Stable Lévy processes, self-similarity and the unit ball

Kyprianou, Andreas E.

doi:10.30757/alea.v15-25

Cited by 27 publications

(21 citation statements)

References 75 publications

(176 reference statements)

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“…the right-continuous extension, enlarged by null sets). For further details about stable processes we refer the reader to Chapter 7 of Bertoin [3], Chapter 3 of Sato [25] or the review article Kyprianou [16].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…To understand why the latter is true, note that the symmetric and oscillatory behaviour of X in the current regime of α ensures that lim sup t→∞ ξ t = ∞ almost surely, irrespective of the point of issue of (ξ, J). This, in turn, means that lim t→∞ t 0 e αξs ds = ∞ in the same almost sure sense; see for example the discussion of the different properties of stable processes and their underlying MAPs in Kyprianou [16]. Consequently, as (ϕ t , t ≥ 0) is the inverse of the aforementioned integrated exponential of ξ, it follows that, for each t ≥ 0, x ∈ R and i ∈ {−1, 1} we have that ϕ t < ∞, P x,i -almost surely.…”

Section: Proofsmentioning

confidence: 92%

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Stable processes conditioned to avoid an interval

Döring

Kyprianou

Weissmann

2020

Stochastic Processes and their Applications

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Conditioning Markov processes to avoid a domain is a classical problem that has been studied in many settings. Ingredients for standard arguments involve the leading order tail asymptotics of the distribution of the first hitting time of the domain of interest and its relation to an underlying harmonic function.In the present article we condition stable processes to avoid intervals. The required tail asymptotics in the stable setting for α ≥ 1 go back to classical work of Blumenthal et al. and Port from the 1960s. For α < 1, we appeal to recent results centred around the so-called deep factorisation of the stable process to compute hitting probabilities and, moreover, to identify the associated harmonic functions for all α ∈ (0, 2). With these in hand, we thus prove that conditioning to avoid an interval is possible in the classical sense and that the resulting process is a Doob h-transform of the stable process killed on entering the aforesaid interval. Appealing to the representation of the conditioned process as a Doob h-transform, we verify that the conditioned process is transient.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Proofsmentioning

confidence: 92%

Stable processes conditioned to avoid an interval

Döring

Kyprianou

Weissmann

2020

Stochastic Processes and their Applications

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“…It follows that the third term of (11) is 0. So it remains to consider the first and the second sumand of (11). With Proposition 6.2 (ii) of Kunita and Watanabe [9] and Corollary 3.2 we see for the first term:…”

Section: Harmonic Functionsmentioning

confidence: 95%

“…Before presenting our results, we introduce the most important definitions. More details can be found, for example, in Chung and Walsh [5], Bertoin [2], Kyprianou [11] or Sato [17].…”

Section: Introductionmentioning

confidence: 99%

Stable processes conditioned to hit an interval continuously from the outside

Döring¹,

Weissmann²

2020

Bernoulli

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“…A Newton potential formula tells us that [15], and hence, after an application of Fubini's theorem for the two spherical integrals and change of variable,…”

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confidence: 99%

Deep Factorisation of the Stable Process III: the View from Radial Excursion Theory and the Point of Closest Reach

2019

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In this paper, we continue our understanding of the stable process from the perspective of the theory of self-similar Markov processes in the spirit of [10,13]. In particular, we turn our attention to the case of d-dimensional isotropic stable process, for d ≥ 2. Using a completely new approach we consider the distribution of the point of closest reach. This leads us to a number of other substantial new results for this class of stable processes. We engage with a new radial excursion theory, never before used, from which we develop the classical Blumenthal-Getoor-Ray identities for first entry/exit into a ball, cf. [3], to the setting of n-tuple laws. We identify explicitly the stationary distribution of the stable process when reflected in its running radial supremum. Moreover, we provide a representation of the Wiener-Hopf factorisation of the MAP that underlies the stable process through the Lamperti-Kiu transform.where the two terms either side of the multiplication sign constitute the two Wiener-Hopf factors. See e.g. Chapter VI in [2] for background. Recall that if Ψ is the characteristic exponent of any Lévy process, then there exist two Bernstein functions κ andκ (see [18] for a definition) such that, up to a multiplicative constant,Identity (1.7) is what we refer to as the Wiener-Hopf factorisation. The left-hand factor codes the range of the running maximum and the right-hand factor codes the range of the running 2 infimum of ξ. It can be checked that both belong to the class of so-called beta subordinators (see [8], as well as some of the discussion later in this paper) and, in particular, have infinite activity. This implies that ξ is regular for both the upper and lower half-lines, which in turn, means that any sphere of radius r > 0 is regular for both its interior and exterior for X. This and the fact that X has càdlàg paths ensures that, denotingthe quantity X G(t) is well defined as the point of closest reach to the origin up to time t in the sense that X G(t)− = X G(t) andThe process (G(t), t ≥ 0) is monotone increasing and hence there is no problem defining G(∞) = lim t→∞ G(t) almost surely. Moreover, as X is transient in the sense of (1.3), it is also clear that, almost surely, G(∞) = G(t) for all t sufficiently large and thatOur first main result provides explicitly the law of X G(∞) .Theorem 1.1 (Point of Closest Reach to the origin). The law of the point of closest reach to the origin is given by P x (X G(∞) ∈ dy) = π −d/2 Γ (d/2) 2 Γ ((d − α)/2) Γ (α/2) (|x| 2 − |y| 2 ) α/2 |x − y| d |y| α dy, 0 < |y| < |x|.

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Stable Lévy processes, self-similarity and the unit ball

Cited by 27 publications

References 75 publications

Stable processes conditioned to avoid an interval

Stable processes conditioned to avoid an interval

Stable processes conditioned to hit an interval continuously from the outside

Deep Factorisation of the Stable Process III: the View from Radial Excursion Theory and the Point of Closest Reach

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