Resolvent intertwining and spectral duality in Markov chains with geometric resetting

[2608.15140] Resolvent intertwining and spectral duality in Markov chains with geometric resetting Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Probability arXiv:2608.15140 (math) [Submitted on 15 Aug 2026] Title: Resolvent intertwining and spectral duality in Markov chains with geometric resetting Authors: Juan Antonio Vega Coso View a PDF of the paper titled Resolvent intertwining and spectral duality in Markov chains with geometric resetting, by Juan Antonio Vega Coso View PDF HTML (experimental) 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 $B_\nu(z)=\kappa(z)\,A_\nu(\sigma(z))$ is equivalent to a single symmetry of the resolvent $R(\gamma)=(I-(1-\gamma)P)^{-1}$: the intertwining relation $[\Delta^2\mathcal{R},R(\gamma)]=0$, where $\mathcal{R}$ is the reflection operator of an involution $\sigma$ and $\Delta=\operatorname{diag}(\sqrt{\kappa(z)})$; equivalently, $\widetilde{\mathcal{T}}=K^{-1/2}\Delta^2\mathcal{R}$ is an involution. This symmetry determines the universal critical value $C^*=1/(1+\sqrt{K})$, with $K=\kappa(z)\kappa(\sigma(z))$, which depends only on the scalar $K$ --- not on the resetting rate $\gamma$, the reset distribution, or the particular chain. We characterize the class of $(\sigma,\kappa)$-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 $\psi(\gamma)=R(\gamma)(b^{(0)}-C^*b)$: its gauge-normalized form $\Delta^{-1}\psi(\gamma)$ is antisymmetric under $\sigma$, it has an exact node at the fixed point of $\sigma$, and it governs the exact sign law $\operatorname{sgn}(C(\pi,\gamma)-C^*) =\operatorname{sgn}\langle\pi,\psi(\gamma)\rangle$. 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. Comments: 17 pages, 3 figures, 5 tables Subjects: Probability (math.PR) MSC classes: 60J10 (Primary), 60J35, 47A10 (Secondary) Cite as: arXiv:2608.15140 [math.PR] (or arXiv:2608.15140v1 [math.PR] for this version) https://doi.org/10.48550/arXiv.2608.15140 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Juan Antonio Vega Coso [ view email ] [v1] Sat, 15 Aug 2026 09:31:30 UTC (30 KB) Full-text links: Access Paper: View a PDF of the paper titled Resolvent intertwining and spectral duality in Markov chains with geometric resetting, by Juan Antonio Vega Coso View PDF HTML (experimental) TeX Source view license Current browse context: math.PR < prev | next > new | recent | 2026-08 Change to browse by: math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from

Related documents

• Resolvent intertwining and spectral duality in Markov chains with geometric resetting#611089
• Markov processes associated with L-p-resolvents and applications to stochastic differential equations on Hilbert space#492803
• Same Evidence, Different Answers: Canonical-Context On-Policy Distillation for Multi-Turn Language Models#238655
• Large deviations for reducible Markov chains#364128
• Remarks on DAG exchangeability and generalized wreath products#1119140