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 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