Variability of paths and differential equations with $BV$-coefficients

Abstract

We define compositions ϕ(X) of Hölder paths X in R n and functions of bounded variation ϕ under a relative condition involving the path and the gradient measure of ϕ. We show the existence and properties of generalized Lebesgue-Stieltjes integrals of compositions ϕ(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 R n driven by Hölder paths and involving coefficients of bounded variation. Examples include equations with discontinuous coefficients driven by paths of two-dimensional fractional Brownian motions.

Sobolev regularity of occupation measures and paths, variability and compositions

“…Earlier results exploiting the same mechanism can be found in [22,33,67,75] and closely related results in [47]. The present article may be seen as a continuation of our results in [43]. One goal is to point out how variability can be discussed in terms of Sobolev regularity of measures: In [43] we had formulated several results under the hypothesis that the measures involved are upper (Ahlfors) regular.…”

“…If X does not spend positive time in N , then t → ϕ(X t ) provides a correct definition of ϕ • X as an element of L 1 (0, T ). To ensure this we use a condition we refer to as variability, [43,Definition 2.1]. It is a relative condition joint on X and ϕ, although we mainly interpret it as a condition on X relative to a fixed BV -function ϕ.…”

“…In [43] we had used variability to define compositions of Hölder paths and BVfunctions. We had applied this result to ensure the existence of generalized Lebesgue-Stieltjes integrals and to solve certain differential systems with BV -coefficients and driven by fractional Brownian paths.…”

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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.

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Regularization of multiplicative SDEs through additive noise

“…This result can be extended to a broader class of non-degenerate diffusion coefficients b 2 by means of a Doss-Sussman transformation, in the style of [2]. Recently, [23] investigated the case b 1 ≡ 0 and b 2 non-degenerate of bounded variation; however, the conditions included therein for wellposedness are fairly specific and require verification for each choice of b 2 .…”