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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Richard Feynman
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Notes
- 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.
w corresponds to the wavenumber k which represents the “spatial” frequency in reciprocal space.
- 3.
The horizontal axis corresponds to time, while the vertical axis denotes the position.
- 4.
For the sake of space economy we use the abbreviation \(\sum \) for \(\sum _{i=1}^{d}\sum _{j=1}^{d}\).
- 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.
In contrast with Sect. 9.3, herein we absorb the coefficient σ 0 in and consider forcing with unit-variance white noise.
- 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.
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.
The positivity of the coefficients is ensured by the fact that α 1, α 3 > 0 and α 2 > 1.
- 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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