Variability of paths and differential equations with $BV$coefficients
Abstract
We define compositions $\varphi(X)$ of Hölder paths $X$ in $\mathbb{R}^n$ and functions of bounded variation $\varphi$ under a relative condition involving the path and the gradient measure of $\varphi$. We show the existence and properties of generalized LebesgueStieltjes integrals of compositions $\varphi(X)$ with respect to a given Hölder path $Y$. These results are then used, together with Doss' transform, to obtain existence and, in a certain sense, uniqueness results for differential equations in $\mathbb{R}^n$ driven by Hölder paths and involving coefficients of bounded variation. Examples include equations with discontinuous coefficients driven by paths of twodimensional fractional Brownian motions.
 Publication:

arXiv eprints
 Pub Date:
 March 2020
 arXiv:
 arXiv:2003.11698
 Bibcode:
 2020arXiv200311698H
 Keywords:

 Mathematics  Probability;
 Mathematics  Analysis of PDEs;
 Mathematics  Functional Analysis;
 31B10;
 34A12;
 34A34 (primary);
 26A33;
 26A42;
 26B30;
 26B35;
 28A78;
 31B99;
 60G22 (secondary)