# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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

##### MSC:
 60J25 Continuous-time Markov processes on general state spaces 60J99 Markov processes 60G44 Martingales with continuous parameter