arXiv subjects
Publications and source records attributed to Laurent Veysseire.
In this paper, we give one possible definition for functions of several variables applied to endomorphisms of finite dimensional C-vector spaces. This definition is consistent with the usual notion of a function of a square matrix. Some basic rules of computation are given, as well as a formula for differentiation of it. A few examples of possible applications is also given at the end of the article.
A nonnegative coarse Ricci curvature for a Markov chain and the existence of an attractive point implies the concentration of the invariant probability measure around this point. The mass outside balls centered at the attractive point, as a function of the radius, decreases at least as fast as the exponential of a double integral of the coarse Ricci curvature. This is exactly the behaviour of the density of the reversible measure for diffusion processes on the real line.
The coarse Ricci curvature for Markov chains is generalized for continuous time. We show that a positive coarse Ricci curvature implies a contraction of the Markov process for the Wasserstein distance between probability measures. This leads to a spectral gap estimate in the reversible case.
A coupling method and an analytic one allow us to prove new lower bounds for the spectral gap of reversible diffusions on compact manifolds. Those bounds are based on the a notion of curvature of the diffusion, like the coarse Ricci curvature or the Bakry--Emery curvature-dimension inequalities. We show that when this curvature is nonnegative, its harmonic mean is a lower bound for the spectral gap.