Fractional pseudodifferential modeling through spatiotemporal duality applied to Geosciences

Abstract potential theory and Dirichlet’s principle constitute the basic elements of the well- known classical theory of Markov processes and Dirichlet forms. In the spatial Gaussian framework, the duality condition allows the derivation of differential models whose solution is a Markov spatial process of finite-order. Gaussian fractional-order pseudodifferential models have been obtained by means of the duality condition in the papers by Ruiz-Medina, Angulo and Anh (2003) and Ruiz-Medina, Anh and Angulo (2004). The present paper extends these results in the spatiotemporal context using the theory of non-local Dirichlet forms, arising in the framework of symmetric Markov processes of pure jump type, allowing a spatiotemporal formulation of the duality condition. This condition leads, in particular, to the derivation in an abstract setting of a fractional-order pseudodifferential representation in space and time for Gaussian random fields, whose continuous spatiotemporal covariance kernel could be non-differentiable in the strong-sense. That is, new classes of spatiotemporal Gaussian processes can be introduced in this framework, including several classes of fractal spatiotemporal processes, as well as the class of spatiotemporal random fields, S/TRF-ν/μ, introduced in Christakos (1991), in connection with Stochastic Partial Differential Equations (SPDE).