On the martingale problem for generators of stable processes with perturbations
The probability measure \(P_ x\) on the Skorokhod space \(D_{[0,\infty)} (R^ d)\) is said to solve the martingale problem for the operator L starting from x if for each \(f\in {\mathcal S}(R^ d) M_ t\!^ f=f(X_ t)-f(x)-\int^{t}_{0}Lf(X_ s)ds\) is a \(P_ x\)-martingale with \(M_ 0\!^ f=0\), where \(X_ t(\omega)=\omega(t)\), \(\omega \in D_{[0,\infty)}(R^ d)\), \(t\geq 0.\) 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 \(L=A^{(\alpha)}+B^{(\alpha)}\), where \(A^{(\alpha)}\) is the generator of a stable process with index \(\alpha\), \(0<\alpha \leq 2\), and \[ B^{(\alpha)}f(x)=\int [f(x+y)-f(x)- I_{\{| y| \leq 1\}}\sum^{d}_{j=1}y_ j\frac{\partial f}{\partial x_ j}(x)]N^{(\alpha)}(x,dy)+ \] \[ \sum^{d}_{j=1}\ell_ j\!^{(\alpha)}(x)\partial f(x)/\partial x_ j. \] The results of \textit{M. Tsuchiya} [Proc. 2nd Japan-USSR Sympos. Probab. Theory, Kyoto 1972, Lect. Notes Math. 330, 490-497 (1973; Zbl 0284.60056)] are improved.
uniqueness of the solution to the martingale problem
scientific article; zbMATH DE number 3850215