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$χ^2$-cut-off phenomenon for Galerkin projections of Fokker-Planck equations with monomial potentials

Benny Avelin, Gerardo Barrera

Abstract

In this manuscript, we establish the existence/non-existence of the cut-off phenomenon for the Langevin-Kolmogorov random dynamics with monomial convex potentials, possible singular, and driven by a Brownian motion with small strength. We consider a truncated $χ^2$-distance, that is, a distance based on Galerkin projections of the eigensystem, and show that not only a refined knowledge of the eigenvalues is needed but also a refined asymptotics of the growth for the eigenfunctions of the Fokker-Planck equations associated to the Langevin-Kolmogorov dynamics. In addition, asymptotics of the mixing times are established.

$χ^2$-cut-off phenomenon for Galerkin projections of Fokker-Planck equations with monomial potentials

Abstract

In this manuscript, we establish the existence/non-existence of the cut-off phenomenon for the Langevin-Kolmogorov random dynamics with monomial convex potentials, possible singular, and driven by a Brownian motion with small strength. We consider a truncated -distance, that is, a distance based on Galerkin projections of the eigensystem, and show that not only a refined knowledge of the eigenvalues is needed but also a refined asymptotics of the growth for the eigenfunctions of the Fokker-Planck equations associated to the Langevin-Kolmogorov dynamics. In addition, asymptotics of the mixing times are established.

Paper Structure

This paper contains 17 sections, 21 theorems, 223 equations, 1 figure.

Table of Contents

  1. Introduction and main results
  2. Outline of the proofs
  3. Scaling argument
  4. The growth of the eigenfunctions
  5. Growth of the Fourier coefficients and mixing times asymptotics for 1/3 < gamma < 1
  6. Proof of the main results: \ref{['thm:main']} and \ref{['thm:main:2']}
  7. Proof of \ref{['thm:main']}
  8. Proof of \ref{['thm:main:2']}
  9. The growth of the Fourier coefficients and asymptotics of the mixing times for 1/3 < gamma < 1
  10. Growth of the Fourier coefficients 1/3 < gamma < 1 and epsilon > 0
  11. Mixing times asymptotics for 1/3 < gamma < 1
  12. The growth of the eigenfunctions
  13. WKB approximation
  14. Growth of the error term Rn
  15. Spectral representation of the chi2--distance to equilibrium
  16. ...and 2 more sections

Key Result

Theorem 1.2

Assume that $\gamma \in (1/3,1)$, $\delta(\varepsilon) = \varepsilon^{\frac{1-\gamma}{1+\gamma}}$, and $n\in \mathbb{N}$. For any $x_0\neq 0$ window cut-off holds true in the sense of eq:def:wcut, where $d_{n,\varepsilon}$ is defined in eq:Ga, $t_\varepsilon$ is defined as the unique solution to in other words, and the time window is given by $w_\varepsilon:= \varepsilon^{\frac{1-\gamma}{1+\gamm

Figures (1)

  • Figure 1: The vector field for the angle ODE \ref{['eq:ODE:theta']} when $\gamma=1/2$ and $g=2$. The blue shaded region depicts the area between $[\theta_l=0,\theta_u]$, and the dotted curve are the asymptotes for the limits $\pi/2$ and $-\pi/2$. The $\omega$-limit is expected to be the set $\{-\pi/2,0,\pi/2\}$.

Theorems & Definitions (43)

  • Remark 1.1: Cut-off phenomenon: window and profile
  • Theorem 1.2: Window cut-off phenomenon for Galerkin projection, $\gamma\in (1/3,1)$
  • Remark 1.3: Explicit asymptotic bounds for the cut-off time for $\gamma\in (1/3,1)$
  • Theorem 1.4: Profile cut-off phenomenon for Galerkin projection, $\gamma\in (0,1/3\rbrack$
  • Remark 1.5: Explicit asymptotic bounds for the cut-off time for $\gamma\in (0,1/3\rbrack$
  • Theorem 1.6: No cut-off phenomenon for Galerkin projection, $\gamma>1$
  • Remark 1.7: Ornstein--Uhlenbeck process
  • Lemma 2.1: Scaling of the spectrum
  • Lemma 2.2: WKB expansion
  • Theorem 2.3: Growth of the eigenfunctions, $\varepsilon=1$ and $\gamma>0$
  • ...and 33 more