Carré Du Champ Operator

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

The operator was introduced in 1969 by 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 $(X,\mathcal,\mu)$ be a σ-finite measure space, $\_$ a Markov semigroup of non-negative operators on $L^2(X,\mu)$, $A$ the infinitesimal generator of $\_$ and $\mathcal$ the algebra of functions in $\mathcal(A)$, i.e. a vector space such that for all $f,g\in \mathcal$ also $fg\in \mathcal$.

Carré du champ operator

The of a Markovian semigroup $\_$ is the operator $\Gamma:\mathcal\times \mathcal\to\mathbb$ defined (following P. A. Meyer) as
:$\Gamma(f,g)=\frac\left(A(fg)-fA(g)-gA(f)\right)$
for all $f,g \in \mathcal$.

Properties

From the definition, it follows that
:$\Gamma(f,g)=\lim\limits_\frac\left(P_t(fg)-P_tfP_tg\right).$

For $f\in\mathcal$ we have $P_t(f^2)\geq (P_tf)^2$ and thus $A(f^2)\geq 2 fAf$ and
:$\Gamma(f):=\Gamma(f,f)\geq 0,\quad \forall f\in\mathcal$
therefore the is positive.

The domain is
:$\mathcal(A):=\left\.$

Remarks

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

Bibliography

*
*{{cite encyclopedia , first=Paul-André, last=Meyer , title=L'Operateur carré du champ , publisher=Springer , encyclopedia=Séminaire de Probabilités X Université de Strasbourg , series=Lecture Notes in Mathematics , volume=511 , place=Berlin, Heidelberg , date=1976 , pages=142–161 , lang=fr , doi=10.1007/BFb0101102, isbn=978-3-540-07681-0

References

Analysis
Probability theory
Functions and mappings