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Estimation of the harmonic metric

Wilczek K.

Zeszyty Naukowe Politechniki Rzeszowskiej. Matematyka

2000

z. 24 [181]

145-153

In this paper some special metric in the given domain is investigated. Using the harmonic center of trilaterals the special harmonic coordinates of points are defined. The harmonic metric is associated with these harmonic coordinates. Some relation between this metric and another known metrics are given.

On harmonic measure for Lévy processes

Sztonyk P.

Probability and Mathematical Statistics

2000

Vol. 20, Fasc. 2

383--390

Let {Xt} be a Lévy process in Rd, d ≥ 2, with infinite Lévy measure. If {Xt} has no Gaussian component, then the process does not hit the boundary of Lipschitz domain S ⊂ Rd at the first exit time of S under mild conditions on {Xt}. The conditions are met, e.g., if {Xt} is rotation invariant.

Exist time and Green function of cone for symmetric stable processes

Kulczycki T.

Probability and Mathematical Statistics

1999

Vol. 19, Fasc. 2

337--374

We obtain estimates of the harmonic measure and the expectation of the exit time of a bounded cone for symmetric α-stable processes Xt in Rd (α ϵ (0, 2), d ≥ 3). This enables us to study the asymptotic behaviour of the corresponding Green function of both bounded and unbounded cones. We also apply our estimates to the problem concerning the exit time τv of the process Xt from the unbounded cone V of angle λ ϵ (0, π/2). We namely obtain upper and lower bounds for the constant p0 = p0 (d, α, λ) such that for all x ϵ V we have Ex (τpV) < ∞ for 0 ≤ p < p0 and Ex (τpV) = ∞ for p > p0.