Brownian sheet

In mathematics, a brownian sheet 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.[1]

(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}s_{l}\wedge t_{l}&{\text{if }}i=j,\\0&{\text{else}}\end{cases}}

for

1\leq i,j\leq d

.[2]

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 one-dimensional brownian motion $X_{t,s}$ on the index set $(t,s)\in [0,\infty )\times [0,\infty )$ .

Literature

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.

References

Walsh, John B. (1986). An introduction to stochastic partial differential equations. Springer Berlin Heidelberg. p. 269. ISBN 978-3-540-39781-6.
Davar Khoshnevisan und Yimin Xiao (2004), Images of the Brownian Sheet, arXiv:math/0409491

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[1] Walsh, John B. (1986). An introduction to stochastic partial differential equations. Springer Berlin Heidelberg. p. 269. ISBN 978-3-540-39781-6.

[2] Davar Khoshnevisan und Yimin Xiao (2004), Images of the Brownian Sheet, arXiv:math/0409491