arXiv2020
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.