Brownian sheet explained

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.^[1]

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,...} t _{n)\in} R ^{n} _{+}

• for the covariance function

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

for

1 \leq i,j\leq d .

Properties

From the definition follows

B( 0 ,t _{2,...,t} _{n)=B(t} _{1,0,...,t} _{n)= … =B(t} _{1,t} _{2,...,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 , inf ty) x [ 0 , inf ty) .

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) = ((|t| + |s| - |t - s|))/(2)} where | ⋅ | is the Euclidean metric on \R ^{n} .^[2]

Existence of abstract Wiener measure

Consider the space

\Theta ^{(n +1)/(2)} ( R ^{n;\R)} of continuous functions of the form f: R ^{n\toR} satisfying$$\backslash lim\backslash limits\_$$

\to \infty

\left(\log(e+|x|)\right)^|f(x)|=0.This space becomes a separable Banach space when equipped with the norm$$\backslash |f\backslash |\_\; :=\; \backslash sup\_\backslash left(\backslash log(e+|x|)\backslash right)^|f(x)|.$$

Notice this space includes densely the space of zero at infinity

C ^{n; R )} _{0 ( R} equipped with the uniform norm, since one can bound the uniform norm with the norm of \Theta ^{(n +1)/(2)} ( R ^{n;\R)} from above through the Fourier inversion theorem.

Let

l{S}'( R ^{n} ; R ) be the space of tempered distributions. One can then show that there exist a suitalbe separable Hilbert space (and Sobolev space)H ^{(n +1)/(2)} ( R ^{n,R)\subseteql{S}'(R} ^{n} ; R ) that is continuously embbeded as a dense subspace in C ^{n; R )} _{0 ( R} and thus also in \Theta ^{(n +1)/(2)} ( R ^{n;R)} and that there exist a probability measure \omega on \Theta ^{(n +1)/(2)} ( R ^{n;R)} such that the triple$$(H^(\backslash mathbb\; R^n;\backslash mathbb),\backslash Theta^(\backslash mathbb\; R^n;\backslash mathbb),\backslash omega)$$is an abstract Wiener space.

A path

\theta \in \Theta ^{(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

• .
• Book: Walsh. John B.. An introduction to stochastic partial differential equations. 1986. Springer Berlin Heidelberg. 978-3-540-39781-6.
• Book: Multiparameter Processes: An Introduction to Random Fields. Davar. Khoshnevisan. Springer. 978-0387954592.

References

1. Book: Walsh. John B.. An introduction to stochastic partial differential equations. 1986. Springer Berlin Heidelberg. 269. 978-3-540-39781-6.
2. Lévy's Brownian motion as a set-indexed process and a related central limit theorem . Mina . Ossiander . Ronald . Pyke. Stochastic Processes and their Applications. 21. 1. 133-145. 1985. 10.1016/0304-4149(85)90382-5.