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 $$\R_{+}$$ 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.

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 $$\R^n$$.

Existence of abstract Wiener measure
Consider the space $$\Theta^{\frac{n+1}{2}}(\mathbb R^n;\R)$$ 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 $$\|f\|_{\Theta^{\frac{n+1}{2}}(\mathbb R^n;\R)} := \sup_{x\in\mathbb R^n}\left(\log(e+|x|)\right)^{-1}|f(x)|.$$

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 $$\Theta^{\frac{n+1}{2}}(\mathbb R^n;\R)$$ 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.