- Source: 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
{\displaystyle X}
be a real vector space,
A
{\displaystyle {\mathcal {A}}}
be σ-algebra that is invariant under translation by vectors
h
∈
X
{\displaystyle h\in X}
, i.e.
A
+
t
h
∈
A
{\displaystyle A+th\in {\mathcal {A}}}
for all
A
∈
A
{\displaystyle A\in {\mathcal {A}}}
and
t
∈
R
{\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
X
{\displaystyle X}
to be a real Hausdorff locally convex space with the Borel or cylindrical σ-algebra
A
{\displaystyle {\mathcal {A}}}
.
For a measure
μ
{\displaystyle \mu }
let
μ
h
(
A
)
:=
μ
(
A
+
h
)
{\displaystyle \mu _{h}(A):=\mu (A+h)}
denote the shifted measure by
h
∈
X
{\displaystyle h\in X}
.
= Fomin differentiability
=A measure
μ
{\displaystyle \mu }
on
(
X
,
A
)
{\displaystyle (X,{\mathcal {A}})}
is Fomin differentiable along
h
∈
X
{\displaystyle h\in X}
if for every set
A
∈
A
{\displaystyle A\in {\mathcal {A}}}
the limit
d
h
μ
(
A
)
:=
lim
t
→
0
μ
(
A
+
t
h
)
−
μ
(
A
)
t
{\displaystyle d_{h}\mu (A):=\lim \limits _{t\to 0}{\frac {\mu (A+th)-\mu (A)}{t}}}
exists. We call
d
h
μ
{\displaystyle d_{h}\mu }
the Fomin derivative of
μ
{\displaystyle \mu }
.
Equivalently, for all sets
A
∈
A
{\displaystyle A\in {\mathcal {A}}}
is
f
μ
A
,
h
:
t
↦
μ
(
A
+
t
h
)
{\displaystyle f_{\mu }^{A,h}:t\mapsto \mu (A+th)}
differentiable in
0
{\displaystyle 0}
.
Properties
The Fomin derivative is again another measure and absolutely continuous with respect to
μ
{\displaystyle \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
)
{\displaystyle d_{h}^{n}=d_{h}(d_{h}^{n-1})}
.
= Skorokhod differentiability
=Let
μ
{\displaystyle \mu }
be a Baire measure and let
C
b
(
X
)
{\displaystyle C_{b}(X)}
be the space of bounded and continuous functions on
X
{\displaystyle X}
.
μ
{\displaystyle \mu }
is Skorokhod differentiable (or S-differentiable) along
h
∈
X
{\displaystyle h\in X}
if a Baire measure
ν
{\displaystyle \nu }
exists such that for all
f
∈
C
b
(
X
)
{\displaystyle f\in C_{b}(X)}
the limit
lim
t
→
0
∫
X
f
(
x
−
t
h
)
−
f
(
x
)
t
μ
(
d
x
)
=
∫
X
f
(
x
)
ν
(
d
x
)
{\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
lim
t
→
0
∫
X
f
(
x
−
t
h
)
−
f
(
x
)
t
μ
(
d
x
)
=
lim
t
→
0
∫
X
f
d
(
μ
t
h
−
μ
t
)
.
{\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
ν
{\displaystyle \nu }
is called the Skorokhod derivative (or S-derivative or weak derivative) of
μ
{\displaystyle \mu }
along
h
∈
X
{\displaystyle h\in X}
and is unique.
= Albeverio-Høegh-Krohn Differentiability
=A measure
μ
{\displaystyle \mu }
is Albeverio-Høegh-Krohn differentiable (or AHK differentiable) along
h
∈
X
{\displaystyle h\in X}
if a measure
λ
≥
0
{\displaystyle \lambda \geq 0}
exists such that
μ
t
h
{\displaystyle \mu _{th}}
is absolutely continuous with respect to
λ
{\displaystyle \lambda }
such that
λ
t
h
=
f
t
⋅
λ
{\displaystyle \lambda _{th}=f_{t}\cdot \lambda }
,
the map
g
:
R
→
L
2
(
λ
)
,
t
↦
f
t
1
/
2
{\displaystyle 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
μ
{\displaystyle \mu }
be a measure with a continuously differentiable Radon-Nikodým density
g
{\displaystyle g}
, then the Fomin derivative is
d
h
μ
(
A
)
=
lim
t
→
0
μ
(
A
+
t
h
)
−
μ
(
A
)
t
=
lim
t
→
0
∫
A
g
(
x
+
t
h
)
−
g
(
x
)
t
d
x
=
∫
A
g
′
(
x
)
d
x
.
{\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.}
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.
References
Kata Kunci Pencarian:
- Differentiable measure
- Differentiation of measures
- Absolute continuity
- Differentiation
- Weak derivative
- Smoothness
- Derivative
- Differentiation of integrals
- Almost everywhere
- Total variation
The Dark Sisters (2023)
Bioskop Online, BioskopKeren, Cinemaindo, Dewanonton, Drakor ID, DrakorIndo, Drama, DramaQu, Dunia21, DutaFilm, Ganool, gudangmovie, Horror, IndoXX1, Indoxxi, KorDramas, LayarKaca21, Layarkaca21 INDOXXI, LK21, LK21 XXI, Nonton drama, Nonton Movie, Pahe.in, PusatFilm21,
Richard Bailey
Concrete Utopia (2023)
Bioskop Online, bioskop21, BioskopKeren, Celebrities, Cinemaindo, Drakor ID, DrakorIndo, Drama, DramaQu, Dunia21, DutaFilm, Ganool, gudangmovie, gudangmovie21, IndoXX1, Indoxxi, KorDramas, LayarKaca21, Nonton drama, Nonton Movie, Pahe.in, PusatFilm21, Science Fiction, Thriller, Korea
Um Tae-hwa
No More Posts Available.
No more pages to load.
