Riccati flow

Wiki card — concept node for exp-families-stability.

(i) Formal statement

A matrix Riccati differential equation on symmetric matrices \(S \in \mathrm{Sym}_d(\mathbb{R})\) has the general form

\[ \dot S \;=\; A + B S + S B^\top - S C S, \tag{Ric-gen} \]

for constant matrices \(A \in \mathrm{Sym}_d\), \(B, C \in \mathbb{R}^{d \times d}\) (with \(C \in \mathrm{Sym}_d\)). Special cases for our problem.

• Heat-flow Riccati (precision coordinates). For \(K_t = \Sigma_t^{-1}\) with \(\Sigma_t = \Sigma_0 + t I\): \[ \boxed{\;\dot K \;=\; -K^2.\;} \tag{Ric-heat} \]
• OU Riccati. With drift \(-x\cdot\nabla\) (invariant \(\gamma_1\)): \[ \dot K \;=\; -K^2 \;-\; 2 K. \tag{Ric-OU} \]
• Kalman filter Riccati. For linear-Gaussian dynamics \(\dot x = F x + \xi\), observation \(y = H x + \eta\): \[ \dot P \;=\; F P + P F^\top \;+\; Q \;-\; P H^\top R^{-1} H P. \tag{Ric-Kalman} \]

Closed-form integration (heat case). \(\dot K = -K^2\) with \(K_0 = K\) has solution \[ K_t \;=\; (K^{-1} + t I)^{-1}, \qquad \Sigma_t = \Sigma_0 + t I. \tag{Ric-sol} \]

(ii) Role in Q1 / Q2 / Q3

• Q2 (main payoff). The quadratic-linear family \(\varphi(x) = (\mathrm{vec}(xx^\top), x)\) produces, via Cole–Hopf, the flow \(\dot K = -K^2\), \(\dot b = -Kb\). The closed-form (Ric-sol) is the explicit reparameterisation \(\theta \mapsto \tilde\theta\) required by Q2.
• Q1 (general closure). Equation (3.5) of invariant-reformulation.md is a multivariate Riccati on \(\theta \in \mathbb{R}^r\): \(\dot\theta_k = \sum_i\alpha_{ik}\theta_i - \sum_{i,j}\gamma_{ijk}\theta_i\theta_j\). The quadratic in \(\theta\) is the fingerprint of the \(\Gamma(\varphi,\varphi^\top)\) term in Cole–Hopf — Riccati is the universal form of the reparameterisation flow when \((\dagger)\) holds.
• Q3. Hamilton–Jacobi on \(f\) ⇔ Riccati on \(\theta\). The HJ-to-Riccati reduction is the finite-dimensional projection of an infinite-dimensional PDE onto the \(r\)-dimensional submanifold \(\mathcal{E}_\varphi\) — projective dynamics.

(iii) References

• Kalman (1960) — A New Approach to Linear Filtering and Prediction Problems. Introduces the matrix Riccati for covariance evolution. doi:10.1115/1.3662552.
• Bakry, Gentil, Ledoux (2014) — §1.11, §2.7 (OU Riccati covariance flow as prototype of \(CD(K,\infty)\) dynamics).
• Morris (1982) — Natural Exponential Families with Quadratic Variance Functions. NEF-QVF classification; classical statistical analog of the Riccati-closed class. doi:10.1214/aos/1176345690.

(iv) Worked miniature — integrating \(\dot K = -K^2\)

Take \(d = 1\), \(K_0 = 1/\sigma_0^2 > 0\). The ODE \(\dot K = -K^2\) is separable: \(dK/K^2 = -dt\), integrating to \(-1/K = -t - 1/K_0\), i.e. \[ K_t \;=\; \frac{K_0}{1 + t K_0} \;=\; \frac{1}{\sigma_0^2 + t}. \]

So \(\sigma_t^2 = \sigma_0^2 + t\), which matches \(\mathcal{N}(0, \sigma_0^2) * \gamma_{\sqrt{t}} = \mathcal{N}(0, \sigma_0^2 + t)\). ✓

Matrix case. For general \(K_0 \succ 0\), \(K_t = (K_0^{-1} + t I)^{-1}\). One way to see this: differentiate \(K_t \cdot (K_0^{-1} + t I) = I\) — \(\dot K_t (K_0^{-1} + tI) + K_t \cdot I = 0\), i.e. \(\dot K_t = -K_t (K_0^{-1} + tI)^{-1} = -K_t \cdot K_t = -K_t^2\). ✓

Information-geometric reading. In \(\eta\)-coordinates (covariance), the flow is linear \(\dot\Sigma = I\); in \(\theta\)-coordinates (precision), the flow is Riccati \(\dot K = -K^2\). The nonlinear transform \(\theta \leftrightarrow \eta\) is the Legendre of the log-partition — Riccati is the dual of a linear flow. This is a universal pattern: any time a quadratic statistic \(\varphi\) is stable, the \(\theta\)-flow is Riccati because the \(\eta\)-flow is affine.

Kalman connection. Kalman (1960)'s covariance propagation for linear- Gaussian dynamics is (Ric-Kalman); the terms \(F P + P F^\top\) come from drift, \(Q\) from process noise, \(-P H^\top R^{-1} H P\) from the innovation (the Gaussian measurement update). Our heat-flow case is Kalman with \(F = 0\), \(Q = I\), no measurement (\(H = 0\)): \(\dot P = I\), equivalently \(\dot K = -K^2\).