The (French for ''square of a field'' operator) is a
bilinear,
symmetric
Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...
operator from
analysis
Analysis (: analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (38 ...
and
probability theory
Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...
. The measures how far an
infinitesimal generator is from being a
derivation
Derivation may refer to:
Language
* Morphological derivation, a word-formation process
* Parse tree or concrete syntax tree, representing a string's syntax in formal grammars
Law
* Derivative work, in copyright law
* Derivation proceeding, a ...
.
The operator was introduced in 1969 by and independently discovered in 1976
by
Jean-Pierre Roth in his doctoral thesis.
The name ''"carré du champ"'' comes from
electrostatics
Electrostatics is a branch of physics that studies slow-moving or stationary electric charges.
Since classical antiquity, classical times, it has been known that some materials, such as amber, attract lightweight particles after triboelectric e ...
.
Carré du champ operator for a Markov semigroup
Let
be a
σ-finite measure space
A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the -algebra) and the method that ...
,
a
Markov semigroup of non-negative operators on
,
the
infinitesimal generator of
and
the algebra of functions in
, i.e. a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
such that for all
also
.
Carré du champ operator
The of a Markovian semigroup
is the operator
defined (following
P. A. Meyer) as
:
for all
.
Properties
From the definition, it follows that
:
For
we have
and thus
and
:
therefore the is positive.
The domain is
:
Remarks
*The definition in Roth's thesis is slightly different.
Bibliography
*
*{{cite encyclopedia , first=Paul-André, last=Meyer , title=L'Operateur carré du champ , publisher=Springer , encyclopedia=Séminaire de Probabilités X Université de Strasbourg , series=Lecture Notes in Mathematics , volume=511 , place=Berlin, Heidelberg , date=1976 , pages=142–161 , lang=fr , doi=10.1007/BFb0101102, isbn=978-3-540-07681-0
References
Analysis
Probability theory
Functions and mappings