arXiv subjects

Juan Antonio Vega Coso

Publications and source records attributed to Juan Antonio Vega Coso.

6 recordsLinked to original sources

arXiv2026

Spectral duality structures and the Fisher--Rao geometry of reset distributions

We study the geometry that spectral duality induces on the simplex of reset distributions for absorbed Markov processes with geometric resetting. In the Fisher--Rao metric, selected by Čencov's theorem, the separatrix $Σ$ of reset-neutral distributions is rigid: under the square-root embedding $π\mapsto2\sqrtπ$ its interior is a totally geodesic subsphere of the positive octant, itself a lower-dimensional Fisher--Rao simplex. The reset response reduces to a finite structure: the response functionals span a subspace whose dimension $r$ equals the number of active orbits of the duality involution, and the local separatrix is its annihilator. At the vertices of the simplex we prove a sign theorem valid for every $r$, which recovers the two-zone phenomenon of the first paper of this series. The invariant $r$ also settles the question of global orientation: for $r=1$, under a scalar sign condition satisfied by the canonical realisation, the sign of the response is fixed by the side of $Σ$; for $r\ge2$ the orientation cannot stay constant, and we give the criterion under which no global sign law holds, with counterexamples due to T. Newton showing that it is met in the abstract class. The biased random walk with multi-site geometric resetting realises the whole picture explicitly.

math.PR↗

arXiv2026

Algebra of spectral duality structures

This paper establishes the algebraic and categorical foundations of spectral duality structures (SDS). We show that the class of all SDS admits the structure of a graded symmetric monoidal category, where the degree $K$ is the fundamental invariant indexing the strata. We define operations of Cartesian product and disjoint union, morphisms preserving involution and weights, and prove that $C^\ast=1/(1+\sqrt K)$ is a functor constant on each stratum. We classify objects up to isomorphism by the combinatorial type $(k,f)$, the degree $K$, and a multiset of inversion classes $[r]=\{r,r^{-1}\}$. In each degree, skeletons are exactly the finite $C_2$-sets over the additive group of the reals with $C_2$ acting by $t\mapsto -t$, which explains the monoidal structure and identifies the classification with the decomposition into orbits. Furthermore, we prove that the response rank is the rank of an explicit matrix built from the amplitudes and the weights, and hence exactly the number of non-trivial pairs $k$ under the full-rank condition, and we connect the categorical structure with the Fisher--Rao geometry of Paper V. The SDS, originally discovered in stochastic resetting problems, is thus an autonomous mathematical object with a rich algebraic structure and a natural geometric realisation.

math.CT↗

arXiv2026

Resolvent intertwining and spectral duality in Markov chains with geometric resetting

We uncover the resolvent origin of the spectral duality governing reset-neutral distributions in Markov chains with geometric resetting. Starting from the abstract conditions of Paper~III, we show that the spectral duality $B_ν(z)=κ(z)\,A_ν(σ(z))$ is equivalent to a single symmetry of the resolvent $R(γ)=(I-(1-γ)P)^{-1}$: the intertwining relation $[Δ^2\mathcal{R},R(γ)]=0$, where $\mathcal{R}$ is the reflection operator of an involution $σ$ and $Δ=\operatorname{diag}(\sqrt{κ(z)})$; equivalently, $\widetilde{\mathcal{T}}=K^{-1/2}Δ^2\mathcal{R}$ is an involution. This symmetry determines the universal critical value $C^*=1/(1+\sqrt{K})$, with $K=κ(z)κ(σ(z))$, which depends only on the scalar $K$ --- not on the resetting rate $γ$, the reset distribution, or the particular chain. We characterize the class of $(σ,κ)$-reversible chains, encompassing both the biased random walk and genuinely non-homogeneous dynamics sharing the same $C^*$; a Doob $h$-transform realizes the duality $K\mapsto1/K$, hence $C^*\mapsto1-C^*$, with fixed point $C^*=1/2$. The orientation field admits the explicit resolvent representation $ψ(γ)=R(γ)(b^{(0)}-C^*b)$: its gauge-normalized form $Δ^{-1}ψ(γ)$ is antisymmetric under $σ$, it has an exact node at the fixed point of $σ$, and it governs the exact sign law $\operatorname{sgn}(C(π,γ)-C^*) =\operatorname{sgn}\langleπ,ψ(γ)\rangle$. Numerical experiments confirm the theory to machine precision. These results establish the operator-theoretic foundation of the spectral duality of Paper~III and provide the bridge to the information-geometric framework of Paper~V.

