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

$\operatorname {cov} (B_{s}^{(i)},B_{t}^{(j)})={\begin{cases}\prod \limits _{l=1}^{n}\operatorname {min} (s_{l},t_{l})&{\text{if }}i=j,\\0&{\text{else}}\end{cases}}$

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

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 $\mathbb {R} ^{n}$.

Existence of abstract Wiener measure

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

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

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

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

Notice this space includes densely the space of zero at infinity $C_{0}(\mathbb {R} ^{n};\mathbb {R} )$ equipped with the uniform norm, since one can bound the uniform norm with the norm of $\Theta ^{\frac {n+1}{2}}(\mathbb {R} ^{n};\mathbb {R} )$ 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 suitalbe 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 $\alpha >1/2$.

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.