Brownian sheet

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 $t$ of a Brownian motion $B_t$ from $\R_{+}$ to $\R_{+}^n$ .

The exact dimension $n$ of the space of the new time parameter varies from authors. We follow John B. Walsh and define the $(n,d)$ -Brownian sheet, while some authors define the Brownian sheet specifically only for $n=2$ , what we call the $(2,d)$ -Brownian sheet.{{cite book|last1=Walsh|first1=John B.|title=An introduction to stochastic partial differential equations|date=1986|publisher=Springer Berlin Heidelberg|pages=269|ISBN=978-3-540-39781-6}}

This definition is due to Nikolai Chentsov, there exist a slightly different version due to Paul Lévy.

(n,d)-Brownian sheet

A $d$ -dimensional gaussian process $B=(B_t,t\in \mathbb{R}_+^n)$ is called a $(n,d)$ -Brownian sheet if

it has zero mean, i.e. $\mathbb{E}[B_t]=0$ for all $t=(t_1,\dots t_n)\in \mathbb{R}_+^n$
for the covariance function

:: $\operatorname{cov}(B_s^{(i)},B_t^{(j)})=\begin{cases}
\prod\limits_{l=1}^n \operatorname{min} (s_l,t_l) & \text{if }i=j,\\
0 &\text{else}
\end{cases}$

: for $1\leq i,j\leq d$ .{{citation|arxiv=math/0409491|author=Davar Khoshnevisan und Yimin Xiao|date=2004|title=Images of the Brownian Sheet}}

= Properties =

From the definition follows

: $B(0,t_2,\dots,t_n)=B(t_1,0,\dots,t_n)=\cdots=B(t_1,t_2,\dots,0)=0$

almost surely.

= Examples =

$(1,1)$ -Brownian sheet is the Brownian motion in $\mathbb{R}^1$ .
$(1,d)$ -Brownian sheet is the Brownian motion in $\mathbb{R}^d$ .
$(2,1)$ -Brownian sheet is a multiparametric Brownian motion $X_{t,s}$ with index set $(t,s)\in [0,\infty)\times [0,\infty)$ .

= Lévy's definition of the multiparametric Brownian motion =

In Lévy's definition one replaces the covariance condition above with the following condition

:: $\operatorname{cov}(B_s,B_t)=\frac{(|t|+|s|-|t-s|)}{2}$

where $|\cdot|$ is the Euclidean metric on $\R^n$ .{{cite journal|title = Lévy's Brownian motion as a set-indexed process and a related central limit theorem |first1=Mina |last1=Ossiander |first2=Ronald |last2=Pyke|journal = Stochastic Processes and their Applications|volume = 21|number=1|pages = 133-145|year=1985|doi=10.1016/0304-4149(85)90382-5}}

Existence of abstract Wiener measure

Consider the space $\Theta^{\frac{n+1}{2}}(\mathbb R^n;\R)$ of continuous functions of the form $f:\mathbb R^n\to\mathbb R$ satisfying

$\lim\limits_{|x|\to \infty}\left(\log(e+|x|)\right)^{-1}|f(x)|=0.$

This space becomes a separable Banach space when equipped with the norm

$\|f\|_{\Theta^{\frac{n+1}{2}}(\mathbb R^n;\R)} := \sup_{x\in\mathbb R^n}\left(\log(e+|x|)\right)^{-1}|f(x)|.$

Notice this space includes densely the space of zero at infinity $C_0(\mathbb{R}^n;\mathbb{R})$ equipped with the uniform norm, since one can bound the uniform norm with the norm of $\Theta^{\frac{n+1}{2}}(\mathbb R^n;\R)$ from above through the Fourier inversion theorem.

Let $\mathcal{S}'(\mathbb{R}^{n};\mathbb{R})$ be the space of tempered distributions. One can then show that there exist a suitable separable Hilbert space (and Sobolev space)

: $H^\frac{n+1}{2}(\mathbb R^n,\mathbb R)\subseteq \mathcal{S}'(\mathbb{R}^{n};\mathbb{R})$

that is continuously embbeded as a dense subspace in $C_0(\mathbb{R}^n;\mathbb{R})$ and thus also in $\Theta^{\frac{n+1}{2}}(\mathbb R^n;\mathbb{R})$ and that there exist a probability measure $\omega$ on $\Theta^{\frac{n+1}{2}}(\mathbb R^n;\mathbb{R})$ such that the triple

$(H^{\frac{n+1}{2}}(\mathbb R^n;\mathbb{R}),\Theta^{\frac{n+1}{2}}(\mathbb R^n;\mathbb{R}),\omega)$

is an abstract Wiener space.

A path $\theta \in \Theta^{\frac{n+1}{2}}(\mathbb{R}^n;\mathbb{R})$ is $\omega$ -almost surely

Hölder continuous of exponent $\alpha \in (0,1/2)$
nowhere Hölder continuous for any $\alpha> 1/2$ .{{citation|first=Daniel|last=Stroock|authorlink=Daniel Stroock|title=Probability theory: an analytic view|publisher=Cambridge|year=2011|edition=2nd|page=349-352}}

This handles of a Brownian sheet in the case $d=1$ . For higher dimensional $d$ , the construction is similar.

Literature

{{citation|first=Daniel|last=Stroock|authorlink=Daniel Stroock|title=Probability theory: an analytic view|publisher=Cambridge|year=2011|edition=2nd}}.
{{cite book|last1=Walsh|first1=John B.|title=An introduction to stochastic partial differential equations|date=1986|publisher=Springer Berlin Heidelberg|ISBN=978-3-540-39781-6}}
{{cite book|title= Multiparameter Processes: An Introduction to Random Fields|first1=Davar|last1=Khoshnevisan|publisher=Springer|ISBN=978-0387954592}}

References

Category:Wiener process

Category:Robert Brown (botanist, born 1773)