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On the Exact Asymptotics of Exit Time From a Cone of an Isotropic α-Self-Similar Markov Process with a Skew-Product Structure

Palmowski Zbigniew, Wang Longmin

Probability and Mathematical Statistics

2021

Vol. 41, Fasc. 1

25--38

In this paper we identify the asymptotic tail of the distribution of the exit time τC from a cone C of an isotropic α-self-similar Markov process Xt with a skew-product structure, that is, Xt is a product of its radial process and an independent time changed angular component θt. Under some additional regularity assumptions, the angular process θt killed on exiting the cone C has a transition density that can be expressed in terms of a complete set of orthogonal eigenfunctions with corresponding eigenvalues of an appropriate generator. Using this fact and some asymptotic properties of the exponential functional of a killed Lévy process related to the Lamperti representation of the radial process, we prove that Px(τC> t) ~h(x)t-k1 as t→∞ for h and k1 identified explicitly. The result extends the work of De Blassie (1988) and Bañuelos and Smits (1997) concerning the Brownian motion.

Captivity of mean-field particle systems and the related exit problems

Tugaut J.

Probability and Mathematical Statistics

2015

Vol. 35, Fasc. 1

143--159

A mean-field system is a weakly interacting system of N particles in Rd confined by an external potential. The aim of this work is to establish a simple result about the exit problem of mean-field systems from some domains when the number of particles goes to infinity. More precisely, we prove the existence of some subsets of RdN such that the probability of leaving these sets before any T > 0 is arbitrarily small by taking N large enough. On the one hand, we show that the number of steady states in the small-noise limit is arbitrarily large with a sufficiently large number of particles. On the other hand, using the long-time convergence of the hydrodynamical limit, we identify the steady states as N goes to infinity with the invariant probabilities of the McKean-Vlasov diffusion so that some steady states in the small-noise limit are not steady states in the large N limit.

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.