On self-attracting \(d\)-dimensional random walks
Let \(\{X_t\}_{t\geq 0}\) be a symmetric, nearest-neighbour random walk on \(\mathbb{Z}^d\) with exponential holding times of expectation \(1/d\), starting at the origin. For a potential \(V : \mathbb{Z}^d\to [0,\infty)\) with finite and nonempty support, transformed path measures \(\widehat P_T\), \(T>0\), are suitably defined. The authors prove that if \(d=1\) or if the self-attraction is sufficiently strong, then \(|X_t|_\infty\) has an exponential moment under \(\widehat P_t\) which is uniformly bounded for \(T>0\) and \(t\in [0,T]\). The main result is that \(\{X_t\}_{t\geq 0}\) under suitable subsequences of \(\{\widehat P_T\}_{T>0}\) behaves for large \(T\) asymptotically like a mixture of space-inhomogeneous ergodic random walks. For special cases like a sufficiently strong Dirac-type interaction, the authors even prove convergence of the transformed path measures and the law of \(X_T\) as well as the law of the empirical measure \(L_T\) under \(\{\widehat P_T\}_{T>0}\). The Donsker-Varadhan large deviation theory plays a crucial role in the authors' approach.
scientific article; zbMATH DE number 1018401
scientific article