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Inverse subordination of excessive functions. (English) Zbl 0813.60071

Freidlin, Mark I. (ed.), The Dynkin Festschrift. Markov processes and their applications. In celebration of Eugene B. Dynkin’s 70th birthday. Boston, MA: Birkhäuser. Prog. Probab. 34, 111-131 (1994).
Let \((X_ t)\) be a right process with semigroup \((P_ t)\), let \((M_ t)\) be a multiplicative functional (in the usual sense, it takes values in \([0,1])\). For simplicity, assume here that \(M_ 0 = 1\). We associate with \(M\) the corresponding semigroup \(Q_ t \leq P_ t\), the class of \(Q\)-excessive functions (larger than that of \(P\)-excessive ones) and the sub-Markovian kernels \[ P^ q_ M = \mathbb{E}^ \bullet \int^ \infty_ 0 - e^{-qt} f(X_ t) dM_ t \] (no \(q\) in the notation means \(q = 0)\). It is well-known that \(P_ M\) preserves \(P\)-excessive functions \(u\) and that \(u-P_ Mu\) is \(Q\)-excessive. The problem handled here is: conversely, given \(u\geq 0\) such that \(u-P_ Mu\) is \(Q\)-excessive, is it \(P\)-excessive? If \(M\) never vanishes, the answer is yes, and the easy proof is given as a starting point. But in case \(M\) may vanish, the problem is delicate, and its answer may be negative unless \(M\) vanishes by jumping to 0. The approach gives several interesting by-products, among which a formula giving the iterates of \(P^ q_ M\).
For the entire collection see [Zbl 0808.00010].

MSC:

60J57 Multiplicative functionals and Markov processes
60J45 Probabilistic potential theory
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