Spatial Disease Propagation With Hubs

Question: How can disease propagation through common hubs be rigorously modeled using random bipartite geometric (RBG) graphs, and what are the key theoretical insights regarding: | | the degree statistics of agents and hubs under general connection functions, | | the conditions under which a critical hub density exists or vanishes depending on the support of the connection function, and | | the impact of dispersion in the connection function on percolation thresholds and large-scale outbreak risk?

| Model spatial disease propagation through **random bipartite geometric (RBG) graphs**. Agents are distributed according to a stationary point process $\Phi$ of intensity $\lambda$, and hubs are distributed according to another point process $\Psi$ of intensity $\mu$. An agent $x \in \Phi$ connects to a hub $y \in \Psi$ with probability | | $$ | \mathbb{P}((x,y)\in E) = f(\|x-y\|), | $$ | | where $f:\mathbb{R}_+ \to [0,1]$ is a non-increasing **connection function**. | | --- | | ## 1. Degree Statistics | | For a typical agent at the origin, the expected degree is given by **Campbell’s theorem**: | | $$ | \mathbb{E}_o \Bigg[\sum_{y\in\Psi} I(o,y)\Bigg] = \mu c_d d \int_0^\infty f(r) r^{d-1} dr, | $$ | | where $c_d$ is the volume of the unit ball in $\mathbb{R}^d$. | Symmetrically, for a typical hub, the mean degree is: | | $$ | \mathbb{E}\big[\deg(h)\big] = \lambda c_d d \int_0^\infty f(r) r^{d-1} dr. | $$ | | For **independent Poisson point processes (PPPs)**, let | | $$ | M = \# \{\text{agents connected to origin}\}, \quad N = \# \{\text{two-edge paths from origin to other agents}\}. | $$ | | Then: | | $$ | \mathbb{E}[N] = \lambda \mu \left( \int_{\mathbb{R}^d} f(\|x\|) dx \right)^2, \tag{1} | $$ | | $$ | \mathbb{E}[M] = \lambda \int_{\mathbb{R}^d} \Big(1 - e^{-\mu \int_{\mathbb{R}^d} f(\|y\|) f(\|x-y\|) dy}\Big) dx. \tag{2} | $$ | | Both $M, N$ are **super-Poisson** random variables, with variances exceeding their means. | | --- | | ## 2. Critical Hub Density and Percolation | | We define the **percolation region**: | | $$ | D = \{ (\lambda,\mu): G(\Phi,\Psi,f)\ \text{percolates} \}. | $$ | | Percolation means the existence of an infinite connected component almost surely. | | ### Theorem 1 (Threshold Existence via Support of $f$) | | * If $f$ has **bounded support** on $[0,a]$, then the critical hub density satisfies | | $$ | \mu_c = \zeta(2a) > 0, | $$ | | where $\zeta(2a)$ is the percolation threshold of the Boolean model with radius $2a$. | | * If $f$ has **unbounded support**, then | | $$ | \mu_c = 0, | $$ | | i.e., *arbitrarily small hub density suffices for percolation if agent density is high enough*. | | This dichotomy shows that **long-distance interactions (unbounded $f$) fundamentally remove the epidemic threshold**. | | --- | | ## 3. Dispersion Effect | | For any $p \in (0,1]$, define the **dispersed connection function**: | | $$ | f_p(r) = p f(\sqrt{d}\,pr). | $$ | | This transformation spreads out connections without changing mean degree: | | $$ | \int_0^\infty f(r) r^{d-1} dr = \int_0^\infty f_p(r) r^{d-1} dr. | $$ | | ### Theorem 2 (Effect of Dispersion) | | For all $f$ and $p \in (0,1]$, | | $$ | \lambda_c^p(\mu) \leq \lambda_c(\mu), \quad \mu_c^p \leq \mu_c. | $$ | | Thus, **increasing dispersion lowers the percolation threshold** and enlarges the percolation region. Epidemiologically, this means allowing agents to travel further (even at lower probability) makes large-scale outbreaks more likely. | | --- | | ## 4. Practical Implications | | * The RBG graph model shows that **hub density thresholds** are not universal: they depend critically on whether human mobility is bounded. | * **Bounding travel distances** (finite support of $f$) guarantees the existence of a threshold below which epidemics cannot spread globally. | * **Allowing rare long-range travel** (infinite support of $f$) destroys this threshold: even sparse hubs can sustain percolation. | * **Increased dispersion** (e.g., flights or long commutes) significantly lowers the epidemic threshold, amplifying outbreak risk. | | --- | | ## Final Synthesis | | The RBG graph framework unifies stochastic geometry and epidemic theory by explicitly linking: | | 1. **Degree distributions** → heterogeneity of individual risk. | 2. **Support of the connection function** → existence vs. absence of epidemic thresholds. | 3. **Dispersion** → sensitivity of outbreak risk to long-range mobility. | | These rigorous results establish that curbing long-distance travel or limiting hub density is mathematically necessary to prevent large-scale disease outbreaks. | |

1. **Model setup**: The answer should correctly describe the RBG graph framework, including the roles of agents, hubs, and the connection function $f$. | 2. **Degree statistics**: The answer should derive or explain the mean degree formulas for agents and hubs using integrals of $f$. | 3. **Percolation threshold condition**: The answer should explain that the existence of a positive hub density threshold depends on whether $f$ has bounded support. | 4. **Critical cases**: The answer should explicitly state the dichotomy: bounded support ⇒ $\mu_c > 0$; unbounded support ⇒ $\mu_c = 0$. | 5. **Dispersion effect**: The answer should address how dispersing the connection function lowers the percolation threshold and enlarges the percolation region. | 6. **Methodological tools**: The answer should mention key analytical tools such as stochastic geometry, Palm calculus, Campbell’s theorem, coupling arguments, or branching processes. | 7. **Practical implications**: The answer should connect the theoretical results to real-world epidemic insights, such as the role of long-distance travel and hub density in determining outbreak risk.

# Background Hints
Airborne infectious diseases spread via physical contact or proximity, and common hubs (e.g., supermarkets, airports) are key to their transmission. Understanding how agent density, hub density, and connectivity patterns jointly affect spread is vital for prevention.

Traditional homogeneous mixture models assume equal contact chances between individuals, with the basic reproduction number as a core indicator, but they fail to capture heterogeneous, location-dependent human interactions. Spatial models (e.g., discrete square lattice Bernoulli percolation, continuum Poisson Boolean models) address this by focusing on proximity-driven interactions and percolation (disease spreading to large components). Poisson random connection models generalize these, with dispersive functions aiding percolation, yet most spatial models overlook hubs.

Some abstract bipartite graph models include hubs but lack geometric dependence. The AB continuum percolation model uses two Poisson point processes but has limited connectivity rules, creating a need for a more general model integrating hubs, geometric dependence, and flexible connection functions.

