Noiseless regularisation by noise

Abstract

We analyse the effect of a generic continuous additive perturbation to the well-posedness of ordinary differential equations. Genericity here is understood in the sense of prevalence. This allows us to discuss these problems in a setting where we do not have to commit ourselves to any restrictive assumption on the statistical properties of the perturbation. The main result is that a generic continuous perturbation renders the Cauchy problem well-posed for arbitrarily irregular vector fields. Therefore we establish regularisation by noise "without probability".

Sobolev regularity of occupation measures and paths, variability and compositions

“…Of course it is to be expected that the present composition results can be generalized further using a refined Fourieranalytic approach. Important recent results close to the present article can be found in [19,27,28,35,36]. In [28] the authors provide a comprehensive study of ρ-irregularity from the point of view of prevalence.…”

“…In [28] the authors provide a comprehensive study of ρ-irregularity from the point of view of prevalence. The notion of ρ-irregularity had been introduced in [19], it quantifies the regularity of occupation measures in a rather complete way and has natural consequences for the mapping properties of averaging operators, [19,27,36], which are integrals of compositions involving shifted paths. In the present article we do not aim at integrals of compositions but at compositions themselves, clearly a different question.…”

Regularization of multiplicative SDEs through additive noise

“…Proof. The case of w sampled as an fBm follows from the results from [16], see for instance Remark 7 or Section 3.3 more in general; indeed for b as above, almost every realisation of w satisfies…”

“…Our approach to the problem follows the ideas introduced in [8], where analytic conditions on w which imply well-posedness for (1.1) with b 2 ≡ 0 and possibly distributional drift b 1 are identified. In recent years, this analytic approach to regularization by noise phenomena has been considerably expanded, see [16,22,21].…”

“…In contrast, in order to define the integral appearing in (1.5) as a Young integral, we need at least to require w to be δ-Hölder continuous with H + δ > 1; without this assumption, it is unclear how to interpret neither (1.2) nor (1.5), even when b is a smooth function. At the same time, it is now clear from [8,16,22] that a strong regularisation effect is expected to hold for especially rough w, i.e. for very small values of δ, thus making the requirement H + δ > 1 too restrictive.…”