谱对偶结构与重置分布的Fisher-Rao几何
Spectral duality structures and the Fisher--Rao geometry of reset distributions
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中文总结 AI 辅助
本文研究具有几何重置的吸收马尔可夫过程重置分布单纯形上的谱对偶结构与Fisher-Rao几何，证明符号定理、解决定向原理，用多站点几何重置随机游走实现该构造，是系列工作第五篇。
中文摘要 AI 辅助
我们研究谱对偶在具有几何重置的吸收马尔可夫过程的重置分布单纯形上诱导的几何。Fisher-Rao度量提供了内蕴几何：在平方根嵌入下，重置中性分岔线Σ成为全测地子球和低维Fisher-Rao单纯形。随后我们将重置响应约化为有限结构：响应泛函ψ(γ)张成子空间V，其维数r等于对偶对合的活跃轨道数，而局部分岔线是其零化子。在单纯形的顶点处，我们证明了对任意r都成立的符号定理，恢复了第一篇论文中的两区现象。不变量r也解决了第三篇论文中猜想的全局定向原理。当r=1时，所有响应泛函共线，且在典范实现满足的标量符号条件下，响应在Σ两侧具有固定符号；当r≥2时，响应张成空间维数至少为2，定向可旋转，我们给出了全局符号律失效的判据，并在抽象类中构造了反例。具有多站点几何重置的有偏随机游走明确实现了整个构造。这是将随机重置与谱理论和信息几何联系起来的系列工作的第五篇论文。
英文摘要
We study the geometry that spectral duality induces on the simplex of reset distributions for absorbed Markov processes with geometric resetting. The Fisher--Rao metric provides the intrinsic geometry: under the square-root embedding, the reset-neutral separatrix $Σ$ becomes a totally geodesic subsphere and a Fisher--Rao simplex of lower dimension. We then reduce the reset response to a finite structure: the response functionals $ψ(γ)$ span a subspace $V$ whose dimension $r$ equals the number of active orbits of the duality involution, while the local separatrix is its annihilator. At the vertices of the simplex we prove a sign theorem valid for every $r$, recovering the two-zone phenomenon of Paper~I. The invariant $r$ also resolves the global orientation principle conjectured in Paper~III. For $r=1$, all response functionals are collinear and, under a scalar sign condition satisfied by the canonical realisation, the response has a fixed sign on each side of $Σ$. For $r\ge2$, the response span has dimension at least two and the orientation can rotate; we give the criterion for the failure of a global sign law and exhibit counterexamples in the abstract class. The biased random walk with multi-site geometric resetting realises the whole construction explicitly. This is the fifth paper in a program connecting stochastic resetting with spectral theory and information geometry.
