Resolvent intertwining and spectral duality in Markov chains with geometric resetting
Abstract
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 Bν(z)=κ(z) Aν(σ(z)) is equivalent to a single symmetry of the resolvent R(γ)=(I−(1−γ)P)−1: the intertwining relation [Δ2ℛ,R(γ)]=0, where ℛ is the reflection operator of an involution σ and Δ=diag(√(κ(z))); equivalently, 𝒯̃=K−1/2Δ2ℛ is an involution. This symmetry determines the universal critical value C*=1/(1+√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↦1/K, hence C*↦1−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−1ψ(γ) is antisymmetric under σ, it has an exact node at the fixed point of σ, and it governs the exact sign law sgn(C(π,γ)−C*) =sgn⟨π,ψ(γ)⟩. 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.
Related
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BibTeX
@misc{coso2026resolvent,
title = {{Resolvent intertwining and spectral duality in Markov chains with geometric resetting}},
author = {Juan Antonio Vega Coso},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.15140},
eprint = {2608.15140},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.15140},
}
RIS
TY - UNPB
TI - Resolvent intertwining and spectral duality in Markov chains with geometric resetting
AU - Juan Antonio Vega Coso
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2608.15140
UR - https://arxiv.org/abs/2608.15140
AB - 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^2ℛ,R(γ)]=0, where ℛ is the reflection operator of an involution σ and Δ=diag(√(κ(z))); equivalently, 𝒯̃=K^−1/2Δ^2ℛ is an involution. This symmetry determines the universal critical value C^*=1/(1+√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↦1/K, hence C^*↦1−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^−1ψ(γ) is antisymmetric under σ, it has an exact node at the fixed point of σ, and it governs the exact sign law sgn(C(π,γ)−C^*) =sgn⟨π,ψ(γ)⟩. 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.
ER -