Carré du champ operator

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

The operator was introduced in 1969 by Template:Ill 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 Template:Lang of a Markovian semigroup $\{P_t{\}}_{t\ge 0}$ is the operator $\mathrm{\Gamma }:𝒜\times 𝒜\to ℝ$ 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

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

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 Template:Lang is positive.

The domain is

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

Remarks

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

Bibliography

• Template:Cite journal
• Template:Cite encyclopedia