Heat kernel of supercritical nonlocal operators with unbounded drifts
Journal de l’École polytechnique — Mathématiques, Volume 9 (2022), pp. 537-579.

Let α(0,2) and d. Consider the following stochastic differential equation (SDE) in d :

dX t =b(t,X t )dt+a(t,X t- )dL t (α) ,X 0 =x,

where L (α) is a d-dimensional rotationally invariant α-stable process, b: + × d d and a: + × d d d are Hölder continuous functions in space, with respective order β,γ(0,1) such that (βγ)+α>1, uniformly in t. Here b may be unbounded. When a is bounded and uniformly elliptic, we show that the unique solution X t (x) of the above SDE admits a continuous density, which enjoys sharp two-sided estimates. We also establish sharp upper-bound for the logarithmic derivative. In particular, we cover the whole supercritical range α(0,1). Our proof is based on ad hoc parametrix expansions and probabilistic techniques.

Soit α(0,2) et d. Considérons l’équation différentielle stochastique (EDS) suivante dans d  :

dX t =b(t,X t )dt+a(t,X t- )dL t (α) ,X 0 =x,

L (α) est un processus α-stable isotrope de dimension d, b: + × d d et a: + × d d d sont des fonctions Hölder continues en espace, d’indices respectifs β,γ(0,1) tels que (βγ)+α>1, uniformément en t. En particulier b peut être non bornée. Lorsque a est bornée et uniformément elliptique, nous montrons que la solution X t (x) de l’EDS admet une densité continue, que l’on peut encadrer, à constante multiplicative près, par une même quantité. Nous obtenons également une borne supérieure précise pour la dérivée logarithmique de la densité. En particulier, nous traitons complètement le régime surcritique α(0,1). Notre approche se base sur des développements parametrix ad hoc et des techniques probabilistes.

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DOI: 10.5802/jep.189
Classification: 35K08, 60H10, 60G52
Keywords: Supercritical stable SDE, heat kernel estimates, logarithmic derivative, parametrix, regularized flows
Mot clés : EDS stable surcritique, estimées de noyau de la chaleur, dérivée logarithmique, parametrix, flots régularisés
Stéphane Menozzi 1; Xicheng Zhang 2

1 Laboratoire de Modélisation Mathématique d’Evry (LaMME), UMR CNRS 8071, Université d’Evry Val d’Essonne (Université Paris Saclay) 23 Boulevard de France 91037 Evry, France & Laboratory of Stochastic Analysis, HSE Pokrovsky Blvd, 11, Moscow, Russian Federation
2 School of Mathematics and Statistics, Wuhan University Wuhan, Hubei 430072, P.R. China
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Stéphane Menozzi; Xicheng Zhang. Heat kernel of supercritical nonlocal operators with unbounded drifts. Journal de l’École polytechnique — Mathématiques, Volume 9 (2022), pp. 537-579. doi : 10.5802/jep.189. https://jep.centre-mersenne.org/articles/10.5802/jep.189/

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