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Looking in a number of mathematical papers dealing with Markov semigroups and heat kernels, very often the Carrè du Champ operator appears that is defined as a bilinear form based on the infinitesimal generator $L$ (the Laplacian typically).

For two functions $f$ and $g$ the Carrè du Champ operator $\Gamma(f,g)$ is defined as:

$$ \Gamma(f,g) = \frac{1}{2}\left( L(fg) - fLg - gLf \right) $$

My question is what this operator should measure and what is its physical meaning (if there is one, apart from mathematical details).

P.S.

Initially I've doubted that "Carrè du Champ" was the name of someone. Apparently though, a rough translation means something like "square of the field" but I don't understand this choice, if there is a reason.

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