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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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