Entropic Projections and Dominating Points

Generalized entropic projections and dominating points are solutions to convex minimization problems related to conditional laws of large numbers. They appear in many areas of applied mathematics such as statistical physics, information theory, mathematical statistics, ill-posed inverse problems or large deviation theory. By means of convex conjugate duality and functional analysis, criteria are derived for their existence. Representations of the generalized entropic projections are obtained: they are the ``measure component" of some extended entropy minimization problem.Comment: ESAIM P&S (2011) to appea

On the convexity of the entropy along entropic interpolations

Convexity properties of the entropy along displacement interpolations are crucial in the Lott-Sturm-Villani theory of lower bounded curvature of geodesic measure spaces. As discrete spaces fail to be geodesic, an alternate analogous theory is necessary in the discrete setting. Replacing displacement interpolations by entropic ones allows for developing a rigorous calculus, in contrast with Otto's informal calculus. When the underlying state space is a Riemannian manifold, we show that the first and second derivatives of the entropy as a function of time along entropic interpolations are expressed in terms of the standard Bakry-\'Emery operators $\Gamma$ and $\Gamma_2$. On the other hand, in the discrete setting new operators appear. Our approach is probabilistic; it relies on the Markov property and time reversal. We illustrate these calculations by means of Brownian diffusions on manifolds and random walks on graphs. We also give a new unified proof, covering both the manifold and graph cases, of a logarithmic Sobolev inequality in connection with convergence to equilibrium

Girsanov theory under a finite entropy condition

This paper is about Girsanov's theory. It (almost) doesn't contain new results but it is based on a simplified new approach which takes advantage of the (weak) extra requirement that some relative entropy is finite. Under this assumption, we present and prove all the standard results pertaining to the absolute continuity of two continuous-time processes with or without jumps. We have tried to give as much as possible a self-contained presentation. The main advantage of the finite entropy strategy is that it allows us to replace martingale representation results by the simpler Riesz representations of the dual of a Hilbert space (in the continuous case) or of an Orlicz function space (in the jump case)

Noncommutative Khintchine inequalities in interpolation spaces of LpL_p-spaces

We prove noncommutative Khintchine inequalities for all interpolation spaces between $L_p$ and $L_2$ with $p<2$. In particular, it follows that Khintchine inequalities hold in $L_{1,\infty}$. Using a similar method, we find a new deterministic equivalent for the $RC$-norm in all interpolation spaces between $L_p$-spaces which unifies the cases $p > 2$ and $p < 2$. It produces a new proof of Khintchine inequalities for $p<1$ for free variables. To complete the picture, we exhibit counter-examples which show that neither of the usual closed formulas for Khintchine inequalities can work in $L_{2,\infty}$. We also give an application to martingale inequalities.Comment: 33 pages, published versio