On a Connection between Kuznetsov Processes and Quasi-Processes

Fitzsimmons, P. J.

doi:10.1007/978-1-4684-0550-7_5

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Elements Of Probabilistic Potential Theory1

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1989

2022

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Cited by 8 publications

(2 citation statements)

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“…More precisely, one asks for extensions {P z , z ≥ 0} with the Feller property so that in particular P 0 := w-lim z↓0 P z exists in the Skorokhod topology. Both questions have been solved in recent years: In the recurrent regime it was proved by Fitzsimmons [12] and Rivero [29] that there is a unique recurrent self-similar Markov extension (or equivalently a self-similar excursion measure with summable excursion lengths) that leaves zero continuously if and only if E[e λξ 1 ] = 1 for some 0 < λ < α.…”

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confidence: 99%

Real self-similar processes started from the origin

2017

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confidence: 99%

Real self-similar processes started from the origin

2017

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“…Theorem 5 (Fitzsimmons [2]). If Q is a Kuznetsov measure for (P, η) and S is a finite stationary time then the Palm measure of P and S…”

Section: Elements Of Probabilistic Potential Theorymentioning

confidence: 99%

Random Interlacements via Kuznetsov Measures

Dereich¹,

Doering²

2015

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The aim of this note is to give an alternative construction of interlacements -as introduced by Sznitman -which makes use of classical potential theory. In particular, we outline that the intensity measure of an interlacement is known in probabilistic potential theory under the name "approximate Markov chain" or "quasi-process". We provide a simple construction of random interlacements through (unconditioned) two-sided Brownian motions (resp. two-sided random walks) involving Mitro's general construction of Kuznetsov measures and a Palm measures relation due to Fitzsimmons. In particular, we show that random interlacement is a Poisson cloud ('soup') of two-sided random walks (or Brownian motions) started in Lebesgue measure and restricted on being closest to the origin at time T ∈ [0, 1] -modulus time-shift.

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