Remember the key definitions: | | * **Random Bipartite Geometric (RBG) Graph**: $G(\Phi,\Psi,f)$, where $\Phi$ are agents (density $\lambda$), $\Psi$ are hubs (density $\mu$), and edges exist with probability $f(\|x-y\|)$. | * **Connection Function $f$**: a non-increasing function controlling visit probability, possibly bounded or unbounded in support. | * **Degree Statistics**: Mean degree of an agent is | | $$ | \mathbb{E}[\deg(\text{agent})] = \mu c_d d \int_0^\infty f(r) r^{d-1} dr, | $$ | | with symmetric form for hubs. | * **Critical Hub Density $\mu_c$**: the minimal hub density required for the RBG graph to percolate. | * **Dispersed Connection Function**: $f_p(r) = p f(\sqrt{d}\,pr)$, which spreads connections farther while preserving mean degree.

* **Palm calculus & Campbell’s theorem** for deriving mean degrees under point processes. | * **Mass transport principle** to connect agent and hub degree expectations. | * **Coupling arguments** (comparing RBG graphs to Boolean models or Galton–Watson branching processes) to establish percolation thresholds. | * **Probability generating functional (PGFL)** of PPPs to handle edge indicators and variance computations. | * **Comparison between site percolation and bond percolation** to analyze dispersion effects.


On the Popov-Belevitch-Hautus tests for functional observability and output controllability

Question: How can the Popov–Belevitch–Hautus (PBH) tests be generalized to non-diagonalizable linear time-invariant (LTI) systems in order to provide necessary and sufficient conditions for functional observability and output controllability? | | In particular, how can one: | 1. Extend the PBH tests using Jordan decomposition and characterize both properties in the general case? | 2. Establish and formalize the duality between functional observability and output controllability, including minimal algebraic conditions? | 3. Derive theoretical implications for state estimation, control signal design, and eigenspace analysis? | 4. Translate the results into practical design guidelines for observer construction, controller implementation, pole assignment, and optimal sensor/driver placement?

1. Generalized PBH via Jordan decomposition (non-diagonalizable A) — characterization of both properties | Let \mathcal{O}:=O(C,A) and \mathcal{C}:=C(A,B). Write the Jordan form J=PAP^{-1}=\mathrm{diag}(J_1,\dots,J_m), and partition \(\bar C:=CP^{-1}=[C_1~\cdotsC_m]\), \bar F:=FP^{-1}=[F_1\cdots~F_m]. For each eigenvalue \lambda_k, the lead columns of F_k are the columns aligned with the first column of each Jordan block J_{k i}. Then: | • Functional observability (FO) of (C,A;F) holds iff any of the following equivalent conditions holds | \mathrm{rank}\!\begin{bmatrix}\mathcal{O}\\ F\end{bmatrix}=\mathrm{rank}(\mathcal{O})\quad\Longleftrightarrow\quad | \mathrm{rank}\!\begin{bmatrix}\mathcal{O}\\ O(F,A)\end{bmatrix}=\mathrm{rank}(\mathcal{O}). | Moreover, if all nonzero F_k have LI lead columns, FO is also equivalent to the PBH-type rank test (valid for non-diagonalizable A) | \mathrm{rank}\!\begin{bmatrix}\lambda I-A\\ C\\ F\end{bmatrix}=\mathrm{rank}\!\begin{bmatrix}\lambda I-A\\ C\end{bmatrix}\quad \forall\,\lambda\in\mathbb{C}. | When A is diagonalizable, the PBH form above is automatically equivalent to FO (no extra assumption needed). | • Output controllability (OC) of (A,B;F) admits the algebraic test | \mathrm{rank}(F\mathcal{C})=\mathrm{rank}(F)\quad(\text{necessary and sufficient}). | A PBH-type form is | \mathrm{rank}\, \big[\,F(\lambda I-A)\;\; FB\,\big] = \mathrm{rank}(F)\quad \forall\,\lambda\in\mathbb{C}, | which is equivalent to OC if (i) (A,B) is full-state controllable, or (ii) A is diagonalizable; in the general nondiagonalizable case it must be paired with the eigenspace condition stated in item 2) below. ￼ | 2. Duality and minimal algebraic conditions | There is weak duality: FO of (C,A;F) \Rightarrow OC of the dual (A^{\!\top},C^{\!\top};F). Strong duality holds (FO \Leftrightarrow dual OC) iff | F\,\mathrm{Im}(W)\ \perp\ F\,\ker(W), | where W=\int_0^{t_1} e^{A^{\!\top}t}C^{\!\top}Ce^{At}\,dt is the observability Gramian of (C,A). | For OC in the general case, the PBH rank condition must be accompanied by the eigenspace intersection requirement | \ker(\mathcal{C}^{\!\top})\ \cap\ \mathrm{row}(F)\ \cap\ \Big(\bigcup_{i=1}^m E_i\Big)\ \neq\ \varnothing, | where E_i=\{v: A^{\!\top}v=\lambda_i v\}. A convenient sufficient setting is \mathrm{row}(F)\supseteq \ker(\mathcal{C}^{\!\top}), which makes the PBH form necessary and sufficient for OC. ￼ | 3. Implications for estimation, control design, and eigenspace analysis | • Estimation (FO): FO \Leftrightarrow\ \mathrm{row}(F)\subseteq \mathrm{row}(\mathcal{O}). Equivalently, one can reconstruct z(0)=Fx(0) from (y,u) over a finite window; the Jordan-lead-column criterion pinpoints the eigen-Jordan chains that must be “seen” by C and F. In diagonalizable systems, this reduces to simple row-space tests per eigenvalue. | • Control signal design (OC): If W_F(t_1):=F W_c(t_1) F^{\!\top} (with W_c the controllability Gramian) is invertible, then the explicit control | u(t)= -\,B^{\!\top} e^{A^{\!\top}(t_1-t)} F^{\!\top}\, W_F^{-1}(t_1)\,\big(F e^{A t_1}x(0)-z(t_1)\big) | steers z from z(0) to z(t_1). Non-diagonalizable A requires verifying the eigenspace intersection above to ensure the PBH form suffices. | • Eigenspace diagnostics: Failure cases of the PBH-only test (without the intersection condition) occur exactly when F does not “hit” any eigenvector in \ker(\mathcal{C}^{\!\top}); conversely, LI lead columns of \bar F guarantee the FO-PBH equivalence even with Jordan chains. ￼ | 4. Practical design guidelines (observer, controller, pole assignment, sensor/driver placement) | • Observer construction (FO): First check \mathrm{rank}\big([\mathcal{O};F]\big)=\mathrm{rank}(\mathcal{O}). If A has Jordan blocks, verify LI lead columns of \bar F=FP^{-1}; if violated, redesign F (or augment outputs C) to cover the first vectors of each active Jordan chain. | • Controller implementation (OC): Prefer the algebraic test \mathrm{rank}(F\mathcal{C})=\mathrm{rank}(F). If using PBH, ensure either (a) (A,B) is full-state controllable, (b) A is diagonalizable, or (c) enforce \mathrm{row}(F)\supseteq \ker(\mathcal{C}^{\!\top}) (e.g., by shaping F or adding drivers so the intersection condition is satisfied). | • Pole assignment: Apply the FO/OC PBH tests per eigenvalue (or Jordan chain): an unobservable/uncontrollable target component at \lambda cannot be assigned independently—fix C,F (for FO) or B,F (for OC) to include the corresponding lead directions. | • Optimal sensor/driver placement: Use the PBH–Jordan view to pick rows of C (sensors) and rows of F (target functionals) that span lead columns of the relevant blocks (FO), and choose B and F so that \mathrm{row}(F) intersects \ker(\mathcal{C}^{\!\top}) along some eigenvector (OC); in diagonalizable cases, eigenvector coverage per \lambda_k suffices. ￼

