Differentiable measure
In functional analysis and measure theory, a differentiable measure is a measure that has a notion of a derivative. The theory of differentiable measure was introduced by Russian mathematician Sergei Fomin and proposed at the International Congress of Mathematicians in 1966 in Moscow as an infinite-dimensional analog of the theory of distributions.[1] Besides the notion of a derivative of a measure by Sergei Fomin there exists also one by Anatoliy Skorokhod,[2] one by Sergio Albeverio and Raphael Høegh-Krohn, and one by Oleg Smolyanov and Heinrich von Weizsäcker .[3]
Differentiable measure
[edit]Let
- be a real vector space,
- be σ-algebra that is invariant under translation by vectors , i.e. for all and .
This setting is rather general on purpose since for most definitions only linearity and measurability is needed. But usually one chooses to be a real Hausdorff locally convex space with the Borel or cylindrical σ-algebra .
For a measure let denote the shifted measure by .
Fomin differentiability
[edit]A measure on is Fomin differentiable along if for every set the limit
exists. We call the Fomin derivative of .
Equivalently, for all sets is differentiable in .[4]
Properties
[edit]- The Fomin derivative is again another measure and absolutely continuous with respect to .
- Fomin differentiability can be directly extend to signed measures.
- Higher and mixed derivatives will be defined inductively .
Skorokhod differentiability
[edit]Let be a Baire measure and let be the space of bounded and continuous functions on .
is Skorokhod differentiable (or S-differentiable) along if a Baire measure exists such that for all the limit
exists.
In shift notation
The measure is called the Skorokhod derivative (or S-derivative or weak derivative) of along and is unique.[4][5]
Albeverio-Høegh-Krohn Differentiability
[edit]A measure is Albeverio-Høegh-Krohn differentiable (or AHK differentiable) along if a measure exists such that
- is absolutely continuous with respect to such that ,
- the map is differentiable.[4]
Properties
[edit]- The AHK differentiability can also be extended to signed measures.
Example
[edit]Let be a measure with a continuously differentiable Radon-Nikodým density , then the Fomin derivative is
Bibliography
[edit]- Bogachev, Vladimir I. (2010). Differentiable Measures and the Malliavin Calculus. American Mathematical Society. pp. 69–72. ISBN 978-0821849934.
- Smolyanov, Oleg G.; von Weizsäcker, Heinrich (1993). "Differentiable Families of Measures". Journal of Functional Analysis. 118 (2): 454–476. doi:10.1006/jfan.1993.1151.
- Bogachev, Vladimir I. (2010). Differentiable Measures and the Malliavin Calculus. American Mathematical Society. pp. 69–72. ISBN 978-0821849934.
- Fomin, Sergei Vasil'evich (1966). "Differential measures in linear spaces". Proc. Int. Congress of Mathematicians, sec.5. Int. Congress of Mathematicians. Moscow: Izdat. Moskov. Univ.
- Kuo, Hui-Hsiung “Differentiable Measures.” Chinese Journal of Mathematics 2, no. 2 (1974): 189–99. JSTOR 43836023.
References
[edit]- ^ Fomin, Sergei Vasil'evich (1966). "Differential measures in linear spaces". Proc. Int. Congress of Mathematicians, sec.5. Int. Congress of Mathematicians. Moscow: Izdat. Moskov. Univ.
- ^ Skorokhod, Anatoly V. (1974). Integration in Hilbert Spaces. Ergebnisse der Mathematik. Berlin, New-York: Springer-Verlag.
- ^ Bogachev, Vladimir I. (2010). "Differentiable Measures and the Malliavin Calculus". Journal of Mathematical Sciences. 87. Springer: 3577–3731. ISBN 978-0821849934.
- ^ a b c Bogachev, Vladimir I. (2010). Differentiable Measures and the Malliavin Calculus. American Mathematical Society. pp. 69–72. ISBN 978-0821849934.
- ^ Bogachev, Vladimir I. (2021). "On Skorokhod Differentiable Measures". Ukrainian Mathematical Journal. 72: 1163. doi:10.1007/s11253-021-01861-x.