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.
Highlights
We consider the problem of computing the covariance kernel of a random object X given by a stochastic partial differential equation (1)HX := (−λ0 + λ1∆)X = W.The random object W denotes the white noise and ∆ is the Laplacian
We study discretisation schemes of stationary continuous-time stochastic processes on uniform lattices and the convergence of the discretised processes
In order to blur the boundary between discrete and continuous time, we introduce the concept of strong-weak convergence of random objects
Summary
We consider the problem of computing the covariance kernel of a random object X given by a stochastic partial differential equation (1). The random object W denotes the (formal) white noise and ∆ is the (formal) Laplacian. These objects remain formal until their domains are given. Stochastic difference equation, covariance convergence, statistical inversion
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