 Carré du champ operator

For a Markov process generator and suitable real functions, this bilinear expression records the local quadratic energy. Positivity of $\Gamma (f,f)$ follows by differentiating $P_t(f^2)\ge (P_{t}f{)}^2$. For $L=D^2-xD$ it is $f^{\prime }{g}^{\prime }$, giving the Gaussian Dirichlet energy after integration against the invariant measure.

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.