Schauder's estimate for nonlocal kinetic equations and its applications

Abstract

In this paper we develop a new method based on Littlewood-Paley's decomposition and heat kernel estimates of integral form, to establish Schauder's estimate for the following degenerate nonlocal equation in ${\mathbb{R}}^{2d}$ with Hölder coefficients: $${\partial }_{t}u={\mathcal{L}}_{\kappa ;\mathrm{v}}^{(\alpha )}u+b\cdot \nabla u+f,u_0=0,$$ where $u=u(t,x,\mathrm{v})$ and ${\mathcal{L}}_{\kappa ;\mathrm{v}}^{(\alpha )}$ is a nonlocal $\alpha$-stable-like operator with $\alpha \in (1,2)$ and kernel function $\kappa$, which acts on the variable $\mathrm{v}$. As an application, we show the strong well-posedness to the following degenerate stochastic differential equation with Hölder drift $b$: $$\mathrm{d}{Z}_t=b(t,Z_t)\mathrm{d}t+(0,\sigma (t,Z_t)\mathrm{d}{L}_t^{(\alpha )}),Z_0=(x,\mathrm{v})\in {\mathbb{R}}^{2d},$$ where $L_t^{(\alpha )}$ is a $d$-dimensional rotationally invariant and symmetric $\alpha$-stable process with $\alpha \in (1,2)$, and $b:{\mathbb{R}}_{+}\times {\mathbb{R}}^{2d}\to {\mathbb{R}}^{2d}$ is a $(\gamma ,\beta )$-Hölder continuous function in $(x,\mathrm{v})$ with $\gamma \in (\frac{2+\alpha }{2(1+\alpha )},1)$ and $\beta \in (1-\frac{\alpha }{2},1)$, $\sigma :{\mathbb{R}}_{+}\times {\mathbb{R}}^{2d}\to {\mathbb{R}}^d\otimes {\mathbb{R}}^d$ is a Lipschitz function. Moreover, we also show that for almost all $\omega$, the following random transport equation has a unique $C_b^1$-solution: $${\partial }_{t}u(t,x,\omega )+(b(t,x)+L_t^{(\alpha )}(\omega ))\cdot {\nabla }_{x}u(t,x,\omega )=0,u(0,x)=\phi (x),$$ where $\phi \in C_b^1({\mathbb{R}}^d)$ and $b:{\mathbb{R}}_{+}\times {\mathbb{R}}^d\to {\mathbb{R}}^d$ is a bounded continuous function in $(t,x)$ and $\gamma$-order Hölder continuous in $x$ uniformly in $t$ with $\gamma \in (\frac{2+\alpha }{2(1+\alpha )},1)$.

36pages

References in corpus (2)

• Well-posedness of supercritical SDE driven by Lévy processes with irregular drifts
• Hörmander's hypoelliptic theorem for nonlocal operators

Cited by in corpus (2)

• Schauder's estimates for nonlocal equations with singular Lévy measures
• Hölder regularity and gradient estimates Hölder regularity and gradient estimates for SDEs driven by cylindrical $\alpha$-stable processes