On recurrence and transience of self-interacting random walks
Take \(\mu_1,\mu_2,\dots{},\mu_k\) probability measures on \(\mathbb R^d\) with zero mean and supports containing \(d\)-linearly independent vectors of \(\mathbb R^d\). Consider the following (possibly non-Markovian) random walk \((X_n)_{n\in\mathbb N}\): at each time \(i\in\mathbb N\), choose one of the \(k\) measures according to a strategy that depends only on the trajectory \((X_0,\dots{},X_i)\) of the walk up to time \(i\) and perform the next step of the walk according to the chosen measure. The authors deal with the problem of transience VS recurrence for this kind of walks. The main result of the paper states that, if \(k=2\), \(d\geq 3\) and \(\mu_1,\mu_2\) have finite \(2+\beta\) moments, with \(\beta>0\), then \((X_n)_{n\in\mathbb N}\) is transient for any possible choice of the strategy. On the other hand, for \(k=d\), the authors exhibit explicit examples of measures and strategies that produce walks that are recurrent. Note that these two results together give, in particular, a complete picture of the case \(d=3\).
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