Carré du champ operator

The carré du champ operator (French for square of a field operator) is a bilinear, symmetric operator from analysis and probability theory. The carré du champ operator measures how far an infinitesimal generator is from being a derivation.

The operator was introduced in 1969 by Hiroshi Kunita and independently discovered in 1976 by Jean-Pierre Roth in his doctoral thesis.

The name "carré du champ" comes from electrostatics.

Carré du champ operator for a Markov semigroup
Let $$(X,\mathcal{E},\mu)$$ be a σ-finite measure space, $$\{P_t\}_{t\geq 0}$$ a Markov semigroup of non-negative operators on $$L^2(X,\mu)$$, $$A$$ the infinitesimal generator of $$\{P_t\}_{t\geq 0}$$ and $$\mathcal{A}$$ the algebra of functions in $$\mathcal{D}(A)$$, i.e. a vector space such that for all $$f,g\in \mathcal{A}$$ also $$fg\in \mathcal{A}$$.

Carré du champ operator
The carré du champ operator of a Markovian semigroup $$\{P_t\}_{t\geq 0}$$ is the operator $$\Gamma:\mathcal{A}\times \mathcal{A}\to\mathbb{R}$$ defined (following P. A. Meyer) as
 * $$\Gamma(f,g)=\frac{1}{2}\left(A(fg)-fA(g)-gA(f)\right)$$

for all $$f,g \in \mathcal{A}$$.

Properties
From the definition, it follows that
 * $$\Gamma(f,g)=\lim\limits_{t\to 0}\frac{1}{2t}\left(P_t(fg)-P_tfP_tg\right).$$

For $$f\in\mathcal{A}$$ we have $$P_t(f^2)\geq (P_tf)^2$$ and thus $$A(f^2)\geq 2 fAf$$ and
 * $$\Gamma(f):=\Gamma(f,f)\geq 0,\quad \forall f\in\mathcal{A}$$

therefore the carré du champ operator is positive.

The domain is
 * $$\mathcal{D}(A):=\left\{f \in L^2(X,\mu) ;\;\lim\limits_{t\downarrow 0}\frac{P_t f-f}{t}\text{ exists and is in } L^2(X,\mu)\right\}.$$

Remarks

 * The definition in Roth's thesis is slightly different.