For a Markov process generator and suitable real functions, this bilinear expression records the local quadratic energy. Positivity of follows by differentiating . For it is , giving the Gaussian Dirichlet energy after integration against the invariant measure.
If real and belong to the generator domain of a Feller semigroup, Jensen inequality gives . This expression is zero at , so its right derivative is nonnegative, proving the inequality. It is equivalent to positivity of the carré du champ.
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The Carré du champ operator is a mathematical operator that arises in the study of functional inequalities, Markov processes, and the analysis of Dirichlet forms in the context of stochastic processes, particularly in the framework of the theory of diffusion.