ASIP rates as strong averaging rates

Definitions and mathematical statement

Let $(Y,\mu)$ be a probability space and let $T:Y\to Y$ preserve $\mu$. Write $y_n=T^ny$, with $y$ sampled from $\mu$, and let $f:{\mathbb{R}}^q\times Y\to{\mathbb{R}}^q$ be integrable in its second variable. For $\epsilon>0$, compare, from a common initial point $x_0$, \[ x_{n+1}^{\epsilon,y}=x_n^{\epsilon,y}+\epsilon f(x_n^{\epsilon,y},T^ny),\qquad \bar x_{n+1}=\bar x_n+\epsilon\bar f(\bar x_n),\quad \bar f(x)=\int_Y f(x,u)\,d\mu(u). \] The averaging question asks for estimates and limit laws for \[ D_\epsilon=\sup_{0\le n<1/\epsilon}|x_n^{\epsilon,y}-\bar x_n|. \] The accompanying ASIP question asks whether the annealed almost sure invariant principle can have remainder $N^{1/4}\log N$, in place of $N^{1/4+\delta}$ for $\delta>0$. Here an annealed maximal ASIP for a centered observable $b_0:Y\to{\mathbb{R}}^q$ means a coupling, on an extension of the law $y\sim\mu$, with a Brownian motion $W$ of specified covariance, such that \[ S_m=\sum_{j=0}^{m-1}b_0(T^jy),\qquad \max_{m\le N}|S_m-W(m)|=O_\omega(r(N)). \] The random constant in $O_\omega$ is finite almost surely. Neither the observable and covariance in the ASIP question nor dynamical, mixing and regularity assumptions sufficient for it are specified by the source; invariance of $\mu$ alone is not such an assumption.

The supplied aid gives a conditional transfer statement for additive forcing $f(x,y)=a(x)+b(y)$, with $a$ globally Lipschitz of constant $L$, $b$ integrable, and the above ASIP for $b_0=b-\int b\,d\mu$. Set $A=a+\int b\,d\mu$ and define the Brownian-corrected recursion \[ X_0=x_0,\qquad X_{n+1}=X_n+\epsilon A(X_n)+\epsilon\left(W(n+1)-W(n)\right). \] It reports the bound \[ \max_{n\le N}|x_n^{\epsilon,y}-X_n| \le\epsilon e^{L\epsilon N}\max_{m\le N}|S_m-W(m)|. \] For fixed $H>0$ and $N=\lfloor H/\epsilon\rfloor$, this yields respectively $O_\omega(\epsilon^{3/4}\log(1/\epsilon))$ and $O_\omega(\epsilon^{3/4-\delta})$. The single factor $\epsilon$ is reported to be sharp when $a=0$. These are errors relative to the Brownian-corrected recursion, whereas the discrepancy from the averaged recursion has fluctuation scale $\sqrt\epsilon$ when a nondegenerate diffusion limit applies.

For example, conditional on a functional fluctuation theorem \[ \epsilon^{-1/2}\left(x_{\lfloor t/\epsilon\rfloor}^{\epsilon,y}-\bar x_{\lfloor t/\epsilon\rfloor}\right) \ \Longrightarrow\ Z(t)\quad(0\le t\le H) \] with a continuous limiting process in the uniform topology, the corresponding normalized maximal discrepancy converges to $\sup_{0\le t\le H}|Z(t)|$. Identification of $Z$ as a linearized Gaussian diffusion needs additional drift smoothness and fluctuation hypotheses; those hypotheses are not supplied here. Thus the conditional transfer does not establish the requested ASIP rate for general systems or a general limit theorem for $D_\epsilon$.

Short English statement

Ask for an annealed ASIP remainder $N^{1/4}\log N$ and for limit laws of slow dynamics relative to their averaged recursion. The supplied additive-forcing result transfers an assumed ASIP to a Brownian-corrected slow path; existence of that ASIP and general averaging limits require assumptions absent from the record.

Sources

• AIM workshop Stochastic methods for non-equilibrium dynamical systems, Problem 8: https://aimath.org/pastworkshops/noneqdynsysproblems.pdf.
• Local registry data/unsolvedmath/problems.json, permanent ID 20001393, source code catalogue entry; canonical location aim-dynamical-systems-notes.json, notes[50].
• The supplied reconstruction interprets OCR superscripts as $N^{1/4}$ and $N^{1/4+\delta}$, the control character as $\epsilon$, and isolated "$>12$" as a page transition. It is labeled reconstructed and unverified.
• Attached machine-generated attempt 1: partial result, internal rigor label "proved", low novelty confidence; not peer reviewed and requiring independent verification before citation. No separate literature citation is supplied.