TY - JOUR AB - Many trace inequalities can be expressed either as concavity/convexity theorems or as monotonicity theorems. A classic example is the joint convexity of the quantum relative entropy which is equivalent to the Data Processing Inequality. The latter says that quantum operations can never increase the relative entropy. The monotonicity versions often have many advantages, and often have direct physical application, as in the example just mentioned. Moreover, the monotonicity results are often valid for a larger class of maps than, say, quantum operations (which are completely positive). In this paper we prove several new monotonicity results, the first of which is a monotonicity theorem that has as a simple corollary a celebrated concavity theorem of Epstein. Our starting points are the monotonicity versions of the Lieb Concavity and the Lieb Convexity Theorems. We also give two new proofs of these in their general forms using interpolation. We then prove our new monotonicity theorems by several duality arguments. AU - Carlen, Eric A. AU - Zhang, Haonan ID - 12216 JF - Linear Algebra and its Applications KW - Discrete Mathematics and Combinatorics KW - Geometry and Topology KW - Numerical Analysis KW - Algebra and Number Theory SN - 0024-3795 TI - Monotonicity versions of Epstein's concavity theorem and related inequalities VL - 654 ER - TY - JOUR AB - We study the hydrodynamic and hydrostatic limits of the one-dimensional open symmetric inclusion process with slow boundary. Depending on the value of the parameter tuning the interaction rate of the bulk of the system with the boundary, we obtain a linear heat equation with either Dirichlet, Robin or Neumann boundary conditions as hydrodynamic equation. In our approach, we combine duality and first-second class particle techniques to reduce the scaling limit of the inclusion process to the limiting behavior of a single, non-interacting, particle. AU - Franceschini, Chiara AU - Gonçalves, Patrícia AU - Sau, Federico ID - 12281 IS - 2 JF - Bernoulli KW - Statistics and Probability SN - 1350-7265 TI - Symmetric inclusion process with slow boundary: Hydrodynamics and hydrostatics VL - 28 ER - TY - JOUR AB - We consider symmetric partial exclusion and inclusion processes in a general graph in contact with reservoirs, where we allow both for edge disorder and well-chosen site disorder. We extend the classical dualities to this context and then we derive new orthogonal polynomial dualities. From the classical dualities, we derive the uniqueness of the non-equilibrium steady state and obtain correlation inequalities. Starting from the orthogonal polynomial dualities, we show universal properties of n-point correlation functions in the non-equilibrium steady state for systems with at most two different reservoir parameters, such as a chain with reservoirs at left and right ends. AU - Floreani, Simone AU - Redig, Frank AU - Sau, Federico ID - 10797 IS - 1 JF - Annales de l'institut Henri Poincare (B) Probability and Statistics SN - 0246-0203 TI - Orthogonal polynomial duality of boundary driven particle systems and non-equilibrium correlations VL - 58 ER - TY - JOUR AB - We construct a recurrent diffusion process with values in the space of probability measures over an arbitrary closed Riemannian manifold of dimension d≥2. The process is associated with the Dirichlet form defined by integration of the Wasserstein gradient w.r.t. the Dirichlet–Ferguson measure, and is the counterpart on multidimensional base spaces to the modified massive Arratia flow over the unit interval described in V. Konarovskyi and M.-K. von Renesse (Comm. Pure Appl. Math. 72 (2019) 764–800). Together with two different constructions of the process, we discuss its ergodicity, invariant sets, finite-dimensional approximations, and Varadhan short-time asymptotics. AU - Dello Schiavo, Lorenzo ID - 11354 IS - 2 JF - Annals of Probability SN - 0091-1798 TI - The Dirichlet–Ferguson diffusion on the space of probability measures over a closed Riemannian manifold VL - 50 ER - TY - JOUR AB - We study the temporal dissipation of variance and relative entropy for ergodic Markov Chains in continuous time, and compute explicitly the corresponding dissipation rates. These are identified, as is well known, in the case of the variance in terms of an appropriate Hilbertian norm; and in the case of the relative entropy, in terms of a Dirichlet form which morphs into a version of the familiar Fisher information under conditions of detailed balance. Here we obtain trajectorial versions of these results, valid along almost every path of the random motion and most transparent in the backwards direction of time. Martingale arguments and time reversal play crucial roles, as in the recent work of Karatzas, Schachermayer and Tschiderer for conservative diffusions. Extensions are developed to general “convex divergences” and to countable state-spaces. The steepest descent and gradient flow properties for the variance, the relative entropy, and appropriate generalizations, are studied along with their respective geometries under conditions of detailed balance, leading to a very direct proof for the HWI inequality of Otto and Villani in the present context. AU - Karatzas, Ioannis AU - Maas, Jan AU - Schachermayer, Walter ID - 10023 IS - 4 JF - Communications in Information and Systems KW - Markov Chain KW - relative entropy KW - time reversal KW - steepest descent KW - gradient flow SN - 1526-7555 TI - Trajectorial dissipation and gradient flow for the relative entropy in Markov chains VL - 21 ER -