Stochastic differential equations with Sobolev drifts and driven by $\alpha$-stable processes
Annales de l'I.H.P. Probabilités et statistiques, Volume 49 (2013) no. 4, p. 1057-1079

In this article we prove the pathwise uniqueness for stochastic differential equations in ${ℝ}^{d}$ with time-dependent Sobolev drifts, and driven by symmetric $\alpha$-stable processes provided that $\alpha \in \left(1,2\right)$ and its spectral measure is non-degenerate. In particular, the drift is allowed to have jump discontinuity when $\alpha \in \left(\frac{2d}{d+1},2\right)$. Our proof is based on some estimates of Krylov’s type for purely discontinuous semimartingales.

Dans cet article nous prouvons l’existence et l’unicité d’équations différentielles stochastiques dans ${ℝ}^{d}$ avec terme de dérive dépendant du temps dans un espace de Sobolev et dirigées par un processus de Lévy $\alpha$-stable symétrique avec $\alpha \in \left(1,2\right)$ et de mesure spectrale non-dégénérée. En particulier, le terme de dérive peut avoir des discontinuités de saut quand $\alpha \in \left(\frac{2d}{d+1},2\right)$. Notre preuve est basée sur des estimations de type Krylov pour des semimartingales purement discontinues.

DOI : https://doi.org/10.1214/12-AIHP476
Classification:  60H10
Keywords: pathwise uniqueness, symmetric $\alpha$-stable process, Krylov’s estimate, fractional Sobolev space
@article{AIHPB_2013__49_4_1057_0,
author = {Zhang, Xicheng},
title = {Stochastic differential equations with Sobolev drifts and driven by $\alpha$-stable processes},
journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
publisher = {Gauthier-Villars},
volume = {49},
number = {4},
year = {2013},
pages = {1057-1079},
doi = {10.1214/12-AIHP476},
zbl = {1279.60074},
mrnumber = {3127913},
language = {en},
url = {http://www.numdam.org/item/AIHPB_2013__49_4_1057_0}
}

Zhang, Xicheng. Stochastic differential equations with Sobolev drifts and driven by $\alpha$-stable processes. Annales de l'I.H.P. Probabilités et statistiques, Volume 49 (2013) no. 4, pp. 1057-1079. doi : 10.1214/12-AIHP476. http://www.numdam.org/item/AIHPB_2013__49_4_1057_0/

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