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We study the fractional Fokker-Planck-Kolmogorov equation with the fractional index α ∈ [ 1 , 2) and use a vector-valued Calderón-Zygmund theorem to obtain the existence and uniqueness of L p ([ 0 , T ] ; C b α + β (R d)) ∩ W 1 , p ([ 0 , T ] ; C b β (R d)) solution under the assumptions that the drift coefficient and nonhomogeneous term are in L p ([ 0 , T ] ; C b β (R d)) with p ∈ [ α / (α - 1) , + ∞ ] and β ∈ (0 , 1) . As applications, we prove the unique strong solvability as well as Davie's type uniqueness of time inhomogeneous stochastic differential equation with the drift in L p ([ 0 , T ] ; C b β (R d ; R d)) and driven by the α -stable process for β > 1 - α / 2 and p > 2 α / (α + 2 β - 2) . [ABSTRACT FROM AUTHOR]

The goal of this paper is to establish the Lipschitz and W 2 , ∞ estimates for a second-order parabolic PDE ∂ t u (t , x) = 1 2 Δ u (t , x) + f (t , x) on R d with zero initial data and f satisfying a Ladyzhenskaya–Prodi–Serrin type condition. Following the theoretic result, we then give two applications. The first is to discuss the regularity of the stochastic heat equations, and the second is to discuss the Sobolev differentiability of strong solutions to a class of SDEs with singular drift coefficients. [ABSTRACT FROM AUTHOR]

Consider the following stochastic differential equation (SDE): X t = x + ∫ 0 t b (s , X s) d s + ∫ 0 t σ (s , X s) d B s , 0 ≤ t ≤ T , x ∈ R , where { B s } 0 ≤ s ≤ T is a 1-dimensional standard Brownian motion on [ 0 , T ] . Suppose that q ∈ (1 , ∞ ] , p ∈ (1 , ∞) , b = b 1 + b 2 , b 1 ∈ L q (0 , T ; L p (R)) such that 1 / p + 2 / q < 1 and b 2 is bounded measurable, with σ ∈ L ∞ (0 , T ; C u (R)) there being a real number δ > 0 such that σ 2 ≥ δ . Then there exists a weak solution to the above equation. Moreover, (i) if σ ∈ C ([ 0 , T ] ; C u (R)) , all weak solutions have the same probability law on 1-dimensional classical Wiener space on [ 0 , T ] and there is a density associated with the above SDE; (ii) if b 2 = 0 , p ∈ [ 2 , ∞) and σ ∈ L 2 (0 , T ; C b 1 / 2 (R)) , the pathwise uniqueness holds. [ABSTRACT FROM AUTHOR]