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 ${\mathbb{ℝ}}_{+}$ to ${\mathbb{ℝ}}_{+}^n$. 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 $B=(B_t,t\in {\mathbb{ℝ}}_{+}^n)$ 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

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

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 ${\mathbb{ℝ}}^1$.
• $(1,d)$-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 $(t,s)\in [0,\mathrm{\infty })\times [0,\mathrm{\infty })$.

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 ${\mathbb{ℝ}}^n$.

Existence of abstract Wiener measure

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

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 ${\mathrm{\Theta }}^{\frac{n+1}{2}}({\mathbb{ℝ}}^n;\mathbb{ℝ})$ from above through the Fourier inversion theorem.

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

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

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

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

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

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

• 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.