Some Theorems on Stable Processes

In this paper we study the potential theory of symmetric geometric stable processes by realizing them as subordinate Brownian motions with geometric stable subordinators. More precisely, we establish the asymptotic behaviors of the Green function and the Lévy density of symmetric geometric stable processes. The asymptotics of these functions near zero exhibit features that are very different from the ones for stable processes. The Green function behaves near zero as 1/(|x| d log 2 |x|), while the Lévy density behaves like 1/|x| d . We also study the asymptotic behaviors of the Green function and Lévy density of subordinate Brownian motions with iterated geometric stable subordinators. As an application, we establish estimates on the capacity of small balls for these processes, as well as mean exit time estimates from small balls and a Harnack inequality for these processes.

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“…Now we can use Theorem 2.1 of [5] and the dominated convergence theorem to arrive at our conclusion. 2…”

Section: Now Combine This With Theorem 24 To Get Thatmentioning

confidence: 90%

“…Recall that p α (t, x) is the transition density of the symmetric α-stable process in R d . It is well known (see Theorem 2.1 of [5]) that there exist positive constants C 1 and C 2 such that…”

Section: Theorem 47 We Havementioning

confidence: 99%

Potential Theory of Geometric Stable Processes

Šikić

Song

Vondraček

2005

Probab. Theory Relat. Fields

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“…By comparison, this lower estimate entails immediately the good lower bounds under all the L p -norms (1 ≤ p ≤ ∞), since the critical exponent γ = α does not depend on p. A natural (and finer) semi-norm on the set of real càd-làg functions is the strong pvariation in the sense of N. Wiener, which was quite intensively studied by the stochastic community in the midst of the last century. Bretagnolle [2] had obtained a general criterion ensuring that an α-stable Lévy process has a.s. finite p-variation if and only if p > α (see also [12] and [4] for previous results in the symmetric case). More recently, Chistyakov and Galkin [5] proved an interesting embedding theorem which entails that for continuous paths, p-variation and (1/p)-Hölder semi-norm are roughly equivalent notions when p ≥ 1.…”

Section: Introductionmentioning

confidence: 91%

“…In the symmetric case, the if part is not difficult to prove via (9), (10), and Bochner's subordination [4].…”

Section: Introductionmentioning

confidence: 99%

Small Ball Estimates in p-Variation for Stable Processes

Simon

2004

Journal of Theoretical Probability

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Let {Z t , t ≥ 0} be a strictly stable process on R with index α ∈ (0, 2]. We prove that for every p > α, there exists γ = γ(α, p) and K = K(α, p) ∈ (0, +∞) such thatwhere ||Z|| p stands for the strong p-variation of Z on [0,1]. The critical exponent γ(α, p) takes a different shape according as |Z| is a subordinator and p > 1, or not. The small ball constant K(α, p) is explicitly computed when p ≤ 1, and a lower bound on K(α, p) is easily obtained in the general case. In the symmetric case and when p > 2, we can also give an upper bound on K(α, p) in terms of the Brownian small ball constant under the (1/p)-Hölder semi-norm. Along the way, we remark that the positive random variable ||Z|| p p is not necessarily stable when p > 1, which gives a negative answer to an old question of P. E. Greenwood [9].Keywords: Hölder semi-norm -p-variation -Small balls probabilities -Stable processesSubordination. MSC 2000: 60F99, 60G52 IntroductionIn a recent paper [13], a general method was introduced to prove the existence of finite small ball constants for real fractional α-stable processes, under different norms. Whereas the existence result holds with a reasonable level of generality including for example all symmetric α-stable processes -see Theorem 3.1 in [13], the finiteness result (which amounts to a lower estimate on the small ball probabilities -see Theorem 4.1 in [13]) was obtained with the help of wavelet decompositions concerning only continuous processes. The suitable lower estimate for small probabilities of α-stable Lévy processes under the uniform norm is a classical result, which dates back to Taylor [20] and Mogul'skiǐ [14]. By comparison, this lower estimate entails immediately the good lower bounds under all the L p -norms (1 ≤ p ≤ ∞), since the critical exponent γ = α does not depend on p. A natural (and finer) semi-norm on the set of real càd-làg functions is the strong pvariation in the sense of N. Wiener, which was quite intensively studied by the stochastic community in the midst of the last century. Bretagnolle [2] had obtained a general criterion ensuring that an α-stable Lévy process has a.s. finite p-variation if and only if p > α (see also [12] and [4] for previous results in the symmetric case). More recently, Chistyakov 1 and Galkin [5] proved an interesting embedding theorem which entails that for continuous paths, p-variation and (1/p)-Hölder semi-norm are roughly equivalent notions when p ≥ 1. In the discontinuous framework, p-variation seems to be a good substitute for the irrelevant (1/p)-Hölder semi-norm, in studying finer sample path properties. We finally refer to the comprehensive survey of Dudley and Norvaiša [6] for recent developments about this notion, both from the analytical and probabilistic point of view.The purpose of the present paper is to prove the existence of the small deviation constant for strictly (non necessarily symmetric) α-stable processes with respect to the p-variation ||.|| p , when p > α. In other words, we prove thatwhere the critical exponent is given...

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