Asymptotic Behaviour of a Random Walk Killed on a Finite Set

Abstract

running head: random walk killed on a finite set key words: two-dimensional random walk; exterior domain; transition probability; overshoot estimates AMS Subject classification (2010): Primary 60G50, Secondary 60J45. AbstractWe study asymptotic behavior, for large time n, of the transition probability of a two-dimensional random walk killed when entering into a non-empty finite subset A. We show that it behaves like 4ũ A (x)ũ −A (−y)(lg n) −2 p n (y − x) for large n, uniformly in the parabolic regime |x| ∨ |y| = O( √ n), where p n (y − x) is the transition kernel of the random walk (without killing) andũ A is the unique harmonic function in the 'exterior of A' satisfying the boundary conditionũ A (x) ∼ lg |x| at infinity.

The Transition Density of Brownian Motion Killed on a Bounded Set

“…The constants involved in the O term appearing in (13) of course depends on M and the same remark applies to the results that follow. If the boundary ∂A is smooth, then err…”

“…These are to verify (4) in the case x ≤ y ≤ C √ t/ lg t. The proofs for the other cases of (4) need some results from [11] and [12] in addition to those mentioned above. The same method as described above applies to random walks as is discussed in [13].…”

One dimensional random walks killed on a finite set

“…It rests on the results of [12] in which the same problem as the present paper is studied but with A = {0} and an asymptotic form of the transition probability valid uniformly for space variables is obtained. The same method is applied in [13] to higher dimensional random walks to obtain a similar strengthening of Kesten's result. For multidimensional Brownian motions the corresponding problem is studied by [3] for space variables restricted to compact sets and by [14] without any restriction as such.…”

“…where the right side does not depend on the choice of ξ 0 (cf. [13], see also (3.1)). We shall see that…”

“…Remark 3. In [13] it is shown for a two-dimensional walk that the right side of (3.2) is dual-harmonic (i.e. invariant under the transform h(y) → z p(y − z)h(z)) as a function of y ∈ Z, hence does not depend on y due to the fact that the only bounded harmonic functions are constant functions [10, Theorem 24.1], [6,.…”