Spectral decompositions and $\mathbb{L}^2$-operator norms of toy hypocoercive semi-groups

Abstract

For any a ą 0, consider the hypocoercive generators yB x `aB 2 y ´yB y and yB x ´axB y `B2 y ´yB y , respectively for px, yq P R{p2πZq ˆR and px, yq P R ˆR. The goal of the paper is to obtain exactly the L 2 pµ a q-operator norms of the corresponding Markov semi-group at any time, where µ a is the associated invariant measure. The computations are based on the spectral decomposition of the generator and especially on the scalar products of the eigenvectors. The motivation comes from an attempt to find an alternative approach to classical ones developed to obtain hypocoercive bounds for kinetic models.

The hypocoercivity index for the short time behavior of linear time-invariant ODE systems

“…In [14] the short-time decay behavior of a kinetic Fokker-Planck equation on the torus in was computed as 1 − 3 12 + ( 3 ). Again, in Fourier space and by using a Hermite function basis in velocity, this model can be written as an (infinite dimensional) conservative-dissipative system with hypocoercivity index 1 (see §2.1 of [14]). In that paper it was also mentioned that the decay exponent in (2.6) can be seen as some "order of hypocoercivity" of the generator.…”

Hypocoercivity and controllability in linear semi-dissipative Hamiltonian ODEs and DAEs

“…The Hermitian part = diag(−1, 0) is only semi-definite and the semi-dissipative matrix has HC-index = 1. The squared propagator norm, see [8,25], satisfies…”

“…To compare the Algorithms 3 and 4, we consider = , = such that − has finite HC-index . Algorithm 4 constructs a unitary transformation to reduce a matrix pencil to the staircase form (25), whereas Algorithm 3 uses these unitary transformations without computing the staircase form explicitly. Both algorithms partition matrices.…”

Hypoelliptic entropy dissipation for stochastic differential equations

“…In literature, there are many studies to investigate convergence behaviors of degenerate SDEs [2,15,18,25,26,39,13,34,41,42]. These methods have established the convergence rates for the density functions of SDE (1) under different metrics, e.g.…”