Abstract
Let $S^{1},S^{2}$ be independent simple random walks in $\mathbb{Z}^{d}$ ($d=2,3$) started at the origin. We construct two-sided random walk paths conditioned that $S^{1}[0,\infty ) \cap S^{2}[1, \infty ) = \emptyset$ by showing the existence of the following limit:\begin{equation*}\lim _{n \rightarrow \infty } P ( \cdot | S^{1}[0, \tau ^{1} ( n) ] \cap S^{2}[1, \tau ^{2}(n) ] = \emptyset ),\end{equation*}where $\tau^{i}(n) = \inf \{ k \ge 0 : |S^{i} (k) | \ge n \}$. Moreover, we give upper bounds of the rate of the convergence. These are discrete analogues of results for Brownian motion obtained by Lawler.
Highlights
Introduction and Main Results1.1 IntroductionLet S = (S(n)) be a simple random walk in Zd (d = 2, 3) started at the origin
Even though the conditioned Brownian paths were already constructed as in ( 1.6), it is not straightforward to construct it for the simple random walk. Both in [3] and [8], the scaling property of Brownian motion is crucial in the construction and the same arguments cannot be applied for the simple random walk case
We will use the strong approximation of Brownian motion by simple random walk derived from the Skorohod embedding
Summary
We will construct the path in (1.3) by proving the existence of the limit as in ( 1.6) for simple random walk (Theorem 1.2.1). Both in [3] and [8], the scaling property of Brownian motion is crucial in the construction and the same arguments cannot be applied for the simple random walk case To overcome this problem, we will use the strong approximation of Brownian motion by simple random walk derived from the Skorohod embedding. S1 and S2 are far apart, we can use the Skorohod embedding to control the non-intersection probability of simple random walks (see Proposition 3.3.16 for details).
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