Poincar\'{e} and transportation inequalities for Gibbs measures under the Dobrushin uniqueness condition

In in this paper we establish an explicit and sharp estimate of the spectral gap (Poincar\'{e} inequality) and the transportation inequality for Gibbs measures, under the Dobrushin uniqueness condition. Moreover, we give a generalization of the Liggett's $M-\epsilon$ theorem for interacting particle systems.Comment: Published at http://dx.doi.org/10.1214/009117906000000368 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Uniqueness of a pre-generator for C0C_0-semigroup on a general locally convex vector space

The main purpose is to generalize a theorem of Arendt about uniqueness of $C_0$-semigroups from Banach space setting to the general locally convex vector spaces, more precisely, we show that cores are the only domains of uniqueness for $C_0$-semigroups on locally convex spaces. As an application, we find a necessary and sufficient condition for that the mass transport equation has one unique $L^1(\R^d,dx)$ weak solution

Bernstein type's concentration inequalities for symmetric Markov processes

Using the method of transportation-information inequality introduced in \cite{GLWY}, we establish Bernstein type's concentration inequalities for empirical means $\frac 1t \int_0^t g(X_s)ds$ where $g$ is a unbounded observable of the symmetric Markov process $(X_t)$. Three approaches are proposed : functional inequalities approach ; Lyapunov function method ; and an approach through the Lipschitzian norm of the solution to the Poisson equation. Several applications and examples are studied