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 alter… Show more
“…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.…”
Section: Short-time Decay Of Conservative-dissipative Ode Systemsmentioning
We consider the class of conservative-dissipative ODE systems, which is a subclass of Lyapunov stable, linear time-invariant ODE systems. We characterize asymptotically stable, conservativedissipative ODE systems via the hypocoercivity (theory) of their system matrices. Our main result is a concise characterization of the hypocoercivity index (an algebraic structural property of matrices with positive semi-definite Hermitian part introduced in [2]) in terms of the short time behavior of the propagator norm for the associated conservative-dissipative ODE system.
“…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.…”
Section: Short-time Decay Of Conservative-dissipative Ode Systemsmentioning
We consider the class of conservative-dissipative ODE systems, which is a subclass of Lyapunov stable, linear time-invariant ODE systems. We characterize asymptotically stable, conservativedissipative ODE systems via the hypocoercivity (theory) of their system matrices. Our main result is a concise characterization of the hypocoercivity index (an algebraic structural property of matrices with positive semi-definite Hermitian part introduced in [2]) in terms of the short time behavior of the propagator norm for the associated conservative-dissipative ODE system.
“…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…”
Section: Example 3 (Example 52 In [4]mentioning
confidence: 99%
“…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.…”
Section: Comparison Of Algorithm 3 and Algorithmmentioning
For the classes of finite-dimensional linear semi-dissipative Hamiltonian ordinary differential equations or differential-algebraic equations with constant coefficients, stability and hypocoercivity are discussed and related to concepts from control theory. On the basis of staircase forms the solution behavior is characterized. The results are applied to two infinite-dimensional flow problems.
“…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.…”
Section: Consider An Itô Stochastic Differential Equation (Sde) Bymentioning
We study convergence behaviors of degenerate and non-reversible stochastic differential equations. Our method follows a Lyapunov method in probability density space, in which the Lyapunov functional is chosen as a weighted relative Fisher information functional. We construct a weighted Fisher information induced Gamma calculus method with a structure condition. Under this condition, an explicit algebraic tensor is derived to guarantee the convergence rate for the probability density function converging to its invariant distribution. We provide an analytical example for underdamped Langevin dynamics with variable diffusion coefficients.
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