Gaussian deconvolution on $\mathbb R^d$ with application to self-repellent Brownian motion

TL;DR

This paper extends a deconvolution theorem from lattice to continuum settings, providing decay conditions for solutions and applying these results to analyze the critical behavior of self-repellent Brownian motion in high dimensions.

Contribution

It generalizes a recent deconvolution theorem to anisotropic continuum spaces and applies it to establish decay properties of self-repellent Brownian motion.

Findings

Deconvolution G(x) decays as (x·Σ^{-1}x)^{-(d-2)/2} for large |x|

Critical two-point function of self-repellent Brownian motion decays as |x|^{-(d-2)}

Extension of lattice deconvolution results to continuum models

Abstract

We consider the convolution equation $(\delta - J) * G = g$ on $\mathbb R^d$, $d>2$, where $\delta$ is the Dirac delta function and $J,g$ are given functions. We provide conditions on $J, g$ that ensure the deconvolution $G(x)$ to decay as $( x \cdot \Sigma^{-1} x)^{-(d-2)/2}$ for large $|x|$, where $\Sigma$ is a positive-definite diagonal matrix. This extends a recent deconvolution theorem on $\mathbb Z^d$ proved by the author and Slade to the possibly anisotropic, continuum setting while maintaining its simplicity. Our motivation comes from studies of statistical mechanical models on $\mathbb R^d$ based on the lace expansion. As an example, we apply our theorem to a self-repellent Brownian motion in dimensions $d>4$, proving its critical two-point function to decay as $|x|^{-(d-2)}$, like the Green function of the Laplace operator $\Delta$.