Heat kernel of supercritical SDEs with unbounded drifts

Abstract

Let $\alpha \in (0,2)$ and $d\in \mathbb{N}$. Consider the following SDE in ${\mathbb{R}}^d$:$$\mathrm{d}{X}_t=b(t,X_t)\mathrm{d}t+a(t,X_{t-})\mathrm{d}{L}_t^{(\alpha )},X_0=x,$$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 H{ö}lder 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.

References in corpus (9)

• Uniqueness of stable-like processes
• On the Cauchy problem for integro-differential operators in Hölder classes and the uniqueness of the martingale problem
• Dirichlet problem for supercritical nonlocal operators
• Density estimates for sdes driven by tempered stable processes
• Construction and heat kernel estimates of general stable-like Markov processes
• Heat kernels for time-dependent non-symmetric mixed Lévy-type operators
• Semigroup properties of solutions of SDEs driven by L{é}vy processes with independent coordinates
• On weak solution of SDE driven by inhomogeneous singular Lévy noise
• Gradient formula for transition semigroup corresponding to stochastic equation driven by a system of independent Lévy processes

Cited by in corpus (1)

• Multidimensional Stable Driven McKean-Vlasov SDEs with Distributional Interaction Kernel: Critical Thresholds and Related Models