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Article Dans Une Revue Journal de l'École polytechnique — Mathématiques Année : 2022

Heat kernel of supercritical SDEs with unbounded drifts

Résumé

Let $\alpha\in(0,2)$ and $d\in{\mathbb N}$. Consider the following SDE in ${\mathbb R}^d$: $$ {\rm d}X_t=b(t,X_t){\rm d} t+a(t,X_{t-}){\rm d} L^{(\alpha)}_t,\ \ X_0=x, $$ where $L^{(\alpha)}$ is a $d$-dimensional rotationally invariant $\alpha$-stable process, $b:{\mathbb R}_+\times{\mathbb R}^d\to{\mathbb R}^d$ and $a:{\mathbb R}_+\times{\mathbb R}^d\to{\mathbb R}^d\otimes{\mathbb R}^d$ 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 $X_t(x)$ 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 $\alpha\in (0,1) $. Our proof is based on ad hoc parametrix expansions and probabilistic techniques.
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Dates et versions

hal-03088376 , version 1 (26-12-2020)
hal-03088376 , version 2 (07-02-2022)

Identifiants

Citer

Stéphane Menozzi, Xicheng Zhang. Heat kernel of supercritical SDEs with unbounded drifts. Journal de l'École polytechnique — Mathématiques, 2022, 9, pp.537-579. ⟨10.5802/jep.189⟩. ⟨hal-03088376v2⟩
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