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 from to .

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 , what we call the -Brownian sheet.[1]

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

(n,d)-Brownian sheet

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A  -dimensional gaussian process   is called a  -Brownian sheet if

  • it has zero mean, i.e.   for all  
  • for the covariance function
 
for  .[2]

Properties

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From the definition follows

 

almost surely.

Examples

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  •  -Brownian sheet is the Brownian motion in  .
  •  -Brownian sheet is the Brownian motion in  .
  •  -Brownian sheet is a multiparametric Brownian motion   with index set  .

Lévy's definition of the multiparametric Brownian motion

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

 

where   is the Euclidean metric on  .[3]

Existence of abstract Wiener measure

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Consider the space   of continuous functions of the form   satisfying   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.

Let   be the space of tempered distributions. One can then show that there exist a suitalbe separable Hilbert space (and Sobolev space)

 

that is continuously embbeded as a dense subspace in   and thus also in   and that there exist a probability measure   on   such that the triple   is an abstract Wiener space.

A path   is  -almost surely

  • Hölder continuous of exponent  
  • nowhere Hölder continuous for any  .[4]

This handles of a Brownian sheet in the case  . For higher dimensional  , the construction is similar.

See also

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Literature

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  • 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.

References

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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
  3. ^ Ossiander, Mina; Pyke, Ronald (1985). "Lévy's Brownian motion as a set-indexed process and a related central limit theorem". Stochastic Processes and their Applications. 21 (1): 133–145. doi:10.1016/0304-4149(85)90382-5.
  4. ^ Stroock, Daniel (2011), Probability theory: an analytic view (2nd ed.), Cambridge, p. 349-352