The contraction rate in Thompson metric of order-preserving flows on a cone - application to generalized Riccati equations

We give a formula for the Lipschitz constant in Thompson’s part metric of any order-preserving flow on the interior of a (possibly infinite dimensional) closed convex pointed cone. This provides an explicit form of a characterization of Nussbaum concerning non order-preserving flows. As an application of this formula, we show that the flow of the generalized Riccati equation arising in stochastic linear quadratic control is a local contraction on the cone of positive definite matrices and characterize its Lipschitz constant by a matrix inequality. We also show that the same flow is no longer a contraction in other natural Finsler metrics on this cone, including the standard invariant Riemannian metric. This is motivated by a series of contraction properties concerning the standard Riccati equation, established by Bougerol, Liverani, Wojtowski, Lawson, Lee and Lim: we show that some of these properties do, and that some other do not, carry over to the generalized Riccati equation.

💡 Research Summary

The paper establishes an explicit formula for the Lipschitz (contraction) constant of any order‑preserving flow acting on the interior of a closed, convex, pointed cone, when distances are measured with Thompson’s part metric. The authors begin by recalling the definition of Thompson’s metric, which quantifies the logarithmic ratio between two points in the cone interior, and by noting that order‑preserving maps naturally interact with this metric. They then prove that if a flow (\phi_t) is (C^1) and its Jacobian (D\phi_t(x)) maps the cone into itself for every (x) in the interior, the optimal Lipschitz constant (\kappa(t)) with respect to the Thompson metric is given by

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