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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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[1] [Russian Text Ignored] 1979, 94–106
[2] [Russian Text Ignored] 1980, 84–101
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