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Carré du champ operator

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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 [d] 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 , μ ) {\displaystyle (X,{\mathcal {E}},\mu )} be a σ-finite measure space, { P t } t ≥ 0 {\displaystyle \{P_{t}\}_{t\geq 0}} a Markov semigroup of non-negative operators on L 2 ( X , μ ) {\displaystyle L^{2}(X,\mu )} , A {\displaystyle A} the infinitesimal generator of { P t } t ≥ 0 {\displaystyle \{P_{t}\}_{t\geq 0}} and A {\displaystyle {\mathcal {A}}} the algebra of functions in D ( A ) {\displaystyle {\mathcal {D}}(A)} , i.e. a vector space such that for all f , g ∈ A {\displaystyle f,g\in {\mathcal {A}}} also f g ∈ A {\displaystyle fg\in {\mathcal {A}}} .

Carré du champ operator

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

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

for all f , g ∈ A {\displaystyle f,g\in {\mathcal {A}}} .

Properties

From the definition, it follows that

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

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

Γ ( f ) := Γ ( f , f ) ≥ 0 , ∀ f ∈ 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

D ( A ) := { f ∈ L 2 ( X , μ ) ; lim t ↓ 0 P t f − f t  exists and is in  L 2 ( X , μ ) } . {\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.

Bibliography

References

  1. ^ Ledoux, Michel (2000). "The geometry of Markov diffusion generators". Annales de la Faculté des sciences de Toulouse: Mathématiques. Série 6. 9 (2): 312. doi:10.5802/afst.962. hdl:20.500.11850/146400.
  2. Kunita, Hiroshi (1969). "Absolute continuity of Markov processes and generators". Nagoya Mathematical Journal. 36: 1–26. doi:10.1017/S0027763000013106. S2CID 118693611.
  3. ^ Roth, Jean-Pierre (1976). "Opérateurs dissipatifs et semi-groupes dans les espaces de fonctions continues". Annales de l'Institut Fourier. 26 (4): 1–97. doi:10.5802/aif.632.
  4. 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.
  5. 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.
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