1. Defines functional observability and output controllability in terms of (C,A;F) and (A,B;F). | 2. States the rank conditions for functional observability: | \mathrm{rank}\big([\mathcal{O};F]\big)=\mathrm{rank}(\mathcal{O}) and | \mathrm{rank}\big([\mathcal{O};O(F,A)]\big)=\mathrm{rank}(\mathcal{O}). | 3. Clarifies the PBH-type equivalence for FO under the linear independence of lead columns of F_k. | 4. Provides the algebraic test for OC: \mathrm{rank}(F\mathcal{C})=\mathrm{rank}(F). | 5. Establishes the PBH-type test for OC: \mathrm{rank}[F(\lambda I-A)\; FB]=\mathrm{rank}(F), with clear conditions on full-state controllability, diagonalizability, or eigenspace intersection. | 6. Identifies the eigenspace intersection condition: \ker(\mathcal{C}^T)\cap \mathrm{row}(F)\cap(\cup_i E_i)\neq\varnothing. | 7. Explains the weak duality (FO ⇒ dual OC) and the strong duality (FO ⇔ dual OC with Gramian orthogonality). | 8. Links FO and OC to Gramians: W_o for observability, W_c and W_F for controllability. | 9. Demonstrates the role of Jordan decomposition and lead columns in extending PBH to non-diagonalizable systems. | 10. Provides practical implications: observer construction, controller implementation, pole assignment, and sensor/driver placement strategies. | 11. Ensures symbol consistency: \mathcal{O},\mathcal{C}, J, J_k, F_k, E_i all used uniformly. | 12. Clarifies assumptions explicitly (e.g., rank(F)=r, LI of lead columns, row-space inclusion conditions).

1. LTI model with target output: \dot{x}=Ax+Bu,\; y=Cx,\; z=Fx. The challenge is estimating/controlling z when r \ll n. | 2. Functional observability ensures z(0) can be uniquely reconstructed from (y,u) over finite horizon. | 3. Output controllability ensures z(t) can be driven to any final value by some input u(t). | 4. Classical PBH tests cover only full-state observability/controllability and fail for non-diagonalizable A. | 5. Recent progress: generalized PBH for functional observability (using rank and Jordan decomposition), new PBH-type test for output controllability, and explicit duality linking the two ￼.

1. Observability matrix \mathcal{O}(C,A) = [C^T, (CA)^T,\dots,(CA^{n-1})^T]^T. | 2. Controllability matrix \mathcal{C}(A,B) = [B, AB, A^2B, \dots, A^{n-1}B]. | 3. Jordan decomposition: J = PAP^{-1} = \mathrm{diag}(J_1,\dots,J_m). | 4. Each Jordan submatrix J_k = \mathrm{diag}(J_{k1},\dots,J_{km_k}) encodes eigenvalue \lambda_k with algebraic multiplicity n_k and geometric multiplicity m_k. | 5. Lead columns: for each C_k or F_k aligned with J_k, the columns corresponding to the first column of each Jordan block J_{ki}; their linear independence is crucial for PBH equivalence. | 6. Compact notation: \mathcal{O}\equiv \mathcal{O}(C,A),\;\mathcal{C}\equiv \mathcal{C}(A,B)

1. Functional observability ： equivalence of | • \mathrm{rank}\big([\mathcal{O};F]\big)=\mathrm{rank}(\mathcal{O}), | • \mathrm{rank}\big([\mathcal{O};O(F,A)]\big)=\mathrm{rank}(\mathcal{O}), | • PBH-type rank test \mathrm{rank}\begin{bmatrix}\lambda I-A\\ C\\ F\end{bmatrix}=\mathrm{rank}\begin{bmatrix}\lambda I-A\\ C\end{bmatrix} for all \lambda, under LI lead columns assumption. | 2. Output controllability : equivalent to \mathrm{rank}(F\mathcal{C})=\mathrm{rank}(F); PBH form \mathrm{rank}[F(\lambda I-A)\; FB]=\mathrm{rank}(F) holds if (A,B) full-state controllable or with added eigenspace intersection condition. | 3. Eigenspace condition: require \ker(\mathcal{C}^T)\cap\mathrm{row}(F)\cap(\cup_i E_i)\neq \varnothing, ensuring that F intersects eigenvectors in uncontrollable directions. | 4. Duality: weak duality states FO implies dual OC; strong duality requires orthogonality F\,\mathrm{Im}(W)\perp F\,\ker(W) where W is the observability Gramian. | 5. Practical implications: Gramian-based expressions yield explicit observer/controller formulas; Jordan–PBH equivalence guides sensor and actuator placement


On polynomials associated to Voronoi diagrams of point sets and crossing numbers

Question: How can one develop a rigorous framework that connects Voronoi diagrams, circle polynomials, and $(\leq k)$-edge polynomials of point sets to the rectilinear crossing number? In particular, what structural relations among these polynomials allow the derivation of exact formulas for the crossing number through their coefficients and roots, and how can these results be used to characterize when all roots lie on the unit circle and to establish bounds on their location for general point sets?

