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 [math]\displaystyle{ t }[/math] of a Brownian motion [math]\displaystyle{ B_t }[/math] from [math]\displaystyle{ \R_{+} }[/math] to [math]\displaystyle{ \R_{+}^n }[/math]. The exact dimension [math]\displaystyle{ n }[/math] of the space of the new time parameter varies from authors. We follow John B. Walsh and define the [math]\displaystyle{ (n,d) }[/math]-Brownian sheet, while some authors define the Brownian sheet specifically only for [math]\displaystyle{ n=2 }[/math], what we call the [math]\displaystyle{ (2,d) }[/math]-Brownian sheet.

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

(n,d)-Brownian sheet

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

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

[math]\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} }[/math]

for [math]\displaystyle{ 1\leq i,j\leq d }[/math].

Properties

From the definition follows

[math]\displaystyle{ B(0,t_2,\dots,t_n)=B(t_1,0,\dots,t_n)=\cdots=B(t_1,t_2,\dots,0)=0 }[/math]

almost surely.

Examples

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

Lévy's definition of the multiparametric Brownian motion

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

[math]\displaystyle{ \operatorname{cov}(B_s,B_t)=\frac{(|t|+|s|-|t-s|)}{2} }[/math]

where [math]\displaystyle{ |\cdot| }[/math] is the euclidean metric on [math]\displaystyle{ \R^n }[/math].

Existence of abstract Wiener measure

Consider the space [math]\displaystyle{ \Theta^{\frac{n+1}{2}}(\mathbb R^n;\R) }[/math] of continuous functions of the form [math]\displaystyle{ f:\mathbb R^n\to\mathbb R }[/math] satisfying [math]\displaystyle{ \lim\limits_{|x|\to \infty}\left(\log(e+|x|)\right)^{-1}|f(x)|=0. }[/math] This space becomes a separable Banach space when equipped with the norm [math]\displaystyle{ \|f\|_{\Theta^{\frac{n+1}{2}}(\mathbb R^n;\R)} := \sup_{x\in\mathbb R^n}\left(\log(e+|x|)\right)^{-1}|f(x)|. }[/math]

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

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

[math]\displaystyle{ H^\frac{n+1}{2}(\mathbb R^n,\mathbb R)\subseteq \mathcal{S}'(\mathbb{R}^{n};\mathbb{R}) }[/math]

that is continuously embbeded as a dense subspace in [math]\displaystyle{ C_0(\mathbb{R}^n;\mathbb{R}) }[/math] and thus also in [math]\displaystyle{ \Theta^{\frac{n+1}{2}}(\mathbb R^n;\mathbb{R}) }[/math] and that there exist a probability measure [math]\displaystyle{ \omega }[/math] on [math]\displaystyle{ \Theta^{\frac{n+1}{2}}(\mathbb R^n;\mathbb{R}) }[/math] such that the triple [math]\displaystyle{ (H^{\frac{n+1}{2}}(\mathbb R^n;\mathbb{R}),\Theta^{\frac{n+1}{2}}(\mathbb R^n;\mathbb{R}),\omega) }[/math] is an abstract Wiener space.

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

• Hölder continuous of exponent [math]\displaystyle{ \alpha \in (0,1/2) }[/math]
• nowhere Hölder continuous for any [math]\displaystyle{ \alpha\gt 1/2 }[/math].

This handles of a Brownian sheet in the case [math]\displaystyle{ d=1 }[/math]. For higher dimensional [math]\displaystyle{ d }[/math], the construction is similar.

See also

• Gaussian free field

Literature

• 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.