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The nonlocal Fokker–Planck equations for a class of stochastic differential equations with non-Gaussian α -stable Lévy motion in Euclidean space are studied. The existence and uniqueness of weak solution are obtained with rough drift. The solution is shown to be smooth on spatial variable if all derivatives of the drift are bounded. Moreover, the solution is jointly smooth on spatial and time variable if we assume further that the drift grows like a power of logarithm function at infinity. [ABSTRACT FROM AUTHOR]

We study the quantitative unique continuation property of some higher order elliptic operators. We prove a lower bound for nontrivial solutions of the equation ( − Δ ) m u + V ( x ) u = 0 , m ∈ N , V ( x ) is bounded. The bound shows that nontrivial solutions can not decay faster than e − | x | 4 / 3 ln ⁡ | x | at infinity. Moreover, we obtain an improved lower bound of nontrivial solutions for a special forth order elliptic operators in dimension 2, the bound is shown to be essentially sharp by constructing a Meshkov-type example. [ABSTRACT FROM AUTHOR]

For every p belongs to [ 1 , ∞) and n dimensional vector b , it is shown that − Λ α − b ⋅ ∇ generates an analytic semigroup in L p (R n) if 1 ≤ α ≤ 2 , and a Gevrey semigroup of order δ , for any δ > 1 α , in L p (R n) if 0 < α < 1. Moreover, it is proved that the semigroup is not a Gevrey semigroup of order δ with δ ≤ 1 α if 0 < α < 1. This shows that our result is sharp. [ABSTRACT FROM AUTHOR]