}_{[0,\tau]}$ to be $p(\cdot)$, since this "strong convergence" implies the "weak convergence" being asked about in the question. Furthermore, this will then immediately yield a positive answer to the second question about absolute continuity.

Remark. If, in the definition of $A_G$, we replace the condition $$ \text{for all $s,t \in G$, } |s-t| \leq \Delta \ \Rightarrow \ |\omega(s)-\omega(t)| \leq M $$ with a more general condition of the form $$ \text{for all $s$ with $[s,s+\Delta] \subset G$, } \ \left.\big(\omega(\,\cdot\, + s)-\omega(s)\big)\right|_{[0,\Delta]} \in \mathcal{D} $$ for some closed set $\mathcal{D} \subset C_0([0,\Delta],\mathbb{R})$, even if we still require that $\mathbb{P}_G(A_G)>0$ for every compact interval $G \ni 0$, I expect that the limiting measure $\tilde{P}_{[0,\tau]}$ defined in part (A) will not necessarily exist.