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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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