Semi Log-Concave Markov Diffusions
By Patrick Cattiaux, Arnaud Guillin. 2013
63 citations recorded by the source corpus.
In this paper we intend to give a comprehensive approach of functional inequalities for diffusion processes under some ''curvature'' assumptions. Our notion of curvature coincides with the usual $\Gamma_2$ curvature of Bakry and Emery in the case of a (reversible) drifted Brownian motion, but differs for more general diffusion processes. Our approach using simple coupling arguments together with classical stochastic tools, allows us to obtain new results, to recover and to extend already known results, giving in many situations explicit (though non optimal) bounds. In particular, we show new results for gradient/semigroup commutation in the log concave case. Some new convergence to equilibrium in the granular media equation is also exhibited.
Subfields: Applied Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/a3f9281db3137eb7226220f19e801eacc1a2f17a
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In Applied Mathematics
In Geometry and Topology
Citations in the corpus
- Log-Concavity and Strong Log-Concavity: a review (2014, methodology): “For θ ∈ (1/2, 1), φ′′ θ (x) = (− log pθ)′′(x) is bounded below, by some negative value depending on θ, and hence these densities are semi-log-concave in the terminology of Cattiaux and Guillin (2013) who introduce this further generalization of log-concave densities by allowing the constant in the d”