Sharp pathwise nonuniqueness for additive SDEs
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2026
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866911625139716096 |
|---|---|
| author | Hess-Childs, Elias Rowan, Keefer |
| author_facet | Hess-Childs, Elias Rowan, Keefer |
| contents | We construct a family of velocity fields demonstrating the sharpness of the classical Zvonkin--Veretennikov--Davie strong well-posedness by noise regime. We consider stochastic differential equations driven by Brownian noise with drift $u$ and show that for any $α<0$, there exists a velocity field $u \in L^\infty_t C^α_x$ that admits a unique weak solution but does not satisfy pathwise uniqueness (and hence has no strong solutions). This contrasts with the case $α\geq 0$, for which the existence of a unique strong solution is guaranteed. The velocity field construction is random, and the proof essentially uses central limit theorem scaling through the Berry--Esseen theorem. We also give natural extensions to non-Brownian driving noises, including nonuniqueness for arbitrary driving noises with certain Hölder regularities and an analogous sharpness of the strong well-posedness by noise regime for fractional Brownian motions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_23883 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Sharp pathwise nonuniqueness for additive SDEs Hess-Childs, Elias Rowan, Keefer Probability We construct a family of velocity fields demonstrating the sharpness of the classical Zvonkin--Veretennikov--Davie strong well-posedness by noise regime. We consider stochastic differential equations driven by Brownian noise with drift $u$ and show that for any $α<0$, there exists a velocity field $u \in L^\infty_t C^α_x$ that admits a unique weak solution but does not satisfy pathwise uniqueness (and hence has no strong solutions). This contrasts with the case $α\geq 0$, for which the existence of a unique strong solution is guaranteed. The velocity field construction is random, and the proof essentially uses central limit theorem scaling through the Berry--Esseen theorem. We also give natural extensions to non-Brownian driving noises, including nonuniqueness for arbitrary driving noises with certain Hölder regularities and an analogous sharpness of the strong well-posedness by noise regime for fractional Brownian motions. |
| title | Sharp pathwise nonuniqueness for additive SDEs |
| topic | Probability |
| url | https://arxiv.org/abs/2604.23883 |