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Duality theory for self-similar processes. (English) Zbl 0608.60057
Let $$(X_ t,t\geq 0)$$ be a time homogeneous strong Markov process on $${\mathbb{R}}^ n\setminus \{0\}$$ with transition function $$(P_ t(\cdot,\cdot))_{t\geq 0}$$ and with nice sample paths. Assume that (i) for some $$\alpha >0$$, $$P_ t(x,A)=P_{ct}(c^{\alpha}x,c^{\alpha}A)$$ for $$t\geq 0$$, $$x\in {\mathbb{R}}^ n\setminus \{0\}$$, $$A\in {\mathcal B}({\mathbb{R}}^ n\setminus \{0\})$$ and $$c>0$$, and (ii) $$P_ t(x,A)=P_ t(T(x),T(A))$$ for $$T\in {\mathcal O}({\mathbb{R}}^ n)$$ (the group of orthogonal transformations on $${\mathbb{R}}^ n)$$. In short, $$(X_ t,t\geq 0)$$ is taken to be an $$\alpha$$-self similar, rotation invariant Markov process.
It is shown that for the process X there exists another rotation invariant $$\alpha$$-self similar Markov process which is in a weak duality with X with respect to the measure $$| x|^{1/\alpha -n}dx$$. Two characterizations of the dual process are also given.
Reviewer: Lou Jiann-Hua

##### MSC:
 60G99 Stochastic processes 60K99 Special processes 60J99 Markov processes
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