For a set $S$ of $n$ points in general position in the plane, the rectilinear crossing number $cr(S)$, which counts the number of pairwise edge crossings in the complete graph $K_n$ when drawn with straight lines, is closely related to three key polynomials associated with Voronoi diagrams and geometric properties of the point set: | | 1. **Voronoi Polynomial** | The Voronoi polynomial $p_V(z)$ is defined as: | | $$ | p_V(z) = \sum_{k=1}^{n-1} v_k z^{k-1}, | $$ | | where $v_k$ represents the number of vertices in the order-$k$ Voronoi diagram of the point set $S$. The vertices of the Voronoi diagram are linked to the combinatorics of point sets, where the coefficients of this polynomial provide insights into the geometric arrangement of the points in $S$. | | 2. **Circle Polynomial** | The circle polynomial $p_C(z)$ is defined as: | | $$ | p_C(z) = \sum_{k=0}^{n-3} c_k z^k, | $$ | | where $c_k$ counts the number of circles passing through three points of $S$ and enclosing exactly $k$ other points of $S$. The relationship between the circle polynomial and the Voronoi diagram is critical in understanding the spatial structure of the points and their influence on the rectilinear crossing number. | | 3. **$(\leq k)$-Edge Polynomial** | The $(\leq k)$-edge polynomial $p_E(z)$ is defined as: | | $$ | p_E(z) = \sum_{k=0}^{n-3} E_{\leq k} z^k, | $$ | | where $E_{\leq k}$ counts the number of edges in the set $S$ that have at most $k$ points on one side. This polynomial encodes higher-order geometric properties of the set, linking the arrangement of edges to the combinatorial structure of the Voronoi diagram. | | ### **Mathematical Insights and Contributions**: | | * **Crossing Number Expression**: The rectilinear crossing number $cr(S)$ can be derived from the derivatives of the polynomials $p_V(z)$, $p_C(z)$, and $p_E(z)$ evaluated at $z = 1$: | | * For $p_C(z)$: | | $$ | p'_C(1) = \binom{n}{4} + cr(S) | $$ | * For $p_V(z)$: | | $$ | p'_V(1) = \binom{n}{3} + 2 \binom{n}{4} + 2cr(S) | $$ | * For $p_E(z)$: | | $$ | p'_E(1) = 9 \binom{n}{4} - cr(S) | $$ | | These relationships establish a direct connection between the coefficients of the polynomials and the crossing number, allowing for its computation in terms of algebraic quantities. | | * **Roots of the Polynomials**: The roots of the **circle polynomial** $p_C(z)$ and **Voronoi polynomial** $p_V(z)$ are critical in determining the geometric arrangement of the points. In particular, the roots of the Voronoi polynomial lie on the **unit circle** if and only if the set $S$ is in **convex position**. This result is significant because it provides a geometric characterization of the point set's configuration, linking the crossing number to the distribution of roots. | | * **Lower Bound on Root Modulus**: For sets of points in general position, bounds on the modulus of the largest root of the **circle polynomial** $p_C(z)$ and **Voronoi polynomial** $p_V(z)$ are derived. Specifically, for point sets that minimize the rectilinear crossing number $cr(S) = \alpha \cdot \binom{n}{4}$, the largest root of $p_V(z)$ is bounded below by: | | $$ | |z_{\text{max}}| \geq 1 + \frac{(1-\alpha) \pi}{16(n-3)^2} + O\left(\frac{1}{n^4}\right) | $$ | | This provides a lower bound on the root locations, which is essential for understanding the behavior of the crossing number with respect to point set configurations. | | * **Generalization to $(\leq k)$-Edge Polynomial**: A similar approach is applied to the **$(\leq k)$-edge polynomial**, yielding bounds on the largest modulus of its roots. For point sets with crossing number $cr(S) = \alpha \cdot \binom{n}{4}$, the largest root of the $p_E(z)$ polynomial is bounded below by: | | $$ | |z_{\text{max}}| \geq \frac{3 + \alpha}{9 - \alpha} | $$ | | These bounds are key in understanding the intersection between algebraic properties (roots of polynomials) and geometric configurations (the structure of the point set). | | --- | | ### **Conclusion**: | | The three polynomials—**Voronoi polynomial**, **circle polynomial**, and **$(\leq k)$-edge polynomial**—are deeply connected to the **rectilinear crossing number** of point sets. By analyzing their coefficients, derivatives, and roots, one can derive exact formulas for the crossing number and establish geometric conditions for when the roots lie on the unit circle. The results also provide bounds on the modulus of the largest roots, which are crucial for understanding the behavior of crossing numbers in various geometric configurations. | |

"1. Model setup: Does the answer correctly define the Voronoi, circle, and (≤k)-edge polynomials and their relation to point set geometry?", | "2. Derivative formulas: Does the answer provide the derivative identities at z=1 that link these polynomials to cr(S)?", | "3. Root-based relations: Does the answer explain how cr(S) can also be expressed through sums involving the roots of the polynomials?", | "4. Unit circle characterization: Does the answer state that all roots of the Voronoi polynomial lie on the unit circle iff the point set is convex?", | "5. Bounds for non-convex sets: Does the answer provide lower bounds for the modulus of roots when the set is not convex, linked to α = cr(S)/C(n,4)?", | "6. Extension to (≤k)-edge polynomial: Does the answer include results like |z_max| ≥ (3+α)/(9-α) for its roots?", | "7. Methodological tools: Does the answer mention relevant tools such as Aziz–Mohammad identity, palindromic polynomial root theorems, Laguerre and Obrechkoff theorems, Eneström–Kakeya, and analytic bounds?", | "8. Practical implications: Does the answer connect algebraic results to geometric implications for extremal crossing numbers?" | ]

| | * Recall that the **rectilinear crossing number** $cr(S)$ of a point set $S$ is the number of convex quadrilaterals or equivalently the number of edge crossings in the straight-line drawing of $K_n$ on $S$. | * Known results establish relations between **circle counts**, **Voronoi diagram vertices**, and **$(\leq k)$-edges**, which connect combinatorial geometry with crossing numbers. | * Classical approaches: inequalities for $c_k$, relations $v_k = c_{k-1} + c_{k-2}$, and formulas linking sums of $k\cdot c_k$ to binomial coefficients plus crossing numbers. | * Understanding root distributions of polynomials derived from these combinatorial quantities (Voronoi, circle, and edge polynomials) is important for analyzing extremal properties of $cr(S)$. |

* **Voronoi Polynomial**: $p_V(z) = \sum_{k=1}^{n-1} v_k z^{k-1}$, where $v_k$ counts vertices in the order-$k$ Voronoi diagram. | * **Circle Polynomial**: $p_C(z) = \sum_{k=0}^{n-3} c_k z^k$, where $c_k$ counts circles through three points enclosing exactly $k$ others. | * **$(\leq k)$-edge Polynomial**: $p_E(z) = \sum_{k=0}^{n-3} E_{\leq k} z^k$, with $E_{\leq k}$ the number of edges with at most $k$ points to one side. | * **Reciprocal Polynomial**: For a polynomial $P(z) = \sum c_k z^k$, the reciprocal is $P^*(z) = \sum c_k z^{n-k}$, often used in root symmetry arguments. | * **Convex Position Characterization**: All roots of $p_V(z)$ lie on the unit circle if and only if the point set is in convex position.

* Apply **Campbell’s theorem** and combinatorial identities to link coefficients of the polynomials to crossing numbers. | * Use **derivatives at $z=1$** to extract expressions involving $cr(S)$. | * Employ **Aziz–Mohammad identity** to connect polynomial coefficients and roots with crossing number formulas. | * Use **Palindromic polynomial root theorems** to characterize unit circle root distributions in convex configurations. | * Apply **Laguerre’s separation theorem** and **Obrechkoff’s theorem** to bound root locations for non-convex configurations. | * Leverage **Eneström–Kakeya theorem** and **analytic bounds** (e.g., Titchmarsh, Rahman–Schmeisser) to estimate moduli of roots of $p_E(z)$.


Algebraic Geometry codes in the sum-rank metric

Question: How can one construct the first geometric codes in the sum-rank metric, known as linearized Algebraic Geometry codes, using Ore polynomial rings with coefficients from algebraic curve function fields, and how can this construction surpass existing performance bounds? Specifically, how are quotient rings of Ore polynomials defined, how does the sum-rank metric code structure work with evaluation maps, and what are the bounds on the dimension and minimum distance of these codes compared to classical bounds like Goppa’s and Gilbert–Varshamov?

