Differentiable measure

Short description: Measure that has a notion of derivative

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. Besides the notion of a derivative of a measure by Sergei Fomin there exists also one by Anatoliy Skorokhod, one by Sergio Albeverio and Raphael Høegh-Krohn, and one by Oleg Smolyanov and Heinrich von Weizsäcker.

Differentiable measure

Let

• [math]\displaystyle{ X }[/math] be a real vector space,
• [math]\displaystyle{ \mathcal{A} }[/math] be σ-algebra that is invariant under translation by vectors [math]\displaystyle{ h\in X }[/math], i.e. [math]\displaystyle{ A +th\in \mathcal{A} }[/math] for all [math]\displaystyle{ A\in\mathcal{A} }[/math] and [math]\displaystyle{ t\in\R }[/math].

This setting is rather general on purpose since for most definitions only linearity and measurability is needed. But usually one chooses [math]\displaystyle{ X }[/math] to be a real Hausdorff locally convex space with the Borel or cylindrical σ-algebra [math]\displaystyle{ \mathcal{A} }[/math].

For a measure [math]\displaystyle{ \mu }[/math] let [math]\displaystyle{ \mu_h(A):=\mu(A+h) }[/math] denote the shifted measure by [math]\displaystyle{ h\in X }[/math].

Fomin differentiability

A measure [math]\displaystyle{ \mu }[/math] on [math]\displaystyle{ (X,\mathcal{A}) }[/math] is Fomin differentiable along [math]\displaystyle{ h\in X }[/math] if for every set [math]\displaystyle{ A\in\mathcal{A} }[/math] the limit

[math]\displaystyle{ d_{h}\mu(A):=\lim\limits_{t\to 0}\frac{\mu(A+th)-\mu(A)}{t} }[/math]

exists. We call [math]\displaystyle{ d_{h}\mu }[/math] the Fomin derivative of [math]\displaystyle{ \mu }[/math].

Equivalently, for all sets [math]\displaystyle{ A\in\mathcal{A} }[/math] is [math]\displaystyle{ f_{\mu}^{A,h}:t\mapsto \mu(A+th) }[/math] differentiable in [math]\displaystyle{ 0 }[/math].

Properties

• The Fomin derivative is again another measure and absolutely continuous with respect to [math]\displaystyle{ \mu }[/math].
• Fomin differentiability can be directly extend to signed measures.
• Higher and mixed derivatives will be defined inductively [math]\displaystyle{ d^n_{h}=d_{h}(d^{n-1}_{h}) }[/math].

Skorokhod differentiability

Let [math]\displaystyle{ \mu }[/math] be a Baire measure and let [math]\displaystyle{ C_b(X) }[/math] be the space of bounded and continuous functions on [math]\displaystyle{ X }[/math].

[math]\displaystyle{ \mu }[/math] is Skorokhod differentiable (or S-differentiable) along [math]\displaystyle{ h\in X }[/math] if a Baire measure [math]\displaystyle{ \nu }[/math] exists such that for all [math]\displaystyle{ f\in C_b(X) }[/math] the limit

[math]\displaystyle{ \lim\limits_{t\to 0}\int_X\frac{ f(x-th)-f(x)}{t}\mu(dx)=\int_X f(x)\nu(dx) }[/math]

exists.

In shift notation

[math]\displaystyle{ \lim\limits_{t\to 0}\int_X\frac{ f(x-th)-f(x)}{t}\mu(dx)=\lim\limits_{t\to 0}\int_Xf\; d\left(\frac{\mu_{th}-\mu}{t}\right). }[/math]

The measure [math]\displaystyle{ \nu }[/math] is called the Skorokhod derivative (or S-derivative or weak derivative) of [math]\displaystyle{ \mu }[/math] along [math]\displaystyle{ h\in X }[/math] and is unique.

Albeverio-Høegh-Krohn Differentiability

A measure [math]\displaystyle{ \mu }[/math] is Albeverio-Høegh-Krohn differentiable (or AHK differentiable) along [math]\displaystyle{ h\in X }[/math] if a measure [math]\displaystyle{ \lambda\geq 0 }[/math] exists such that

1. [math]\displaystyle{ \mu_{th} }[/math] is absolutely continuous with respect to [math]\displaystyle{ \lambda }[/math] such that [math]\displaystyle{ \lambda_{th}=f_t\cdot \lambda }[/math],
2. the map [math]\displaystyle{ g:\R\to L^2(\lambda),\; t\mapsto f_{t}^{1/2} }[/math] is differentiable.

Properties

• The AHK differentiability can also be extende to signed measures.

Example

Let [math]\displaystyle{ \mu }[/math] be a measure with a continuously differentiable Radon-Nikodým density [math]\displaystyle{ g }[/math], then the Fomin derivative is

[math]\displaystyle{ d_{h}\mu(A)=\lim\limits_{t\to 0}\frac{\mu(A+th)-\mu(A)}{t}=\lim\limits_{t\to 0}\int_A\frac{g(x+th)-g(x)}{t}\mathrm{d}x=\int_A g'(x)\mathrm{d}x. }[/math]

Bibliography

• 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". 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.