Weak Well-Posedness of Multidimensional Stable Driven SDEs in the Critical Case

International audience ; We establish weak well-posedness for critical symmetric stable driven SDEs in R d with additive noise Z, d ≥ 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≥0 stands for a symmetric d-dimensional stable process of order α = 1 defined on some filtered probability space (Ω, F, (F t) t≥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 → R d is continuous and bounded. 1 Above, the generator L of Z writes: Lϕ(x) = p.v. R d \{0} [ϕ(x + z) − ϕ(x)]ν(dz), x ∈ R d , ϕ ∈ C 2 b (R d), ν(dz) = dρ ρ 2μ (dθ), z = ρθ, (ρ, θ) ∈ R * + × S d−1. (1.2) (here ·, · (or ·) and | · | denote respectively the inner product and the norm in R d). In the above equation, ν is the Lévy intensity measure of Z, S d−1 is the unit sphere of R d andμ is a spherical measure on S d−1. It is well know, see e.g. [20] that the Lévy exponent Φ of Z writes as: (1.3) Φ(λ) = E[exp(i λ, Z 1)] = exp − S d−1 | λ, θ |µ(dθ) , λ ∈ R d , where µ = c 1μ , 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 ≥ 1 s.t. (1.4) ∀λ ∈ R d , κ −1 |λ| ≤ S d−1 | λ, θ |µ(dθ) ≤ 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.