The theory of **linearized Algebraic Geometry (AG) codes in the sum-rank metric** introduces a new class of codes that extend the rich interplay between algebraic geometry and coding theory into a noncommutative framework. These codes arise from combining **Ore polynomial rings** with the function fields of algebraic curves, thereby generalizing both classical AG codes (in the Hamming metric) and linearized Reed–Solomon or Gabidulin codes (in rank and sum-rank metrics). Their construction not only provides explicit new families of codes but also yields parameter bounds that surpass existing performance thresholds, including asymptotic improvements over the Gilbert–Varshamov (GV) bound. | | --- | | ## 1. Ore Polynomial Rings and Noncommutative Riemann–Roch Theory | | The foundation of this construction lies in **Ore polynomial rings**. Given a cyclic Galois extension $L/K$ of function fields of curves with automorphism $\Phi$, one considers the Ore polynomial ring $L[T;\Phi]$, where multiplication is twisted by the relation | | $$ | T \cdot a = \Phi(a) T, \quad a \in L. | $$ | | Taking the quotient by relations of the form $T^r - x$, with $x \in K^\times$, yields algebras $D_{L,x} = L[T;\Phi]/(T^r - x)$. These are central simple algebras that behave as noncommutative analogues of function fields. | | Within this setting, one extends **divisor theory** and the **Riemann–Roch framework**. For each divisor on a covering curve $Y$ lying above a base curve $X$, one defines a Riemann–Roch space $\Lambda_{L,x}(E)$ consisting of Ore polynomial elements subject to valuation constraints at places of $Y$. These spaces are finite-dimensional over the base field and admit explicit lower bounds on their dimensions, closely paralleling the classical Riemann–Roch theorem. Thus, the classical geometric machinery of divisors and function spaces is successfully transplanted into the noncommutative Ore polynomial environment. | | --- | | ## 2. Sum-Rank Metric and Evaluation Mechanism | | The **sum-rank metric** interpolates between Hamming and rank metrics. Given vector space tuples $W = (W_1,\dots,W_s)$ and $V = (V_1,\dots,V_s)$, the ambient space | | $$ | \text{Hom}_k(W,V) = \prod_{i=1}^s \text{Hom}_k(W_i, V_i) | $$ | | carries the sum-rank weight, defined as the sum of the ranks of each linear map. A code in this metric is a linear subspace of this ambient space. | | The construction of linearized AG codes proceeds by defining **evaluation maps**. Fixing rational places on the base curve, one evaluates Ore polynomial functions in the Riemann–Roch space at selected points, producing tuples of linear maps in the ambient space. These evaluations, reduced modulo appropriate local parameters, yield matrices that form the codewords. This mechanism is the natural generalization of evaluating rational functions at rational points to obtain classical AG codes, but now within a noncommutative algebraic framework. | | --- | | ## 3. Explicit Construction and Hypotheses for Well-Defined Codes | | A **linearized Algebraic Geometry code** is defined as the image of the multi-evaluation map | | $$ | \alpha : \Lambda_{L,x}(E) \to \prod_{i=1}^s \text{End}_k(V_i), | $$ | | where $E$ is a chosen divisor on the covering curve, and the $V_i$ are local quotient spaces at evaluation places. Several hypotheses guarantee that this construction is well-defined: | | * The algebra $D_{L,x}$ must contain no zero divisors, ensuring injectivity of evaluation. | * Compatibility conditions at evaluation places must hold so that Ore elements can be consistently reduced. | | Under these assumptions, the codes are rigorously defined, and their parameters can be studied with precision. | | --- | | ## 4. Parameter Bounds: Dimension and Minimum Distance | | The central result concerns **parameter bounds**. For a code $C$ constructed from divisor $E$, one obtains: | | * A **dimension bound**: | | $$ | \dim C \;\geq\; r \cdot \deg(E) - r \cdot (g_Y - 1) - \text{local correction terms}, | $$ | | where $g_Y$ is the genus of the covering curve $Y$, and the correction terms depend on ramification and local invariants. | | * A **minimum distance bound**: | | $$ | d \;\geq\; sr - \deg(E), | $$ | | where $s$ is the number of evaluation places and $r$ the extension degree. | | These bounds generalize Goppa’s bounds for classical AG codes in the Hamming metric. Importantly, the codes simultaneously exhibit large dimension and large minimum distance, a hallmark of good code families. | | --- | | ## 5. Comparison with Classical Bounds | | The parameters of linearized AG codes satisfy an analogue of the Singleton inequality in the sum-rank metric. More significantly, in appropriate field settings—such as finite fields of square cardinality—these codes yield families that **asymptotically surpass the Gilbert–Varshamov bound** in the sum-rank metric. This result mirrors the landmark Tsfasman–Vlăduţ–Zink breakthrough for AG codes in the Hamming metric, establishing that geometry-based constructions can again outperform random coding bounds in a broader metric context. | | --- | | ## 6. Relation to Reed–Solomon and Gabidulin Codes | | Linearized AG codes recover known families as special cases: | | * For the projective line as the base curve, the construction reduces to **linearized Reed–Solomon codes**, which are maximum sum-rank distance (MSRD) codes attaining the Singleton bound. | * By varying evaluation, one recovers **Gabidulin codes**, the classical rank-metric analogue of Reed–Solomon codes. | | Compared to these, linearized AG codes provide **longer lengths** over the same finite fields (thanks to curves with many rational points) and require **smaller field sizes** for given dimensions and distances. This overcomes the key limitation of Reed–Solomon-type codes, which demand large field extensions for long code lengths. | | --- | | ## 7. Asymptotic Goodness and Beyond | | By considering towers of curves with many rational points, one constructs infinite families of linearized AG codes with asymptotically positive rate and relative distance, hence proving their **asymptotic goodness**. Even stronger, when the field size is sufficiently large, these codes strictly improve on the random coding GV bound in the sum-rank setting. This demonstrates that geometry-driven methods remain a source of asymptotically optimal constructions in modern coding theory. | | --- | | ## 8. Broader Perspective | | The emergence of linearized Algebraic Geometry codes marks a conceptual advance: | | * They extend AG code theory to noncommutative algebras (Ore rings). | * They unify and generalize classical Reed–Solomon and Gabidulin families. | * They provide explicit and implementable constructions with strong parameter guarantees. | * They surpass probabilistic bounds asymptotically, reaffirming the power of algebraic geometry in coding. | | This development thus positions linearized AG codes as a central object for future research, bridging noncommutative algebra, algebraic geometry, and information theory. | | --- | | |

