Year: 2026
Type: preprint
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 $Δ_{m-1}$ 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 $C^* = 1/(1+\sqrt{K})$, 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 $\mathrm{sgn}(\partial_γC) = \mathrm{sgn}\langleπ-π^*,ψ(γ)\rangle$, 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.
Source: arXiv (Cornell University)
Author: coso, Juan Antonio Vega
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: Universidad de Salamanca
OpenAlex ID: W7168276520