Abstract:The Heisenberg group is one of the simplest sub-Riemannian settings in which we can define non-elliptic Hörmander type generators. We can then consider coercive inequalities associated to such generators. We prove that a certain class of non-trivial Gibbs measures with quadratic interaction potential on an infinite product of Heisenberg groups satisfy logarithmic Sobolev inequalities.

We develop a new Bayesian modelling framework for the class of higher‐order, variable‐memory Markov chains, and introduce an associated collection of methodological tools for exact inference with discrete time series. We show that a version of the context tree weighting alg‐orithm can compute the prior predictive likelihood exa‐ctly (averaged over both models and parameters), and two related algorithms are introduced, which identify the a posteriori most likely models and compute their exact posterior probabilities. All three algorithms are deterministic and have linear‐time complexity. A family of variable‐dimension Markov chain Monte Carlo samplers is also provided, facilitating further exploration of the posterior. The performance of the proposed methods in model selection, Markov order estimation and prediction is illustrated through simulation experiments and real‐world applications with data from finance, genetics, neuroscience and animal communication. The associated algorithms are implemented in the R package BCT.

We study Talagrand concentration and Poincaré type inequalities for unbounded pure jump Markov processes. In particular we focus on processes with degenerate jumps that depend on the past of the whole system, based on the model introduced by Galves and Löcherbach in [19], in order to describe the activity of a biological neural network. As a result we obtain exponential rates of convergence to equilibrium.2010 Mathematics Subject Classification. 60K35, 26D10, 60G99, Key words and phrases. Talagrand inequality, Poincaré inequality, brain neuron networks.

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