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.

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

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

$\mathrm{cov}(B_s^{(i)},B_t^{(j)})=\{\begin{array}{cc}\prod _{l=1}^n\min (s_l,t_l)& \text{if }i=j,\\ 0& \text{else}\end{array}$

for $1\le i,j\le d$.

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

Lévy's definition of the multiparametric Brownian motion

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

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

where $|\cdot |$ is the Euclidean metric on ${ℝ}^n$.

Existence of abstract Wiener measure

Consider the space ${\mathrm{\Theta }}^{\frac{n+1}{2}}({ℝ}^n;ℝ)$ of continuous functions of the form $f:{ℝ}^n\to ℝ$ satisfying $$\underset{|x|\to \mathrm{\infty }}{lim}{(\log (e+|x|))}^{-1}|f(x)|=0.$$ This space becomes a separable Banach space when equipped with the norm $$\Vert f{\Vert }_{{\mathrm{\Theta }}^{\frac{n+1}{2}}({ℝ}^n;ℝ)}:=\underset{x\in {ℝ}^n}{sup}{(\log (e+|x|))}^{-1}|f(x)|.$$

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

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

$H^{\frac{n+1}{2}}({ℝ}^n,ℝ)\subseteq {𝒮}^{\prime }({ℝ}^n;ℝ)$

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

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

• Hölder continuous of exponent $\alpha \in (0,1/2)$
• nowhere Hölder continuous for any $\alpha >1/2$.

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

See also

• Gaussian free field

Literature

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