Let and . Consider the following stochastic differential equation (SDE) in :
where is a -dimensional rotationally invariant -stable process, and are Hölder continuous functions in space, with respective order such that , uniformly in . Here may be unbounded. When is bounded and uniformly elliptic, we show that the unique solution 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 . Our proof is based on ad hoc parametrix expansions and probabilistic techniques.
Soit et . Considérons l’équation différentielle stochastique (EDS) suivante dans :
où est un processus -stable isotrope de dimension , et sont des fonctions Hölder continues en espace, d’indices respectifs tels que , uniformément en . En particulier peut être non bornée. Lorsque est bornée et uniformément elliptique, nous montrons que la solution 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 . Notre approche se base sur des développements parametrix ad hoc et des techniques probabilistes.
Accepted:
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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
@article{JEP_2022__9__537_0, 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}, volume = {9}, year = {2022}, doi = {10.5802/jep.189}, language = {en}, url = {https://jep.centre-mersenne.org/articles/10.5802/jep.189/} }
TY - JOUR AU - Stéphane Menozzi AU - Xicheng Zhang TI - Heat kernel of supercritical nonlocal operators with unbounded drifts JO - Journal de l’École polytechnique — Mathématiques PY - 2022 SP - 537 EP - 579 VL - 9 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.189/ DO - 10.5802/jep.189 LA - en ID - JEP_2022__9__537_0 ER -
%0 Journal Article %A Stéphane Menozzi %A Xicheng Zhang %T Heat kernel of supercritical nonlocal operators with unbounded drifts %J Journal de l’École polytechnique — Mathématiques %D 2022 %P 537-579 %V 9 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.189/ %R 10.5802/jep.189 %G en %F JEP_2022__9__537_0
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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