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A deviation inequality for non-reversible Markov processes. (English) Zbl 0972.60003
One studies the probability of deviation of the empirical mean from its real mean \[ P_{\nu}\left(\left|\frac{1}{t}\int_{0}^{t}V(X_{s}) ds-\int_{E}V d\mu\right|>r\right). \] Here \(X\) is a conservative càdlàg Markov process with values in a Polish space \(E\) with \(\nu\) as initial measure and \(p_{t}(x,dy)\) as transition probability, \(\mu\) is an invariant and ergodic probability measure on \(E\) with respect to \(p_{t}\) and \(V\) is a real \(\mu\)-integrable function. When the Markov process is \(\mu\)-reversible, J.-D. Deuschel and D. W. Stroock [“Large deviations” (1989; Zbl 0705.60029)] proved a large deviation estimation for bounded \(V\) [for general \(V\) see the author, J. Funct. Anal. 123, No. 1, 202-231 (1994; Zbl 0798.60067)]. In the present paper the author proposes an extended and strengthened inequality. The proof is done using the dissipative criterion of Lumer-Philips for the contraction semigroup. The inequality is in terms of the symmetrized Dirichlet form (used in the definition of the action functional) [see also G. Ben Arous and J.-D. Deuschel, Commun. Pure Appl. Math. 47, No. 6, 843-860 (1994; Zbl 0811.58065)]. A more explicit version is obtained in the case where the logarithmic Sobolev inequality holds [see also M. Ledoux, in: Séminaire de probabilités XXXIII. Lect. Notes Math. 1709, 120-216 (1999; Zbl 0957.60016) for similar questions].

MSC:
60F10 Large deviations
60J25 Continuous-time Markov processes on general state spaces
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