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.^[1]

The operator was introduced in 1969^[2] by Hiroshi Kunita [d] and independently discovered in 1976^[3] 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 ${\displaystyle (X,{\mathcal {E}},\mu )}$ be a σ-finite measure space, ${\displaystyle \{P_{t}\}_{t\geq 0}}$ a Markov semigroup of non-negative operators on ${\displaystyle L^{2}(X,\mu )}$, ${\displaystyle A}$ the infinitesimal generator of ${\displaystyle \{P_{t}\}_{t\geq 0}}$ and ${\displaystyle {\mathcal {A}}}$ the algebra of functions in ${\displaystyle {\mathcal {D}}(A)}$, i.e. a vector space such that for all ${\displaystyle f,g\in {\mathcal {A}}}$ also ${\displaystyle fg\in {\mathcal {A}}}$.

Carré du champ operator

The carré du champ operator of a Markovian semigroup ${\displaystyle \{P_{t}\}_{t\geq 0}}$ is the operator ${\displaystyle \Gamma$ :{\mathcal {A}}\times {\mathcal {A}}\to \mathbb {R} } defined (following P. A. Meyer) as

${\displaystyle \Gamma (f,g)={\frac {1}{2}}\left(A(fg)-fA(g)-gA(f)\right)}$

for all ${\displaystyle f,g\in {\mathcal {A}}}$.^[4][5]

Properties

From the definition, it follows that^[1]

${\displaystyle \Gamma (f,g)=\lim \limits _{t\to 0}{\frac {1}{2t}}\left(P_{t}(fg)-P_{t}fP_{t}g\right).}$

For ${\displaystyle f\in {\mathcal {A}}}$ we have ${\displaystyle P_{t}(f^{2})\geq (P_{t}f)^{2}}$ and thus ${\displaystyle A(f^{2})\geq 2fAf}$ and

${\displaystyle \Gamma (f):=\Gamma (f,f)\geq 0,\quad \forall f\in {\mathcal {A}}}$

therefore the carré du champ operator is positive.

The domain is

${\displaystyle {\mathcal {D}}(A):=\left\{f\in L^{2}(X,\mu );\;\lim \limits _{t\downarrow 0}{\frac {P_{t}f-f}{t}}{\text{ exists and is in }}L^{2}(X,\mu )\right\}.}$

Remarks

• The definition in Roth's thesis is slightly different.^[3]

Bibliography

• Ledoux, Michel (2000). "The geometry of Markov diffusion generators". Annales de la Faculté des Sciences de Toulouse: Mathématiques. Série 6. 9 (2): 305–366. doi:10.5802/afst.962. hdl:20.500.11850/146400.
• Meyer, Paul-André (1976). "L'Operateur carré du champ". Séminaire de Probabilités X Université de Strasbourg. Lecture Notes in Mathematics (in French). Vol. 511. Berlin, Heidelberg: Springer. pp. 142–161. doi:10.1007/BFb0101102. ISBN 978-3-540-07681-0.