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
of a Brownian motion
The exact dimension
of the space of the new time parameter varies from authors.
We follow John B. Walsh and define the
-Brownian sheet, while some authors define the Brownian sheet specifically only for
[1] This definition is due to Nikolai Chentsov, there exist a slightly different version due to Paul Lévy.
-dimensional gaussian process
is called a
-Brownian sheet if From the definition follows almost surely.
In Lévy's definition one replaces the covariance condition above with the following condition where
is the Euclidean metric on
of continuous functions of the form
satisfying
lim
This space becomes a separable Banach space when equipped with the norm
Notice this space includes densely the space of zero at infinity
equipped with the uniform norm, since one can bound the uniform norm with the norm of
from above through the Fourier inversion theorem.
be the space of tempered distributions.
One can then show that there exist a suitable separable Hilbert space (and Sobolev space) that is continuously embbeded as a dense subspace in
and that there exist a probability measure
ω
) , ω )
is an abstract Wiener space.
ω
-almost surely This handles of a Brownian sheet in the case
For higher dimensional
, the construction is similar.