1. Defines the algebraic setup with integers \(r,d,m\) (where \(r=md\)), the Galois extension \(L_0/K\), the construction of \(L\), the automorphism \(\Phi\), the Ore polynomial ring \(L[T;\Phi]\), the quotient algebra \(D_{L,x}\), and the reduced norm map. | | 2. States the Laurent series framework: \(O_K=k[[t]], K=k((t))\), the extension of \(v_t\) to \(L_0\), the valuation maps \(w_{j,x}\), the subring \(\Lambda_{L,x}\), and the conditions for \(\gamma_u\) to induce homomorphisms. | | 3. Describes the algebraic curve background: the function field \(K=k(X)\), divisor group \(\mathrm{Div}(X)\), Riemann–Roch spaces, the Galois cover \(\pi:Y\to X\), valuation extensions to \(Y\), and the definition of \(\rho_p\). | | 4. Defines the noncommutative Riemann–Roch space \(\Lambda_{L,x}(E)\), its decomposition into classical spaces \(L_Y(E_i)\), and the derivation of the dimension lower bound. | | 5. States the prerequisites for code construction: rational places \(p_1,\dots,p_s\), the zero-divisor-free condition for \(D_{L,x}\), the norm hypotheses at places, and the setup of the ambient space \(H=\prod_i \mathrm{End}_k(V_i)\). | | 6. Defines the evaluation map \(\alpha:\Lambda_{L,x}(E)\to H\) via the homomorphisms \(\gamma_u\), \(\varepsilon\), and reductions, and introduces the code \(C=\mathrm{Im}(\alpha)\). | | 7. Proves injectivity of \(\alpha\) under the zero-divisor-free assumption, derives the minimum distance bound \(d\ge sr-\deg_Y(E)\), the dimension bound from Riemann–Roch analysis, and formulates the Singleton-type defect. | | 8. Describes the isotrivial cover construction with \(Y=\mathrm{Spec}(\ell)\times_{\mathrm{Spec}(k)}X\), its unramified property, genus relation, and the specialization to linearized Reed–Solomon codes attaining MSRD. | | 9. Establishes the asymptotic performance framework: towers of curves with many rational points, construction of code families \((C_n)\), definitions of rate and relative distance, derivation of asymptotic bounds, and comparison with the sum-rank Gilbert–Varshamov bound. |

| 1. In the Hamming metric, Reed–Solomon codes achieve optimal trade-offs between length, dimension, and distance, but their construction requires large finite fields when long codes are needed. | 2. In the rank and sum-rank metrics, Gabidulin codes and linearized Reed–Solomon codes play analogous roles, but they inherit the same limitation: long codes cannot be constructed without dramatically enlarging the field size. | 3. Algebraic Geometry codes in the Hamming setting provided a breakthrough by evaluating functions on algebraic curves with many rational points, allowing longer codes over smaller fields and even producing asymptotic improvements over random coding bounds. | 4. In contrast, for the rank and sum-rank metrics, no geometric constructions had been available, so there was no analogue of the rich curve-based methods known in the Hamming case. This left open the problem of finding codes in these metrics that combine structural elegance with strong asymptotic parameters. |

| 1. Ore Polynomial Rings: Noncommutative rings consisting of polynomials \(L[T;\Phi]\), where multiplication is twisted by an automorphism \(\Phi\). Quotients of the form \(L[T;\Phi]/(T^r - x)\) provide a natural algebraic environment in which to encode curve-based information. | 2. Noncommutative Riemann–Roch Spaces: Vector spaces of Ore polynomial elements constrained by valuations relative to divisors on algebraic curves. They generalize the classical Riemann–Roch spaces and allow one to transport divisor theory into the noncommutative setting. | 3. Sum-Rank Metric: A distance measure defined on tuples of linear maps or matrices. The weight of a tuple is the sum of the individual ranks, and the code distance is induced from this weight. | 4. Linearized Algebraic Geometry Codes: Codes constructed by evaluating elements from noncommutative Riemann–Roch spaces at selected places of a curve, producing tuples of endomorphisms. These codes generalize Reed–Solomon codes (Hamming metric) and Gabidulin codes (rank metric) into the sum-rank setting. |

| 1. Algebraic Tools from Ore Rings: Properties of Ore polynomial quotients ensure that evaluation maps are well-defined and injective, avoiding zero divisors and guaranteeing consistent code construction. | 2. Dimension Bounds via Riemann–Roch: A generalized Riemann–Roch theorem applied in the noncommutative framework provides lower bounds on the dimension of code spaces. | 3. Distance Analysis through Evaluation Maps: Studying kernels of evaluation morphisms and reduced norm arguments establishes lower bounds on the minimum distance of codewords. | 4. Asymptotic Constructions from Curves: Using towers of algebraic curves with many rational points allows the formation of infinite sequences of codes whose parameters eventually beat the Gilbert–Varshamov bound in the sum-rank metric. |


Variance Decay Property for Filter Stability

Question: How can one rigorously establish the stability of nonlinear filters in continuous-time hidden Markov models with white noise observations, going beyond classical ergodicity and observability arguments, to prove asymptotic forgetting of initial conditions in the χ²-divergence sense? Specifically, under what precise conditions does stability hold, how does variance decay serve as the central mechanism, and how can advanced tools such as minimum variance duality transformations, Poincaré-type inequalities for nonlinear filters, and stochastic observability analysis be used to derive the results?

