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

Abstract

We consider the convolution equation $(δ- J) * G = g$ on $\mathbb R^d$, $d>2$, where $δ$ 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 $( x \cdot Σ^{-1} x)^{-(d-2)/2}$ for large $|x|$, where $Σ$ 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 $Δ$.