Constructing continuous stationary covariances as limits of the      second-order stochastic difference equations

Lassi Roininen; Petteri Piiroinen; Markku Lehtinen

doi:10.3934/ipi.2013.7.611

Article Contents

2013, Volume 7, Issue 2: 611-647. Doi: 10.3934/ipi.2013.7.611

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Constructing continuous stationary covariances as limits of the second-order stochastic difference equations

1.
University of Oulu, Sodankylä Geophysical Observatory, Tähteläntie 62, FI-99600 Sodankylä
2.
University of Helsinki, Department of Mathematics and Statistics, Gustaf Hällströmin katu 2b, FI-00014 University of Helsinki
3.
University of Oulu, Sodankylä Geophysical Observatory, Sodankylä

Received: October 31, 2011

Revised: July 31, 2012

Published: April 2013

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Abstract

In Bayesian statistical inverse problems the a priori probability distributions are often given as stochastic difference equations. We derive a certain class of stochastic partial difference equations by starting from second-order stochastic partial differential equations in one and two dimensions. We discuss discretisation schemes on uniform lattices of these stationary continuous-time stochastic processes and convergence of the discrete-time processes to the continuous-time processes. A special emphasis is given to an analytical calculation of the covariance kernels of the processes. We find a representation for the covariance kernels in a simple parametric form with controllable parameters: correlation length and variance. In the discrete-time processes the discretisation step is also given as a parameter. Therefore, the discrete-time covariances can be considered as discretisation-invariant. In the two-dimensional cases we find rotation-invariant and anisotropic representations of the difference equations and the corresponding continuous-time covariance kernels.

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Mathematics Subject Classification: Primary: 65Q10, 60G10; Secondary: 42A38.

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