On the problem of exit from cycles for simulated annealing processes. A backward equation approach (Q1296611)

Let \(X_t\) be a (time-)inhomogeneous Markov chain on a finite set \(S\) with transition rates \(q_{ij}(t)=p_{ij}\exp\{-U(i,j)T^{-1}(t)\}\) if \(j\neq i\), where \(P=(p_{ij})\) is an irreducible neighborhood choosing matrix with nonnegative entries, \(U(i,j)\) is an arbitrary function from \(S\times S\) to \([O,\infty)\) and \(T(t)>0\) is a suitable temperature function converging to 0. The paper studies the exit distribution \(P_{t,i}(X_\tau=a)\) and the mean exit time \(E_{t,i}\tau\) as \(t\to\infty\), where \(\tau\) is the exit time from a cycle containing \(i\).