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On the martingale problem for generators of stable processes with perturbations. (English) Zbl 0535.60063
The probability measure $P\sb x$ on the Skorokhod space $D\sb{[0,\infty)} (R\sp d)$ is said to solve the martingale problem for the operator L starting from x if for each $f\in {\cal S}(R\sp d) M\sb t\!\sp f=f(X\sb t)-f(x)-\int\sp{t}\sb{0}Lf(X\sb s)ds$ is a $P\sb x$-martingale with $M\sb 0\!\sp f=0$, where $X\sb t(\omega)=\omega(t)$, $\omega \in D\sb{[0,\infty)}(R\sp d)$, $t\ge 0.$ Using the theory of singular integrals, the author has found general conditions for the uniqueness of the solution to the martingale problem for the operators $L=A\sp{(\alpha)}+B\sp{(\alpha)}$, where $A\sp{(\alpha)}$ is the generator of a stable process with index $\alpha$, $0<\alpha \le 2$, and $$ B\sp{(\alpha)}f(x)=\int [f(x+y)-f(x)- I\sb{\{\vert y\vert \le 1\}}\sum\sp{d}\sb{j=1}y\sb j\frac{\partial f}{\partial x\sb j}(x)]N\sp{(\alpha)}(x,dy)+ $$ $$ \sum\sp{d}\sb{j=1}\ell\sb j\!\sp{(\alpha)}(x)\partial f(x)/\partial x\sb j. $$ The results of {\it M. Tsuchiya} [Proc. 2nd Japan-USSR Sympos. Probab. Theory, Kyoto 1972, Lect. Notes Math. 330, 490-497 (1973; Zbl 0284.60056)] are improved.
Reviewer: B.Grigelionis

60J25Continuous-time Markov processes on general state spaces
60J99Markov processes
60G44Martingales with continuous parameter