Carré du champ operator


The carré du champ 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 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 carré du champ operator of a Markovian semigroup is the operator defined as
for all.

Properties

From the definition, it follows that
For we have and thus and
therefore the carré du champ operator is positive.
The domain is

Remarks

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