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,
• ${\mathcal {A}}$ be σ-algebra that is invariant under translation by vectors $h\in X$, i.e. $A+th\in {\mathcal {A}}$ for all $A\in {\mathcal {A}}$ 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 ${\mathcal {A}}$.

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

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

exists. We call $d_{h}\mu$ the Fomin derivative of $\mu$.

Equivalently, for all sets $A\in {\mathcal {A}}$ 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

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

exists.

In shift notation

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

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 $\lambda \geq 0$ 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

$d_{h}\mu (A)=\lim \limits _{t\to 0}{\frac {\mu (A+th)-\mu (A)}{t}}=\lim \limits _{t\to 0}\int _{A}{\frac {g(x+th)-g(x)}{t}}\mathrm {d} x=\int _{A}g'(x)\mathrm {d} x.$

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.