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 be a σ-finite measure space, a Markov semigroup of non-negative operators on , the infinitesimal generator of and the algebra of functions in , i.e. a vector space such that for all also .
Carré du champ operator The of a Markovian semigroup is the operator defined (following P. A. Meyer) as
for all .
Properties From the definition, it follows that
For we have and thus and
therefore the is positive.
The domain is
Remarks The definition in Roth's thesis is slightly different.
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