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 ${\Delta }_{m-1}$ carries a foliation by level sets of the coupling functional $C$, organized around a critical manifold $\Sigma$. Under four structural conditions (S1)-(S4) on the coupling functional, we establish the existence and explicit characterization of the separatrix $\Sigma$, derive the invariant value $C^{\ast }=1/(1+\sqrt{K})$, 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 $\mathrm{sgn}({\partial }_{\gamma }C)=\mathrm{sgn}⟨\pi -{\pi }^{\ast },\psi (\gamma )⟩$, where $\psi (\gamma )$ 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.

arXiv comment15 pages, 5 figures. v2: global orientation conjecture withdrawn; corrections to Lemma 3.1, Theorem 3.3 and Proposition 5.8; remark on independence of the resetting mechanism (following the revised Paper I); numerical data of Test 2 regenerated; references updated