One dimensional random walks killed on a finite set

Abstract

running head: random walk killed on a finite set key words: exterior domain; transition probability; escape from a finite set; hitting probability of a finite set; potential theory AMS Subject classification (2010): Primary 60G50, Secondary 60J45. AbstractWe study the transition probability,

Scaling Limits of Random Walk Bridges Conditioned to Avoid a Finite Set

“…We present known results taken mainly from [12], [13] and [14] and prove some related results. In the rest of this paper the letters x, y, z and w always denote the integers representing states of the walk.…”

“…e.g., [14, (2.7)]). It is observed in [14] that S b N conditioned on event (2.8) either comes near to A but still avoids it or clears A by one long jump that becomes indefinitely large as N → ∞ according as the above infinite series is convergent or divergent. This would convince one that if the series are convergent there appears a continuous process in the limit.…”

“…14) is proved also in case β = 3 but by a different approach in the next paragraph 4.2.2.) (4.13) follows immediately from Corollary 3.1(a,b).…”

“…We consider the both of cases σ < ∞ and σ = ∞, but usually suppose σ < ∞ unless the contrary is stated explicitly when we discuss the problem for the case σ = ∞. In [14] the present author obtained a precise asymptotic form of transition probability of the walk S killed on a finite nonempty set A (in case σ < ∞). In the present paper we are interested in the behaviour of S b N k := S k + b N , k = 1, .…”

“…In [14] it is observed that if E[|S 1 | 3 ; S 1 < 0] < ∞, then the walk thus conditioned "continuously" transits from the positive half line to the negative half still avoiding A, while if E[|S 1 | 3 ; S 1 < 0] = ∞ it clears A by one "long jump". In this paper we observe that this is reflected to the scaling limit.…”