Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Uniform subelliptic estimates for degenerating Fokker-Planck equations

Hart F. Smith

TL;DR

The paper studies uniform long-time bounds for subelliptic Fokker-Planck equations arising from Lindblad-type open quantum dynamics with diffusion only in momentum variables. It builds a nonisotropic parametrix on a step-2 nilpotent Lie group model, derives explicit fundamental solutions and a robust kernel calculus, and proves global $L^p$ bounds for the associated semigroup $e^{tQ}$ that are uniform in the small parameters $\epsilon$ and $\gamma$. The authors extend the elliptic diffusion techniques to subelliptic diffusion by scaling the parametrix with $\epsilon$ and obtaining sharp operator bounds for the resulting kernels, enabling both local smoothing and global-in-time control. They then apply these estimates to extend results on the quantum-classical correspondence, showing trace-norm and Hilbert-Schmidt closeness between quantum and classical evolutions in the subelliptic setting, under appropriate symbol-class and coupling assumptions. Overall, the work broadens the scope of semiclassical and microlocal analysis in open quantum systems by establishing uniform subelliptic estimates and their implications for quantum dynamics.

Abstract

We expand upon recent work of Hernandez-Ranard-Riedel, Galkowski-Zworski, and Li, by proving long time bounds for solutions to certain Fokker-Planck equations with subelliptic diffusion term. We consider the case where the jump operators in the Lindbladian are linear functions of $x$, and place an assumption which implies that the Hörmander condition holds for the resulting Fokker-Planck equation. By constructing a suitable parametrix for this equation we show that semiclassical derivative estimates established in the above works for elliptic diffusion also hold for subelliptic diffusion, with global bounds in $L^p$ for all $1\le p\le \infty$.

Uniform subelliptic estimates for degenerating Fokker-Planck equations

TL;DR

The paper studies uniform long-time bounds for subelliptic Fokker-Planck equations arising from Lindblad-type open quantum dynamics with diffusion only in momentum variables. It builds a nonisotropic parametrix on a step-2 nilpotent Lie group model, derives explicit fundamental solutions and a robust kernel calculus, and proves global bounds for the associated semigroup that are uniform in the small parameters and . The authors extend the elliptic diffusion techniques to subelliptic diffusion by scaling the parametrix with and obtaining sharp operator bounds for the resulting kernels, enabling both local smoothing and global-in-time control. They then apply these estimates to extend results on the quantum-classical correspondence, showing trace-norm and Hilbert-Schmidt closeness between quantum and classical evolutions in the subelliptic setting, under appropriate symbol-class and coupling assumptions. Overall, the work broadens the scope of semiclassical and microlocal analysis in open quantum systems by establishing uniform subelliptic estimates and their implications for quantum dynamics.

Abstract

We expand upon recent work of Hernandez-Ranard-Riedel, Galkowski-Zworski, and Li, by proving long time bounds for solutions to certain Fokker-Planck equations with subelliptic diffusion term. We consider the case where the jump operators in the Lindbladian are linear functions of , and place an assumption which implies that the Hörmander condition holds for the resulting Fokker-Planck equation. By constructing a suitable parametrix for this equation we show that semiclassical derivative estimates established in the above works for elliptic diffusion also hold for subelliptic diffusion, with global bounds in for all .
Paper Structure (7 sections, 16 theorems, 121 equations)

This paper contains 7 sections, 16 theorems, 121 equations.

Table of Contents

  1. Introduction
  2. The model operator and the case $\epsilon=1$
  3. The Fokker-Planck equation
  4. The case $0<\epsilon\le 1$
  5. Operator bounds
  6. Proof of Theorem \ref{['thm:main']}
  7. Applications to classical-quantum correspondence

Key Result

Theorem 1.1

Assume that $p=\frac{1}{2}|\xi|^2+V(x)$ with real $V$ satisfying eqn:Vbounds, and $Q$ is of the form eqn:Qdef, where eqn:cholesky holds with non-singular $B$. Then for all $N\in\mathbb{Z}_+$ there is $C_N$ such that, with $u=e^{tQ}u_0$ and $\epsilon=\sqrt{\gamma h/2}$, for all $1\le p\le\infty$, and Additionally, for all $N\in\mathbb{Z}_+$ and $T>0$ there is $C_{N,T}$ so that where $C_{N,T}=\math

Theorems & Definitions (29)

  • Theorem 1.1
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Corollary 2.4
  • proof
  • Theorem 2.5
  • proof
  • ...and 19 more