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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.

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 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.

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.

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]):

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.

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]):

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 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.