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]. 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 Lévy processes, and we use it to show the existence of Stieltjes type integrals involving compositions.
Funding Statement
The research of MH was supported in part by the DFG IRTG 2235 ‘Searching for the regular in the irregular: Analysis of singular and random systems’ and by the DFG CRC 1283, ‘Taming uncertainty and profiting from randomness and low regularity in analysis, stochastics and their applications’. JMT acknowledges support by the Academy of Finland and the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreements no. 741487 and no. 818437).
Acknowledgments
We warmly thank Lucio Galeati for kindly bringing the works [31, 30] to our attention and the anonymous referees for their masterful advice.
Citation
Michael Hinz. Jonas M. Tölle. Lauri Viitasaari. "Sobolev regularity of occupation measures and paths, variability and compositions." Electron. J. Probab. 27 1 - 29, 2022. https://doi.org/10.1214/22-EJP797
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