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 ${\displaystyle t}$ of a Brownian motion ${\displaystyle B_{t}}$ from ${\displaystyle \mathbb {R} _{+}}$ to ${\displaystyle \mathbb {R} _{+}^{n}}$.

The exact dimension ${\displaystyle n}$ of the space of the new time parameter varies from authors. We follow John B. Walsh and define the ${\displaystyle (n,d)}$-Brownian sheet, while some authors define the Brownian sheet specifically only for ${\displaystyle n=2}$, what we call the ${\displaystyle (2,d)}$-Brownian sheet.^[1]

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

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

• it has zero mean, i.e. ${\displaystyle \mathbb {E} [B_{t}]=0}$ for all ${\displaystyle t=(t_{1},\dots t_{n})\in \mathbb {R} _{+}^{n}}$
• for the covariance function

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

for ${\displaystyle 1\leq i,j\leq d}$.^[2]

From the definition follows

${\displaystyle B(0,t_{2},\dots ,t_{n})=B(t_{1},0,\dots ,t_{n})=\cdots =B(t_{1},t_{2},\dots ,0)=0}$

almost surely.

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

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

${\displaystyle \operatorname {cov} (B_{s},B_{t})={\frac {(|t|+|s|-|t-s|)}{2}}}$

where ${\displaystyle |\cdot |}$ is the Euclidean metric on ${\displaystyle \mathbb {R} ^{n}}$.^[3]

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

Notice this space includes densely the space of zero at infinity ${\displaystyle C_{0}(\mathbb {R} ^{n};\mathbb {R} )}$ equipped with the uniform norm, since one can bound the uniform norm with the norm of ${\displaystyle \Theta ^{\frac {n+1}{2}}(\mathbb {R} ^{n};\mathbb {R} )}$ from above through the Fourier inversion theorem.

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

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

that is continuously embbeded as a dense subspace in ${\displaystyle C_{0}(\mathbb {R} ^{n};\mathbb {R} )}$ and thus also in ${\displaystyle \Theta ^{\frac {n+1}{2}}(\mathbb {R} ^{n};\mathbb {R} )}$ and that there exist a probability measure ${\displaystyle \omega }$ on ${\displaystyle \Theta ^{\frac {n+1}{2}}(\mathbb {R} ^{n};\mathbb {R} )}$ such that the triple $${\displaystyle (H^{\frac {n+1}{2}}(\mathbb {R} ^{n};\mathbb {R} ),\Theta ^{\frac {n+1}{2}}(\mathbb {R} ^{n};\mathbb {R} ),\omega )}$$ is an abstract Wiener space.

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

• Hölder continuous of exponent ${\displaystyle \alpha \in (0,1/2)}$
• nowhere Hölder continuous for any ${\displaystyle \alpha >1/2}$.^[4]

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

• Gaussian free field

• 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.