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Zbl 0802.60092
Concordet, D.
Estimation of the density of simulated annealing. (Estimation de la densité du recuit simulé.) (French)
[J] Ann. Inst. Henri Poincaré, Probab. Stat. 30, No.2, 265-302 (1994). ISSN 0246-0203

The author considers the problem of estimating the density of a symmetric annealing process $(X(t)$, $t\in\bbfR$, $t\geq 0)$ on a finite set or a compact manifold $E$, with generator of the form $$L\sb \beta f= \Delta f- \beta\nabla U\cdot \nabla f, \qquad f\in \bbfC\sp \infty(E).$$ The method considers weak Sobolev inequalities for studying the Radon-Nikodym derivative $h\sb t$ of $\nu\sb t$ (the law of the process) relatively to the instantaneous invariant measure $\mu\sb{\beta(t)}$. Choosing cooling schedules of the form $\beta(t)= \Gamma\sp{-1} \log (1+t)$, where $\Gamma>\gamma$ (the highest energy barrier), it is shown that $$\exp(-K \log(t)\sp{5n}/ t\sp{1-(\gamma/ \Gamma)})\leq h\sb t(x,y)\leq \exp(K' \log(t)\sp{5n +1}/ t\sp{1- (\gamma/ \Gamma)}), \qquad \text{as } t\to\infty,$$ when $E$ has dimension $n\in\bbfN$, and $$\exp (-K \log(t)\sp 2/ t\sp{1- (\gamma/ \Gamma)})\leq h\sb t (x,y)\leq \exp(K' \log (t)\sp 3/ t\sp{1- (\gamma/ \Gamma)}), \qquad \text{as } t\to\infty,$$ when $E$ is a finite set, where the constants $K$ and $K'$ only depend on $E$, $U$ and $\Gamma$.
[C.Mazza (Fribourg)]

MSC 2000:: *60K35 Interacting random processes
60F10 Large deviations
60J20 Appl. of discrete Markov processes
62M99 Inference from stochastic processes

Keywords: spectral gap; symmetric annealing process; compact manifold; weak Sobolev inequalities; Radon-Nikodym derivative; instantaneous invariant measure

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