Accurate Estimates for the Exponential Decay of Semigroups with Non‐Self‐Adjoint Generators

Nier, Francis

doi:10.1002/9781118577608.ch15

Cited by 2 publications

(3 citation statements)

References 31 publications

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“…The purpose of this note is to apply the spectral results of [13], [14] together with the resolvent bounds obtained in [3], [4], to establish an expansion for the evolution semigroup associated to a class of semiclassical non-selfadjoint magnetic Schrödinger operators, in the limit of large times. It is well known that the exponential decay of a contraction semigroup on a Hilbert space is closely connected to the resolvent estimates for the corresponding semigroup generator, see [9], [19], and the references given there. Restricting the attention to the case of generators given by semiclassical pseudodifferential operators on R n , relevant here, let us recall roughly some general resolvent bounds provided by the abstract operator theory, following [5], [24].…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Semigroup expansions for non-selfadjoint Schrödinger operators

Bellis

Hitrik

2019

Journal of Functional Analysis

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Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Semigroup expansions for non-selfadjoint Schrödinger operators

Bellis

Hitrik

2019

Journal of Functional Analysis

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“…This is typical for non-symmetric Fokker-Planck equations, and it is due to the non-orthogonality of the eigenfunctions of L (cf. §5 below; and [30] for a closely related discussion of L 2 -estimates for semigroups). Due to the applied regularisation and the above proof, we expect that the multiplicative constants in (4.26) and (4.27) are not sharp.…”

Section: And the Above Inequality Readsmentioning

confidence: 99%

“…The quest for these sharp constants for non-symmetric semigroups (particularly in L 2 -estimates) is an active research area (cf. [30]).…”

Section: Sharpness Of the Decay Ratementioning

confidence: 99%

Sharp entropy decay for hypocoercive and non-symmetric Fokker-Planck equations with linear drift

Arnold¹,

Jan²

2014

Preprint

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We investigate the existence of steady states and exponential decay for hypocoercive Fokker-Planck equations on the whole space with drift terms that are linear in the position variable. For this class of equations, we first establish that hypoellipticity of its generator and confinement of the system is equivalent to the existence of a unique normalised steady state. These two conditions also imply hypocoercivity, i.e. exponential convergence of the solution to equilibrium.Since the standard entropy method does not apply to degenerate parabolic equations, we develop a new modified entropy method (based on a modified, non-degenerate entropy dissipation-like functional) to prove this exponential decay in relative entropy (logarithmic till quadratic) -with a sharp rate. Furthermore, we compute the spectrum and eigenspaces of the generator as well as flow-invariant manifolds of Gaussian functions.Next, we extend our method to kinetic Fokker-Planck equations with a class of non-quadratic potentials. And, finally, we apply this new method to non-symmetric, uniformly parabolic Fokker-Planck equations with linear drift. At least in 2D this always yields the sharp exponential envelopes for the entropy function. In this case, we obtain even a sharp multiplicative constant in the decay estimate for the non-symmetric semigroup. Contents 1. Introduction 1 2. Existence of solutions and positivity 4 3. Existence of a steady state, decomposition of the generator L 8 4. Entropy method, explicit decay rate 13 5. Spectral analysis and flow-invariant manifolds 25 6. Sharpness of the decay rate 30 7. Kinetic Fokker-Planck equation 34 8. Non-degenerate, non-symmetric Fokker-Planck equations 37 9. Appendix 39 References 44

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Accurate Estimates for the Exponential Decay of Semigroups with Non‐Self‐Adjoint Generators

Cited by 2 publications

References 31 publications

Semigroup expansions for non-selfadjoint Schrödinger operators

Semigroup expansions for non-selfadjoint Schrödinger operators

Sharp entropy decay for hypocoercive and non-symmetric Fokker-Planck equations with linear drift

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