Alternative constructions of a harmonic function for a random walk in a cone

Abstract

For a random walk killed at leaving a cone we suggest two new constructions of a positive harmonic function. These constructions allow one to remove a quite strong extendability assumption, which has been imposed in our previous paper (Denisov and Wachtel, 2015, Random walks in cones). As a consequence, all the limit results from that paper remain true for cones which are either convex or star-like and C 2 .

Discrete harmonic functions in Lipschitz domains

“…Applying her general results to random walk in a convex cone she deduced the uniqueness (up to a multiplicative constant) of the harmonic function constructed by Denisov and Wachtel in [11] under some moment condition on the jumps. Alternative constructions of this harmonic function are proposed by Denisov and Wachtel in [12]. These new constructions allow them to remove quite restrictive extendability assumption imposed in [11].…”

Martin boundary of random walks in convex cones

“…Denisov and Wachtel proved [16,18] the existence of a positive harmonic function V : K → R + defined by…”

“…In this paper, we do not require the existence of a bigger cone K ′ with ∂K \ {0} ⊂ int(K ′ ), such that the réduite u can be extended to a harmonic function on K ′ . This necessary condition in [16] is removed in [18] under the moment assumption (M 1).…”

Conditioned limit theorems for hyperbolic dynamical systems

“…Most of this work is based on the Wiener-Hopf factorization and various factorization identities. Recently Denisov and Wachtel [7,8] have studied the setting of random walks in cones and have developed a new approach based on the construction of the harmonic function thus avoiding the use of the Wiener-Hopf factorization. Following this method, in the case of dependent random variables very recent progress was made in [15,13], where conditioned integral limit theorems for products of random matrices and for Markov chains satisfying spectral gap properties have been obtained.…”