Sobolev regularity of occupation measures and paths, variability and compositions

TL;DR

This paper establishes fractional Sobolev regularity results for compositions of paths with bounded variation functions, allowing for discontinuous paths and applications to Gaussian and Lévy processes.

Contribution

It introduces a new Sobolev regularity result for path compositions under relaxed conditions, extending previous work to include discontinuous paths and broader stochastic processes.

Findings

Proves fractional Sobolev regularity for composed paths with BV functions.

Allows for discontinuous paths in Sobolev regularity analysis.

Demonstrates existence of Stieltjes integrals involving such compositions.

Abstract

We prove a result on the fractional Sobolev regularity of composition of paths of low fractional Sobolev regularity with functions of bounded variation. The result relies on the notion of variability, proposed by us in the previous article [43, arXiv:2003.11698]. Here we work under relaxed hypotheses, formulated in terms of Sobolev norms, and we can allow discontinuous paths, which is new. The result applies to typical realizations of certain Gaussian or L\'evy processes, and we use it to show the existence of Stieltjes type integrals involving compositions.