Schauder estimates for nonlocal kinetic equations and applications

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 R^2d with Hölder coefficients: ∂_tu=L_κ;v^(α)u+b⋅∇u+f,u₀=0, where u=u(t,x,v) and L_κ;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: dZ_t=b(t,Z_t)dt+(0,σ(t,Z_t)dL_t^(α)),Z₀=(x,v)∈R^2d, where L_t^(α) is a d-dimensional rotationally invariant and symmetric α-stable process with α∈(1,2), and b:R₊×R^2d→R^2d is a (γ,β)-order Hölder continuous function in (x,v) with [Formula presented] and [Formula presented], σ:R₊×R^2d→R^d⊗R^d is a Lipschitz function. Moreover, we also show that for almost all ω, the following random transport equation has a unique C_b¹-solution: ∂_tu(t,x,ω)+(b(t,x)+L_t^(α)(ω))⋅∇_xu(t,x,ω)=0,u(0,x)=φ(x), where φ∈C_b¹(R^d) and b:R₊×R^d→R^d is a bounded continuous function of (t,x) and γ-order Hölder continuous in x uniformly in t with [Formula presented].