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. 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

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

This setting is rather general on purpose since for most definitions only linearity and measurability is needed. But usually one chooses $X$ to be a real Hausdorff locally convex space with the Borel or cylindrical σ-algebra ${𝒜}$. For a measure $\mu$ let ${\mu }_h(A):=\mu (A+h)$ denote the shifted measure by $h\in X$.

Fomin differentiability

A measure $\mu$ on $(X,{𝒜})$ is Fomin differentiable along $h\in X$ if for every set $A\in {𝒜}$ the limit

$d_h\mu (A):=\lim {\mathrm{\backslash limits}}_{t\to 0}\frac{\mu (A+th)-\mu (A)}{t}$

exists. We call $d_h\mu$ the Fomin derivative of $\mu$. Equivalently, for all sets $A\in {𝒜}$ is $f_{\mu }^{A,h}:t\mapsto \mu (A+th)$ differentiable in $0$.

Properties

• The Fomin derivative is again another measure and absolutely continuous with respect to $\mu$.
• Fomin differentiability can be directly extend to signed measures.
• Higher and mixed derivatives will be defined inductively $d_h^n=d_h(d_h^{n-1})$.

Skorokhod differentiability

Let $\mu$ be a Baire measure and let $C_b(X)$ be the space of bounded and continuous functions on $X$. $\mu$ is Skorokhod differentiable (or S-differentiable) along $h\in X$ if a Baire measure $\nu$ exists such that for all $f\in C_b(X)$ the limit

$\lim {\mathrm{\backslash limits}}_{t\to 0}\underset{X}{\int }\frac{f(x-th)-f(x)}{t}\mu (dx)=\underset{X}{\int }f(x)\nu (dx)$

exists. In shift notation

$\lim {\mathrm{\backslash limits}}_{t\to 0}\underset{X}{\int }\frac{f(x-th)-f(x)}{t}\mu (dx)=\lim {\mathrm{\backslash limits}}_{t\to 0}\underset{X}{\int }fd\left(\frac{{\mu }_{th}-\mu }{t}\right).$

The measure $\nu$ is called the Skorokhod derivative (or S-derivative or weak derivative) of $\mu$ along $h\in X$ and is unique.

Albeverio-Høegh-Krohn Differentiability

A measure $\mu$ is Albeverio-Høegh-Krohn differentiable (or AHK differentiable) along $h\in X$ if a measure $\lambda \ge 0$ exists such that

1. ${\mu }_{th}$ is absolutely continuous with respect to $\lambda$ such that ${\lambda }_{th}=f_t\cdot \lambda$,
2. the map $g:\mathbb{ℝ}\to L^2(\lambda ),t\mapsto f_t^{1/2}$ is differentiable.

Properties

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

Example

Let $\mu$ be a measure with a continuously differentiable Radon-Nikodým density $g$, then the Fomin derivative is

$d_h\mu (A)=\lim {\mathrm{\backslash limits}}_{t\to 0}\frac{\mu (A+th)-\mu (A)}{t}=\lim {\mathrm{\backslash limits}}_{t\to 0}\underset{A}{\int }\frac{g(x+th)-g(x)}{t}\mathrm{d}x=\underset{A}{\int }{g}^{\prime }(x)\mathrm{d}x.$

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". 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.