We consider the formal SDE

\(\textrm{d} X_t = b(t,X_t)\textrm{d} t + \textrm{d} Z_t, \qquad X_0 = x \in \mathbb {R}^d, (\text {E})\)

where \(b\in L^r ([0,T],\mathbb {B}_{p,q}^\beta (\mathbb {R}^d,\mathbb {R}^d))\) is a time-inhomogeneous Besov drift and \(Z_t\) is a symmetric d-dimensional \(\alpha \)-stable process, \(\alpha \in (1,2)\), whose spectral measure is absolutely continuous w.r.t. the Lebesgue measure on the sphere. Above, \(L^r\) and \(\mathbb {B}_{p,q}^\beta \) respectively denote Lebesgue and Besov spaces. We show that, when \(\beta > \frac{1-\alpha + \frac{\alpha }{r} + \frac{d}{p}}{2}\), the martingale solution associated with the formal generator of (E) admits a density which enjoys two-sided heat kernel bounds as well as gradient estimates w.r.t. the backward variable. Our proof relies on a suitable mollification of the singular drift aimed at using a Duhamel-type expansion. We then use a normalization method combined with Besov space properties (thermic characterization, duality and product rules) to derive estimates.

我们考虑正式的SDE

\(\textrm{d} X_t = b(t,X_t)\textrm{d} t + \textrm{d} Z_t, \qquad X_0 = x \in \mathbb {R}^d, (\text {E} )\)

其中\(b\in L^r ([0,T],\mathbb {B}_{p,q}^\beta (\mathbb {R}^d,\mathbb {R}^d))\)是时间非均匀贝索夫漂移，\(Z_t\)是对称d维\(\alpha \)稳定过程，\(\alpha \in (1,2)\)，其谱测量绝对连续 wrt球面上的勒贝格测度。上面，\(L^r\)和\(\mathbb {B}_{p,q}^\beta \)分别表示勒贝格空间和贝索夫空间。我们证明，当\(\beta > \frac{1-\alpha + \frac{\alpha }{r} + \frac{d}{p}}{2}\)时，与形式相关的鞅解(E) 的生成器承认具有两侧热核边界的密度以及关于后向变量的梯度估计。我们的证明依赖于对奇异漂移的适当缓和，旨在使用杜哈梅尔型展开。然后，我们使用归一化方法结合贝索夫空间属性（热表征、对偶性和乘积规则）来得出估计值。