WIAS Preprint No. 2901, (2021)

Positivity and polynomial decay of energies for square-field operators


  • Stephan, Artur
    ORCID: 0000-0001-9871-3946
  • Stephan, Holger
    ORCID: 0000-0002-6024-5355

2020 Mathematics Subject Classification

  • 60J25 39B62 35B40 46B42 35Q84 47D07


  • Square-field operator, carré du champs operator, Gamma calculus, Markov generator, Markov operator, Banach lattice positivity, quadratic energies, asymptotic polynomial decay, normal




We show that for a general Markov generator the associated square-field (or carré du champs) operator and all their iterations are positive. The proof is based on an interpolation between the operators involving the generator and their semigroups, and an interplay between positivity and convexity on Banach lattices. Positivity of the square-field operators allows to define a hierarchy of quadratic and positive energy functionals which decay to zero along solutions of the corresponding evolution equation. Assuming that the Markov generator satisfies an operator-theoretic normality condition, the sequence of energies is log-convex. In particular, this implies polynomial decay in time for the energy functionals along solutions.

Download Documents