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Spartan Random Fields and Langevin Equations

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Random Fields for Spatial Data Modeling

Part of the book series: Advances in Geographic Information Science ((AGIS))

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Abstract

In this chapter we look at Spartan random fields from a different perspective. Our first goal is to show that solutions of linear stochastic partial differential (Langevin) equations are random fields with rational spectral densities [694]. In addition, the respective covariance function is the Green’s function of a suitable (i.e., derivable from Langevin equation) partial differential equation. Finally, the joint dependence of random fields that satisfy Langevin equations driven by a Gaussian white noise process can be expressed in terms of an exponential Gibbs-Boltzmann pdf; the latter has a quadratic energy function that involves local (i.e., based on low-order field derivatives) terms.

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Notes

  1. 1.

    To simplify the notation and avoid confusing ω with the angular frequency w, we replace X(t; ω) with x(t), where the latter represents the trajectory for a specific realization.

  2. 2.

    w corresponds to the wavenumber k which represents the “spatial” frequency in reciprocal space.

  3. 3.

    The horizontal axis corresponds to time, while the vertical axis denotes the position.

  4. 4.

    For the sake of space economy we use the abbreviation \(\sum \) for \(\sum _{i=1}^{d}\sum _{j=1}^{d}\).

  5. 5.

    Note that \(\mathcal {L}\) is the operator acting on the covariance, while \(\mathcal {L}^{\ast }\) is the operator acting on the random field.

  6. 6.

    In contrast with Sect. 9.3, herein we absorb the coefficient σ 0 in and consider forcing with unit-variance white noise.

  7. 7.

    The Green’s function equation is sometimes expressed as \(\mathcal {L} [G(\mathbf {s} - {\mathbf {s}}')] = -\delta (\mathbf {s} - \mathbf {s}')\), i.e., with a negative sign in front of the delta function.

  8. 8.

    In the time series literature the autocovariance function is usually denoted by the symbol γ, which we have reserved for the variogram as is common in spatial statistics.

  9. 9.

    The positivity of the coefficients is ensured by the fact that α 1, α 3 > 0 and α 2 > 1.

  10. 10.

    The equation for the realizations x t holds for Xt(ω) as well.

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Hristopulos, D.T. (2020). Spartan Random Fields and Langevin Equations. In: Random Fields for Spatial Data Modeling. Advances in Geographic Information Science. Springer, Dordrecht. https://doi.org/10.1007/978-94-024-1918-4_9

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