Quelques problèmes associés au processus de Donsker-Varadhan
Forward-backward martingale decomposition and compactness results for additive functionals of stationary ergodic Markov processes
Estimate of spectral gap for continuous gas
A deviation inequality for non-reversible Markov processes
Transportation inequalities for stochastic differential equations of pure jumps
For stochastic differential equations of pure jumps, though the Poincaré inequality does not hold in general, we show that 1 transportation inequalities hold for its invariant probability measure and for its process-level law on right continuous paths space in the 1-metric or in uniform metrics, under the dissipative condition. Several applications to concentration inequalities are given.
Lipschitzian norm estimate of one-dimensional Poisson equations and applications
By direct calculus we identify explicitly the lipschitzian norm of the solution of the Poisson equation in terms of various norms of , where is a Sturm–Liouville operator or generator of a non-singular diffusion in an interval. This allows us to obtain the best constant in the 1-Poincaré inequality (a little stronger than the Cheeger isoperimetric inequality) and some sharp transportation–information inequalities and concentration inequalities for empirical means. We conclude with several illustrative...
Poincaré and log-Sobolev inequality for stationary Gaussian processes and moving average processes
For stationary Gaussian processes, we obtain the necessary and sufficient conditions for Poincaré inequality and log-Sobolev inequality of process-level and provide the sharp constants. The extension to moving average processes is also presented, as well as several concrete examples.
Large deviation principles for Markov processes via phi-Sobolev inequalities.
Identification of the rate function for large deviations of an irreducible Markov chain.
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