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Connection of hypocoercivity and hypocontractivity via the Cayley transform

Anton Arnold, Stefan Egger, Volker Mehrmann, Eduard A. Nigsch

TL;DR

This work builds a bridge between hypocoercivity and hypocontractivity for linear semi-dissipative continuous-time and semi-contractive discrete-time evolution on a Hilbert space via the Cayley transform. It provides a streamlined proof of the CT short-time decay with an explicit leading-term constant and extends the results to the DT setting, showing index preservation under Cayley and establishing hypocontractivity as a high-fidelity finite-difference proxy for the CT short-time decay. By proving the HC-index equals the dHC-index under the Cayley map (including scaled Cayley transforms), the paper unifies CT/DT decay behavior and informs numerical discretization and error estimates. It further develops maximally coercive/contractive representations through Lyapunov transforms and analyzes their influence on numerical solvers, offering practical guidance for choosing representations to improve stability and accuracy.

Abstract

The concepts of hypocoercivity and hypocontractivity and their relationship are studied for semi-dissipative continuous-time and discrete-time evolution equations in a Hilbert space setting. New proofs for the characterization of the short-time decay of the solution from the initial value are presented, that in particular characterize the constants in the leading terms of the solution when expanded in time. Maximally coercive/contractive representations of hypocoercive and hypocontractive semi-dissipative systems are presented, as well as the effect of different representations on the error estimates for the numerical solution.

Connection of hypocoercivity and hypocontractivity via the Cayley transform

TL;DR

This work builds a bridge between hypocoercivity and hypocontractivity for linear semi-dissipative continuous-time and semi-contractive discrete-time evolution on a Hilbert space via the Cayley transform. It provides a streamlined proof of the CT short-time decay with an explicit leading-term constant and extends the results to the DT setting, showing index preservation under Cayley and establishing hypocontractivity as a high-fidelity finite-difference proxy for the CT short-time decay. By proving the HC-index equals the dHC-index under the Cayley map (including scaled Cayley transforms), the paper unifies CT/DT decay behavior and informs numerical discretization and error estimates. It further develops maximally coercive/contractive representations through Lyapunov transforms and analyzes their influence on numerical solvers, offering practical guidance for choosing representations to improve stability and accuracy.

Abstract

The concepts of hypocoercivity and hypocontractivity and their relationship are studied for semi-dissipative continuous-time and discrete-time evolution equations in a Hilbert space setting. New proofs for the characterization of the short-time decay of the solution from the initial value are presented, that in particular characterize the constants in the leading terms of the solution when expanded in time. Maximally coercive/contractive representations of hypocoercive and hypocontractive semi-dissipative systems are presented, as well as the effect of different representations on the error estimates for the numerical solution.
Paper Structure (14 sections, 31 theorems, 198 equations, 2 figures)

This paper contains 14 sections, 31 theorems, 198 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Continuous-time systems: new proof of short-time decay
  4. Equivalence of hypocoercivity index and hypocontractivity index via the Cayley transform
  5. Continuous-time systems
  6. Discrete-time dynamical systems
  7. Relation between hypocoercivity and hypocontractivity via the Cayley transform
  8. Hypocontractivity as finite difference approximation of the short-time decay in hypocoercive systems
  9. An illustrating example
  10. Proof of Theorem \ref{['th:FinDiff-limit']}
  11. Maximally coercive/contractive representations
  12. Continuous-time Lyapunov transformations
  13. Discrete-time Lyapunov transformations
  14. The effect of a Lyapunov transformation on the numerical solution

Key Result

Theorem 2.3

Let $-B_c\in \mathcal{B}(\mathcal{H})$ be semi-dissipative. Then $B_c$ is hypocoercive (with hypocoercivity index $m_{HC}\in\mathbb{N}_0$) if and only if for some $a,\,c>0$. In this case, necessarily $a=2m_{HC}+1$. For matrices $B_c$, the remainder term in cont-decaycan be improved to $\mathcal{O}(t^{a+1})$.

Figures (2)

  • Figure 1: Example with HC-index 1: Approximation of the propagator norm $\|e^{-B_ct}\|$ (blue) by its 3rd order Taylor expansion about $t=0$ (red), and the norm of iterates of the scaled Cayley transform, $\|B_d(\tau)^k\|$ with step size $\tau=\frac{1}{2}$ (circles). Note the extension of the latter to negative indices, using odd symmetry. This is needed for constructing the symmetric finite difference of the grid function $\phi(\tau)$, centered at $k=0$.
  • Figure 2: Example with HC-index 2: Approximation of the propagator norm $\|e^{-B_ct}\|$ (blue) by its 5th order Taylor expansion about $t=0$ (red), and the norm of iterates of the scaled Cayley transform, $\|B_d(\tau)^k\|$ with step size $\tau=\frac{1}{2}$ (circles). Note the extension of the latter to negative indices, using odd symmetry. This is needed for constructing the symmetric finite difference of the grid function $\phi(\tau)$, centered at $k=0$.

Theorems & Definitions (68)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • proof
  • Remark 3.4
  • ...and 58 more