Separatrix structure and the geometry of reset distributions
Abstract
We study the geometry of reset distributions for absorbed Markov processes with geometric resetting, working at an abstract level that isolates the structural mechanism underlying reset-neutral invariance. We show that the spectral duality endows the simplex of reset distributions with a non-trivial spectral response geometry: the simplex Δm−1 carries a foliation by level sets of the coupling functional C, organized around a critical manifold Σ. Under four structural conditions (S1)-(S4) on the coupling functional, we establish the existence and explicit characterization of the separatrix Σ, derive the invariant value C* = 1/(1+√K), identify a projective structure in the spectral coefficients, and prove a global sign principle in the two-site case. In the canonical realization, the biased random walk with multi-site geometric resetting, numerical evidence shows that sgn(∂γC) = sgn⟨π−π*,ψ(γ)⟩, where ψ(γ) is the linear response functional at the separatrix; this global orientation does not, however, follow from (S1)-(S4) alone, and is dynamical rather than algebraic information. This is the third paper in a program connecting stochastic resetting with spectral theory and information geometry.
Related
- Spectral duality structures and the Fisher--Rao geometry of reset distributions
- Spectral Duality and Reset-Neutral Distributions in Random Walks with Multi-Site Geometric Resetting
- Learning the Geometry of Admissible Hypotheses through Inductive Bias in Training Distributions
- Dually flat structure of von Mises-Fisher distributions
- Separatrix splitting in a multiferroic neuromorphic oscillator: toward Arnold diffusion
BibTeX
@misc{coso2026separatrix,
title = {{Separatrix structure and the geometry of reset distributions}},
author = {Juan Antonio Vega coso},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.10717},
eprint = {2607.10717},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.10717},
}
RIS
TY - UNPB
TI - Separatrix structure and the geometry of reset distributions
AU - Juan Antonio Vega coso
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2607.10717
UR - https://arxiv.org/abs/2607.10717
AB - We study the geometry of reset distributions for absorbed Markov processes with geometric resetting, working at an abstract level that isolates the structural mechanism underlying reset-neutral invariance. We show that the spectral duality endows the simplex of reset distributions with a non-trivial spectral response geometry: the simplex Δ_m−1 carries a foliation by level sets of the coupling functional C, organized around a critical manifold Σ. Under four structural conditions (S1)-(S4) on the coupling functional, we establish the existence and explicit characterization of the separatrix Σ, derive the invariant value C^* = 1/(1+√K), identify a projective structure in the spectral coefficients, and prove a global sign principle in the two-site case. In the canonical realization, the biased random walk with multi-site geometric resetting, numerical evidence shows that sgn(∂_γC) = sgn⟨π−π^*,ψ(γ)⟩, where ψ(γ) is the linear response functional at the separatrix; this global orientation does not, however, follow from (S1)-(S4) alone, and is dynamical rather than algebraic information. This is the third paper in a program connecting stochastic resetting with spectral theory and information geometry.
ER -