Hitting distribution to a quadrant of two-dimensional random walk

Abstract

Let H L (ζ,η) be the probability that a two-dimensional simple random walk starting at ξ hits the third quadrant L for the first time at η. The main objective of this paper is to investigate the asymptotic behavior of H L (ζ,η).It is especially proved that there exists a constant C o such that for ξ e Z 2 \L and / e N, \H L (ξ, (-/, 0)) -A L (f, (-/,0)where h L (ξ, •) is the density of the hitting distribution to the third quadrant of two-dimensional standard Brownian motion starting at f. This estimate is sharp at least in the sense that the powers -2/3 and -5/3 can not be improved.

The hitting distributions of a half real line for two-dimensional random walks

“…The time reversed version of (14) shows that 1/ (−x+1)r is the correct order in this bound. The method of [2] also gives an estimate of H − x (s) (for simple walk), which is better than (8) and that of [4] but not sharp near the edge (cf. [13]) as in (9) or (10).…”

“…: a * (y) is equal to the potential function of the one-dimensional walk S (2) n if y =0 (cf. [10]) and a * (0)=1.…”

The Hitting Distribution of a Line Segment for Two-Dimensional Random Walks

“…If w = −(1 − x 2 + y 2 ) + i2xy and φ = π − arg w ∈ (−π/2, π/2), then |f (z)| 2 = (x + |w| 1/2 sin φ) 2 + (y + |w| 1/2 cos φ) 2 and we see that y −1 (|f (z)| 2 − 1) → 2/ √ 1 − x 2 . In view of (39) this shows that where cos θ t = t with θ t ∈ (0, π) for −1 < t < 1.…”

“…For a rectangle with a side on the real axis Lawler and Limic [9] give an explicit expression for the hitting distribution of its boundary for a simple random walk started inside it and, by taking limits, derive from it the corresponding ones for a half-infinite strip and a quadrant. For a quadrant of the plane, one half of it split along its diagonal line and the complements of these regions as well Fukai [2] obtains very detailed evaluations of the hitting distributions by exploiting the properties special to simple random walk. Throughout this paper we suppose that the walk S n is irreducible, E 0 [S 1 ] = 0 and E 0 [|S 1 | 2+δ ] < ∞ either for δ = 0 or for some δ > 1/2;…”

Mathematical Model for Hiv and Cd4+ Cells Dynamics in Vivo

“…Fukai [2] gives an analysis of two-dimensional paths that actually enter the third quadrant. And Shimura [4] analyzes asymptotic behaviour of paths exitng a quadrant.…”