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 $\alpha$ = 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 $\alpha$ = 1 defined on some
filtered probability space ($\Omega$, 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$\Phi$(x) = p.v. R d \{0} [$\Phi$(x + z) -- $\Phi$(x)]$\nu$(dz), x $\in$ R d , $\Phi$ $\in$ C 2
b (R d), $\nu$(dz) = d$\rho$ $\rho$ 2$\mu$ (d$\theta$), z = $\rho$$\theta$,
($\rho$, $\theta$) $\in$ R * + x S d--1. (1.2) (here $\times$, $\times$ (or
$\times$) and | $\times$ | denote respectively the inner product and the norm
in R d). In the above equation, $\nu$ is the L{\'e}vy intensity measure of Z, S
d--1 is the unit sphere of R d and$\mu$ is a spherical measure on S d--1. It is
well know, see e.g. [20] that the L{\'e}vy exponent $\Phi$ of Z writes as:
(1.3) $\Phi$($\lambda$) = E[exp(i $\lambda$, Z 1)] = exp -- S d--1 | $\lambda$,
$\theta$ |$\mu$(d$\theta$) , $\lambda$ $\in$ R d , where $\mu$ = c 1$\mu$ , for
a positive constant c 1 , is the so-called spectral measure of Z. We will
assume some non-degeneracy conditions on $\mu$. Namely we introduce assumption
(ND) There exists $\kappa$ $\ge$ 1 s.t. (1.4) $\forall$$\lambda$ $\in$ R d ,
$\kappa$ --1 |$\lambda$| $\le$ S d--1 | $\lambda$, $\theta$ |$\mu$(d$\theta$)
$\le$ $\kappa$|$\lambda$|. 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.
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