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Knotting of one-dimensional Feller processes. (English) Zbl 0532.60066
Suppose we are given two linear operators $$A_ 0$$, $$A_ 1$$, defined on continuous functions on the intervals $$[r_ 0,r),(r,r_ 1]$$, resp., such that after imposing suitable boundary conditions at r they generate Feller processes on $$[r_ 0,r]$$ and $$[r,r_ 1]$$, resp. In the paper a description of all the infinitesimal generators A of Feller processes on $$[r_ 0,r_ 1]$$ is given which ”coincide” on $$[r_ 0,r) ((r,r_ 1])$$ with $$A_ 0 (A_ 1$$, resp.). It turns out that by means of $$A_ 0, A_ 1$$ a ”maximal” operator $$\tilde A$$ on $$[r_ 0,r_ 1]$$ can be introduced, such that all the infinitesimal generators A are given by a ”knotting condition” at r of the form $(*)\quad \sigma(Af)(r)+\pi_ 0f'(r-)-\pi_ 1f'(r+)+\kappa f(r)+\int^{r_ 1}_{r_ 0}((f(r)- f(x))/| r-x| dq(x)=0,$ $$\sigma$$, $$\pi_ 0$$, $$\pi_ 1$$, $$\kappa$$, q nonnegative, not all 0. These results are closely related to two papers by B. I. Kopytko, Stochastic processes in problems of mathematical physics, Collect. sci. Works, 94-106 (1979; Zbl 0439.60080) and Probability methods of infinite-dimensional analysis, Collect. sci. Works, 84-101 (1980; Zbl 0463.60065); special cases of the condition (*) arise at Feller’s ”shunts” and in connection with the skew Brownian motion.

##### MSC:
 60J35 Transition functions, generators and resolvents
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##### References:
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