Uniqueness of stable processes with drift (Q2796737)

Let \(d\geq 1\) and \(\alpha\in(0,1)\). The authors study \(d\)-dimensional rotationally symmetric \(\alpha\)-stable processes. The fractional Laplace operator denoted as \(-(-\Delta)^{\alpha/2}\) is a generator of any of such processes. Some special Kato class of functions is introduced, and the operator \(L_b=-(-\Delta)^{\alpha/2}+b\nabla\) with some \(b\) from this Kato class is created. It is proved that the martingale problem for this operator and for any initial point \(x\in \mathbb{R}^d\) is well-posed in a sense that it has a unique solution. It is established also that the equation \(dX_t=dY_t+b(X_t)dt\) has a unique weak solution for any \(d\)-dimensional rotationally symmetric \(\alpha\)-stable process \(Y\) and any initial point \(x\in \mathbb{R}^d\).