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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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