Nonlocal elliptic equation in Hölder space and the martingale problem

Ling, Chengcheng; Zhao, Guohuan

doi:10.1016/j.jde.2022.01.025

Cited by 7 publications

(2 citation statements)

References 16 publications

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“…Without any structure on the drift, a similar and consistent description of the dynamics for the weak solutions of (1.1) in the multidimensional setting is obtained in [CdRM22a] for β > 1−α 2 . The case of a non-trivial diffusion coefficient was investigated in [LZ22] with the same thresholds. Note that, in the present work, we chose to work with a trivial diffusion coefficient as the most delicate issue is the handling of the singular drift (see Remark 15 in [CdRM22a] for the handling of a non trivial diffusion coefficient in a Duhamel expansion).…”

Section: Scope Of This Papermentioning

confidence: 99%

Heat kernel estimates for stable-driven SDEs with distributional drift

Fitoussi¹

2023

Preprint

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We consider the formal SDE) is a time-inhomogeneous Besov drift and Z t is a symmetric d-dimensional α-stable process, α ∈ (1, 2), whose spectral measure is absolutely continuous w.r.t. the Lebesgue measure on the sphere. Above, L r and B β p,q respectively denote Lebesgue and Besov spaces. We show that, when β >, the martingale solution associated with the formal generator of (E) admits a density which enjoys two-sided heat kernel bounds as well as gradient estimates w.r.t. the backward variable. Our proof relies on a suitable mollification of the singular drift aimed at using Duhamel expansion. We then use a normalization method combined with Besov space properties (thermic characterization, duality and product rules) to derive estimates.

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Section: Scope Of This Papermentioning

confidence: 99%

Heat kernel estimates for stable-driven SDEs with distributional drift

Fitoussi¹

2023

Preprint

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“…The first contribution in this direction was one-dimensional and made by [1]. Further work was done by [16], [7], and [15], the latter even beyond the so-called Young regime. In these works the Brownian motion is replaced by a Lévy α-stable process, which produces the regularization by noise.…”

Section: Introductionmentioning

confidence: 99%

Path-dependent SDEs with jumps and irregular drift: well-posedness and Dirichlet properties

Bandini¹,

Russo²

2022

Preprint

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We discuss a concept of path-dependent SDE with distributional drift with possible jumps. We interpret it via a suitable martingale problem, for which we provide existence and uniqueness. The corresponding solutions are expected to be Dirichlet processes, nevertheless we give examples of solutions which do not fulfill this property. In the second part of the paper we indeed state and prove significant new results on the class of Dirichlet processes.Résumé. Nous introduisons un concept d'EDS dépendant de la trajectoire avec drift distributionnel et avec sauts. On s'attend que les solutions correspondantes soient des processus de Dirichlet; néanmoins nous exhibons des exemples de solutions ne vérifiant pas cette propriété. Dans la seconde partie de l'article nous prouvons par ailleurs de nouveaux résultats significatifs sur la classe des processus de Dirichlet.

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Strong regularization by Brownian noise propagating through a weak Hörmander structure

Raynal

Honoré

Menozzi

2022

Probab. Theory Relat. Fields

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We establish strong uniqueness for a class of degenerate SDEs of weak Hörmander type under suitable Hölder regularity conditions for the associated drift term. Our approach relies on the Zvonkin transform which requires to exhibit good smoothing properties of the underlying parabolic PDE with rough, here Hölder, drift coefficients and source term. Such regularizing effects are established through a perturbation technique (forward parametrix approach) which also heavily relies on appropriate duality properties on Besov spaces.For the method employed, we exhibit some sharp thresholds on the Hölder exponents for the strong uniqueness to hold.

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Nonlocal elliptic equation in Hölder space and the martingale problem

Cited by 7 publications

References 16 publications

Heat kernel estimates for stable-driven SDEs with distributional drift

Heat kernel estimates for stable-driven SDEs with distributional drift

Path-dependent SDEs with jumps and irregular drift: well-posedness and Dirichlet properties

Strong regularization by Brownian noise propagating through a weak Hörmander structure

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