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

Abstract

We consider the convolution equation $(\delta -J)\ast G=g$ 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 $\Delta$.

13 pages

References in corpus (2)

• Application of the lace expansion to the ${\phi }^4$ model
• Gaussian deconvolution and the lace expansion