Quadratic forms and Sobolev spaces of fractional order

TL;DR

This paper investigates quadratic functionals related to fractional Sobolev spaces, demonstrating their equivalence to standard seminorms and providing a framework for discrete approximations with applications to Markov jump processes.

Contribution

It introduces a novel class of quadratic functionals involving inhomogeneous double cones, proving their equivalence to standard Sobolev seminorms and establishing a general scheme for discrete approximations.

Findings

Quadratic functionals are comparable to standard Sobolev seminorms.

A general scheme for discrete approximations of nonlocal quadratic forms is developed.

Applications to Markov jump processes are discussed.

Abstract

We study quadratic functionals on $L^2(\mathbb{R}^d)$ that generate seminorms in the fractional Sobolev space $H^s(\mathbb{R}^d)$ for $0 < s < 1$. The functionals under consideration appear in the study of Markov jump processes and, independently, in recent research on the Boltzmann equation. The functional measures differentiability of a function $f$ in a similar way as the seminorm of $H^s(\mathbb{R}^d)$. The major difference is that differences $f(y) - f(x)$ are taken into account only if $y$ lies in some double cone with apex at $x$ or vice versa. The configuration of double cones is allowed to be inhomogeneous without any assumption on the spatial regularity. We prove that the resulting seminorm is comparable to the standard one of $H^s(\mathbb{R}^d)$. The proof follows from a similar result on discrete quadratic forms in $\mathbb{Z}^d$, which is our second main result. We establish a general scheme for discrete approximations of nonlocal quadratic forms. Applications to Markov jump processes are discussed.