Carré du champ operator

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

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

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

From the definition, it follows that^[1]

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

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

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

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

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