We address the problem of proving stability of nonlinear filters in continuous-time hidden Markov models (HMMs) with white noise observations. Stability means asymptotic forgetting of the initial condition, measured here in the χ²-divergence sense. The proof combines minimum variance duality, a newly defined Poincaré inequality (PI) for nonlinear filters, and the variance decay property. | | Step 1. Mathematical Setup | • Hidden process: X_t is a Feller-Markov process on state space S with infinitesimal generator A. | • Observation process: | dZ_t = h(X_t)\,dt + dW_t, | where h: S \to \mathbb{R}^m is the observation function and W_t is an m-dimensional Brownian motion, independent of X_t. | • Filter definition: | For f \in C_b(S), | \pi_t(f) := \mathbb{E}[f(X_t) \mid \mathcal{Z}_t], | where \mathcal{Z}_t = \sigma\{Z_s: 0 \leq s \leq t\}. | The process \pi_t is called the nonlinear filter. | • Stability question: Suppose the filter is initialized with two different priors \mu,\nu \in \mathcal{P}(S) with \mu \ll \nu. Denote the two filters \pi_t^\mu and \pi_t^\nu. Stability means: | \mathbb{E}_\mu\big[\chi^2(\pi_T^\mu \,\|\, \pi_T^\nu)\big] \xrightarrow[T \to \infty]{} 0. | | Step 2. Likelihood Ratio and Backward Map | | Define the likelihood ratio at time T: | \gamma_T(x) := \frac{d\pi_T^\mu}{d\pi_T^\nu}(x). | This random variable is \mathcal{Z}_T-measurable. | | Introduce the backward map: | y_0(x) := \mathbb{E}_\nu[\gamma_T(X_T)\mid X_0=x]. | • y_0 is deterministic and satisfies \nu(y_0)=1. | • Crucially, it connects the terminal likelihood ratio \gamma_T back to the initial state. | | Variance bound: | \left(\mathbb{E}\mu\big[\chi^2(\pi_T^\mu \| \pi_T^\nu)\big]\right)^2 | \;\leq\; \mathrm{Var}\nu(y_0(X_0)) \cdot \chi^2(\mu \| \nu). | | Thus, to prove stability, it suffices to show | \mathrm{Var}_\nu(y_0(X_0)) \xrightarrow[T\to\infty]{} 0. | | Step 3. Variance Decay Property | | From the definition of the backward map and Jensen’s inequality, | \mathrm{Var}\nu(y_0(X_0)) \;\leq\; \mathrm{Var}\nu(\gamma_T(X_T)). | | This is the variance decay property: the variance at the initial time (backward-projected) is always smaller than or equal to the variance at the terminal time. | | This property shows that variance contracts along the backward map. To obtain exponential decay, we need a structural inequality that quantifies this contraction — the Poincaré inequality. | | Step 4. Embedding into BSDE Optimal Control | | The variance decay is formalized by embedding the backward map into a backward stochastic differential equation (BSDE): | -dY_t(x) = \Big[(A Y_t)(x) + h(x)^\top(U_t + V_t(x))\Big] dt - V_t(x)^\top dZ_t, | \quad Y_T(x) = \gamma_T(x). | | The associated cost functional is | J(U) = \mathrm{Var}\nu(Y_0(X_0)) + \mathbb{E}\nu\left[\int_0^T \ell(Y_t,V_t,U_t;X_t) dt\right], | with running cost \ell(y,v,u;x) = \Gamma y(x) + \|u+v(x)\|^2, where \Gamma is the carré du champ operator of A. | | This control problem reveals that variance decay is equivalent to optimality of the controlled BSDE system. | | Step 5. Poincaré Inequality for Nonlinear Filters | | For a Markov process with invariant distribution \bar\mu, the classical Poincaré inequality is: | \bar\mu(\Gamma f) \geq c \,\mathrm{Var}_{\bar\mu}(f), | where c>0 is the Poincaré constant. | | In the nonlinear filter setting, an analogous inequality is defined for the dual BSDE system: | \mathbb{E}\rho\Big[\int_0^\tau \pi_t(\Gamma Y_t) + \|V_t\|^2 + |V\rho(h,Y_t)|^2 \, dt \Big] \;\geq\; \beta_\rho^\tau \,\mathrm{Var}\rho(Y_0(X_0)). | This inequality ensures that variance contracts at rate \beta\rho^\tau, giving a filter-specific Poincaré constant c_\rho. | | Step 6. Exponential Decay of χ²-Divergence | | From the variance decay inequality and the PI, one derives: | \mathbb{E}\mu[\chi^2(\pi_T^\mu \| \pi_T^\nu)] \;\leq\; \frac{1}{a}\, e^{-c(T-\tau)} \,\chi^2(\mu\|\nu), | where a = \mathrm{ess\,inf}{x\in S} \gamma_0(x). | | Thus, χ²-divergence decays exponentially, proving stability. | | By known inequalities: | \chi^2 \text{-stability} \;\;\implies\;\; \text{KL-stability, total variation stability, and \(L^2\)-stability}. | | Step 7. Connection to Model Properties | | The positivity of the Poincaré constant is tied to structural properties of the HMM: | 1. Ergodicity of the hidden process: ensures variance contracts due to mixing in the state dynamics. | 2. Observability of the observation model: ensures distinct states produce distinct observation statistics. | 3. Non-degenerate noise: ensures information in the observations is not lost. | | In finite-state HMMs: | • Ergodicity (irreducibility + aperiodicity of the generator), | • Observability (observation matrix full rank), or | • Detectability (all unobservable modes are stable) | → each implies c_\rho > 0, hence stability. | | Step 8. Broader Interpretation | • The variance decay property generalizes the Kalman filter’s covariance decay to nonlinear filtering. | • The new PI unifies previously separate ergodic and observable cases. | • The framework yields not only qualitative stability but also explicit convergence rates, important for algorithm design (e.g., particle filters, reinforcement learning in partially observed systems). | | Final Conclusion | | By reformulating nonlinear filtering through minimum variance duality and BSDE control, introducing the variance decay property, and defining a Poincaré inequality for nonlinear filters, we obtain a complete framework for stability. | | Result: Under conditions of ergodicity of the hidden Markov process, observability under white-noise measurements, and non-degenerate noise distributions, the χ²-divergence between filters with different priors decays exponentially. Therefore, nonlinear filters are stable in χ²-divergence, and consequently also in KL divergence, total variation, and L^2. | | This provides rigorous conditions and proofs ensuring asymptotic forgetting of initial conditions and long-term reliability of nonlinear filtering

1. Defines unnormalized filter $\sigma^\rho_t$ and normalized filter $\pi^\rho_t = \sigma^\rho_t(f)/\sigma^\rho_t(1)$, introduces conditional covariance $V^\rho_t(f,g)$ and variance $V^\rho_t(f)$, and states the carré du champ operator $\Gamma$ with its role in local variance. | 2. Lists key mathematical tools (Itô–Wentzell formula, Banach–Alaoglu theorem, Dominated Convergence Theorem) and clarifies their relevance to subsequent proofs. | 3. Derives the conditional variance SDE in two ways: (i) Hamiltonian/optimal control via momentum process $P_t$ with Itô’s formula and cross-term cancellation, (ii) Kushner–Stratonovich equation specialized to $g=Y_t$ with innovation process substitution, confirming both yield the same SDE. | 4. Proves the likelihood ratio property by assuming $\pi^\rho_\tau(F)=c$, invoking measure equivalence $\tilde P^\rho\sim P^\rho$, applying uniqueness of Itô representation to show $U_t^{(\text{opt})}=0$, and deriving conditional expectation form of $Y_t(x)$, concluding $Y_t$ is a likelihood ratio with $\pi^\nu_t(Y_t)=1$. | 5. Establishes the Poincaré constant’s existence and continuity: verifies norm identity, boundedness of infimizing sequence, weak convergence (Banach–Alaoglu), convexity for minimality, compactness of $\mathcal{L}_0$ for strong convergence, and norm equivalence for upper/lower bounds leading to continuity. | 6. States the Markov BSDE with terminal condition $Y_T(x)=\gamma_T(x)$, proves consistency and independence from $\mathcal{Z}_t$, rewrites energy functional with $\mathbb{E}^{\pi^\nu_t}$, and derives inequality $\mathbb{E}^\nu[V^\nu_{t+\tau}(Y_{t+\tau}) \mid \mathcal{Z}_t] \ge e^{\tau c_{\pi^\nu_t}} V^\nu_t(Y_t)$. | 7. Validates variance decay under two scenarios: (i) uniformly integrable likelihood ratios via $\epsilon$–$K$ decomposition and DCT, (ii) vanishing Poincaré constants implying Dirac convergence of the filter and a.s. variance collapse, with both showing $\mathbb{E}^\nu[e^{-\tau C_N}V^\nu_T(\gamma_T)]\to0$. | 8. Confirms positivity of the Poincaré constant by contradiction ($c_\rho=0$ ⇒ $\beta^\rho_\tau=0$), showing impossibility under (i) ergodic processes, (ii) observable HMMs, and (iii) detectable HMMs, each yielding contradiction to $\mathrm{var}^\rho(Y_0)=1$. | 9. Proves filter stability for detectable HMMs in three steps: (i) boundedness of $\gamma_T$ ensures uniform integrability of $V^\nu_T(\gamma_T)$, (ii) variance decay with $C_N\to\infty$ plus Vitali theorem and contradiction argument shows a.s. convergence, (iii) total variation norm linked to $\mathbb{E}^\nu[|\gamma_T(X_T)-1|\mid\mathcal{Z}_T]$ bounded by $\sqrt{V^\nu_T(\gamma_T)}$, yielding $\|\pi^\mu_T-\pi^\nu_T\|_{TV}\to0$. | 10. Ensures consistency of notation ($\rho,\mu,\nu,\pi^\rho_t,\sigma^\rho_t,V^\rho_t,\Gamma,\beta^\rho_\tau$) and assumptions (equivalent initial measures, finite state space, Lipschitz coefficients), and validates generality by recovering Kalman filter stability as a special case. | |

