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( T o p )
1
( n , d ) - B r o w n i a n s h e e t
T o g g l e ( n , d ) - B r o w n i a n s h e e t s u b s e c t i o n
1 . 1
P r o p e r t i e s
1 . 2
E x a m p l e s
1 . 3
L é v y ' s d e f i n i t i o n o f t h e m u l t i p a r a m e t r i c B r o w n i a n m o t i o n
2
E x i s t e n c e o f a b s t r a c t W i e n e r m e a s u r e
3
S e e a l s o
4
L i t e r a t u r e
5
R e f e r e n c e s
T o g g l e t h e t a b l e o f c o n t e n t s
B r o w n i a n s h e e t
2 l a n g u a g e s
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P r i n t / e x p o r t
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A p p e a r a n c e
F r o m W i k i p e d i a , t h e f r e e e n c y c l o p e d i a
This definition is due to Nikolai Chentsov , there exist a slightly different version due to Paul Lévy .
(n,d)-Brownian sheet
[ edit ]
A
d
{\displaystyle d}
-dimensional gaussian process
B
=
(
B
t
,
t
∈
R
+
n
)
{\displaystyle B=(B_{t},t\in \mathbb {R} _{+}^{n})}
is called a
(
n
,
d
)
{\displaystyle (n,d)}
-Brownian sheetif
it has zero mean, i.e.
E
[
B
t
]
=
0
{\displaystyle \mathbb {E} [B_{t}]=0}
for all
t
=
(
t
1
,
…
t
n
)
∈
R
+
n
{\displaystyle t=(t_{1},\dots t_{n})\in \mathbb {R} _{+}^{n}}
for the covariance function
cov
(
B
s
(
i
)
,
B
t
(
j
)
)
=
{
∏
l
=
1
n
min
(
s
l
,
t
l
)
if
i
=
j
,
0
else
{\displaystyle \operatorname {cov} (B_{s}^{(i )},B_{t}^{(j )})={\begin{cases}\prod \limits _{l=1}^{n}\operatorname {min} (s_{l},t_{l})&{\text{if }}i=j,\\0&{\text{else}}\end{cases}}}
for
1
≤
i
,
j
≤
d
{\displaystyle 1\leq i,j\leq d}
.[2]
Properties
[ edit ]
From the definition follows
B
(
0
,
t
2
,
…
,
t
n
)
=
B
(
t
1
,
0
,
…
,
t
n
)
=
⋯
=
B
(
t
1
,
t
2
,
…
,
0
)
=
0
{\displaystyle B(0,t_{2},\dots ,t_{n})=B(t_{1},0,\dots ,t_{n})=\cdots =B(t_{1},t_{2},\dots ,0)=0}
almost surely.
Examples
[ edit ]
(
1
,
1
)
{\displaystyle (1,1)}
-Brownian sheet is the Brownian motion in
R
1
{\displaystyle \mathbb {R} ^{1}}
.
(
1
,
d
)
{\displaystyle (1,d)}
-Brownian sheet is the Brownian motion in
R
d
{\displaystyle \mathbb {R} ^{d}}
.
(
2
,
1
)
{\displaystyle (2,1)}
-Brownian sheet is a multiparametric Brownian motion
X
t
,
s
{\displaystyle X_{t,s}}
with index set
(
t
,
s
)
∈
[
0
,
∞
)
×
[
0
,
∞
)
{\displaystyle (t,s)\in [0,\infty )\times [0,\infty )}
.
Lévy's definition of the multiparametric Brownian motion
[ edit ]
In Lévy's definition one replaces the covariance condition above with the following condition
cov
(
B
s
,
B
t
)
=
(
|
t
|
+
|
s
|
−
|
t
−
s
|
)
2
{\displaystyle \operatorname {cov} (B_{s},B_{t})={\frac {(|t|+|s|-|t-s|)}{2}}}
where
|
⋅
|
{\displaystyle |\cdot |}
is the Euclidean metric on
R
n
{\displaystyle \mathbb {R} ^{n}}
.[3]
Existence of abstract Wiener measure
[ edit ]
Consider the space
Θ
n
+
1
2
(
R
n
;
R
)
{\displaystyle \Theta ^{\frac {n+1}{2}}(\mathbb {R} ^{n};\mathbb {R} )}
of continuous functions of the form
f
:
R
n
→
R
{\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} }
satisfying
lim
|
x
|
→
∞
(
log
(
e
+
|
x
|
)
)
−
1
|
f
(
x
)
|
=
0.
{\displaystyle \lim \limits _{|x|\to \infty }\left(\log(e+|x|)\right)^{-1}|f(x )|=0.}
This space becomes a separable Banach space when equipped with the norm
‖
f
‖
Θ
n
+
1
2
(
R
n
;
R
)
:=
sup
x
∈
R
n
(
log
(
e
+
|
x
|
)
)
−
1
|
f
(
x
)
|
.
