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On the Bochner subordination of exit laws

Hmissi M., Maaouia W.

Opuscula Mathematica

2011

Vol. 31, no. 2

195-207

Let P = (Pt)t≥0 be a sub-Markovian semigroup on L2(m), let β = (βt)t≥0 be a Bochner subordinator and let Pβ = (Pβ(t ))t≥0 be the subordinated semigroup of P by means of β, i.e. Pβ(s):= ∫∞(0) Pr βs(dr). Let φ:= (φt)t>0 be a P-exit law, i.e. Ptφs = φs+t, s,t>0 and let φβ(t):= ∫∞(0)φs βt(ds). Then φβ:= (φβ(t)t>0 is a Pβ-exit law whenever it lies in L2(m). This paper is devoted to the converse problem when β is without drift. We prove that a Pβ-exit law ψ:= (ψt)t>0 is subordinated to a (unique) P-exit law φ (i.e. ψ= φ β) if and only if (Ptu)t>0 ⊂ D(Aβ), where u = ∫∞(0)e-s ψ sds and Aβ, is the L2(m)-generator of Pβ.

Small deviation of subordinated processes over compact sets

Linde W., Zipfel P.

Probability and Mathematical Statistics

2008

Vol. 28, Fasc. 2

281--304

Let A =(A(t)t≥0 be a subordinator. Given a compact set K ⊂[(0;∞) we prove two-sided estimates for the covering numbers of the random set {A(t) : t ∈ K} which depend on the Laplace exponent Φ of A and on the covering numbers of K. This extends former results in the case K = [0; 1]. Using this we find the behavior of the small deviation probabilities for subordinated processes(WH(A(t))tЄK, whereWH is a fractional Brownian motion with Hurst index 0 < H < 1. The results are valid in the quenched as well as in the annealed case. In particular, those questions are investigated for Gamma processes. Here some surprising new phenomena appear. As application of the general results we find the behavior of log P(suptЄK |Zα(t)| < ε) as ε→ 0 for the α-stable Lévy motion Zα. For example, if K is a self-similar set with Hausdorff dimension D > 0, then this behavior is of order −ε−αD in complete accordance with the Gaussian case α = 2.