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On the local time of multiparameter symmmetric stable processes. Regularity and limit theorem in Besov spaces

Boufoussi B., Dozzi M.

Probability and Mathematical Statistics

2003

Vol. 23, Fasc. 2

369--387

Let X = (Xz, z ϵ TN = [0, l]N) be a symmetric α-stable process, 1 < α ≤ 2. Based on a Kolmogorov type continuity theorem we show Hölder conditions in Lp-norms for the local time of X with respect to the space and time variables, by distinguishing the cases where the time variables do or do not meet the axes. Weak convergence of the occupation integral is proved.

On the exit time of alpha-stable process

Lewandowski M.

Probability and Mathematical Statistics

2003

Vol. 23, Fasc. 1

209--215

In this paper we investigate the probability that α-stable Lévy process stays in convex body up to time t. This can be optimally estimated from below by the same probability but of the rotationally invariant process.

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.