Differentiable measure

Let

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

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 $\backslash mathcal\{A\}$.

For a measure $\backslash mu$ let $\backslash mu\_h(A):=\backslash mu(A+h)$ denote the shifted measure by $h\backslash in\; X$.

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

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

exists. We call $d\_\{h\}\backslash mu$ the Fomin derivative of $\backslash mu$.

Equivalently, for all sets $A\backslash in\backslash mathcal\{A\}$ is $f\_\{\backslash mu\}^\{A,h\}:t\backslash mapsto\; \backslash mu(A+th)$ differentiable in $0$.{{cite book|first=Vladimir I.|last=Bogachev|date=2010|title=Differentiable Measures and the Malliavin Calculus|publisher=American Mathematical Society|pages=69–72|isbn=978-0821849934}}

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

Let $\backslash mu$ be a Baire measure and let $C\_b(X)$ be the space of bounded and continuous functions on $X$.

$\backslash mu$ is Skorokhod differentiable (or S-differentiable) along $h\backslash in\; X$ if a Baire measure $\backslash nu$ exists such that for all $f\backslash in\; C\_b(X)$ the limit

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

exists.

In shift notation

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

The measure $\backslash nu$ is called the Skorokhod derivative (or S-derivative or weak derivative) of $\backslash mu$ along $h\backslash in\; X$ and is unique.{{cite journal|first=Vladimir I. |last=Bogachev |title=On Skorokhod Differentiable Measures|journal=Ukrainian Mathematical Journal|volume=72|page=1163|date=2021|doi=10.1007/s11253-021-01861-x}}

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

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

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

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

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

• {{cite book|first=Vladimir I.|last=Bogachev|date=2010|title=Differentiable Measures and the Malliavin Calculus|publisher=American Mathematical Society|pages=69–72|isbn=978-0821849934}}
• {{cite journal|title=Differentiable Families of Measures|journal=Journal of Functional Analysis|volume=118|number=2|pages=454–476|date=1993|doi=10.1006/jfan.1993.1151|first1=Oleg G.|last1=Smolyanov|first2=Heinrich|last2=von Weizsäcker|doi-access=free}}
• {{cite book|first=Vladimir I.|last=Bogachev|date=2010|title=Differentiable Measures and the Malliavin Calculus|publisher=American Mathematical Society|pages=69–72|isbn=978-0821849934}}
• {{cite conference |first=Sergei Vasil'evich|last=Fomin|title=Differential measures in linear spaces |conference=Int. Congress of Mathematicians|book-title=Proc. Int. Congress of Mathematicians, sec.5|publisher=Izdat. Moskov. Univ.|place=Moscow|date=1966}}
• Kuo, Hui-Hsiung “Differentiable Measures.” Chinese Journal of Mathematics 2, no. 2 (1974): 189–99. {{JSTOR|43836023}}.

Category:Functional analysis

Category:Measure theory