math.PR↗

arXiv2026

Axioms and internal rigidity of spectral duality structures

Spectral duality structures (SDS) explain why a distinguished family of initial distributions makes the absorption probabilities of resetting processes insensitive to the resetting mechanism. Here we study their axioms. We prove that the six defining conditions are mutually independent and determine what each one contributes. The key tool is a structure theorem valid for an arbitrary permutation and an arbitrary positive weight: every insensitive distribution is supported on a union of cycles balanced at a common level, and the level of a cycle is fixed by the geometric mean of the weight on it. From it, positivity selects the admissible root, the constant orbit product is equivalent to the existence of an insensitive distribution of full support, and the involution is equivalent to the response rank being equal to the number of free weight parameters. Without the duality relation, insensitivity of a single distribution is cheap and generically isolated; duality turns it into structural neutrality of the separatrix. The full-rank condition splits into two layers: a weaker condition is equivalent to maximal response rank, while full rank also excludes insensitivity at any other level. We then show that a single SDS is internally rigid: compatible semigroup structures force a constant weight, compatible abelian group structures obey a precise arithmetic condition, congruences form a simple lattice, and the automorphism group is computed explicitly. Finally, the axioms do not determine the orientation of the response; in the two-site case we give the exact condition under which the geometric and analytic sign criteria agree.

math.PR↗

arXiv2026

Spectral Duality and Reset-Neutral Distributions in Random Walks with Multi-Site Geometric Resetting

We study the gambler's ruin problem for a biased random walk on $\{0,1,\dots,a\}$ under multi-site geometric resetting: at each step the walker is reset with probability $γ\in(0,1)$ to a position drawn from a distribution $π$ over $m$ interior sites. Using renewal theory, we derive a closed-form expression for the ruin probability, showing that the whole effect of $π$ is encoded in a single scalar, the coupling constant $C(π,γ)$. A spectral analysis via the Doob symmetrization reveals its structure. Our central result is a general criterion, valid for any absorbed Markov chain with a spectral decomposition, for the existence of a reset-neutral distribution $π^*$ with $C(π^*,γ)$ independent of $γ$. The criterion is a spectral duality condition: an involution $σ$ on the reset sites and mode-independent weights $κ$ with $B_ν(z)=κ(z)A_ν(σ(z))$ for all modes $ν$, and a common pair product $κ(z)κ(σ(z))=K$. The invariant value is then $C^*=1/(1+\sqrt{K})$. For the biased random walk the condition is equivalent to the geometric symmetry $z_i+z_i'=a$, and $C^*$ is the classical ruin probability from the midpoint, for any domain size, number of reset sites, bias and resetting rate. Since the duality holds step by step in time, the critical distributions are reset-neutral for every resetting mechanism independent of the walk under which absorption is almost sure. Algebraic and Monte Carlo verifications confirm the theory to machine and statistical precision, and numerical illustrations reveal a phase-like structure in the space of reset distributions, with $π^*$ acting as a separatrix between monotone regimes.

math.PR↗

arXiv2026

Channel Duality and Midpoint Invariance in Random Walks with Resetting

We study absorption of a biased nearest-neighbor random walk on a finite interval that is reset to its initial position. The central contribution is an exact duality between the two absorption channels, which holds mode by mode in the spectral representation and step by step in time, together with the invariance it produces: when the interval has even length, the ruin probability from its midpoint equals the classical reset-free value for every law of the reset times independent of the walk, provided absorption occurs almost surely. Away from the midpoint, the same duality shows that every such resetting mechanism favors the nearer boundary. For geometric resetting we obtain the ruin probability in closed form and prove that its response to the resetting probability changes sign at the midpoint and nowhere else, for every bias; when the interval has odd length the sign change falls between the two central sites. For a diffusion with drift the midpoint invariance holds for every resetting mechanism, and under Poissonian resetting the sign change is again unique. Channel duality is thus the mechanism behind the invariance and behind its robustness to the resetting protocol and to the passage to the continuum. All analytic results are checked against an independent linear solve and against simulation.

math.PR↗