INTEGRABILITY OF EXIT TIMES

KRZYSZTOF BOGDAN (JOINT WORK WITH RODRIGO BAÑUELOS)

For d = 2, 3, … and 0 < β < 1, we define the parabola–shaped region in R^{d}

P_{β} = {x = (x_{1}, x̃) : x_{1} > 0 , x̃ ∈ R^{d−1} , |x̃| < x_{1}^{β}}.

Let 0 < α < 2. By {X_{t}} we denote the isotropic α-stable R^{d}-valued Lévy process. Let τ_{β} be the first exit time of {X_{t}} from P_{β}. Let p ≥ 0. The main result of [1] is that E_{x}τ_{β}^{p} < ∞ for (some, hence for all) x ∈ P_{β} if and only if p < p_{0}, where the critical exponent of integrability is

p_{0} = \frac{(d − 1)(1 − β) + α}{αβ}.

When P_{β} is replaced by an open cone, situation becomes less explicit, but the corresponding critical exponent of integrability of the first exit time can be expresses in terms of the cone’s Martin kernel with pole at infinity. This is the subject of [2].

I will discuss results and methods of both papers. References to the corresponding results for the Brownian motion are in [3, 4].

References

[1] Rodrigo Bañuelos and Krzysztof Bogdan. Symmetric stable processes in parabola–shaped regions. to appear in Proc. Amer. Math. Soc.

[2] Rodrigo Bañuelos and Krzysztof Bogdan. Symmetric stable processes in cones. to appear in Potential Analysis 21 (2004), no. 3.

[3] Rodrigo Bañuelos and Robert G. Smits. Brownian motion in cones. Probab. Theory Related Fields, 108(3):299–319, 1997.

[4] Mikhail Lifshits and Zhan Shi. The first exit time of Brownian motion from a parabolic domain. Bernoulli, 8(6):745–765, 2002.

Polish Academy of Sciences and Wrocław University of Technology, Wybrzeże Wyspiańskiego 27, 50–370 Wrocław, Poland

  E-mail address: bogdan@im.pwr.wroc.pl

2000 Mathematics Subject Classification. 60J45, 31C05, 60G51.

Key words and phrases. cone, parabola, stable process, exit time.