2 Kernel theory (paper §4–5)
For a driftless, finite-range, reversible walk on \(\mathbb Z\) with measure bounds \(c_1 \le m \le c_2\), exit rate \(\le \Lambda \) and nearest-neighbour conductance \(\ge \delta {\gt} 0\): \(\sup _r \mathbb P(X_t = r) \le C/\sqrt{1+t}\) with \(C = C(c_1,c_2,\delta ,\Lambda ,\varrho )\) explicit. Carries the faithful a-priori hypothesis \(p \le 1\); the full Nash/CKS argument runs on the exponential semigroup (TypeDDecouplingCKS.free_bound), with \(p\) identified with the semigroup kernel by a weighted-\(\ell ^1\) Grönwall uniqueness argument.
The 1D Agmon bound, the discrete Nash inequality \(\| f\| _2^6 \le 4\| f\| _1^4\| \nabla f\| _2^2\), the ODE iteration \(u' \le -\kappa u^3 \Rightarrow u \le 1/\sqrt{2\kappa t}\), and the pointwise assembly.
The forward generator of a finite-range walk with bounded exit rates is a bounded operator on \(\ell ^1(\mathbb Z)\) (\(\| A\| \le 2\Lambda \)); its exponential semigroup satisfies positivity, mass conservation, Chapman–Kolmogorov, reversibility, and the energy identity \(u' = -2\mathcal E\) (continued in the CKS files).
For the relative walk with sticky origin (split rate \(\nu _{\mathrm{sp}} = 2q^2(1-q^2)\), merge rate \(1-q^2\)), uniformly over \(q \in [q_0,1)\) in the window \(\nu _{\mathrm{sp}} t \le K\): \(\mathbb P(R_t = r) \le C(K,q_0)/\sqrt{1+t} + \delta _{r,0}e^{-\nu _{\mathrm{sp}}t}\). Sorry-free: the excursion/renewal representation enters as the single documented hypothesis bundle hrenew (occupation, zero-decomposition, renewal integral); all convolution assemblies and the \(q\)-uniform constant are proved.
For independent integer-valued \(Y_1,\dots ,Y_n\): \(\sup _x \mathbb P(\sum _j Y_j = x) \le C\big(\sum _j(1-\mathcal Q(Y_j))\big)^{-1/2}\) with universal \(C\). Proved from scratch (lattice Esseen + product-over-gaps), core TypeDDecoupling.KR.KR_abstract.
Conditionally on the unsigned skeleton, the sum coordinate’s largest atom is \(\le C(\delta )/\sqrt{1+M}\); the ambiguous-jump count dominates a Poisson and \(\mathbb P(M {\lt} t/2) \le e^{-ct}\) with \(c = (1-\log 2)/2\). (Statement strengthened: universal constant.)
If \(\omega (\lambda ) = \int _0^\infty e^{-\lambda t}p(t)\, dt \sim c\lambda ^{-\rho }\) as \(\lambda \downarrow 0\) with \(p \ge 0\), then \(\int _0^s p \sim c\, s^\rho /\Gamma (\rho +1)\) as \(s \to \infty \). (Tauberian direction, general \(\rho {\gt} 0\); core TypeDDecouplingKaramata.tauberian_isEquivalent.)
For the recurrent reversible walk agreeing off a finite set with the symmetric rate-\(a\) walk: \(\tau _r(s) \sim m(r)\sqrt{s/(\pi a)}\).
The same-species two-particle dual kernel obeys \(p_t(\xi ,\xi ') \le C(q)/(1+t)\). Now unconditional: the kernel is synthesised as the Schütz reflection series asepKernel q := Bethe.asepReflect 1 q^2, and the bound follows from TypeDDecoupling.Bethe.asepReflect_decay with no extra hypothesis.
The exclusion contact condition forces the reflection amplitude \(S = -\frac{r_R-(r_R+r_L)z_2+r_Lz_1z_2}{r_R-(r_R+r_L)z_1+r_Lz_1z_2}\); the geometric reflection series (ratio \(\rho = r_L/r_R\)) satisfies the free equation off contact, the boundary identity, and decays like \(C/(1+t)\).