Carré du champ operator

Short description: Operator in analysis and probability theory

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

Carré du champ operator

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

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

for all $f,g\in {𝒜}$.

Properties

From the definition, it follows that

$\mathrm{\Gamma }(f,g)=\lim {}_{t\to 0}\frac{1}{2t}(P_t(fg)-P_{t}f{P}_{t}g).$

For $f\in {𝒜}$ we have $P_t(f^2)\ge (P_{t}f{)}^2$ and thus $A(f^2)\ge 2fAf$ and

$\mathrm{\Gamma }(f):=\mathrm{\Gamma }(f,f)\ge 0,\mathrm{\forall }f\in {𝒜}$

therefore the carré du champ operator is positive.

The domain is

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

Remarks

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

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. http://www.numdam.org/item/AFST_2000_6_9_2_305_0/. 
• Meyer, Paul-André (1976). "L'Operateur carré du champ". Séminaire de Probabilités X Université de Strasbourg. Lecture Notes in Mathematics. 511. Berlin, Heidelberg: Springer. pp. 142–161. doi:10.1007/BFb0101102. ISBN 978-3-540-07681-0.