{\displaystyle \|f\|_{\Theta ^{\frac {n+1}{2}}(\mathbb {R} ^{n};\mathbb {R} )}:=\sup _{x\in \mathbb {R} ^{n}}\left(\log(e+|x|)\right)^{-1}|f(x )|.}
Notice this space includes densely the space of zero at infinity
C
0
(
R
n
;
R
)
{\displaystyle C_{0}(\mathbb {R} ^{n};\mathbb {R} )}
equipped with the uniform norm, since one can bound the uniform norm with the norm of
Θ
n
+
1
2
(
R
n
;
R
)
{\displaystyle \Theta ^{\frac {n+1}{2}}(\mathbb {R} ^{n};\mathbb {R} )}
from above through the Fourier inversion theorem .
Let
S
′
(
R
n
;
R
)
{\displaystyle {\mathcal {S}}'(\mathbb {R} ^{n};\mathbb {R} )}
be the space of tempered distributions . One can then show that there exist a suitalbe separable Hilbert space (and Sobolev space )
H
n
+
1
2
(
R
n
,
R
)
⊆
S
′
(
R
n
;
R
)
{\displaystyle H^{\frac {n+1}{2}}(\mathbb {R} ^{n},\mathbb {R} )\subseteq {\mathcal {S}}'(\mathbb {R} ^{n};\mathbb {R} )}
that is continuously embbeded as a dense subspace in
C
0
(
R
n
;
R
)
{\displaystyle C_{0}(\mathbb {R} ^{n};\mathbb {R} )}
and thus also in
Θ
n
+
1
2
(
R
n
;
R
)
{\displaystyle \Theta ^{\frac {n+1}{2}}(\mathbb {R} ^{n};\mathbb {R} )}
and that there exist a probability measure
ω
{\displaystyle \omega }
on
Θ
n
+
1
2
(
R
n
;
R
)
{\displaystyle \Theta ^{\frac {n+1}{2}}(\mathbb {R} ^{n};\mathbb {R} )}
such that the triple
(
H
n
+
1
2
(
R
n
;
R
)
,
Θ
n
+
1
2
(
R
n
;
R
)
,
ω
)
{\displaystyle (H^{\frac {n+1}{2}}(\mathbb {R} ^{n};\mathbb {R} ),\Theta ^{\frac {n+1}{2}}(\mathbb {R} ^{n};\mathbb {R} ),\omega )}
is an abstract Wiener space .
A path
θ
∈
Θ
n
+
1
2
(
R
n
;
R
)
{\displaystyle \theta \in \Theta ^{\frac {n+1}{2}}(\mathbb {R} ^{n};\mathbb {R} )}
is
ω
{\displaystyle \omega }
-almost surely
Hölder continuous of exponent
α
∈
(
0
,
1
/
2
)
{\displaystyle \alpha \in (0,1/2)}
nowhere Hölder continuous for any
α
>
1
/
2
{\displaystyle \alpha >1/2}
.[4]
This handles of a Brownian sheet in the case
d
=
1
{\displaystyle d=1}
. For higher dimensional
d
{\displaystyle d}
, the construction is similar.
See also
[ edit ]
Literature
[ edit ]
Stroock, Daniel (2011), Probability theory: an analytic view (2nd ed.), Cambridge .
Walsh, John B. (1986). An introduction to stochastic partial differential equations . Springer Berlin Heidelberg. ISBN 978-3-540-39781-6 .
Khoshnevisan, Davar. Multiparameter Processes: An Introduction to Random Fields . Springer. ISBN 978-0387954592 .
References
[ edit ]
^ Walsh, John B. (1986). An introduction to stochastic partial differential equations . Springer Berlin Heidelberg. p. 269. ISBN 978-3-540-39781-6 .
^ Davar Khoshnevisan und Yimin Xiao (2004), Images of the Brownian Sheet , arXiv :math/0409491
^ Ossiander, Mina; Pyke, Ronald (1985). "Lévy's Brownian motion as a set-indexed process and a related central limit theorem". Stochastic Processes and their Applications . 21 (1 ): 133–145. doi :10.1016/0304-4149(85 )90382-5 .
^ Stroock, Daniel (2011), Probability theory: an analytic view (2nd ed.), Cambridge, p. 349-352
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=Brownian_sheet&oldid=1230197950 "
C a t e g o r i e s :
● W i e n e r p r o c e s s
● R o b e r t B r o w n ( b o t a n i s t , b o r n 1 7 7 3 )
● T h i s p a g e w a s l a s t e d i t e d o n 2 1 J u n e 2 0 2 4 , a t 0 9 : 0 8 ( U T C ) .
● T e x t i s a v a i l a b l e u n d e r t h e C r e a t i v e C o m m o n s A t t r i b u t i o n - S h a r e A l i k e L i c e n s e 4 . 0 ;
a d d i t i o n a l t e r m s m a y a p p l y . B y u s i n g t h i s s i t e , y o u a g r e e t o t h e T e r m s o f U s e a n d P r i v a c y P o l i c y . W i k i p e d i a ® i s a r e g i s t e r e d t r a d e m a r k o f t h e W i k i m e d i a F o u n d a t i o n , I n c . , a n o n - p r o f i t o r g a n i z a t i o n .
● P r i v a c y p o l i c y
● A b o u t W i k i p e d i a
● D i s c l a i m e r s
● C o n t a c t W i k i p e d i a
● C o d e o f C o n d u c t
● D e v e l o p e r s
● S t a t i s t i c s
● C o o k i e s t a t e m e n t
● M o b i l e v i e w