an:03984245
Zbl 0608.60057
Vuolle-Apiala, J.; Graversen, S. E.
Duality theory for self-similar processes
EN
Ann. Inst. Henri Poincar??, Probab. Stat. 22, 323-332 (1986).
00152049
1986
j
60G99 60K99 60J99
self similarity; rotation invariant Markov process; characterizations of the dual process
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.
Lou Jiann-Hua