Carré du champ operator

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]