Heat kernel of supercritical nonlocal operators with unbounded drifts

Stéphane Menozzi; Xicheng Zhang

doi:10.5802/jep.189

Heat kernel of supercritical nonlocal operators with unbounded drifts
[Noyau de la chaleur pour des EDS surcritiques à dérive non bornée]

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

Journal de l’École polytechnique — Mathématiques, Tome 9 (2022), pp. 537-579.

Résumé
Abstract

Soit $α \in (0, 2)$ et $d \in ℕ$ . Considérons l’équation différentielle stochastique (EDS) suivante dans $ℝ^{d}$ :

d X_{t} = b (t, X_{t}) d t + a (t, X_{t -}) d L_{t}^{(α)}, X_{0} = x,

où $L^{(α)}$ est un processus $α$ -stable isotrope de dimension $d$ , $b : ℝ_{+} \times ℝ^{d} \to ℝ^{d}$ et $a : ℝ_{+} \times ℝ^{d} \to ℝ^{d} \otimes ℝ^{d}$ sont des fonctions Hölder continues en espace, d’indices respectifs $β, γ \in (0, 1)$ tels que $(β \land γ) + α > 1$ , 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 $α \in (0, 1)$ . Notre approche se base sur des développements parametrix ad hoc et des techniques probabilistes.

Let $α \in (0, 2)$ and $d \in ℕ$ . Consider the following stochastic differential equation (SDE) in $ℝ^{d}$ :

d X_{t} = b (t, X_{t}) d t + a (t, X_{t -}) d L_{t}^{(α)}, X_{0} = x,

where $L^{(α)}$ is a $d$ -dimensional rotationally invariant $α$ -stable process, $b : ℝ_{+} \times ℝ^{d} \to ℝ^{d}$ and $a : ℝ_{+} \times ℝ^{d} \to ℝ^{d} \otimes ℝ^{d}$ are Hölder continuous functions in space, with respective order $β, γ \in (0, 1)$ such that $(β \land γ) + α > 1$ , 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 $α \in (0, 1)$ . Our proof is based on ad hoc parametrix expansions and probabilistic techniques.

Reçu le : 2021-01-05
Accepté le : 2022-01-18
Publié le : 2022-03-03

DOI : 10.5802/jep.189

Classification : 35K08, 60H10, 60G52
Keywords: Supercritical stable SDE, heat kernel estimates, logarithmic derivative, parametrix, regularized flows
Mots-clés : EDS stable surcritique, estimées de noyau de la chaleur, dérivée logarithmique, parametrix, flots régularisés

Affiliations des auteurs :

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

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {St\'ephane Menozzi and Xicheng Zhang},
     title = {Heat kernel of supercritical nonlocal operators with unbounded drifts},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {537--579},
     publisher = {\'Ecole polytechnique},
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     year = {2022},
     doi = {10.5802/jep.189},
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Stéphane Menozzi; Xicheng Zhang. Heat kernel of supercritical nonlocal operators with unbounded drifts. Journal de l’École polytechnique — Mathématiques, Tome 9 (2022), pp. 537-579. doi : 10.5802/jep.189. https://jep.centre-mersenne.org/articles/10.5802/jep.189/

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