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Simultaneous almost-sure Ray–Knight-type coupling for all times

Construct a coupling between the local time process {L_t: t ≥ 0} of the constant-speed continuous-time simple random walk on the wired graph D_N ∪ {ρ} (started at ρ) and an R^{V∪{ρ}}-valued process {h(t): t ≥ 0} such that: (i) for every t ≥ 0, h(t) has the law of the Discrete Gaussian Free Field h^V on V = D_N with Dirichlet boundary at ρ; (ii) for every t ≥ 0, the sigma-algebras generated by {h(s): s ≤ t} and {L_{t+u} − L_t: u ≥ 0} are independent; and (iii) for all r, t ≥ 0, the identity ½(h(r)+√(2r))^2 − L_r = ½(h(t)+√(2t))^2 − L_t holds almost surely.

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Background

The Second Ray–Knight theorem gives, for each fixed time t, a coupling in distribution between the local time profile L_t and two independent copies of the DGFF, yielding the identity L_t + ½ h2 = ½(tilde h + √(2t))2. However, the construction provided is non-constructive and specific to each fixed t, relying on conditional sampling, and thus does not yield a single coupling that is valid simultaneously for all times.

A simultaneous almost-sure coupling satisfying natural independence of increments and a conserved quantity across times would provide a powerful pathwise representation of the local time–DGFF connection and simplify technical arguments in subsequent results on exceptional points and time parametrizations.

References

However, since the construction of the signs of \tilde h+\sqrt{2t}, which is what sampling from the conditional measure (2.49) is really about, is non-constructive, a question remains whether an almost-sure coupling can be constructed simultaneously for all times. Is there a coupling of the local time~{L_t\colon t\ge0} (sampled under~P\varrho) and an R{V\cup{\varrho}}-valued cag process {h(t)\colon t\ge0} such that (1) \forall t\ge0\colon\,\, h(t)\,\,\,hV, (2) \forall t\ge0\colon\,{h(s)\colon s\le t} and {L_{t+u}-L_t\colon u\ge0} are independent, (3) for all r,t\ge0, \frac12\bigl(h(r)+\sqrt{2r}\bigr)2-L_r\,\,=\,\,\frac12\bigl(h(t)+\sqrt{2t}\,\bigr)2 - L_t,\quad \text{a.s.} hold true?

— Extremal properties of the random walk local time (2502.09853 - Biskup, 14 Feb 2025) in Question following Eq. (2.55), end of Section “Proof of the Second Ray–Knight Theorem” (Lecture 2)