Heat Kernel Estimates for Stable-driven SDEs with Distributional Drift
Springer Science and Business Media LLC - Trang 1-31 - 2023
Tóm tắt
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.
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