Frederic Herau, Johannes Sjoestrand, Christiaan C. Stolk
We study some accurate semiclassical resolvent estimates for operators that are neither selfadjoint nor elliptic, and applications to the Cauchy problem. In particular we get a precise description of the spectrum near the imaginary axis and precise resolvent estimates inside the pseudo-spectrum. We apply our results to the Kramers-Fokker-Planck operator.
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We consider operators of Kramers-Fokker-Planck type in the semi-classical limit such that the exponent of the associated Maxwellian is a Morse function with two local minima and a saddle point. Under suitable additional assumptions we establish the complete asymptotics of the exponentially small splitting between the first two eigenvalues.
We study operators of Kramers-Fokker-Planck type in the semiclassical limit, assuming that the exponent of the associated Maxwellian is a Morse function with a finite number $n_0$ of local minima. Under suitable additional assumptions, we show that the first $n_0$ eigenvalues are real and exponentially small, and establish the complete semiclassical asymptotics for these eigenvalues.
Spectral properties of the Kramers-Fokker-Planck operator with a long-range potential
本文研究带长程势的Kramers-Fokker-Planck算子的谱性质，证明阈值是特征值唯一可能的聚点、例外集外极限吸收原理成立，并刻画了离散与嵌入非阈值特征值对应特征函数的不同衰减速率。
For quickly decreasing potentials with one position variable, the first threshold zero is always a resonance of the Kramers-Fokker-Planck operator. In this article we study low-energy spectral properties of the operator and calculate large time asymptotics of solutions in terms of the Maxwellian.
In continuation of a former work by the first author with F. Nier (2009) and of a more recent work by the second author on the torus (2019), we consider the Kramers-Fokker-Planck operator (KFP) with an external electromagnetic field on R d. We show a maximal type estimate on this operator using a nilpotent approach for vector field polynomial operators and induced representations of a nilpotent graded Lie algebra. This estimate leads to an optimal characterization of the domain of the closure of the KFP operator and a criterion for the compactness of the resolvent.
We consider Fokker-Planck type differential operators associated with general Langevin processes admitting a Gibbs stationary distribution. Under assumptions insuring suitable resolvent estimates, we prove Eyring-Kramers formulas for the bottom of the spectrum of these operators in the low temperature regime. Our approach is based on the construction of sharp Gaussian quasimodes which avoids supersymmetry or PT-symmetry assumptions.
In this work, we use scattering method to study the Kramers-Fokker-Planck equation with a potential whose gradient tends to zero at the infinity. For short-range potentials in dimension three, we show that complex eigenvalues do not accumulate at low-energies and establish the low-energy resolvent asymptotics. This combined with high energy pseudospectral estimates valid in more general situations gives the large-time asymptotics of the solution in weighted $L^2$ spaces.
In the first part of this work, we consider second order supersymmetric differential operators in the semiclassical limit, including the Kramers-Fokker-Planck operator, such that the exponent of the associated Maxwellian $ϕ$ is a Morse function with two local minima and one saddle point. Under suitable additional assumptions of dynamical nature, we establish the long time convergence to the equilibrium for the associated heat semigroup, with the rate given by the first non-vanishing, exponentially small, eigenvalue. In the second part of the paper, we consider the case when the function $ϕ$ ha…
Hypocoercivity and metastability of degenerate KFP equations at low temperature
本文研究了退化KFP方程在低温下的hypocoercivity与metastability，通过构造精确的高斯quasimodes证明了半经典情况下谱底的Eyring-Kramers公式并量化了谱间隙。
WKB for semiclassical operators: How to fly over caustics (and more)
本文基于微局部分析近期进展，统一阐述了改进的Maslov-WKB方法，严谨证明了半经典算子本征值的Bohr-Sommerfeld-Einstein-Brittle-Keller量子化条件，并回顾了应用与扩展。
We consider the stochastic quantization of a quartic double-well energy functional in the semiclassical regime and derive optimal asymptotics for the exponentially small splitting of the ground state energy. Our result provides an infinite-dimensional version of some sharp tunneling estimates known in finite dimensions for semiclassical Witten Laplacians in degree zero. From a stochastic point of view it proves that the $L^2$ spectral gap of the stochastic one-dimensional Allen-Cahn equation in finite volume satisifies a Kramers-type formula in the limit of vanishing noise. We work with finite…
This text is a survey of recent results obtained by the author and collaborators on different problems for non-self-adjoint operators. The topics are: Kramers-Fokker-Planck type operators, spectral asymptotics in two dimensions and Weyl asymptotics for the eigenvalues of non-self-adjoint operators with small random perturbations. In the introduction we also review the notion of pseudo-spectrum and its relation to non-self-adjoint spectral problems.
Sharp spectral gap of adaptive Langevin dynamics
本文研究自适应 Langevin 动力学的谱间隙问题，利用超协紧技术与 WKB 方法构造高斯准本征函数，证明低温下谱底部的 Eyring-Kramers 公式。
We study the spectrum of the semiclassical Witten Laplacian $Δ_{f}$ associated to a smooth function $f$ on ${\mathbb R}^d$. We assume that $f$ is a confining Morse--Bott function. Under this assumption we show that $Δ_{f}$ admits exponentially small eigenvalues separated from the rest of the spectrum. Moreover, we establish Eyring-Kramers formula for these eigenvalues. Our approach is based on microlocal constructions of quasimodes near the critical submanifolds.
Sharp spectral gap for some degenerate Witten Laplacians
本文通过适应WKB方法构造退化高斯准模，证明了某些非莫尔塞势下温特拉普拉西昂的谱间隙性并量化了谱间隙。
We study resolvents and spectral projections for quadratic differential operators under an assumption of partial ellipticity. We establish exponential-type resolvent bounds for these operators, including Kramers-Fokker-Planck operators with quadratic potentials. For the norms of spectral projections for these operators, we obtain complete asymptotic expansions in dimension one, and for arbitrary dimension, we obtain exponential upper bounds and the rate of exponential growth in a generic situation. We furthermore obtain a complete characterization of those operators with orthogonal spectral pr…
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