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Estimation of the density of simulated annealing. (Estimation de la densité du recuit simulé.) (French) Zbl 0802.60092
The author considers the problem of estimating the density of a symmetric annealing process \((X(t)\), \(t\in\mathbb{R}\), \(t\geq 0)\) on a finite set or a compact manifold \(E\), with generator of the form \[ L_ \beta f= \Delta f- \beta\nabla U\cdot \nabla f, \qquad f\in \mathbb{C}^ \infty(E). \] The method considers weak Sobolev inequalities for studying the Radon-Nikodym derivative \(h_ t\) of \(\nu_ t\) (the law of the process) relatively to the instantaneous invariant measure \(\mu_{\beta(t)}\). Choosing cooling schedules of the form \(\beta(t)= \Gamma^{-1} \log (1+t)\), where \(\Gamma>\gamma\) (the highest energy barrier), it is shown that \[ \exp(-K \log(t)^{5n}/ t^{1-(\gamma/ \Gamma)})\leq h_ t(x,y)\leq \exp(K' \log(t)^{5n +1}/ t^{1- (\gamma/ \Gamma)}), \qquad \text{as } t\to\infty, \] when \(E\) has dimension \(n\in\mathbb{N}\), and \[ \exp (-K \log(t)^ 2/ t^{1- (\gamma/ \Gamma)})\leq h_ t (x,y)\leq \exp(K' \log (t)^ 3/ t^{1- (\gamma/ \Gamma)}), \qquad \text{as } t\to\infty, \] when \(E\) is a finite set, where the constants \(K\) and \(K'\) only depend on \(E\), \(U\) and \(\Gamma\).
Reviewer: C.Mazza (Fribourg)

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60F10 Large deviations
60J20 Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.)
62M99 Inference from stochastic processes
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