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Sub-diffusive regimes for long range self-interacting path measures

Volker Betz, Tobias Schmidt, Mark Sellke

TL;DR

The paper analyzes Brownian paths perturbed by long-range self-interactions and identifies regimes where the mean square displacement grows sub-diffusively or remains localized as time grows. It develops a robust framework based on Gaussian correlation inequalities to compare the perturbed measures with hierarchical Gaussian path measures, enabling precise upper bounds on $\mathbb{E}[|x_T|^2]$ that depend on the temporal decay exponent $\xi$ and the spatial nonlinearity exponent $\gamma$. For Gaussian spatial interactions, sub-diffusion occurs for $2<\xi<3$ while localization arises for $1<\xi<2$, with the regimes shifting by $\gamma/2$ in the general case, and a regime of logarithmic fluctuations emerges when the coupling scales slowly with $T$ (e.g., $\alpha \sim \log T$). The authors present two complementary proof strategies: (i) a recursive Gaussian confinement yielding an explicit fixed-point exponent $\frac{2}{\gamma}(\xi-2)$, and (ii) a hierarchical decomposition that achieves localization through multi-scale control of averaged increments, providing a versatile toolkit for long-range self-interacting path measures and connections to polaron-type models.

Abstract

We study Brownian motion perturbed by a long range self-interaction. We provide variance bounds in terms of the spatial interaction strength and the order of time decay.

Sub-diffusive regimes for long range self-interacting path measures

TL;DR

The paper analyzes Brownian paths perturbed by long-range self-interactions and identifies regimes where the mean square displacement grows sub-diffusively or remains localized as time grows. It develops a robust framework based on Gaussian correlation inequalities to compare the perturbed measures with hierarchical Gaussian path measures, enabling precise upper bounds on that depend on the temporal decay exponent and the spatial nonlinearity exponent . For Gaussian spatial interactions, sub-diffusion occurs for while localization arises for , with the regimes shifting by in the general case, and a regime of logarithmic fluctuations emerges when the coupling scales slowly with (e.g., ). The authors present two complementary proof strategies: (i) a recursive Gaussian confinement yielding an explicit fixed-point exponent , and (ii) a hierarchical decomposition that achieves localization through multi-scale control of averaged increments, providing a versatile toolkit for long-range self-interacting path measures and connections to polaron-type models.

Abstract

We study Brownian motion perturbed by a long range self-interaction. We provide variance bounds in terms of the spatial interaction strength and the order of time decay.

Paper Structure

This paper contains 11 sections, 24 theorems, 215 equations, 2 figures.

Table of Contents

  1. Introduction and main results
  2. Motivation and Results
  3. Connection to polaron models and related work
  4. Main idea of the proof and outline
  5. Hierarchical path measures
  6. (GCI), FKG-GCI and quadratic form domination
  7. Variance decomposition
  8. Proof of Theorem \ref{['thm:sub_diffusive_regime']}
  9. Proof of Theorem \ref{['thm:main_thm_2']}: recursive Gaussian confinement
  10. Proof of Theorem \ref{['thm:main_thm_3']}: Hierarchical decomposition
  11. Closed-form solution for periodic Gaussian measures

Key Result

Theorem 1.1

Let $0 \le \xi < 3$ and $\alpha > 0$. Then, there exists a constant $C>0$ such that for all measures as in equ:most_basic_measure: For $\xi \in (0,1)$ we even find

Figures (2)

  • Figure 1: Summary of our findings for general $\gamma\in (0,2)$. Compared to the Gaussian case, all intervals shift to the left by $\gamma/2$. The blue region indicates a transition to logarithmic fluctuations whenever $\alpha$ scales like $\log(T)$. Compare this to the Gaussian case, for which logarithmic fluctuations only occur for the case $\xi=2$. Note that we only show upper bounds on the variance in all of our results.
  • Figure 2: Dyadic decomposition of the path. The sum of a node's children equals the node's value. At each level, we keep all $\overline{s}$ terms, since those can be estimated directly. This yields a decomposition of $\sigma_{T-1}$ into $\log(T)$ terms.

Theorems & Definitions (49)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 2.1: GCI
  • proof
  • Lemma 2.2: Se24
  • Proposition 2.3
  • proof
  • Lemma 2.4
  • ...and 39 more