AI 中文总结
该研究揭示带几何重置的马尔可夫链中谱对偶性的预解起源，确定通用临界值$C^*=1/(1+\sqrt{K})$，刻画$(\sigma,\kappa)$可逆链类别，经数值实验验证理论并建立算子理论基础。
AI 中文摘要
我们揭示了带几何重置的马尔可夫链中控制重置中性分布的谱对偶性的预解起源。从《论文III》的抽象条件出发，我们证明谱对偶性$B_\nu(z)=\kappa(z)\\,A_\nu(\sigma(z))$等价于预解$R(\gamma)=(I-(1-\gamma)P)^{-1}$的单一对称性：交织关系$[\Delta^2\mathcal{R},R(\gamma)]=0$，其中$\mathcal{R}$是对合$\sigma$的反射算子，$\Delta=\operatorname{diag}(\sqrt{\kappa(z)})$；等价地，$\widetilde{\mathcal{T}}=K^{-1/2}\Delta^2\mathcal{R}$是一个对合。该对称性确定了通用临界值$C^*=1/(1+\sqrt{K})$，其中$K=\kappa(z)\kappa(\sigma(z))$，它仅依赖标量$K$，与重置率$\gamma$、重置分布或特定链无关。我们刻画了$(\sigma,\kappa)$可逆链的类别，涵盖了有偏随机游走和具有相同$C^*$的真正非齐次动力学；Doob h-变换实现了对偶性$K\mapsto1/K$，因此$C^*\mapsto1-C^*$，不动点为$C^*=1/2$。方向场具有显式预解表示$\psi(\gamma)=R(\gamma)(b^{(0)}-C^*b)$：其规范归一化形式$\Delta^{-1}\psi(\gamma)$在$\sigma$下是反对称的，在$\sigma$的不动点处有一个精确节点，且它控制精确符号律$\operatorname{sgn}(C(\pi,\gamma)-C^*) =\operatorname{sgn}\langle\pi,\psi(\gamma)\rangle$。数值实验证实了该理论达到机器精度。这些结果为《论文III》的谱对偶性奠定了算子理论基础，并为《论文V》的信息几何框架提供了桥梁。
英文摘要
We uncover the resolvent origin of the spectral duality governing reset-neutral distributions in Markov chains with geometric resetting. Starting from the abstract conditions of Paper~III, we show that the spectral duality $B_ν(z)=κ(z)\,A_ν(σ(z))$ is equivalent to a single symmetry of the resolvent $R(γ)=(I-(1-γ)P)^{-1}$: the intertwining relation $[Δ^2\mathcal{R},R(γ)]=0$, where $\mathcal{R}$ is the reflection operator of an involution $σ$ and $Δ=\operatorname{diag}(\sqrt{κ(z)})$; equivalently, $\widetilde{\mathcal{T}}=K^{-1/2}Δ^2\mathcal{R}$ is an involution. This symmetry determines the universal critical value $C^*=1/(1+\sqrt{K})$, with $K=κ(z)κ(σ(z))$, which depends only on the scalar $K$ --- not on the resetting rate $γ$, the reset distribution, or the particular chain. We characterize the class of $(σ,κ)$-reversible chains, encompassing both the biased random walk and genuinely non-homogeneous dynamics sharing the same $C^*$; a Doob $h$-transform realizes the duality $K\mapsto1/K$, hence $C^*\mapsto1-C^*$, with fixed point $C^*=1/2$. The orientation field admits the explicit resolvent representation $ψ(γ)=R(γ)(b^{(0)}-C^*b)$: its gauge-normalized form $Δ^{-1}ψ(γ)$ is antisymmetric under $σ$, it has an exact node at the fixed point of $σ$, and it governs the exact sign law $\operatorname{sgn}(C(π,γ)-C^*) =\operatorname{sgn}\langleπ,ψ(γ)\rangle$. Numerical experiments confirm the theory to machine precision. These results establish the operator-theoretic foundation of the spectral duality of Paper~III and provide the bridge to the information-geometric framework of Paper~V.
Comments17 pages, 3 figures, 5 tables
