Intersections of random walks with random sets
Let \(X_n\), \(n\in\mathbb N\), with \(X_0=0\), be a symmetric random walk on \(\mathbb Z^d\) with killing rate \(\beta\in [0,1)\) and \(A_k\), \(k\in\mathbb Z\), be a random sequence in \(\mathbb Z^d\), satisfying certain conditions roughly similar to those defining a symmetric random walk with time set \(\mathbb Z\) and position 0 at \(t=0\). It is assumed that \((X_n\), \(n\in\mathbb N)\) and \((A_k\), \(k\in\mathbb Z)\) are independent. The paper studies the events that the random walk \(X_n\) does not hit the random sets \(\{A_k\),\ \(k\ge m\}.\) A number of general relations is proved connecting expectations of the form \(E(UV)\) where \(U\) and \(V\) are indicators of the above events or random variables of the form \(\sum_{m}\sum_{k}P(X_m=A_ k\mid \mathfrak G)\) where \(\mathfrak G\) is the \(\sigma\)-field generated by the \(A_i\). From these relations bounds are derived for the probability that the random walk \(X_n\) does not hit the nonnegative X-axis, when \(d=2\), \(d=3\). Let \(X_n\), \(Y_n\), \(Z_n\) be independent simple random walks on \(\mathbb Z^2\) or \(\mathbb Z^3\). Bounds are also derived for \(P(X_ i\not\in B\), \(1\leq i\leq N)\), where \(B\) is the union of the ranges of \(X_i\) and \(Y_i\), \(i=0,\ldots,N\).
scientific article; zbMATH DE number 4113688