发表机构
Universidad de Salamanca（萨拉曼卡大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究具有几何重置的吸收马尔可夫过程的重置分布几何，通过谱对偶赋予其非平凡几何。在特定条件下建立分离线相关性质，推导不变值等，线性泛函成全局定向场，有偏随机游走提供实现，是系列论文第三篇。
AI 中文摘要
我们研究具有几何重置的吸收马尔可夫过程的重置分布几何，在抽象层面隔离重置中性不变性的结构机制。我们表明谱对偶赋予重置分布单纯形非平凡的谱响应几何：单纯形\(\Delta_{m - 1}\)由耦合泛函\(C\)的水平集构成叶状结构，围绕作为重置响应全局定向边界的临界流形\(\Sigma\)组织。在耦合泛函的四个结构条件（S1） - （S4）下，我们建立了分离线\(\Sigma\)的存在性和明确特征，推导不变值\(C^* = 1/(1 + \sqrt{K})\)，识别谱系数中的射影结构，并在两点情况证明全局符号原理。线性泛函\(\psi(\gamma)\)作为全局定向场出现：数值证据表明对于所有\(\pi \in \Delta_{m - 1}^\circ\)，\(\operatorname{sgn}(\partial_\gamma C) = \operatorname{sgn}\langle\pi - \pi^*,\psi(\gamma)\rangle\)。具有多站点几何重置的有偏随机游走提供了一个典型实现。这是将随机重置与谱理论和信息几何联系起来的系列论文中的第三篇。
英文摘要
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.
Comments15 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 v2: global orientation conjecture withdrawn; corrections to Lemma 3.1, Theorem 3.3 and Proposition 5.8; remark on independence of the resetting mechanism (following the revised Paper I); numerical data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
