Penalisation of the symmetric random walk by several functions of the supremum (Q2869218)

Let \(X_0, X_1, \ldots\) be a symmetric simple random walk on the one-dimensional integers, and let \(\mathcal{F}_n = \sigma (X_0, \dots, X_n)\) be the accompanying filtration. Let \(G_0, G_1, \ldots\) be an \((\mathcal{F}_n)\)-adapted process taking non-negative real values; one views \(G_p\) as a functional of \(X_0, \dots, X_p\). The author shows that several functionals, including some of the form \(G_p = \varphi ( \max_{k \leq p} X_k )\), may be associated with an \((\mathcal{F}_n)\)-adapted martingale \(M_n \geq 0\) such that, for all \(n\) and all \(\Lambda_n \in \mathcal{F}_n\), NEWLINE\[NEWLINE Q ( \Lambda_n ) := \lim_{p \to \infty} \frac{ \operatorname{E} [ G_p \mathbf{1} \{ \Lambda_n \} ]}{\operatorname{E} [ G_p ]} = \operatorname{E} [ M_n \mathbf{1} \{ \Lambda_n \} ] NEWLINE\]NEWLINE is well defined. Such a functional thus induces a probability measure \(Q\) on events in \(\cup_n \mathcal{F}_n\). The author describes the behaviour of \(X_0, X_1, \ldots\) under the ``penalized'' measure \(Q\); this is in several cases described in terms of a Bessel-like random walk, i.e., a nearest-neighbour random walk with mean drift at \(x\) of order \(1/x\).