The stability of nonlinear filters is a central problem in stochastic systems and signal processing. A nonlinear filter estimates the hidden state of a system based on noisy observations. In continuous-time hidden Markov models (HMMs) with white noise observations, the filter evolves as a conditional distribution over states. The main difficulty lies in whether the filter can forget its initial condition: if started with an incorrect prior distribution, does the filter eventually converge to the one started with the correct prior as time grows? | | In the linear Gaussian case (Kalman filter), stability is well understood. The Riccati equation governing the error covariance shows that under mild observability and controllability conditions, the filter error variance decays and stability follows. | | For nonlinear filters, however, the situation is far more subtle. Two mechanisms are known to support stability: | 1. Ergodicity of the hidden Markov process, which ensures the hidden state mixes over time and thus smooths out dependence on initial conditions. | 2. Observability of the noisy measurements, which allows the filter to correct for incorrect priors because the observations are informative enough to distinguish states. | | Classical results treated these two cases separately. Ergodicity arguments alone fail when the signal is not mixing but the observations are informative, while observability arguments alone fail when the observations are weak but the hidden process mixes strongly. This created a long-standing gap: a unified and quantitative framework for stability of nonlinear filters was missing.

1. Nonlinear Filter | For f \in C_b(S), the nonlinear filter is the conditional expectation | \pi_t(f) := \mathbb{E}[f(X_t) \mid \mathcal{Z}_t], | where X_t is the hidden process, and \mathcal{Z}_t is the observation σ-algebra. It evolves according to the Kushner–Stratonovich equation under Novikov’s condition. | | | | 2. Filter Stability in χ²-Divergence | | Given two priors \mu, \nu with \mu \ll \nu, the filters are stable in χ²-divergence if | \mathbb{E}_\mu\big[\chi^2(\pi_T^\mu \,\|\, \pi_T^\nu)\big] \xrightarrow[T\to\infty]{} 0. | This is the central criterion, since χ²-stability implies stability in KL divergence, total variation, and L^2. | | | | 3. Backward Map (Definition by the authors) | | For likelihood ratio \gamma_T(x) = \tfrac{d\pi_T^\mu}{d\pi_T^\nu}(x), the backward map is defined as | y_0(x) := \mathbb{E}_\nu[\gamma_T(X_T)\mid X_0=x]. | This maps the random terminal likelihood ratio back to a deterministic function at initial time, and its variance controls stability. | | | | 4. Variance Decay Property (Author’s innovation) | | The variance decay property asserts | \mathrm{Var}\nu(y_0(X_0)) \leq \mathrm{Var}\nu(\gamma_T(X_T)). | This inequality shows that variance contracts along the backward map. Exponential decay of χ²-divergence is obtained when combined with the Poincaré inequality. | | | | 5. Minimum Variance Duality | | A dual formulation of the filtering problem: the nonlinear filter can be analyzed through a backward stochastic differential equation (BSDE) optimal control problem, where minimizing variance corresponds to optimality of the control system. | | | | 6. Poincaré Inequality for Nonlinear Filters (Author’s innovation) | | A new functional inequality tailored for nonlinear filters: | \mathbb{E}\rho[\Gamma f] \geq c\rho \,\mathrm{Var}\rho(f), | where \Gamma is the carré du champ operator, \rho is a distribution, and c\rho > 0 is the Poincaré constant. This inequality links filter stability directly to ergodicity, observability, and noise non-degeneracy. | | | | 7. BSDE Optimal Control Formulation | | The backward map Y_t satisfies the BSDE: | -dY_t(x) = \Big[(A Y_t)(x) + h(x)^\top(U_t + V_t(x))\Big]dt - V_t(x)^\top dZ_t, | \quad Y_T(x) = \gamma_T(x). | This connects filtering stability with stochastic control optimization. | | | | 8. Stability Metrics and Implications | • KL divergence: D(\mu\|\nu). | • Total variation: \|\mu-\nu\|_{\mathrm{TV}}. | • L^2-stability: mean-square convergence. | • Relations: χ²-stability ⇒ KL ⇒ TV ⇒ L^2. | |

### 1. Preliminaries | | * Unnormalized filter $\sigma^\rho_t$ vs. normalized filter $\pi^\rho_t$. | * Conditional covariance $V^\rho_t(f,g)$ as variance measure under the filter. | * Carré du Champ operator $\Gamma$ captures local variance from the generator $\mathcal{A}$. | * Itô–Wentzell formula enables dynamics of processes $Y_t(X_t)$. | | ### 2. Conditional Variance SDE | | * Target: obtain SDE for $V^\rho_t(f,Y_t)$. | * Path 1 (Optimal Control): use Hamiltonian duality, momentum process $P_t$, and Itô’s rule. | * Path 2 (Filter Dynamics): apply Kushner–Stratonovich to covariance definition. | * Both approaches yield the same SDE, confirming variance decay dynamics. | | ### 3. Likelihood Ratio Connection | | * Show BSDE solution $Y_t$ acts as likelihood ratio for the filter. | * Uniqueness of Itô representation forces optimal control $ U_t^{(\text{opt})}=0$. | * Hence $Y_t(x)=\mathbb{E}^\rho[F(X_T)\mid \mathcal{Z}_t, X_t=x]$, proving likelihood ratio property. | | ### 4. Poincaré Constant | | * Define $\beta^\rho_\tau$ as minimal energy to sustain unit variance. | * Existence follows from weak compactness (Banach–Alaoglu). | * Continuity established via norm equivalence across measures. | * Provides quantitative bound linking energy functional and conditional variance. | | ### 5. Markov Property and Inequality | | * Optimal control BSDE inherits Markov structure. | * Energy functional conditional expectation reduces to a Poincaré inequality. | * Leads to key inequality: expected future variance is exponentially bounded by present variance. | | ### 6. Variance Decay Theorem | | * Case 1: Uniformly integrable likelihood ratio → variance decays in expectation. | * Case 2: Vanishing Poincaré constants → variance tends to zero almost surely. | * Core message: filter variance always decays under these structural conditions. | | ### 7. Positivity of Poincaré Constant | | * For ergodic, observable, or detectable HMMs, assume $c_\rho=0$ leads to contradiction. | * Therefore $c_\rho>0$, guaranteeing non-trivial variance decay. | | ### 8. Filter Stability Theorem | | * With $c_\rho>0$, conditional variance $V^\nu_T(\gamma_T)\to 0$. | * This implies convergence of $\pi^\mu_T$ to $\pi^\nu_T$ in total variation norm. | | ### 9. Key Assumptions | | * Equivalent initial measures ensure likelihood ratio well-defined. | * Finite-state space simplifies Γ operator and TV distance. | * Lipschitz coefficients guarantee unique solvability and boundedness. | | ### 10. Linear Case Sanity Check | | * For Gaussian linear systems, framework reduces to classical Riccati equation. | * Poincaré constant coincides with observability criterion. | * Confirms general theory recovers Kalman filter stability. | |