Differentiable measure

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Let

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

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

For a measure ${\displaystyle \mu }$ let ${\displaystyle \mu _{h}(A):=\mu (A+h)}$ denote the shifted measure by ${\displaystyle h\in X}$.

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

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

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

Equivalently, for all sets ${\displaystyle A\in {\mathcal {A}}}$ is ${\displaystyle f_{\mu }^{A,h}:t\mapsto \mu (A+th)}$ differentiable in ${\displaystyle 0}$.^[4]

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

Let ${\displaystyle \mu }$ be a Baire measure and let ${\displaystyle C_{b}(X)}$ be the space of bounded and continuous functions on ${\displaystyle X}$.

${\displaystyle \mu }$ is Skorokhod differentiable (or S-differentiable) along ${\displaystyle h\in X}$ if a Baire measure ${\displaystyle \nu }$ exists such that for all ${\displaystyle f\in C_{b}(X)}$ the limit

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

exists.

In shift notation

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

The measure ${\displaystyle \nu }$ is called the Skorokhod derivative (or S-derivative or weak derivative) of ${\displaystyle \mu }$ along ${\displaystyle h\in X}$ and is unique.^[4][5]

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

1. ${\displaystyle \mu _{th}}$ is absolutely continuous with respect to ${\displaystyle \lambda }$ such that ${\displaystyle \lambda _{th}=f_{t}\cdot \lambda }$,
2. the map ${\displaystyle g:\mathbb {R} \to L^{2}(\lambda ),\;t\mapsto f_{t}^{1/2}}$ is differentiable.^[4]

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