We provide a generalization of Ueno's inequality for n-step transition probabilities of Markov chains in a general state space. Our result is relevant to the study of adaptive control problems and approximation problems in the theory of discrete-time Markov decision processes and stochastic games.
We recover the Navier–Stokes equation as the incompressible limit of a stochastic lattice gas in which particles are allowed to jump over a mesoscopic scale. The result holds in any dimension assuming the existence of a smooth solution of the Navier–Stokes equation in a fixed time interval. The proof does not use nongradient methods or the multi-scale analysis due to the long range jumps.
We present a finite dimensional version of the logarithmic Sobolev inequality for heat kernel measures of non-negatively curved diffusion operators that contains and improves upon the Li-Yau parabolic inequality. This new inequality is of interest already in Euclidean space for the standard Gaussian measure. The result may also be seen as an extended version of the semigroup commutation properties under curvature conditions. It may be applied to reach optimal Euclidean logarithmic Sobolev inequalities...
This paper deals with the relationship between two-dimensional parameter Gaussian random fields verifying a particular Markov property and the solutions of stochastic differential equations. In the non Gaussian case some diffusion conditions are introduced, obtaining a backward equation for the evolution of transition probability functions.