Gluing blockwise martingale decompositions across seams

Definitions and mathematical statement

The source asks whether function spaces can yield martingale decompositions and concatenations for some class of hyperbolic maps. It specifies no maps, invariant/reference measures, function spaces or concatenation rule. The aid supplies an abstract seam-gluing statement on a common decreasing filtration; it does not identify a complete hyperbolic-map application.

Let $\mathcal F_0\supseteq\mathcal F_1\supseteq\cdots$ be sigma-algebras on one probability space, and choose block times $0=t_0<t_1<\cdots$. Suppose $Y_t,D_t\in L^2(\mathcal F_t)$, with ${\mathbb{E}}[D_t\mid\mathcal F_{t+1}]=0$, and for every block $j$ there are $H_t^{(j)}\in L^2(\mathcal F_t)$, $t_j\leq t\leq t_{j+1}$, such that \[ Y_t=D_t+H_{t+1}^{(j)}-H_t^{(j)}\qquad(t_j\leq t<t_{j+1}). \] At a seam $\tau=t_j$, $j\geq1$, define \[ J_\tau=H_\tau^{(j-1)}-H_\tau^{(j)}\in L^2(\mathcal F_\tau),\qquad A_{\tau,t}={\mathbb{E}}[J_\tau\mid\mathcal F_t]-{\mathbb{E}}[J_\tau\mid\mathcal F_{t+1}]\quad(t\geq\tau). \] The future reverse-Doob increments have conditional mean zero. Define one global reverse-martingale-difference sequence by \[ \widetilde D_t=D_t+\sum_{\substack{\tau\text{ a seam}\\0<\tau\leq t}}A_{\tau,t}. \] Each sum is finite, and ${\mathbb{E}}[\widetilde D_t\mid\mathcal F_{t+1}]=0$.

For a horizon $N\geq1$, let $j_N$ be the block containing time $N-1$. The exact finite-horizon identity reported by the aid can be written \[ \sum_{t=0}^{N-1}Y_t =\sum_{t=0}^{N-1}\widetilde D_t+R_N, \] \[ R_N=H_N^{(j_N)}-H_0^{(0)} +\sum_{\substack{\tau\text{ a seam}\\0<\tau<N}} {\mathbb{E}}[J_\tau\mid\mathcal F_N]. \] Thus each seam mismatch contributes only its still-unforgotten future conditional expectation; all its intervening projection increments are absorbed in $\widetilde D_t$. A seam exactly at $N$ is not included, since the horizon ends with the preceding block.

A precise summable-memory sufficient condition is: both outer correctors have $L^2$ norm at most $C$, uniformly in $N$, and there is a nonnegative sequence $k(r)$, $r\geq1$, with $\sum_rk(r)<\infty$ such that \[ \|{\mathbb{E}}[J_\tau\mid\mathcal F_N]\|_2\leq k(N-\tau)\quad(0<\tau<N). \] Since there is at most one seam at each integer time, this gives \[ \sup_N\|R_N\|_2\leq2C+\sum_{r\geq1}k(r)<\infty. \] In particular $R_N/\sqrt N\to0$ in $L^2$, even with one seam per time. This permits transfer of a separately established martingale limit theorem; it is not itself a CLT without its further variance and Lindeberg-type hypotheses. The record does not state an exact norm-kernel convention, so the displayed version makes its requirements explicit.

The aid's persistent-tail obstruction can be expressed with a single Rademacher variable $Z$, $\Pr(Z=1)=\Pr(Z=-1)=1/2$, and $\mathcal F_t=\sigma(Z)$ for every $t$. Give every time its own block, take $Y_t=Z,D_t=0$, and choose that block's initial corrector zero and terminal corrector $Z$. Every local corrector has norm at most one, but all $N-1$ internal seams equal $Z$, and the outer endpoint also equals $Z$. Every projection increment is zero, while \[ R_N=(N-1)Z+Z=NZ,\qquad N^{-1/2}\sum_{t<N}Y_t=\sqrt N\,Z \] is not tight. Thus uniformly bounded local correctors alone do not control concatenation; decay of conditional seam memory is a separate requirement.

Short English statement

The aid glues blockwise reverse-martingale decompositions by projecting seam mismatches into the common future filtration. Summable conditional seam memory gives a uniform remainder, whereas persistent aligned seams can accumulate linearly despite bounded local correctors.

Sources

• Local registry: permanent UnsolvedMath ID $20001406$, exact public code catalogue entry; aim-dynamical-systems-notes.json, notes[63].
• AIM workshop "Stochastic methods for non-equilibrium dynamical systems", Problem 21: https://aimath.org/pastworkshops/noneqdynsysproblems.pdf. The exact question asks about function spaces, martingale decompositions and concatenations for hyperbolic maps. No map class, proposer, function-space assumptions or authoritative literature reference is supplied; the literature field is empty.
• Research attempt 1: partial_result, status at run partially_solved, rigor label proved, novelty confidence low, classification PARTIAL-PROGRESS. It reports the future-projected seam identity, summable-memory remainder and persistent-tail Rademacher obstruction as a candidate gluing synthesis. The current registry status is partially_solved. No separate publication citation is supplied.