In mathematics , a Brownian sheet or multiparametric Brownian motion is a multiparametric generalization of the Brownian motion to a Gaussian random field . This means we generalize the "time" parameter
t
{\displaystyle t}
of a Brownian motion
B
t
{\displaystyle B_{t}}
from
R
+
{\displaystyle \mathbb {R} _{+}}
to
R
+
n
{\displaystyle \mathbb {R} _{+}^{n}}
.
The exact dimension
n
{\displaystyle n}
of the space of the new time parameter varies from authors. We follow John B. Walsh and define the
(
n
,
d
)
{\displaystyle (n,d)}
-Brownian sheet, while some authors define the Brownian sheet specifically only for
n
=
2
{\displaystyle n=2}
, what we call the
(
2
,
d
)
{\displaystyle (2,d)}
-Brownian sheet.[1]
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 sheet if
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