Time reversal of random walks in \({\mathbb{R}}^ d\) (Q811015)

Consider a random walk \(\{S_ n\}\) on \({\mathbb{R}}^ d\) with step distribution concentrated on a countable set. Assume that a pseudo-order \(\triangleleft\) is defined on \({\mathbb{R}}^ d\) and let \(\tau\) be the first n such that \(S_ n\triangleleft S_ k\) for \(k=0,...,n-1\). Independent copies of the time-reversed segment \((0,S_{\tau -1}-S_{\tau},...,S_ 1-S_{\tau},-S_{\tau})\) can be glued together in such a way that a Markov chain results. Formulas are given for the associated transition function, these are specialized to pseudo-orders of the form \[ x\triangleleft y\quad \Leftrightarrow \quad <y-x,a_ i>\geq 0\text{ for } i=1,...,m, \] where \(<\cdot,\cdot >\) denotes inner product and \(a_ 1,...,a_ m\in {\mathbb{R}}^ d\) are fixed vectors.