论文标题
在关键情况下，多维稳定驱动SDE的适应性较弱
Weak Well-Posedness of Multidimensional Stable Driven SDEs in the Critical Case
论文作者
论文摘要
我们建立了较弱的适合对称稳定稳定驱动的SDE的较弱的稳定性SDE，d $ \ ge $1。即，我们研究了驱动过程Z的稳定索引z的稳定索引为$α$ = 1，这与具有连续和边界的系数B的漂移项的顺序完全相对应。特别是，当zt =（z 1 t，..，z d t）和z 1，。 ..，z d是独立的一维cauchy过程。我们的方法依赖于稳定操作员的l p估算物，并使用扰动论点。 1。问题和主要结果的陈述，我们有兴趣证明与以下SDE相关的Martingale问题的良好性：（1.1）X T = X + T 0 B（X S）DS + Z T，其中（Z S $ \ GE $ 0立场是对符号d-d-dimensional stable Process of Serve $α$ = 1 $α$ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ nist（f）的$ temect（f）（fom）（f），$ c（fom） ，p）（参见[2]和其中的参考文献）在唯一的连续性和有限性的假设下，对向量值系数B：（c）漂移B：R D $ \ rightarrow $ r D是连续且有限的。上面的1，z的发电机l写入：l $φ$（x）= p.v. r d \ {0} [$φ$（x + z） - $φ$（x）] $ν$（dz），x $ \ in $ r d，$ r d，$φ$ $ \ in $ c 2 b（r d），$ν$（dz $（dz）= d $ $ρ$ $ρ$ p $ 2 $ 2 $ $（$ t $ tub） $ \ in $ r * + x s d--1。 （1.2）（此处$ \ times $，$ \ times $（或$ \ times $）和| $ \ times $ |分别表示内部产品和r d中的标准）。在上面的方程式中，$ν$是z，s d - 1的l {é} vy强度度量是r d和$μ$的单位球体，是S d--1的球形测量。知道了，请参阅例如[20] z的l {é} vy endent $φ$ wert as：（1.3）$φ$（$λ$）= e [exp（i $λ$，z 1）] = exp - s d--1 | $λ$，$θ$ | $μ$（d $θ$），$λ$ $ \ in $ r d，其中$μ$ = c 1 $μ$，对于正常数c 1，是z的频谱度量。我们将假定$μ$的一些非排行条件。即，我们引入假设（ND）存在$κ$ $ \ ge $ 1 S.T. （1.4）$ \ forall$$λ$ $ \ in $ r d，$κ$ -1 | $λ$ | $ \ le $ s d--1 | $λ$，$θ$ | $μ$（d $θ$）$ \ le $ $ $ $κ$ | $λ$ |。 1此处假定B的界限是为了技术简单性。我们的方法可以适用于适当的本地化论点，即具有线性增长的漂移B。
We establish weak well-posedness for critical symmetric stable driven SDEs in R d with additive noise Z, d $\ge$ 1. Namely, we study the case where the stable index of the driving process Z is $α$ = 1 which exactly corresponds to the order of the drift term having the coefficient b which is continuous and bounded. In particular, we cover the cylindrical case when Zt = (Z 1 t ,. .. , Z d t) and Z 1 ,. .. , Z d are independent one dimensional Cauchy processes. Our approach relies on L p-estimates for stable operators and uses perturbative arguments. 1. Statement of the problem and main results We are interested in proving well-posedness for the martingale problem associated with the following SDE: (1.1) X t = x + t 0 b(X s)ds + Z t , where (Z s) s$\ge$0 stands for a symmetric d-dimensional stable process of order $α$ = 1 defined on some filtered probability space ($Ω$, F, (F t) t$\ge$0 , P) (cf. [2] and the references therein) under the sole assumptions of continuity and boundedness on the vector valued coefficient b: (C) The drift b : R d $\rightarrow$ R d is continuous and bounded. 1 Above, the generator L of Z writes: L$Φ$(x) = p.v. R d \{0} [$Φ$(x + z) -- $Φ$(x)]$ν$(dz), x $\in$ R d , $Φ$ $\in$ C 2 b (R d), $ν$(dz) = d$ρ$ $ρ$ 2$μ$ (d$θ$), z = $ρ$$θ$, ($ρ$, $θ$) $\in$ R * + x S d--1. (1.2) (here $\times$, $\times$ (or $\times$) and | $\times$ | denote respectively the inner product and the norm in R d). In the above equation, $ν$ is the L{é}vy intensity measure of Z, S d--1 is the unit sphere of R d and$μ$ is a spherical measure on S d--1. It is well know, see e.g. [20] that the L{é}vy exponent $Φ$ of Z writes as: (1.3) $Φ$($λ$) = E[exp(i $λ$, Z 1)] = exp -- S d--1 | $λ$, $θ$ |$μ$(d$θ$) , $λ$ $\in$ R d , where $μ$ = c 1$μ$ , for a positive constant c 1 , is the so-called spectral measure of Z. We will assume some non-degeneracy conditions on $μ$. Namely we introduce assumption (ND) There exists $κ$ $\ge$ 1 s.t. (1.4) $\forall$$λ$ $\in$ R d , $κ$ --1 |$λ$| $\le$ S d--1 | $λ$, $θ$ |$μ$(d$θ$) $\le$ $κ$|$λ$|. 1 The boundedness of b is here assumed for technical simplicity. Our methodology could apply, up to suitable localization arguments, to a drift b having linear growth.
