Carré du champ operator
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.[1]
The operator was introduced in 1969[2] by Hiroshi Kunita ({{{2}}}) and independently discovered in 1976[3] 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 [math]\displaystyle{ (X,\mathcal{E},\mu) }[/math] be a σ-finite measure space, [math]\displaystyle{ \{P_t\}_{t\geq 0} }[/math] a Markov semigroup of non-negative operators on [math]\displaystyle{ L^2(X,\mu) }[/math], [math]\displaystyle{ A }[/math] the infinitesimal generator of [math]\displaystyle{ \{P_t\}_{t\geq 0} }[/math] and [math]\displaystyle{ \mathcal{A} }[/math] the algebra of functions in [math]\displaystyle{ \mathcal{D}(A) }[/math], i.e. a vector space such that for all [math]\displaystyle{ f,g\in \mathcal{A} }[/math] also [math]\displaystyle{ fg\in \mathcal{A} }[/math].
Carré du champ operator
The carré du champ operator of a Markovian semigroup [math]\displaystyle{ \{P_t\}_{t\geq 0} }[/math] is the operator [math]\displaystyle{ \Gamma:\mathcal{A}\times \mathcal{A}\to\mathbb{R} }[/math] defined (following P. A. Meyer) as
- [math]\displaystyle{ \Gamma(f,g)=\frac{1}{2}\left(A(fg)-fA(g)-gA(f)\right) }[/math]
for all [math]\displaystyle{ f,g \in \mathcal{A} }[/math].[4][5]
Properties
From the definition, it follows that[1]
- [math]\displaystyle{ \Gamma(f,g)=\lim\limits_{t\to 0}\frac{1}{2t}\left(P_t(fg)-P_tfP_tg\right). }[/math]
For [math]\displaystyle{ f\in\mathcal{A} }[/math] we have [math]\displaystyle{ P_t(f^2)\geq (P_tf)^2 }[/math] and thus [math]\displaystyle{ A(f^2)\geq 2 fAf }[/math] and
- [math]\displaystyle{ \Gamma(f):=\Gamma(f,f)\geq 0,\quad \forall f\in\mathcal{A} }[/math]
therefore the carré du champ operator is positive.
The domain is
- [math]\displaystyle{ \mathcal{D}(A):=\left\{f \in L^2(X,\mu) ;\;\lim\limits_{t\downarrow 0}\frac{P_t f-f}{t}\text{ existists and is in } L^2(X,\mu)\right\}. }[/math]
Remarks
- The definition in Roth's thesis is slightly different.[3]
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.
References
- ↑ 1.0 1.1 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. http://www.numdam.org/item/AFST_2000_6_9_2_305_0/.
- ↑ Kunita, Hiroshi (1969). "Absolute continuity of Markov processes and generators". Nagoya Mathematical Journal 36: 1–26. doi:10.1017/S0027763000013106. https://projecteuclid.org/journals/nagoya-mathematical-journal/volume-36/issue-none/Absolute-continuity-of-Markov-processes-and-generators/nmj/1118797793.full.
- ↑ 3.0 3.1 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. http://www.numdam.org/item/AIF_1976__26_4_1_0/.
- ↑ 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.
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