Sobolev regularity of occupation measures and paths, variability and 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 Lévy processes, and we use it to show the existence of Stieltjes type integrals involving compositions.

32 pages, 78 references

References in corpus (5)

• Joint continuity of the local times of fractional Brownian sheets
• Variability of paths and differential equations with $BV$-coefficients
• Regularity of Local times associated to Volterra-Lévy processes and path-wise regularization of stochastic differential equations
• Stochastic Differential Equations with Discontinuous Diffusions
• Prevalence of $\rho$-irregularity and related properties

Cited by in corpus (3)

• Variability of paths and differential equations with $BV$-coefficients
• On the existence and regularity of local times
• Variability and the existence of rough integrals with irregular coefficients