Journal de Mathématiques Pures et Appliquées ( IF 1.885 ) Pub Date : 2020-06-29 , DOI: 10.1016/j.matpur.2020.06.003
Zimo Hao; Mingyan Wu; Xicheng Zhang

In this paper we develop a new method based on Littlewood-Paley's decomposition and heat kernel estimates in integral form, to establish Schauder's estimate for the following degenerate nonlocal equation in R2d with Hölder coefficients:∂tu=Lκ;v(α)u+b⋅∇u+f,u0=0, where u=u(t,x,v) and Lκ;v(α) is a nonlocal α-stable-like operator with α∈(1,2) and kernel function κ, which acts on the variable v. As an application, we show the strong well-posedness to the following degenerate stochastic differential equation with Hölder drift b:dZt=b(t,Zt)dt+(0,σ(t,Zt)dLt(α)),Z0=(x,v)∈R2d, where Lt(α) is a d-dimensional rotationally invariant and symmetric α-stable process with α∈(1,2), and b:R+×R2d→R2d is a (γ,β)-order Hölder continuous function in (x,v) with γ∈(2+α2(1+α),1) and β∈(1−α2,1), σ:R+×R2d→Rd⊗Rd is a Lipschitz function. Moreover, we also show that for almost all ω, the following random transport equation has a unique Cb1-solution:∂tu(t,x,ω)+(b(t,x)+Lt(α)(ω))⋅∇xu(t,x,ω)=0,u(0,x)=φ(x), where φ∈Cb1(Rd) and b:R+×Rd→Rd is a bounded continuous function of (t,x) and γ-order Hölder continuous in x uniformly in t with γ∈(2+α2(1+α),1).

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