Heat kernel of supercritical nonlocal operators with unbounded drifts

Menozzi, Stéphane; Zhang, Xicheng

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

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

$$\mathrm{d}{X}_t=b(t,X_t)\mathrm{d}t+a(t,X_{t-})\mathrm{d}{L}_t^{\left(\alpha \right)},X_0=x,$$

où $L^{\left(\alpha \right)}$ est un processus $\alpha$-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 $\beta ,\gamma \in (0,1)$ tels que $(\beta \wedge \gamma )+\alpha >1$, uniformément en $t$. En particulier $b$ peut être non bornée. Lorsque $a$ est bornée et uniformément elliptique, nous montrons que la solution $X_t\left(x\right)$ 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 $\alpha \in (0,1)$. Notre approche se base sur des développements parametrix ad hoc et des techniques probabilistes.

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

$$\mathrm{d}{X}_t=b(t,X_t)\mathrm{d}t+a(t,X_{t-})\mathrm{d}{L}_t^{\left(\alpha \right)},X_0=x,$$

where $L^{\left(\alpha \right)}$ is a $d$-dimensional rotationally invariant $\alpha$-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 $\beta ,\gamma \in (0,1)$ such that $(\beta \wedge \gamma )+\alpha >1$, uniformly in $t$. Here $b$ may be unbounded. When $a$ is bounded and uniformly elliptic, we show that the unique solution $X_t\left(x\right)$ 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 $\alpha \in (0,1)$. Our proof is based on ad hoc parametrix expansions and probabilistic techniques.

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

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