Consider the following SDE $$ d X_t=b(X_t)\,d t+d L_t, $$ where $L_t$ is the symmetric $\alpha$-stable process. The corresponding generator is given by $$ L=\Delta^{\alpha/2}+b\cdot\nabla. $$ Is the solution for the martingale problem of $L$ equivalent to the weak solution of the above SDE? (For Brownian motion case, it is well known. But for jump Levy noise, I couldn't find any reference books.)
Many thanks for the help!